diff --git a/doc/LectureNotes/DataFiles/cancer.dot b/doc/LectureNotes/DataFiles/cancer.dot index 590609fe5..69f5a4045 100644 --- a/doc/LectureNotes/DataFiles/cancer.dot +++ b/doc/LectureNotes/DataFiles/cancer.dot @@ -10,13 +10,13 @@ edge [fontname="helvetica"] ; 2 -> 3 ; 4 [label="gini = 0.0\nsamples = 239\nvalue = [[239, 0]\n[0, 239]]", fillcolor="#e58139"] ; 3 -> 4 ; -5 [label="worst compactness <= 0.085\ngini = 0.444\nsamples = 3\nvalue = [[2, 1]\n[1, 2]]", fillcolor="#fdf6f0"] ; +5 [label="mean radius <= 12.265\ngini = 0.444\nsamples = 3\nvalue = [[2, 1]\n[1, 2]]", fillcolor="#fdf6f0"] ; 3 -> 5 ; 6 [label="gini = 0.0\nsamples = 1\nvalue = [[0, 1]\n[1, 0]]", fillcolor="#e58139"] ; 5 -> 6 ; 7 [label="gini = 0.0\nsamples = 2\nvalue = [[2, 0]\n[0, 2]]", fillcolor="#e58139"] ; 5 -> 7 ; -8 [label="worst texture <= 29.455\ngini = 0.397\nsamples = 11\nvalue = [[8, 3]\n[3, 8]]", fillcolor="#fae9dd"] ; +8 [label="mean texture <= 20.84\ngini = 0.397\nsamples = 11\nvalue = [[8, 3]\n[3, 8]]", fillcolor="#fae9dd"] ; 2 -> 8 ; 9 [label="gini = 0.0\nsamples = 8\nvalue = [[8, 0]\n[0, 8]]", fillcolor="#e58139"] ; 8 -> 9 ; @@ -30,11 +30,11 @@ edge [fontname="helvetica"] ; 11 -> 13 ; 14 [label="worst texture <= 20.645\ngini = 0.202\nsamples = 167\nvalue = [[19, 148]\n[148, 19]]", fillcolor="#f0b68c"] ; 0 -> 14 [labeldistance=2.5, labelangle=-45, headlabel="False"] ; -15 [label="worst radius <= 17.74\ngini = 0.375\nsamples = 16\nvalue = [[12, 4]\n[4, 12]]", fillcolor="#f9e3d4"] ; +15 [label="worst area <= 964.4\ngini = 0.375\nsamples = 16\nvalue = [[12, 4]\n[4, 12]]", fillcolor="#f9e3d4"] ; 14 -> 15 ; 16 [label="gini = 0.0\nsamples = 11\nvalue = [[11, 0]\n[0, 11]]", fillcolor="#e58139"] ; 15 -> 16 ; -17 [label="worst smoothness <= 0.106\ngini = 0.32\nsamples = 5\nvalue = [[1, 4]\n[4, 1]]", fillcolor="#f6d5bd"] ; +17 [label="concavity error <= 0.016\ngini = 0.32\nsamples = 5\nvalue = [[1, 4]\n[4, 1]]", fillcolor="#f6d5bd"] ; 15 -> 17 ; 18 [label="gini = 0.0\nsamples = 1\nvalue = [[1, 0]\n[0, 1]]", fillcolor="#e58139"] ; 17 -> 18 ; @@ -42,7 +42,7 @@ edge [fontname="helvetica"] ; 17 -> 19 ; 20 [label="mean concave points <= 0.049\ngini = 0.088\nsamples = 151\nvalue = [[7, 144]\n[144, 7]]", fillcolor="#ea985d"] ; 14 -> 20 ; -21 [label="compactness error <= 0.016\ngini = 0.48\nsamples = 15\nvalue = [[6, 9]\n[9, 6]]", fillcolor="#ffffff"] ; +21 [label="concave points error <= 0.01\ngini = 0.48\nsamples = 15\nvalue = [[6, 9]\n[9, 6]]", fillcolor="#ffffff"] ; 20 -> 21 ; 22 [label="gini = 0.0\nsamples = 9\nvalue = [[0, 9]\n[9, 0]]", fillcolor="#e58139"] ; 21 -> 22 ; diff --git a/doc/LectureNotes/DataFiles/cancer.png b/doc/LectureNotes/DataFiles/cancer.png index 7afaf0ca0..98cb944f9 100644 Binary files a/doc/LectureNotes/DataFiles/cancer.png and b/doc/LectureNotes/DataFiles/cancer.png differ diff --git a/doc/LectureNotes/_build/.doctrees/chapter1.doctree b/doc/LectureNotes/_build/.doctrees/chapter1.doctree index d91d09b7f..071bfa47b 100644 Binary files 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Exercises week 36\n", + "**September 4-8, 2023**\n", + "\n", + "Date: **Deadline is Sunday September 10 at midnight**" + ] + }, + { + "cell_type": "markdown", + "id": "b84d6b13", + "metadata": { + "editable": true + }, + "source": [ + "## Overarching aims of the exercises this week\n", + "\n", + "This set of exercises form an important part of the first project. The\n", + "analytical exercises deal with the material covered last week on the\n", + "mathematical interpretations of ordinary least squares and of Ridge\n", + "regression. The numerical exercises can be seen as a continuation of\n", + "exercise 3 from week 35, with the inclusion of Ridge regression. This\n", + "material enters also the discussions of the first project." + ] + }, + { + "cell_type": "markdown", + "id": "96c9c28e", + "metadata": { + "editable": true + }, + "source": [ + "## Exercise 1: Analytical exercises\n", + "\n", + "The aim here is to derive the expression for the optimal parameters\n", + "using Ridge regression. Furthermore, using the singular value\n", + "decomposition, we will analyze the difference between the ordinary\n", + "least squares approach and Ridge regression.\n", + "\n", + "The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the\n", + "optimization problem" + ] + }, + { + "cell_type": "markdown", + "id": "439f1456", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b51e09f7", + "metadata": { + "editable": true + }, + "source": [ + "which we can also write as" + ] + }, + { + "cell_type": "markdown", + "id": "02c45981", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cc0e91ea", + "metadata": { + "editable": true + }, + "source": [ + "where we have used the definition of a norm-2 vector, that is" + ] + }, + { + "cell_type": "markdown", + "id": "b5805f35", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6e3095bf", + "metadata": { + "editable": true + }, + "source": [ + "By minimizing the above equation with respect to the parameters\n", + "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n", + "parameters $\\boldsymbol{\\beta}$.\n", + "\n", + "We can add a regularization parameter $\\lambda$ by\n", + "defining a new cost function to be optimized, that is" + ] + }, + { + "cell_type": "markdown", + "id": "da90fe04", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1a106e07", + "metadata": { + "editable": true + }, + "source": [ + "which leads to the Ridge regression minimization problem. One can require as part of the optimization problem \n", + "that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n", + "a finite number larger than zero. We will not implement that here." + ] + }, + { + "cell_type": "markdown", + "id": "3917877b", + "metadata": { + "editable": true + }, + "source": [ + "### a) Expression for Ridge regression\n", + "\n", + "Show that the optimal parameters" + ] + }, + { + "cell_type": "markdown", + "id": "78226f28", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "951dfffa", + "metadata": { + "editable": true + }, + "source": [ + "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that" + ] + }, + { + "cell_type": "markdown", + "id": "21d2770e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ec212498", + "metadata": { + "editable": true + }, + "source": [ + "with $t$ a finite positive number. \n", + "\n", + "The ordinary least squares result is" + ] + }, + { + "cell_type": "markdown", + "id": "4ffabf6c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f97a6f45", + "metadata": { + "editable": true + }, + "source": [ + "### b) The singular value decomposition\n", + "\n", + "Use the singular value decomposition of an n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)" + ] + }, + { + "cell_type": "markdown", + "id": "8761ed23", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "92f8479e", + "metadata": { + "editable": true + }, + "source": [ + "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n", + "$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n", + "$n\\times p$ matrix which contains the ingular values only. This material was discussed during the lectures of week 35.\n", + "\n", + "Show that you can write the \n", + "OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix $\\boldsymbol{U}$ as" + ] + }, + { + "cell_type": "markdown", + "id": "9df91bda", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\tilde{\\boldsymbol{y}}_{\\mathrm{OLS}}=\\boldsymbol{X}\\boldsymbol{\\beta} = \\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e6de0312", + "metadata": { + "editable": true + }, + "source": [ + "For Ridge regression, show that the corresponding equation is" + ] + }, + { + "cell_type": "markdown", + "id": "8e09d132", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\tilde{\\boldsymbol{y}}_{\\mathrm{Ridge}}=\\boldsymbol{X}\\boldsymbol{\\beta}_{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{\\Sigma}^2\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a9c924ab", + "metadata": { + "editable": true + }, + "source": [ + "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$ from the SVD of the matrix $\\boldsymbol{X}$. \n", + "\n", + "Give an interpretation of the results. [Section 3.4 of Hastie et al's textbook gives a good discussion of the above results](https://link.springer.com/book/10.1007/978-0-387-84858-7)." + ] + }, + { + "cell_type": "markdown", + "id": "3b9328a1", + "metadata": { + "editable": true + }, + "source": [ + "## Exercise 2: Adding Ridge Regression\n", + "\n", + "This exercise is a continuation of exercise 3 from week 35, see . We will use the same function to\n", + "generate our data set, still staying with a simple function $y(x)$\n", + "which we want to fit using linear regression, but now extending the\n", + "analysis to include the Ridge regression method.\n", + "\n", + "In this exercise you need to include the same elements from last week, that is\n", + "1. scale your data by subtracting the mean value from each column in the design matrix.\n", + "\n", + "2. perform a split of the data in a training set and a test set.\n", + "\n", + "The addition to the analysis this time is the introduction of the hyperparameter $\\lambda$ when introducing Ridge regression.\n", + "\n", + "Extend the code from exercise 3 from [week 35](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html) to include Ridge regression with the hyperparameter $\\lambda$. The optimal parameters $\\hat{\\beta}$ for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations" + ] + }, + { + "cell_type": "markdown", + "id": "5b54b7b4", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a7ebe31b", + "metadata": { + "editable": true + }, + "source": [ + "The ordinary least squares result you encoded last week is given by" + ] + }, + { + "cell_type": "markdown", + "id": "874e0dd3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9adcc39f", + "metadata": { + "editable": true + }, + "source": [ + "Use these results to compute the mean squared error for ordinary least\n", + "squares and Ridge regression first for a polynomial of degree five\n", + "with $n=100$ data points and five selected values of\n", + "$\\lambda=[0.0001,0.001, 0.01,0.1,1.0]$. Compute thereafter the mean\n", + "squared error for the same values of $\\lambda$ for polynomials of degree ten\n", + "and $15$. Discuss your results for the training MSE and test MSE with\n", + "Ridge regression and ordinary least squares." + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/LectureNotes/_build/html/_sources/week36.ipynb b/doc/LectureNotes/_build/html/_sources/week36.ipynb new file mode 100644 index 000000000..991b0f568 --- /dev/null +++ b/doc/LectureNotes/_build/html/_sources/week36.ipynb @@ -0,0 +1,3414 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "b49fd9eb", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "138fe268", + "metadata": { + "editable": true + }, + "source": [ + "# Week 36: Statistical interpretation of Linear Regression and Resampling techniques\n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **September 4-8, 2023**" + ] + }, + { + "cell_type": "markdown", + "id": "c20f461b", + "metadata": { + "editable": true + }, + "source": [ + "## Plans for week 36\n", + "\n", + "* Material for the active learning sessions on Tuesday and Wednesday\n", + "\n", + " * Summary from last week on discussion of SVD, Ridge and Lasso linear regression.\n", + "\n", + " * Recommended Reading: Hastie et al chapter 3, see \n", + "\n", + " * Presentation and discussion of first project\n", + "\n", + "* Material for the lecture on Thursday September 7\n", + "\n", + " * Linear Regression and links with Statistics, Resampling methods\n", + "\n", + " * Recommended Reading: Goodfellow et al chapter 3 on probability theory, see URL:\"\"\n", + "\n", + " * See also Murphy, sections 2.4 (Gaussian distributions) and 3.2 (Bayesian Statistics, basis)" + ] + }, + { + "cell_type": "markdown", + "id": "55f0d080", + "metadata": { + "editable": true + }, + "source": [ + "## Material for the active learning sessions Tuesday and Wednesday\n", + "\n", + "The material here contains a summary from last Week and discussion of SVD, Ridge and Lasso regression with examples" + ] + }, + { + "cell_type": "markdown", + "id": "69ce62a6", + "metadata": { + "editable": true + }, + "source": [ + "## Linear Regression and the SVD\n", + "\n", + "We used the SVD to analyse the matrix to invert in ordinary lineat regression" + ] + }, + { + "cell_type": "markdown", + "id": "a9aaa130", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{V}^T.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2b1c0902", + "metadata": { + "editable": true + }, + "source": [ + "Since the matrices here have dimension $p\\times p$, with $p$ corresponding to the singular values, we defined last week the matrix" + ] + }, + { + "cell_type": "markdown", + "id": "093e3a8c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma} = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c0102cd2", + "metadata": { + "editable": true + }, + "source": [ + "where the tilde-matrix $\\tilde{\\boldsymbol{\\Sigma}}$ is a matrix of dimension $p\\times p$ containing only the singular values $\\sigma_i$, that is" + ] + }, + { + "cell_type": "markdown", + "id": "084138a1", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\tilde{\\boldsymbol{\\Sigma}}=\\begin{bmatrix} \\sigma_0 & 0 & 0 & \\dots & 0 & 0 \\\\\n", + " 0 & \\sigma_1 & 0 & \\dots & 0 & 0 \\\\\n", + "\t\t\t\t 0 & 0 & \\sigma_2 & \\dots & 0 & 0 \\\\\n", + "\t\t\t\t 0 & 0 & 0 & \\dots & \\sigma_{p-2} & 0 \\\\\n", + "\t\t\t\t 0 & 0 & 0 & \\dots & 0 & \\sigma_{p-1} \\\\\n", + "\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cb8ed4f2", + "metadata": { + "editable": true + }, + "source": [ + "meaning we can write" + ] + }, + { + "cell_type": "markdown", + "id": "d7c1c21b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2\\boldsymbol{V}^T.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "791e9c6d", + "metadata": { + "editable": true + }, + "source": [ + "Multiplying from the right with $\\boldsymbol{V}$ (using the orthogonality of $\\boldsymbol{V}$) we get" + ] + }, + { + "cell_type": "markdown", + "id": "924b8081", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{V}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "949f6658", + "metadata": { + "editable": true + }, + "source": [ + "## What does it mean?\n", + "\n", + "This means the vectors $\\boldsymbol{v}_i$ of the orthogonal matrix $\\boldsymbol{V}$\n", + "are the eigenvectors of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ with eigenvalues\n", + "given by the singular values squared, that is" + ] + }, + { + "cell_type": "markdown", + "id": "46a4a83c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{v}_i=\\boldsymbol{v}_i\\sigma_i^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ce019aa6", + "metadata": { + "editable": true + }, + "source": [ + "In other words, each non-zero singular value of $\\boldsymbol{X}$ is a positive\n", + "square root of an eigenvalue of $\\boldsymbol{X}^T\\boldsymbol{X}$. It means also that\n", + "the columns of $\\boldsymbol{V}$ are the eigenvectors of\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$. Since we have ordered the singular values of\n", + "$\\boldsymbol{X}$ in a descending order, it means that the column vectors\n", + "$\\boldsymbol{v}_i$ are hierarchically ordered by how much correlation they\n", + "encode from the columns of $\\boldsymbol{X}$. \n", + "\n", + "Note that these are also the eigenvectors and eigenvalues of the\n", + "Hessian matrix.\n", + "\n", + "If we now recall the definition of the covariance matrix (not using\n", + "Bessel's correction) we have" + ] + }, + { + "cell_type": "markdown", + "id": "c662d90f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{X}]=\\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7df45c97", + "metadata": { + "editable": true + }, + "source": [ + "meaning that every squared non-singular value of $\\boldsymbol{X}$ divided by $n$ (\n", + "the number of samples) are the eigenvalues of the covariance\n", + "matrix. Every singular value of $\\boldsymbol{X}$ is thus a positive square\n", + "root of an eigenvalue of $\\boldsymbol{X}^T\\boldsymbol{X}$. If the matrix $\\boldsymbol{X}$ is\n", + "self-adjoint, the singular values of $\\boldsymbol{X}$ are equal to the\n", + "absolute value of the eigenvalues of $\\boldsymbol{X}$." + ] + }, + { + "cell_type": "markdown", + "id": "f0741872", + "metadata": { + "editable": true + }, + "source": [ + "## And finally $\\boldsymbol{X}\\boldsymbol{X}^T$\n", + "\n", + "For $\\boldsymbol{X}\\boldsymbol{X}^T$ we found" + ] + }, + { + "cell_type": "markdown", + "id": "05530ccb", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{U}^T.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3048f911", + "metadata": { + "editable": true + }, + "source": [ + "Since the matrices here have dimension $n\\times n$, we have" + ] + }, + { + "cell_type": "markdown", + "id": "9e3c2de1", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\Sigma}\\boldsymbol{\\Sigma}^T = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\boldsymbol{0}\\\\ \\end{bmatrix}=\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "491923c1", + "metadata": { + "editable": true + }, + "source": [ + "leading to" + ] + }, + { + "cell_type": "markdown", + "id": "77d1dbe5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\boldsymbol{U}^T.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "fb7e2323", + "metadata": { + "editable": true + }, + "source": [ + "Multiplying with $\\boldsymbol{U}$ from the right gives us the eigenvalue problem" + ] + }, + { + "cell_type": "markdown", + "id": "c9cca57a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U}=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "04e4b486", + "metadata": { + "editable": true + }, + "source": [ + "It means that the eigenvalues of $\\boldsymbol{X}\\boldsymbol{X}^T$ are again given by\n", + "the non-zero singular values plus now a series of zeros. The column\n", + "vectors of $\\boldsymbol{U}$ are the eigenvectors of $\\boldsymbol{X}\\boldsymbol{X}^T$ and\n", + "measure how much correlations are contained in the rows of $\\boldsymbol{X}$.\n", + "\n", + "Since we will mainly be interested in the correlations among the features\n", + "of our data (the columns of $\\boldsymbol{X}$, the quantity of interest for us are the non-zero singular\n", + "values and the column vectors of $\\boldsymbol{V}$." + ] + }, + { + "cell_type": "markdown", + "id": "2b3c22fb", + "metadata": { + "editable": true + }, + "source": [ + "## Code for SVD and Inversion of Matrices\n", + "\n", + "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n", + "The simple answer is to use the linear algebra function for pseudoinvers, that is" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "12637a17", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "Ainv = np.linlag.pinv(A)" + ] + }, + { + "cell_type": "markdown", + "id": "904adaf8", + "metadata": { + "editable": true + }, + "source": [ + "Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "6b5465dc", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "# SVD inversion\n", + "def SVDinv(A):\n", + " ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).\n", + " SVD is numerically more stable than the inversion algorithms provided by\n", + " numpy and scipy.linalg at the cost of being slower.\n", + " '''\n", + " U, s, VT = np.linalg.svd(A)\n", + " print('test U')\n", + " print( (np.transpose(U) @ U - U @np.transpose(U)))\n", + " print('test VT')\n", + " print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n", + "\n", + "\n", + " D = np.zeros((len(U),len(VT)))\n", + " D = np.diag(s)\n", + " UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)\n", + " return np.matmul(V,np.matmul(invD,UT))\n", + "\n", + "\n", + "#X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])\n", + "# Non-singular square matrix\n", + "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n", + "print(X)\n", + "A = np.transpose(X) @ X\n", + "# Brute force inversion\n", + "B = np.linalg.inv(A) # here we could use np.linalg.pinv(A)\n", + "C = SVDinv(A)\n", + "print(np.abs(B-C))" + ] + }, + { + "cell_type": "markdown", + "id": "a80dcee4", + "metadata": { + "editable": true + }, + "source": [ + "## Inverse of Rectangular Matrix\n", + "\n", + "Although our matrix to invert $\\boldsymbol{X}^T\\boldsymbol{X}$ is a square matrix, our matrix may be singular. \n", + "\n", + "The pseudoinverse is the generalization of the matrix inverse for square matrices to\n", + "rectangular matrices where the number of rows and columns are not equal.\n", + "\n", + "It is also called the the Moore-Penrose Inverse after two independent discoverers of the method or the Generalized Inverse.\n", + "It is used for the calculation of the inverse for singular or near singular matrices and for rectangular matrices.\n", + "\n", + "Using the SVD we can obtain the pseudoinverse of a matrix $\\boldsymbol{A}$ (labeled here as $\\boldsymbol{A}_{\\mathrm{PI}}$)" + ] + }, + { + "cell_type": "markdown", + "id": "0c4052d3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{A}_{\\mathrm{PI}}= \\boldsymbol{V}\\boldsymbol{D}_{\\mathrm{PI}}\\boldsymbol{U}^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "185d433f", + "metadata": { + "editable": true + }, + "source": [ + "where $\\boldsymbol{D}_{\\mathrm{PI}}$ can be calculated by creating a diagonal matrix from $\\boldsymbol{\\Sigma}$ where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "418f8797", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "# SVD inversion\n", + "def SVDinv(A):\n", + " U, s, VT = np.linalg.svd(A)\n", + " # reciprocals of singular values of s\n", + " d = 1.0 / s\n", + " # create m x n D matrix\n", + " D = np.zeros(A.shape)\n", + " # populate D with n x n diagonal matrix\n", + " D[:A.shape[1], :A.shape[1]] = np.diag(d)\n", + " UT = np.transpose(U)\n", + " V = np.transpose(VT)\n", + " return np.matmul(V,np.matmul(D.T,UT))\n", + "\n", + "\n", + "A = np.array([ [0.3, 0.4], [0.5, 0.6], [0.7, 0.8],[0.9, 1.0]])\n", + "print(A)\n", + "# Brute force inversion of super-collinear matrix\n", + "B = np.linalg.pinv(A)\n", + "print(B)\n", + "# Compare our own algorithm with pinv\n", + "C = SVDinv(A)\n", + "print(np.abs(C-B))" + ] + }, + { + "cell_type": "markdown", + "id": "bf5051ea", + "metadata": { + "editable": true + }, + "source": [ + "As you can see from this example, our own decomposition based on the SVD agrees with the pseudoinverse algorithm provided by **Numpy**." + ] + }, + { + "cell_type": "markdown", + "id": "202d1f90", + "metadata": { + "editable": true + }, + "source": [ + "## Ridge and LASSO Regression\n", + "\n", + "Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is \n", + "our optimization problem is" + ] + }, + { + "cell_type": "markdown", + "id": "df0a8718", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4ebd1d20", + "metadata": { + "editable": true + }, + "source": [ + "or we can state it as" + ] + }, + { + "cell_type": "markdown", + "id": "40df3859", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "980001f5", + "metadata": { + "editable": true + }, + "source": [ + "where we have used the definition of a norm-2 vector, that is" + ] + }, + { + "cell_type": "markdown", + "id": "699b1198", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ff01ef41", + "metadata": { + "editable": true + }, + "source": [ + "## From OLS to Ridge and Lasso\n", + "\n", + "By minimizing the above equation with respect to the parameters\n", + "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n", + "parameters $\\boldsymbol{\\beta}$. We can add a regularization parameter $\\lambda$ by\n", + "defining a new cost function to be optimized, that is" + ] + }, + { + "cell_type": "markdown", + "id": "d7b9188f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "778790dc", + "metadata": { + "editable": true + }, + "source": [ + "which leads to the Ridge regression minimization problem where we\n", + "require that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n", + "a finite number larger than zero. We do not include such a constraints in the discussions here.\n", + "\n", + "By defining" + ] + }, + { + "cell_type": "markdown", + "id": "38dd7428", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6ed3deb0", + "metadata": { + "editable": true + }, + "source": [ + "we have a new optimization equation" + ] + }, + { + "cell_type": "markdown", + "id": "e0860754", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "66751140", + "metadata": { + "editable": true + }, + "source": [ + "which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. \n", + "\n", + "Here we have defined the norm-1 as" + ] + }, + { + "cell_type": "markdown", + "id": "e1fa2bdb", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\vert\\vert \\boldsymbol{x}\\vert\\vert_1 = \\sum_i \\vert x_i\\vert.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "32131982", + "metadata": { + "editable": true + }, + "source": [ + "## Deriving the Ridge Regression Equations\n", + "\n", + "Using the matrix-vector expression for Ridge regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have" + ] + }, + { + "cell_type": "markdown", + "id": "c1979796", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\boldsymbol{\\beta}^T\\boldsymbol{\\beta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d7eb56d7", + "metadata": { + "editable": true + }, + "source": [ + "and \n", + "taking the derivatives with respect to $\\boldsymbol{\\beta}$ we obtain then\n", + "a slightly modified matrix inversion problem which for finite values\n", + "of $\\lambda$ does not suffer from singularity problems. We obtain\n", + "the optimal parameters" + ] + }, + { + "cell_type": "markdown", + "id": "7c42f326", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6496e5c0", + "metadata": { + "editable": true + }, + "source": [ + "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that" + ] + }, + { + "cell_type": "markdown", + "id": "837f7c04", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "55e9395f", + "metadata": { + "editable": true + }, + "source": [ + "with $t$ a finite positive number." + ] + }, + { + "cell_type": "markdown", + "id": "a943f34a", + "metadata": { + "editable": true + }, + "source": [ + "## Note on Scikit-Learn\n", + "\n", + "Note well that a library like **Scikit-Learn** does not include the $1/n$ factor in the expression for the mean-squared error. If you include it, the optimal parameter $\\beta$ becomes" + ] + }, + { + "cell_type": "markdown", + "id": "63a227a9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+n\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8be2bd7b", + "metadata": { + "editable": true + }, + "source": [ + "In our codes where we compare our own codes with **Scikit-Learn**, we do thus not include the $1/n$ factor in the cost function." + ] + }, + { + "cell_type": "markdown", + "id": "35509652", + "metadata": { + "editable": true + }, + "source": [ + "## Comparison with OLS\n", + "When we compare this with the ordinary least squares result we have" + ] + }, + { + "cell_type": "markdown", + "id": "2fbe21d0", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "71e986f8", + "metadata": { + "editable": true + }, + "source": [ + "which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$.\n", + "\n", + "We see that Ridge regression is nothing but the standard OLS with a\n", + "modified diagonal term added to $\\boldsymbol{X}^T\\boldsymbol{X}$. The consequences, in\n", + "particular for our discussion of the bias-variance tradeoff are rather\n", + "interesting. We will see that for specific values of $\\lambda$, we may\n", + "even reduce the variance of the optimal parameters $\\boldsymbol{\\beta}$. These topics and other related ones, will be discussed after the more linear algebra oriented analysis here." + ] + }, + { + "cell_type": "markdown", + "id": "3a5e5f40", + "metadata": { + "editable": true + }, + "source": [ + "## SVD analysis\n", + "\n", + "Using our insights about the SVD of the design matrix $\\boldsymbol{X}$ \n", + "We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\\boldsymbol{U}$ as" + ] + }, + { + "cell_type": "markdown", + "id": "752b6ba2", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\tilde{\\boldsymbol{y}}_{\\mathrm{OLS}}=\\boldsymbol{X}\\boldsymbol{\\beta} =\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4e724327", + "metadata": { + "editable": true + }, + "source": [ + "For Ridge regression this becomes" + ] + }, + { + "cell_type": "markdown", + "id": "b15c2c90", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\tilde{\\boldsymbol{y}}_{\\mathrm{Ridge}}=\\boldsymbol{X}\\boldsymbol{\\beta}_{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{\\Sigma}^2\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "82e70fa5", + "metadata": { + "editable": true + }, + "source": [ + "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$ from the SVD of the matrix $\\boldsymbol{X}$." + ] + }, + { + "cell_type": "markdown", + "id": "3394e2e7", + "metadata": { + "editable": true + }, + "source": [ + "## Interpreting the Ridge results\n", + "\n", + "Since $\\lambda \\geq 0$, it means that compared to OLS, we have" + ] + }, + { + "cell_type": "markdown", + "id": "722ffa86", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda} \\leq 1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "720ed773", + "metadata": { + "editable": true + }, + "source": [ + "Ridge regression finds the coordinates of $\\boldsymbol{y}$ with respect to the\n", + "orthonormal basis $\\boldsymbol{U}$, it then shrinks the coordinates by\n", + "$\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}$. Recall that the SVD has\n", + "eigenvalues ordered in a descending way, that is $\\sigma_i \\geq\n", + "\\sigma_{i+1}$.\n", + "\n", + "For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods." + ] + }, + { + "cell_type": "markdown", + "id": "3b57864f", + "metadata": { + "editable": true + }, + "source": [ + "## More interpretations\n", + "\n", + "For the sake of simplicity, let us assume that the design matrix is orthonormal, that is" + ] + }, + { + "cell_type": "markdown", + "id": "8f36ff27", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{X}=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1} =\\boldsymbol{I}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "70adc9ce", + "metadata": { + "editable": true + }, + "source": [ + "In this case the standard OLS results in" + ] + }, + { + "cell_type": "markdown", + "id": "eb4a995f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\beta}^{\\mathrm{OLS}} = \\boldsymbol{X}^T\\boldsymbol{y}=\\sum_{i=0}^{n-1}\\boldsymbol{u}_i\\boldsymbol{u}_i^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e3cd4b92", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "bbf9b1de", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{I}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\left(1+\\lambda\\right)^{-1}\\boldsymbol{\\beta}^{\\mathrm{OLS}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e30eff48", + "metadata": { + "editable": true + }, + "source": [ + "that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\\lambda$, and\n", + "the Ridge estimator converges to zero when the hyperparameter goes to\n", + "infinity.\n", + "\n", + "We will come back to more interpreations after we have gone through some of the statistical analysis part. \n", + "\n", + "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", + "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended." + ] + }, + { + "cell_type": "markdown", + "id": "9a8f8e2e", + "metadata": { + "editable": true + }, + "source": [ + "## Deriving the Lasso Regression Equations\n", + "\n", + "Using the matrix-vector expression for Lasso regression, we have the following **cost** function" + ] + }, + { + "cell_type": "markdown", + "id": "3cb7928c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b1c6c813", + "metadata": { + "editable": true + }, + "source": [ + "Taking the derivative with respect to $\\boldsymbol{\\beta}$ and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicity)" + ] + }, + { + "cell_type": "markdown", + "id": "da336775", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{d \\vert \\beta\\vert}{d \\beta}=\\mathrm{sgn}(\\beta)=\\left\\{\\begin{array}{cc} 1 & \\beta > 0 \\\\-1 & \\beta < 0, \\end{array}\\right.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2e0de141", + "metadata": { + "editable": true + }, + "source": [ + "we have that the derivative of the cost function is" + ] + }, + { + "cell_type": "markdown", + "id": "40985ec4", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{X},\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}=-\\frac{2}{n}\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})+\\lambda sgn(\\boldsymbol{\\beta})=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "31091d7b", + "metadata": { + "editable": true + }, + "source": [ + "and reordering we have" + ] + }, + { + "cell_type": "markdown", + "id": "e81d965e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta}+\\lambda sgn(\\boldsymbol{\\beta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9984b3a4", + "metadata": { + "editable": true + }, + "source": [ + "This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor $2/n$ in a redefinition of the parameter $\\lambda$. We will solve this type of problems using libraries like **scikit-learn**." + ] + }, + { + "cell_type": "markdown", + "id": "21ec3768", + "metadata": { + "editable": true + }, + "source": [ + "## Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression\n", + "\n", + "Let us assume that our design matrix is given by unit (identity) matrix, that is a square diagonal matrix with ones only along the\n", + "diagonal. In this case we have an equal number of rows and columns $n=p$.\n", + "\n", + "Our model approximation is just $\\tilde{\\boldsymbol{y}}=\\boldsymbol{\\beta}$ and the mean squared error and thereby the cost function for ordinary least sqquares (OLS) is then (we drop the term $1/n$)" + ] + }, + { + "cell_type": "markdown", + "id": "6be0ec63", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "44c7278c", + "metadata": { + "editable": true + }, + "source": [ + "and minimizing we have that" + ] + }, + { + "cell_type": "markdown", + "id": "918a8cc3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\beta}_i^{\\mathrm{OLS}} = y_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3949af51", + "metadata": { + "editable": true + }, + "source": [ + "## Ridge Regression\n", + "\n", + "For Ridge regression our cost function is" + ] + }, + { + "cell_type": "markdown", + "id": "aa856d18", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\beta_i^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "bd30d1d4", + "metadata": { + "editable": true + }, + "source": [ + "and minimizing we have that" + ] + }, + { + "cell_type": "markdown", + "id": "142de535", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\beta}_i^{\\mathrm{Ridge}} = \\frac{y_i}{1+\\lambda}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1efee0c3", + "metadata": { + "editable": true + }, + "source": [ + "## Lasso Regression\n", + "\n", + "For Lasso regression our cost function is" + ] + }, + { + "cell_type": "markdown", + "id": "a6db019a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\vert\\beta_i\\vert=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\sqrt{\\beta_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0f38deda", + "metadata": { + "editable": true + }, + "source": [ + "and minimizing we have that" + ] + }, + { + "cell_type": "markdown", + "id": "7f9738a5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "-2\\sum_{i=0}^{p-1}(y_i-\\beta_i)+\\lambda \\sum_{i=0}^{p-1}\\frac{(\\beta_i)}{\\vert\\beta_i\\vert}=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4e8460f8", + "metadata": { + "editable": true + }, + "source": [ + "which leads to" + ] + }, + { + "cell_type": "markdown", + "id": "ae49ddce", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}_i^{\\mathrm{Lasso}} = \\left\\{\\begin{array}{ccc}y_i-\\frac{\\lambda}{2} &\\mathrm{if} & y_i> \\frac{\\lambda}{2}\\\\\n", + " y_i+\\frac{\\lambda}{2} &\\mathrm{if} & y_i< -\\frac{\\lambda}{2}\\\\\n", + "\t\t\t\t\t\t\t 0 &\\mathrm{if} & \\vert y_i\\vert\\le \\frac{\\lambda}{2}\\end{array}\\right.\\\\.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1efb96d3", + "metadata": { + "editable": true + }, + "source": [ + "Plotting these results shows clearly that Lasso regression suppresses (sets to zero) values of $\\beta_i$ for specific values of $\\lambda$. Ridge regression reduces on the other hand the values of $\\beta_i$ as function of $\\lambda$." + ] + }, + { + "cell_type": "markdown", + "id": "1a030291", + "metadata": { + "editable": true + }, + "source": [ + "## Yet another Example\n", + "\n", + "Let us assume we have a data set with outputs/targets given by the vector" + ] + }, + { + "cell_type": "markdown", + "id": "af486213", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{y}=\\begin{bmatrix}4 \\\\ 2 \\\\3\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "87e5470d", + "metadata": { + "editable": true + }, + "source": [ + "and our inputs as a $3\\times 2$ design matrix" + ] + }, + { + "cell_type": "markdown", + "id": "182f9f51", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}=\\begin{bmatrix}2 & 0\\\\ 0 & 1 \\\\ 0 & 0\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "edc0ee3d", + "metadata": { + "editable": true + }, + "source": [ + "meaning that we have two features and two unknown parameters $\\beta_0$ and $\\beta_1$ to be determined either by ordinary least squares, Ridge or Lasso regression." + ] + }, + { + "cell_type": "markdown", + "id": "673c64a0", + "metadata": { + "editable": true + }, + "source": [ + "## The OLS case\n", + "\n", + "For ordinary least squares (OLS) we know that the optimal solution is" + ] + }, + { + "cell_type": "markdown", + "id": "c0554476", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cc9c761b", + "metadata": { + "editable": true + }, + "source": [ + "Inserting the above values we obtain that" + ] + }, + { + "cell_type": "markdown", + "id": "b5c84ed1", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\begin{bmatrix}2 \\\\ 2\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a8d2ba86", + "metadata": { + "editable": true + }, + "source": [ + "The code which implements this simpler case is presented after the discussion of Ridge and Lasso." + ] + }, + { + "cell_type": "markdown", + "id": "9e407326", + "metadata": { + "editable": true + }, + "source": [ + "## The Ridge case\n", + "\n", + "For Ridge regression we have" + ] + }, + { + "cell_type": "markdown", + "id": "36ce4758", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8665e33b", + "metadata": { + "editable": true + }, + "source": [ + "Inserting the above values we obtain that" + ] + }, + { + "cell_type": "markdown", + "id": "16dbcd82", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\begin{bmatrix}\\frac{8}{4+\\lambda} \\\\ \\frac{2}{1+\\lambda}\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "642f7768", + "metadata": { + "editable": true + }, + "source": [ + "There is normally a constraint on the value of $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2$ via the parameter $\\lambda$.\n", + "Let us for simplicity assume that $\\beta_0^2+\\beta_1^2=1$ as constraint. This will allow us to find an expression for the optimal values of $\\beta$ and $\\lambda$.\n", + "\n", + "To see this, let us write the cost function for Ridge regression." + ] + }, + { + "cell_type": "markdown", + "id": "1784c79c", + "metadata": { + "editable": true + }, + "source": [ + "## Writing the Cost Function\n", + "\n", + "We define the MSE without the $1/n$ factor and have then, using that" + ] + }, + { + "cell_type": "markdown", + "id": "72b7600f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}\\boldsymbol{\\beta}=\\begin{bmatrix} 2\\beta_0 \\\\ \\beta_1 \\\\0 \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "84ca3171", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\beta_0^2+\\beta_1^2),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d31aa45e", + "metadata": { + "editable": true + }, + "source": [ + "and taking the derivative with respect to $\\beta_0$ we get" + ] + }, + { + "cell_type": "markdown", + "id": "1298ca6d", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_0=\\frac{8}{4+\\lambda},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "fa2455d6", + "metadata": { + "editable": true + }, + "source": [ + "and for $\\beta_1$ we obtain" + ] + }, + { + "cell_type": "markdown", + "id": "8cd50073", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_1=\\frac{2}{1+\\lambda},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1f7e12c3", + "metadata": { + "editable": true + }, + "source": [ + "Using the constraint for $\\beta_0^2+\\beta_1^2=1$ we can constrain $\\lambda$ by solving" + ] + }, + { + "cell_type": "markdown", + "id": "a79821fd", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\left(\\frac{8}{4+\\lambda}\\right)^2+\\left(\\frac{2}{1+\\lambda}\\right)^2=1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2ac13531", + "metadata": { + "editable": true + }, + "source": [ + "which gives $\\lambda=4.571$ and $\\beta_0=0.933$ and $\\beta_1=0.359$." + ] + }, + { + "cell_type": "markdown", + "id": "eed25f68", + "metadata": { + "editable": true + }, + "source": [ + "## Lasso case\n", + "\n", + "For Lasso we need now, keeping a constraint on $\\vert\\beta_0\\vert+\\vert\\beta_1\\vert=1$, to take the derivative of the absolute values of $\\beta_0$\n", + "and $\\beta_1$. This gives us the following derivatives of the cost function" + ] + }, + { + "cell_type": "markdown", + "id": "f74d71ec", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\vert\\beta_0\\vert+\\vert\\beta_1\\vert),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f0797067", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_0}=-4(4-2\\beta_0)+\\lambda\\mathrm{sgn}(\\beta_0)=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3cc5d5e0", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "04250e5c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_1}=-2(2-\\beta_1)+\\lambda\\mathrm{sgn}(\\beta_1)=0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2b24d2fa", + "metadata": { + "editable": true + }, + "source": [ + "We have now four cases to solve besides the trivial cases $\\beta_0$ and/or $\\beta_1$ are zero, namely\n", + "1. $\\beta_0 > 0$ and $\\beta_1 > 0$,\n", + "\n", + "2. $\\beta_0 > 0$ and $\\beta_1 < 0$,\n", + "\n", + "3. $\\beta_0 < 0$ and $\\beta_1 > 0$,\n", + "\n", + "4. $\\beta_0 < 0$ and $\\beta_1 < 0$." + ] + }, + { + "cell_type": "markdown", + "id": "aa950760", + "metadata": { + "editable": true + }, + "source": [ + "## The first Case\n", + "\n", + "If we consider the first case, we have then" + ] + }, + { + "cell_type": "markdown", + "id": "475389ed", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "-4(4-2\\beta_0)+\\lambda=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "086558d9", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "1a3b2ba5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "-2(2-\\beta_1)+\\lambda=0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c928bc46", + "metadata": { + "editable": true + }, + "source": [ + "which yields" + ] + }, + { + "cell_type": "markdown", + "id": "809a58f7", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_0=\\frac{16+\\lambda}{8},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3b937bf4", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "6e5b2527", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_1=\\frac{4+\\lambda}{2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c5f88a45", + "metadata": { + "editable": true + }, + "source": [ + "Using the constraint on $\\beta_0$ and $\\beta_1$ we can then find the optimal value of $\\lambda$ for the different cases. We leave this as an exercise to you." + ] + }, + { + "cell_type": "markdown", + "id": "c57753b4", + "metadata": { + "editable": true + }, + "source": [ + "## Simple code for solving the above problem\n", + "\n", + "Here we set up the OLS, Ridge and Lasso functionality in order to study the above example. Note that here we have opted for a set of values of $\\lambda$, meaning that we need to perform a search in order to find the optimal values.\n", + "\n", + "First we study and compare the OLS and Ridge results. The next code compares all three methods." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "392f218d", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "\n", + "def R2(y_data, y_model):\n", + " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "\n", + "# A seed just to ensure that the random numbers are the same for every run.\n", + "# Useful for eventual debugging.\n", + "\n", + "X = np.array( [ [ 2, 0], [0, 1], [0,0]])\n", + "y = np.array( [4, 2, 3])\n", + "\n", + "\n", + "# matrix inversion to find beta\n", + "OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y\n", + "print(OLSbeta)\n", + "# and then make the prediction\n", + "ytildeOLS = X @ OLSbeta\n", + "print(\"Training MSE for OLS\")\n", + "print(MSE(y,ytildeOLS))\n", + "ypredictOLS = X @ OLSbeta\n", + "\n", + "# Repeat now for Ridge regression and various values of the regularization parameter\n", + "I = np.eye(2,2)\n", + "# Decide which values of lambda to use\n", + "nlambdas = 100\n", + "MSEPredict = np.zeros(nlambdas)\n", + "lambdas = np.logspace(-4, 4, nlambdas)\n", + "for i in range(nlambdas):\n", + " lmb = lambdas[i]\n", + " Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y\n", + "# print(Ridgebeta)\n", + " # and then make the prediction\n", + " ypredictRidge = X @ Ridgebeta\n", + " MSEPredict[i] = MSE(y,ypredictRidge)\n", + "# print(MSEPredict[i])\n", + " # Now plot the results\n", + "plt.figure()\n", + "plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Train')\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('MSE')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "dd134ab1", + "metadata": { + "editable": true + }, + "source": [ + "We see here that we reach a plateau. What is actually happening?" + ] + }, + { + "cell_type": "markdown", + "id": "8d9a61db", + "metadata": { + "editable": true + }, + "source": [ + "## With Lasso Regression" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "afba7c4a", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from sklearn import linear_model\n", + "\n", + "def R2(y_data, y_model):\n", + " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "\n", + "# A seed just to ensure that the random numbers are the same for every run.\n", + "# Useful for eventual debugging.\n", + "\n", + "X = np.array( [ [ 2, 0], [0, 1], [0,0]])\n", + "y = np.array( [4, 2, 3])\n", + "\n", + "\n", + "# matrix inversion to find beta\n", + "OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y\n", + "print(OLSbeta)\n", + "# and then make the prediction\n", + "ytildeOLS = X @ OLSbeta\n", + "print(\"Training MSE for OLS\")\n", + "print(MSE(y,ytildeOLS))\n", + "ypredictOLS = X @ OLSbeta\n", + "\n", + "# Repeat now for Ridge regression and various values of the regularization parameter\n", + "I = np.eye(2,2)\n", + "# Decide which values of lambda to use\n", + "nlambdas = 100\n", + "MSERidgePredict = np.zeros(nlambdas)\n", + "MSELassoPredict = np.zeros(nlambdas)\n", + "lambdas = np.logspace(-4, 4, nlambdas)\n", + "for i in range(nlambdas):\n", + " lmb = lambdas[i]\n", + " Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y\n", + " print(Ridgebeta)\n", + " # and then make the prediction\n", + " ypredictRidge = X @ Ridgebeta\n", + " MSERidgePredict[i] = MSE(y,ypredictRidge)\n", + " RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n", + " RegLasso.fit(X,y)\n", + " ypredictLasso = RegLasso.predict(X)\n", + " print(RegLasso.coef_)\n", + " MSELassoPredict[i] = MSE(y,ypredictLasso)\n", + "# Now plot the results\n", + "plt.figure()\n", + "plt.plot(np.log10(lambdas), MSERidgePredict, 'r--', label = 'MSE Ridge Train')\n", + "plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Train')\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('MSE')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "4d63db0b", + "metadata": { + "editable": true + }, + "source": [ + "## Another Example, now with a polynomial fit" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "81115144", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn import linear_model\n", + "\n", + "def R2(y_data, y_model):\n", + " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "\n", + "# A seed just to ensure that the random numbers are the same for every run.\n", + "# Useful for eventual debugging.\n", + "np.random.seed(3155)\n", + "\n", + "x = np.random.rand(100)\n", + "y = 2.0+5*x*x+0.1*np.random.randn(100)\n", + "\n", + "# number of features p (here degree of polynomial\n", + "p = 3\n", + "# The design matrix now as function of a given polynomial\n", + "X = np.zeros((len(x),p))\n", + "X[:,0] = 1.0\n", + "X[:,1] = x\n", + "X[:,2] = x*x\n", + "# We split the data in test and training data\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", + "\n", + "# matrix inversion to find beta\n", + "OLSbeta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train\n", + "print(OLSbeta)\n", + "# and then make the prediction\n", + "ytildeOLS = X_train @ OLSbeta\n", + "print(\"Training MSE for OLS\")\n", + "print(MSE(y_train,ytildeOLS))\n", + "ypredictOLS = X_test @ OLSbeta\n", + "print(\"Test MSE OLS\")\n", + "print(MSE(y_test,ypredictOLS))\n", + "\n", + "# Repeat now for Lasso and Ridge regression and various values of the regularization parameter\n", + "I = np.eye(p,p)\n", + "# Decide which values of lambda to use\n", + "nlambdas = 100\n", + "MSEPredict = np.zeros(nlambdas)\n", + "MSETrain = np.zeros(nlambdas)\n", + "MSELassoPredict = np.zeros(nlambdas)\n", + "MSELassoTrain = np.zeros(nlambdas)\n", + "lambdas = np.logspace(-4, 4, nlambdas)\n", + "for i in range(nlambdas):\n", + " lmb = lambdas[i]\n", + " Ridgebeta = np.linalg.inv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n", + " # include lasso using Scikit-Learn\n", + " RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n", + " RegLasso.fit(X_train,y_train)\n", + " # and then make the prediction\n", + " ytildeRidge = X_train @ Ridgebeta\n", + " ypredictRidge = X_test @ Ridgebeta\n", + " ytildeLasso = RegLasso.predict(X_train)\n", + " ypredictLasso = RegLasso.predict(X_test)\n", + " MSEPredict[i] = MSE(y_test,ypredictRidge)\n", + " MSETrain[i] = MSE(y_train,ytildeRidge)\n", + " MSELassoPredict[i] = MSE(y_test,ypredictLasso)\n", + " MSELassoTrain[i] = MSE(y_train,ytildeLasso)\n", + "\n", + "# Now plot the results\n", + "plt.figure()\n", + "plt.plot(np.log10(lambdas), MSETrain, label = 'MSE Ridge train')\n", + "plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Test')\n", + "plt.plot(np.log10(lambdas), MSELassoTrain, label = 'MSE Lasso train')\n", + "plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Test')\n", + "\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('MSE')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "e076a968", + "metadata": { + "editable": true + }, + "source": [ + "## Material for lecture Thursday September 7" + ] + }, + { + "cell_type": "markdown", + "id": "db822c65", + "metadata": { + "editable": true + }, + "source": [ + "## Linking the regression analysis with a statistical interpretation\n", + "\n", + "We will now couple the discussions of ordinary least squares, Ridge\n", + "and Lasso regression with a statistical interpretation, that is we\n", + "move from a linear algebra analysis to a statistical analysis. In\n", + "particular, we will focus on what the regularization terms can result\n", + "in. We will amongst other things show that the regularization\n", + "parameter can reduce considerably the variance of the parameters\n", + "$\\beta$.\n", + "\n", + "The\n", + "advantage of doing linear regression is that we actually end up with\n", + "analytical expressions for several statistical quantities. \n", + "Standard least squares and Ridge regression allow us to\n", + "derive quantities like the variance and other expectation values in a\n", + "rather straightforward way.\n", + "\n", + "It is assumed that $\\varepsilon_i\n", + "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n", + "independent, i.e.:" + ] + }, + { + "cell_type": "markdown", + "id": "fc4a06b6", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*} \n", + "\\mbox{Cov}(\\varepsilon_{i_1},\n", + "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n", + "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2. \\end{array} \\right.\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "43299fb3", + "metadata": { + "editable": true + }, + "source": [ + "The randomness of $\\varepsilon_i$ implies that\n", + "$\\mathbf{y}_i$ is also a random variable. In particular,\n", + "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n", + "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n", + "non-random scalar. To specify the parameters of the distribution of\n", + "$\\mathbf{y}_i$ we need to calculate its first two moments. \n", + "\n", + "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n", + "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n", + "row number $i$ and perform a sum over all values $p$." + ] + }, + { + "cell_type": "markdown", + "id": "6cba00c7", + "metadata": { + "editable": true + }, + "source": [ + "## Assumptions made\n", + "\n", + "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n", + "that there exists a function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n", + "which describe our data" + ] + }, + { + "cell_type": "markdown", + "id": "8c85f61d", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a1901c32", + "metadata": { + "editable": true + }, + "source": [ + "We approximate this function with our model from the solution of the linear regression equations, that is our\n", + "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with" + ] + }, + { + "cell_type": "markdown", + "id": "c59e8272", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8cba4bae", + "metadata": { + "editable": true + }, + "source": [ + "## Expectation value and variance\n", + "\n", + "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$" + ] + }, + { + "cell_type": "markdown", + "id": "5e18ebb5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*} \n", + "\\mathbb{E}(y_i) & =\n", + "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n", + "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "10d86c58", + "metadata": { + "editable": true + }, + "source": [ + "while\n", + "its variance is" + ] + }, + { + "cell_type": "markdown", + "id": "efba1ba9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n", + "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n", + "[\\mathbb{E}(y_i)]^2 \\\\ & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n", + "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n", + "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n", + "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n", + "\\ast} \\, \\beta)^2 \\\\ & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n", + "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n", + "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n", + "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n", + "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2. \n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a3eccdab", + "metadata": { + "editable": true + }, + "source": [ + "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n", + "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD)." + ] + }, + { + "cell_type": "markdown", + "id": "70b901b4", + "metadata": { + "editable": true + }, + "source": [ + "## Expectation value and variance for $\\boldsymbol{\\beta}$\n", + "\n", + "With the OLS expressions for the optimal parameters $\\boldsymbol{\\hat{\\beta}}$ we can evaluate the expectation value" + ] + }, + { + "cell_type": "markdown", + "id": "752b72d2", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "108aba79", + "metadata": { + "editable": true + }, + "source": [ + "This means that the estimator of the regression parameters is unbiased.\n", + "\n", + "We can also calculate the variance\n", + "\n", + "The variance of the optimal value $\\boldsymbol{\\hat{\\beta}}$ is" + ] + }, + { + "cell_type": "markdown", + "id": "65801f3b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{eqnarray*}\n", + "\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n", + "\\\\\n", + "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}]^{T} \\}\n", + "\\\\\n", + "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} \\, \\mathbf{Y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{Y} \\, \\mathbf{Y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "\\\\\n", + "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n", + "% \\\\\n", + "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n", + "\\\\\n", + "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n", + "\\end{eqnarray*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "faf8da78", + "metadata": { + "editable": true + }, + "source": [ + "where we have used that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n", + "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", + "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n", + "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n", + "variance of the estimate of the $j$-th regression coefficient:\n", + "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $. This may be used to\n", + "construct a confidence interval for the estimates.\n", + "\n", + "In a similar way, we can obtain analytical expressions for say the\n", + "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n", + "when we employ Ridge regression, allowing us again to define a confidence interval. \n", + "\n", + "It is rather straightforward to show that" + ] + }, + { + "cell_type": "markdown", + "id": "1c71921e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cfb57087", + "metadata": { + "editable": true + }, + "source": [ + "We see clearly that \n", + "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n", + "\n", + "We can also compute the variance as" + ] + }, + { + "cell_type": "markdown", + "id": "be4ad159", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f5e11f36", + "metadata": { + "editable": true + }, + "source": [ + "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n", + "\n", + "With this, we can compute the difference" + ] + }, + { + "cell_type": "markdown", + "id": "dbf2700f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0f6d760d", + "metadata": { + "editable": true + }, + "source": [ + "The difference is non-negative definite since each component of the\n", + "matrix product is non-negative definite. \n", + "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below." + ] + }, + { + "cell_type": "markdown", + "id": "ce614956", + "metadata": { + "editable": true + }, + "source": [ + "## Deriving OLS from a probability distribution\n", + "\n", + "Our basic assumption when we derived the OLS equations was to assume\n", + "that our output is determined by a given continuous function\n", + "$f(\\boldsymbol{x})$ and a random noise $\\boldsymbol{\\epsilon}$ given by the normal\n", + "distribution with zero mean value and an undetermined variance\n", + "$\\sigma^2$.\n", + "\n", + "We found above that the outputs $\\boldsymbol{y}$ have a mean value given by\n", + "$\\boldsymbol{X}\\hat{\\boldsymbol{\\beta}}$ and variance $\\sigma^2$. Since the entries to\n", + "the design matrix are not stochastic variables, we can assume that the\n", + "probability distribution of our targets is also a normal distribution\n", + "but now with mean value $\\boldsymbol{X}\\hat{\\boldsymbol{\\beta}}$. This means that a\n", + "single output $y_i$ is given by the Gaussian distribution" + ] + }, + { + "cell_type": "markdown", + "id": "086419d8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y_i\\sim \\mathcal{N}(\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta}, \\sigma^2)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a228dcf4", + "metadata": { + "editable": true + }, + "source": [ + "## Independent and Identically Distrubuted (iid)\n", + "\n", + "We assume now that the various $y_i$ values are stochastically distributed according to the above Gaussian distribution. \n", + "We define this distribution as" + ] + }, + { + "cell_type": "markdown", + "id": "f04f6b76", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(y_i, \\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "642936a8", + "metadata": { + "editable": true + }, + "source": [ + "which reads as finding the likelihood of an event $y_i$ with the input variables $\\boldsymbol{X}$ given the parameters (to be determined) $\\boldsymbol{\\beta}$.\n", + "\n", + "Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event $\\boldsymbol{y}$ as the product of the single events, that is we have" + ] + }, + { + "cell_type": "markdown", + "id": "4f137023", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{y},\\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}=\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "48fdd9f7", + "metadata": { + "editable": true + }, + "source": [ + "We will write this in a more compact form reserving $\\boldsymbol{D}$ for the domain of events, including the ouputs (targets) and the inputs. That is\n", + "in case we have a simple one-dimensional input and output case" + ] + }, + { + "cell_type": "markdown", + "id": "ae36f857", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d312d461", + "metadata": { + "editable": true + }, + "source": [ + "In the more general case the various inputs should be replaced by the possible features represented by the input data set $\\boldsymbol{X}$. \n", + "We can now rewrite the above probability as" + ] + }, + { + "cell_type": "markdown", + "id": "59ea8b57", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b9be32b5", + "metadata": { + "editable": true + }, + "source": [ + "It is a conditional probability (see below) and reads as the likelihood of a domain of events $\\boldsymbol{D}$ given a set of parameters $\\boldsymbol{\\beta}$." + ] + }, + { + "cell_type": "markdown", + "id": "b378b2c9", + "metadata": { + "editable": true + }, + "source": [ + "## Maximum Likelihood Estimation (MLE)\n", + "\n", + "In statistics, maximum likelihood estimation (MLE) is a method of\n", + "estimating the parameters of an assumed probability distribution,\n", + "given some observed data. This is achieved by maximizing a likelihood\n", + "function so that, under the assumed statistical model, the observed\n", + "data is the most probable. \n", + "\n", + "We will assume here that our events are given by the above Gaussian\n", + "distribution and we will determine the optimal parameters $\\beta$ by\n", + "maximizing the above PDF. However, computing the derivatives of a\n", + "product function is cumbersome and can easily lead to overflow and/or\n", + "underflowproblems, with potentials for loss of numerical precision.\n", + "\n", + "In practice, it is more convenient to maximize the logarithm of the\n", + "PDF because it is a monotonically increasing function of the argument.\n", + "Alternatively, and this will be our option, we will minimize the\n", + "negative of the logarithm since this is a monotonically decreasing\n", + "function.\n", + "\n", + "Note also that maximization/minimization of the logarithm of the PDF\n", + "is equivalent to the maximization/minimization of the function itself." + ] + }, + { + "cell_type": "markdown", + "id": "6ddc5457", + "metadata": { + "editable": true + }, + "source": [ + "## A new Cost Function\n", + "\n", + "We could now define a new cost function to minimize, namely the negative logarithm of the above PDF" + ] + }, + { + "cell_type": "markdown", + "id": "bb0f5795", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta}=-\\log{\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})}=-\\sum_{i=0}^{n-1}\\log{p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c969ba54", + "metadata": { + "editable": true + }, + "source": [ + "which becomes" + ] + }, + { + "cell_type": "markdown", + "id": "630a5421", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta}=\\frac{n}{2}\\log{2\\pi\\sigma^2}+\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "49969ea1", + "metadata": { + "editable": true + }, + "source": [ + "Taking the derivative of the *new* cost function with respect to the parameters $\\beta$ we recognize our familiar OLS equation, namely" + ] + }, + { + "cell_type": "markdown", + "id": "93b3d19b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right) =0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f61d06bf", + "metadata": { + "editable": true + }, + "source": [ + "which leads to the well-known OLS equation for the optimal paramters $\\beta$" + ] + }, + { + "cell_type": "markdown", + "id": "8c25aa29", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}!\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "add3ac78", + "metadata": { + "editable": true + }, + "source": [ + "Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics." + ] + }, + { + "cell_type": "markdown", + "id": "2cb62527", + "metadata": { + "editable": true + }, + "source": [ + "## More basic Statistics and Bayes' theorem\n", + "\n", + "A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry.\n", + "Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.\n", + "\n", + "Assume we have two domains of events $X=[x_0,x_1,\\dots,x_{n-1}]$ and $Y=[y_0,y_1,\\dots,y_{n-1}]$.\n", + "\n", + "We define also the likelihood for $X$ and $Y$ as $p(X)$ and $p(Y)$ respectively.\n", + "The likelihood of a specific event $x_i$ (or $y_i$) is then written as $p(X=x_i)$ or just $p(x_i)=p_i$. \n", + "\n", + "**Union of events is given by.**" + ] + }, + { + "cell_type": "markdown", + "id": "3704ebb3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X \\cup Y)= p(X)+p(Y)-p(X \\cap Y).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "76f1d92c", + "metadata": { + "editable": true + }, + "source": [ + "**The product rule (aka joint probability) is given by.**" + ] + }, + { + "cell_type": "markdown", + "id": "31ef87a3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X \\cup Y)= p(X,Y)= p(X\\vert Y)p(Y)=p(Y\\vert X)p(X),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9f87e53d", + "metadata": { + "editable": true + }, + "source": [ + "where we read $p(X\\vert Y)$ as the likelihood of obtaining $X$ given $Y$.\n", + "\n", + "If we have independent events then $p(X,Y)=p(X)p(Y)$." + ] + }, + { + "cell_type": "markdown", + "id": "b465eba1", + "metadata": { + "editable": true + }, + "source": [ + "## Marginal Probability\n", + "\n", + "The marginal probability is defined in terms of only one of the set of variables $X,Y$. For a discrete probability we have" + ] + }, + { + "cell_type": "markdown", + "id": "b522f15f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X)=\\sum_{i=0}^{n-1}p(X,Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert Y=y_i)p(Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert y_i)p(y_i).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a8bf634f", + "metadata": { + "editable": true + }, + "source": [ + "## Conditional Probability\n", + "\n", + "The conditional probability, if $p(Y) > 0$, is" + ] + }, + { + "cell_type": "markdown", + "id": "51e8bfb6", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)}=\\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "bfb3c814", + "metadata": { + "editable": true + }, + "source": [ + "## Bayes' Theorem\n", + "\n", + "If we combine the conditional probability with the marginal probability and the standard product rule, we have" + ] + }, + { + "cell_type": "markdown", + "id": "dfff0c7c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cad0634f", + "metadata": { + "editable": true + }, + "source": [ + "which we can rewrite as" + ] + }, + { + "cell_type": "markdown", + "id": "1de5e07a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X\\vert Y)= \\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}=\\frac{p(Y\\vert X)p(X)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "43f1714e", + "metadata": { + "editable": true + }, + "source": [ + "which is Bayes' theorem. It allows us to evaluate the uncertainty in in $X$ after we have observed $Y$. We can easily interchange $X$ with $Y$." + ] + }, + { + "cell_type": "markdown", + "id": "166406ab", + "metadata": { + "editable": true + }, + "source": [ + "## Interpretations of Bayes' Theorem\n", + "\n", + "The quantity $p(Y\\vert X)$ on the right-hand side of the theorem is\n", + "evaluated for the observed data $Y$ and can be viewed as a function of\n", + "the parameter space represented by $X$. This function is not\n", + "necesseraly normalized and is normally called the likelihood function.\n", + "\n", + "The function $p(X)$ on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.\n", + "\n", + "Let us try to illustrate Bayes' theorem through an example." + ] + }, + { + "cell_type": "markdown", + "id": "d547f8dd", + "metadata": { + "editable": true + }, + "source": [ + "## Example of Usage of Bayes' theorem\n", + "\n", + "Let us suppose that you are undergoing a series of mammography scans in\n", + "order to rule out possible breast cancer cases. We define the\n", + "sensitivity for a positive event by the variable $X$. It takes binary\n", + "values with $X=1$ representing a positive event and $X=0$ being a\n", + "negative event. We reserve $Y$ as a classification parameter for\n", + "either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a positive thing).\n", + "\n", + "We let $Y=1$ represent the the case of having breast cancer and $Y=0$ as not.\n", + "\n", + "Let us assume that if you have breast cancer, the test will be positive with a probability of $0.8$, that is we have" + ] + }, + { + "cell_type": "markdown", + "id": "f848f875", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X=1\\vert Y=1) =0.8.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8c506872", + "metadata": { + "editable": true + }, + "source": [ + "This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of $80\\%$ for having cancer.\n", + "It is however not correct, as the following Bayesian analysis shows." + ] + }, + { + "cell_type": "markdown", + "id": "317558b2", + "metadata": { + "editable": true + }, + "source": [ + "## Doing it correctly\n", + "\n", + "If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number.\n", + "Let us assume that the prior probability in the population as a whole is" + ] + }, + { + "cell_type": "markdown", + "id": "8f8f8c5c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(Y=1) =0.004.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "acb7607d", + "metadata": { + "editable": true + }, + "source": [ + "We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have" + ] + }, + { + "cell_type": "markdown", + "id": "c5866ceb", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(X=1\\vert Y=0) =0.1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e3648f6b", + "metadata": { + "editable": true + }, + "source": [ + "Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute" + ] + }, + { + "cell_type": "markdown", + "id": "0b2cef16", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(Y=1\\vert X=1)=\\frac{p(X=1\\vert Y=1)p(Y=1)}{p(X=1\\vert Y=1)p(Y=1)+p(X=1\\vert Y=0)p(Y=0)}=\\frac{0.8\\times 0.004}{0.8\\times 0.004+0.1\\times 0.996}=0.031.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "94233096", + "metadata": { + "editable": true + }, + "source": [ + "That is, in case of a positive test, there is only a $3\\%$ chance of having breast cancer!" + ] + }, + { + "cell_type": "markdown", + "id": "91284f6d", + "metadata": { + "editable": true + }, + "source": [ + "## Bayes' Theorem and Ridge and Lasso Regression\n", + "\n", + "Hitherto we have discussed Ridge and Lasso regression in terms of a\n", + "linear analysis. This may to many of you feel rather technical and\n", + "perhaps not that intuitive. The question is whether we can develop a\n", + "more intuitive way of understanding what Ridge and Lasso express.\n", + "\n", + "Before we proceed let us perform a Ridge, Lasso and OLS analysis of a polynomial fit." + ] + }, + { + "cell_type": "markdown", + "id": "3debb528", + "metadata": { + "editable": true + }, + "source": [ + "## Test Function for what happens with OLS, Ridge and Lasso\n", + "\n", + "We will play around with a study of the values for the optimal\n", + "parameters $\\boldsymbol{\\beta}$ using OLS, Ridge and Lasso regression. For\n", + "OLS, you will notice as function of the noise and polynomial degree,\n", + "that the parameters $\\beta$ will fluctuate from order to order in the\n", + "polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS.\n", + "\n", + "For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "a4cf2a42", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn import linear_model\n", + "\n", + "def R2(y_data, y_model):\n", + " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "# Make data set.\n", + "n = 10000\n", + "x = np.random.rand(n)\n", + "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)\n", + "\n", + "Maxpolydegree = 5\n", + "X = np.zeros((len(x),Maxpolydegree))\n", + "X[:,0] = 1.0\n", + "\n", + "for polydegree in range(1, Maxpolydegree):\n", + " for degree in range(polydegree):\n", + " X[:,degree] = x**(degree)\n", + "\n", + "\n", + "# We split the data in test and training data\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", + "\n", + "# matrix inversion to find beta\n", + "OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train\n", + "print(OLSbeta)\n", + "ypredictOLS = X_test @ OLSbeta\n", + "print(\"Test MSE OLS\")\n", + "print(MSE(y_test,ypredictOLS))\n", + "# Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn\n", + "# Decide which values of lambda to use\n", + "nlambdas = 4\n", + "MSERidgePredict = np.zeros(nlambdas)\n", + "MSELassoPredict = np.zeros(nlambdas)\n", + "lambdas = np.logspace(-3, 1, nlambdas)\n", + "for i in range(nlambdas):\n", + " lmb = lambdas[i]\n", + " # Make the fit using Ridge and Lasso\n", + " RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n", + " RegRidge.fit(X_train,y_train)\n", + " RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n", + " RegLasso.fit(X_train,y_train)\n", + " # and then make the prediction\n", + " ypredictRidge = RegRidge.predict(X_test)\n", + " ypredictLasso = RegLasso.predict(X_test)\n", + " # Compute the MSE and print it\n", + " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n", + " MSELassoPredict[i] = MSE(y_test,ypredictLasso)\n", + " print(lmb,RegRidge.coef_)\n", + " print(lmb,RegLasso.coef_)\n", + "# Now plot the results\n", + "plt.figure()\n", + "plt.plot(np.log10(lambdas), MSERidgePredict, 'b', label = 'MSE Ridge Test')\n", + "plt.plot(np.log10(lambdas), MSELassoPredict, 'r', label = 'MSE Lasso Test')\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('MSE')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "26270da6", + "metadata": { + "editable": true + }, + "source": [ + "How can we understand this?" + ] + }, + { + "cell_type": "markdown", + "id": "d8f0e20a", + "metadata": { + "editable": true + }, + "source": [ + "## Invoking Bayes' theorem\n", + "\n", + "Using Bayes' theorem we can gain a better intuition about Ridge and Lasso regression. \n", + "\n", + "For ordinary least squares we postulated that the maximum likelihood for the doamin of events $\\boldsymbol{D}$ (one-dimensional case)" + ] + }, + { + "cell_type": "markdown", + "id": "6c93bc4b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8c8b6cc8", + "metadata": { + "editable": true + }, + "source": [ + "is given by" + ] + }, + { + "cell_type": "markdown", + "id": "f76e3a40", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "af323d99", + "metadata": { + "editable": true + }, + "source": [ + "In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set $\\boldsymbol{\\beta}$ given a domain of events $\\boldsymbol{D}$? That is, how can we define the posterior probability" + ] + }, + { + "cell_type": "markdown", + "id": "49a0988c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "38568afe", + "metadata": { + "editable": true + }, + "source": [ + "Bayes' theorem comes to our rescue here since (omitting the normalization constant)" + ] + }, + { + "cell_type": "markdown", + "id": "c3d0eeb9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})\\propto p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})p(\\boldsymbol{\\beta}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1d0ba152", + "metadata": { + "editable": true + }, + "source": [ + "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!" + ] + }, + { + "cell_type": "markdown", + "id": "f936c1ce", + "metadata": { + "editable": true + }, + "source": [ + "## Ridge and Bayes\n", + "\n", + "With the posterior probability defined by a likelihood which we have\n", + "already modeled and an unknown prior, we are now ready to make\n", + "additional models for the prior.\n", + "\n", + "We can, based on our discussions of the variance of $\\boldsymbol{\\beta}$ and the mean value, assume that the prior for the values $\\boldsymbol{\\beta}$ is given by a Gaussian with mean value zero and variance $\\tau^2$, that is" + ] + }, + { + "cell_type": "markdown", + "id": "dd10cea6", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2534eb1d", + "metadata": { + "editable": true + }, + "source": [ + "Our posterior probability becomes then (omitting the normalization factor which is just a constant)" + ] + }, + { + "cell_type": "markdown", + "id": "104af119", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{\\beta\\vert\\boldsymbol{D})}=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5e8e89cf", + "metadata": { + "editable": true + }, + "source": [ + "We can now optimize this quantity with respect to $\\boldsymbol{\\beta}$. As we\n", + "did for OLS, this is most conveniently done by taking the negative\n", + "logarithm of the posterior probability. Doing so and leaving out the\n", + "constants terms that do not depend on $\\beta$, we have" + ] + }, + { + "cell_type": "markdown", + "id": "467910cd", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{2\\tau^2}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "16f15265", + "metadata": { + "editable": true + }, + "source": [ + "and replacing $1/2\\tau^2$ with $\\lambda$ we have" + ] + }, + { + "cell_type": "markdown", + "id": "1cd22464", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "90676d1e", + "metadata": { + "editable": true + }, + "source": [ + "which is our Ridge cost function! Nice, isn't it?" + ] + }, + { + "cell_type": "markdown", + "id": "2082d057", + "metadata": { + "editable": true + }, + "source": [ + "## Lasso and Bayes\n", + "\n", + "To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution ([Laplace in this case](https://en.wikipedia.org/wiki/Laplace_distribution)) with zero mean value, that is" + ] + }, + { + "cell_type": "markdown", + "id": "544b15d3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "592804c7", + "metadata": { + "editable": true + }, + "source": [ + "Our posterior probability becomes then (omitting the normalization factor which is just a constant)" + ] + }, + { + "cell_type": "markdown", + "id": "c11e6e06", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "fdc6e9b5", + "metadata": { + "editable": true + }, + "source": [ + "Taking the negative\n", + "logarithm of the posterior probability and leaving out the\n", + "constants terms that do not depend on $\\beta$, we have" + ] + }, + { + "cell_type": "markdown", + "id": "a0e8571d", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{\\tau}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "84eca415", + "metadata": { + "editable": true + }, + "source": [ + "and replacing $1/\\tau$ with $\\lambda$ we have" + ] + }, + { + "cell_type": "markdown", + "id": "0924d8d4", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7270fef7", + "metadata": { + "editable": true + }, + "source": [ + "which is our Lasso cost function!" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/LectureNotes/_build/html/chapter1.html b/doc/LectureNotes/_build/html/chapter1.html index aa83d5f68..743ba1f0e 100644 --- a/doc/LectureNotes/_build/html/chapter1.html +++ b/doc/LectureNotes/_build/html/chapter1.html @@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output" Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression +
  • + + Exercises week 36 + +
    • +
  • + + Week 36: Statistical interpretation of Linear Regression and Resampling techniques + +
    • @@ -984,13 +994,13 @@ example of the functionality of Scikit-Learn.

    The intercept alpha: 
- [2.11366595]
+ [2.05177695]
 Coefficient beta : 
- [[4.95136998]]
+ [[5.05971574]]
 Mean squared error: 0.28
-Variance score: 0.89
+Variance score: 0.88
 Mean squared log error: 0.01
-Mean absolute error: 0.41
+Mean absolute error: 0.45
    _images/chapter1_19_1.png @@ -1090,7 +1100,7 @@ a linear \(x\)-dependence we s
    _images/chapter1_33_0.png -
    0.004999999999999996
+
    0.004999999999999997
    diff --git a/doc/LectureNotes/_build/html/chapter10.html b/doc/LectureNotes/_build/html/chapter10.html index dd6974658..8421d3a58 100644 --- a/doc/LectureNotes/_build/html/chapter10.html +++ b/doc/LectureNotes/_build/html/chapter10.html @@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output" Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression +
  • + + Exercises week 36 + +
    • +
  • + + Week 36: Statistical interpretation of Linear Regression and Resampling techniques + +
    • @@ -1301,7 +1311,7 @@ the Hadamard product, meaning element-wise multiplication.

    Old accuracy on training data: 0.1440501043841336
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1635,7 +1645,7 @@ Lambda =  10.0
 Accuracy score on test set:  0.19166666666666668
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1644,7 +1654,7 @@ Lambda =  1e-05
 Accuracy score on test set:  0.10555555555555556
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1653,7 +1663,7 @@ Lambda =  0.0001
 Accuracy score on test set:  0.08611111111111111
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1662,7 +1672,7 @@ Lambda =  0.001
 Accuracy score on test set:  0.10555555555555556
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1671,7 +1681,7 @@ Lambda =  0.01
 Accuracy score on test set:  0.08888888888888889
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1680,7 +1690,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.08611111111111111
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1689,7 +1699,7 @@ Lambda =  1.0
 Accuracy score on test set:  0.08888888888888889
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1698,11 +1708,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.09166666666666666
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1711,11 +1721,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1724,11 +1734,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1737,11 +1747,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1750,11 +1760,11 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1763,7 +1773,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -1772,11 +1782,11 @@ Lambda =  1.0
 Accuracy score on test set:  0.10555555555555556
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1785,11 +1795,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1798,11 +1808,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1811,11 +1821,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1824,11 +1834,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1837,11 +1847,11 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1850,11 +1860,11 @@ Lambda =  0.1
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1863,11 +1873,11 @@ Lambda =  1.0
 Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
    
 
    
@@ -1920,15 +1930,15 @@ Accuracy score on test set:  0.07777777777777778
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
    
 
    
@@ -2187,27 +2197,26 @@ Accuracy score on test set:  0.9861111111111112
 
    Learning rate  =  0.01
 Lambda =  0.1
 Accuracy score on test set:  0.9888888888888889
+
+Learning rate  =  0.01
+Lambda =  1.0
+Accuracy score on test set:  0.9722222222222222
 
    
 
    
 
    Learning rate  =  0.01
-Lambda =  1.0
-Accuracy score on test set:  0.9722222222222222
-
-Learning rate  =  0.01
 Lambda =  10.0
 Accuracy score on test set:  0.9527777777777777
-
    
-
    
-
    Learning rate  =  0.1
-Lambda =  1e-05
-Accuracy score on test set:  0.9027777777777778
 
 Learning rate  =  0.1
-Lambda =  0.0001
-Accuracy score on test set:  0.8583333333333333
+Lambda =  1e-05
+Accuracy score on test set:  0.9027777777777778
 
    
 
    
 
    Learning rate  =  0.1
+Lambda =  0.0001
+Accuracy score on test set:  0.8583333333333333
+
+Learning rate  =  0.1
 Lambda =  0.001
 Accuracy score on test set:  0.8722222222222222
 
    
@@ -2219,31 +2228,30 @@ Accuracy score on test set:  0.9055555555555556
 Learning rate  =  0.1
 Lambda =  0.1
 Accuracy score on test set:  0.8805555555555555
+
+Learning rate  =  0.1
+Lambda =  1.0
+Accuracy score on test set:  0.8722222222222222
 
    Learning rate  =  0.1
-Lambda =  1.0
-Accuracy score on test set:  0.8722222222222222
-
-Learning rate  =  0.1
 Lambda =  10.0
 Accuracy score on test set:  0.8666666666666667
-
    -
    -
    Learning rate  =  1.0
+
+Learning rate  =  1.0
 Lambda =  1e-05
 Accuracy score on test set:  0.08611111111111111
 
 Learning rate  =  1.0
 Lambda =  0.0001
 Accuracy score on test set:  0.10555555555555556
-
-Learning rate  =  1.0
-Lambda =  0.001
-Accuracy score on test set:  0.10555555555555556
 
    Learning rate  =  1.0
+Lambda =  0.001
+Accuracy score on test set:  0.10555555555555556
+
+Learning rate  =  1.0
 Lambda =  0.01
 Accuracy score on test set:  0.17777777777777778
 
@@ -2263,13 +2271,13 @@ Accuracy score on test set:  0.09444444444444444
 Learning rate  =  10.0
 Lambda =  1e-05
 Accuracy score on test set:  0.17222222222222222
+
+Learning rate  =  10.0
+Lambda =  0.0001
+Accuracy score on test set:  0.11666666666666667
 
    Learning rate  =  10.0
-Lambda =  0.0001
-Accuracy score on test set:  0.11666666666666667
-
-Learning rate  =  10.0
 Lambda =  0.001
 Accuracy score on test set:  0.10555555555555556
 
diff --git a/doc/LectureNotes/_build/html/chapter11.html b/doc/LectureNotes/_build/html/chapter11.html
index 767cc874d..6f0a465b2 100644
--- a/doc/LectureNotes/_build/html/chapter11.html
+++ b/doc/LectureNotes/_build/html/chapter11.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    •
    @@ -2610,43 +2620,6 @@ Using TensorFlow results in a much better execution time. Try it!

    19 x = tuple(args[i] for i in argnum) ---> 20 return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs) -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py:57, in jacobian(fun, x) - 47 @unary_to_nary - 48 def jacobian(fun, x): - 49 """ - 50 Returns a function which computes the Jacobian of `fun` with respect to - 51 positional argument number `argnum`, which must be a scalar or array. Unlike - (...) - 55 (out1, out2, ...) then the Jacobian has shape (out1, out2, ..., in1, in2, ...). - 56 """ ----> 57 vjp, ans = _make_vjp(fun, x) - 58 ans_vspace = vspace(ans) - 59 jacobian_shape = ans_vspace.shape + vspace(x).shape - -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:10, in make_vjp(fun, x) - 8 def make_vjp(fun, x): - 9 start_node = VJPNode.new_root() ----> 10 end_value, end_node = trace(start_node, fun, x) - 11 if end_node is None: - 12 def vjp(g): return vspace(x).zeros() - -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py:10, in trace(start_node, fun, x) - 8 with trace_stack.new_trace() as t: - 9 start_box = new_box(x, t, start_node) ----> 10 end_box = fun(start_box) - 11 if isbox(end_box) and end_box._trace == start_box._trace: - 12 return end_box._value, end_box._node - -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py:15, in unary_to_nary.<locals>.nary_operator.<locals>.nary_f.<locals>.unary_f(x) - 13 else: - 14 subargs = subvals(args, zip(argnum, x)) ----> 15 return fun(*subargs, **kwargs) - -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py:20, in unary_to_nary.<locals>.nary_operator.<locals>.nary_f(*args, **kwargs) - 18 else: - 19 x = tuple(args[i] for i in argnum) ----> 20 return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs) - File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py:61, in jacobian(fun, x) 59 jacobian_shape = ans_vspace.shape + vspace(x).shape 60 grads = map(vjp, ans_vspace.standard_basis()) @@ -2680,23 +2653,31 @@ Using TensorFlow results in a much better execution time. Try it!

    22 for parent, ingrad in zip(node.parents, ingrads): 23 outgrads[parent] = add_outgrads(outgrads.get(parent), ingrad) -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:67, in defvjp.<locals>.vjp_argnums.<locals>.<lambda>(g) - 64 raise NotImplementedError( - 65 "VJP of {} wrt argnum 0 not defined".format(fun.__name__)) - 66 vjp = vjpfun(ans, *args, **kwargs) ----> 67 return lambda g: (vjp(g),) - 68 elif L == 2: - 69 argnum_0, argnum_1 = argnums +File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:78, in defvjp.<locals>.vjp_argnums.<locals>.<lambda>(g) + 76 vjp_0 = vjp_0_fun(ans, *args, **kwargs) + 77 vjp_1 = vjp_1_fun(ans, *args, **kwargs) +---> 78 return lambda g: (vjp_0(g), vjp_1(g)) + 79 else: + 80 vjps = [vjps_dict[argnum](ans, *args, **kwargs) for argnum in argnums] -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:82, in <lambda>(g) - 80 defvjp(anp.log10, lambda ans, x : lambda g: g / x / anp.log(10)) - 81 defvjp(anp.log1p, lambda ans, x : lambda g: g / (x + 1)) ----> 82 defvjp(anp.sin, lambda ans, x : lambda g: g * anp.cos(x)) - 83 defvjp(anp.cos, lambda ans, x : lambda g: - g * anp.sin(x)) - 84 defvjp(anp.tan, lambda ans, x : lambda g: g / anp.cos(x) **2) +File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:660, in unbroadcast_f.<locals>.<lambda>(g) + 658 def unbroadcast_f(target, f): + 659 target_meta = anp.metadata(target) +--> 660 return lambda g: unbroadcast(f(g), target_meta) -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_boxes.py:27, in ArrayBox.__mul__(self, other) ----> 27 def __mul__(self, other): return anp.multiply(self, other) +File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:35, in <lambda>(g) + 30 # ----- Binary ufuncs ----- + 32 defvjp(anp.add, lambda ans, x, y : unbroadcast_f(x, lambda g: g), + 33 lambda ans, x, y : unbroadcast_f(y, lambda g: g)) + 34 defvjp(anp.multiply, lambda ans, x, y : unbroadcast_f(x, lambda g: y * g), +---> 35 lambda ans, x, y : unbroadcast_f(y, lambda g: x * g)) + 36 defvjp(anp.subtract, lambda ans, x, y : unbroadcast_f(x, lambda g: g), + 37 lambda ans, x, y : unbroadcast_f(y, lambda g: -g)) + 38 defvjp(anp.divide, lambda ans, x, y : unbroadcast_f(x, lambda g: g / y), + 39 lambda ans, x, y : unbroadcast_f(y, lambda g: - g * x / y**2)) + +File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_boxes.py:35, in ArrayBox.__rmul__(self, other) +---> 35 def __rmul__(self, other): return anp.multiply(other, self) File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py:45, in primitive.<locals>.f_wrapped(*args, **kwargs) 43 argnums = tuple(argnum for argnum, _ in boxed_args) @@ -2711,29 +2692,13 @@ Using TensorFlow results in a much better execution time. Try it!

    35 .format(fun_name, parent_argnums)) ---> 36 self.vjp = vjpmaker(parent_argnums, value, args, kwargs) -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:76, in defvjp.<locals>.vjp_argnums(argnums, ans, args, kwargs) - 73 except KeyError: - 74 raise NotImplementedError( - 75 "VJP of {} wrt argnums 0, 1 not defined".format(fun.__name__)) ----> 76 vjp_0 = vjp_0_fun(ans, *args, **kwargs) - 77 vjp_1 = vjp_1_fun(ans, *args, **kwargs) - 78 return lambda g: (vjp_0(g), vjp_1(g)) - -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:34, in <lambda>(ans, x, y) - 30 # ----- Binary ufuncs ----- - 32 defvjp(anp.add, lambda ans, x, y : unbroadcast_f(x, lambda g: g), - 33 lambda ans, x, y : unbroadcast_f(y, lambda g: g)) ----> 34 defvjp(anp.multiply, lambda ans, x, y : unbroadcast_f(x, lambda g: y * g), - 35 lambda ans, x, y : unbroadcast_f(y, lambda g: x * g)) - 36 defvjp(anp.subtract, lambda ans, x, y : unbroadcast_f(x, lambda g: g), - 37 lambda ans, x, y : unbroadcast_f(y, lambda g: -g)) - 38 defvjp(anp.divide, lambda ans, x, y : unbroadcast_f(x, lambda g: g / y), - 39 lambda ans, x, y : unbroadcast_f(y, lambda g: - g * x / y**2)) - -File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:659, in unbroadcast_f(target, f) - 658 def unbroadcast_f(target, f): ---> 659 target_meta = anp.metadata(target) - 660 return lambda g: unbroadcast(f(g), target_meta) +File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:56, in defvjp.<locals>.vjp_argnums(argnums, ans, args, kwargs) + 53 argnums = kwargs.get('argnums', count()) + 54 vjps_dict = {argnum : translate_vjp(vjpmaker, fun, argnum) + 55 for argnum, vjpmaker in zip(argnums, vjpmakers)} +---> 56 def vjp_argnums(argnums, ans, args, kwargs): + 57 L = len(argnums) + 58 # These first two cases are just optimizations KeyboardInterrupt:
    diff --git a/doc/LectureNotes/_build/html/chapter12.html b/doc/LectureNotes/_build/html/chapter12.html index c85ed04ce..08ef4f16d 100644 --- a/doc/LectureNotes/_build/html/chapter12.html +++ b/doc/LectureNotes/_build/html/chapter12.html @@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output" Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression +
  • + + Exercises week 36 + +
    • +
  • + + Week 36: Statistical interpretation of Linear Regression and Resampling techniques + +
    • diff --git a/doc/LectureNotes/_build/html/chapter13.html b/doc/LectureNotes/_build/html/chapter13.html index 430d7f1de..6b8dd6b5b 100644 --- a/doc/LectureNotes/_build/html/chapter13.html +++ b/doc/LectureNotes/_build/html/chapter13.html @@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output" Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression +
  • + + Exercises week 36 + +
    • +
  • + + Week 36: Statistical interpretation of Linear Regression and Resampling techniques + +
    • diff --git a/doc/LectureNotes/_build/html/chapter2.html b/doc/LectureNotes/_build/html/chapter2.html index 20579950c..224ea1380 100644 --- a/doc/LectureNotes/_build/html/chapter2.html +++ b/doc/LectureNotes/_build/html/chapter2.html @@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output" Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression +
  • + + Exercises week 36 + +
    • +
  • + + Week 36: Statistical interpretation of Linear Regression and Resampling techniques + +
    • @@ -1233,10 +1243,10 @@ covariance matrix through the np.linalg.eig() function.

    -
    0.008947823579448178
-4.14822021080294
-[[0.73947737 2.16006961]
- [2.16006961 7.24782786]]
+
    0.01354598394614281
+4.037503978471253
+[[0.95927495 2.85834701]
+ [2.85834701 9.70233292]]
    @@ -1273,10 +1283,10 @@ a more brute force way. Here we scale the mean values for each column of the des
    -
    0.08657894597048958
-2.219222942590059
-[[1.        0.6343356]
- [0.6343356 1.       ]]
+
    0.09309965420717024
+1.6546329613199204
+[[1.         0.57867898]
+ [0.57867898 1.        ]]
    @@ -1306,30 +1316,30 @@ this matrix we easily see that it is a positive definite matrix.

    -
    [[ 0.29724806  0.26804287]
- [-0.15626984 -1.36853738]
- [-0.77070756 -2.13536532]
- [-0.45372697 -3.1582408 ]
- [ 0.52580392  2.72567956]
- [-0.86515815 -1.35704388]
- [-0.73738602 -2.12933164]
- [-0.10486183  1.06292011]
- [ 1.75670484  5.27381733]
- [ 0.50835355  0.81805914]]
+
    [[-0.49352496 -2.38394242]
+ [ 0.18849928  0.73454039]
+ [-1.58104393 -5.16350368]
+ [ 0.34695383  0.23472346]
+ [ 0.95953339  2.95819409]
+ [ 1.31331481  3.59914165]
+ [ 0.14846308  1.1180677 ]
+ [ 0.26022531  0.23496851]
+ [-0.12178678 -0.11868087]
+ [-1.02063403 -1.21350883]]
           0         1
-0  0.297248  0.268043
-1 -0.156270 -1.368537
-2 -0.770708 -2.135365
-3 -0.453727 -3.158241
-4  0.525804  2.725680
-5 -0.865158 -1.357044
-6 -0.737386 -2.129332
-7 -0.104862  1.062920
-8  1.756705  5.273817
-9  0.508354  0.818059
+0 -0.493525 -2.383942
+1  0.188499  0.734540
+2 -1.581044 -5.163504
+3  0.346954  0.234723
+4  0.959533  2.958194
+5  1.313315  3.599142
+6  0.148463  1.118068
+7  0.260225  0.234969
+8 -0.121787 -0.118681
+9 -1.020634 -1.213509
           0         1
-0  1.000000  0.915549
-1  0.915549  1.000000
+0  1.000000  0.949087
+1  0.949087  1.000000
    @@ -1386,37 +1396,37 @@ this matrix we easily see that it is a positive definite matrix.

         0         1         2         3         4         5         6         7   \
 0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
-1   0.0  0.086925  0.087377  0.087520  0.087878  0.088164  0.079680  0.079580   
-2   0.0  0.087377  0.089240  0.088074  0.089063  0.089924  0.079827  0.080060   
-3   0.0  0.087520  0.088074  0.094227  0.094252  0.094139  0.089441  0.088974   
-4   0.0  0.087878  0.089063  0.094252  0.094610  0.094812  0.089040  0.088778   
-5   0.0  0.088164  0.089924  0.094139  0.094812  0.095315  0.088488  0.088425   
-6   0.0  0.079680  0.079827  0.089441  0.089040  0.088488  0.087315  0.086524   
-7   0.0  0.079580  0.080060  0.088974  0.088778  0.088425  0.086524  0.085876   
-8   0.0  0.079499  0.080295  0.088502  0.088506  0.088348  0.085716  0.085210   
-9   0.0  0.079448  0.080548  0.088037  0.088238  0.088275  0.084906  0.084541   
-10  0.0  0.071751  0.071438  0.082839  0.082089  0.081194  0.082509  0.081481   
-11  0.0  0.071392  0.071284  0.082139  0.081533  0.080780  0.081559  0.080641   
-12  0.0  0.071071  0.071164  0.081471  0.081007  0.080395  0.080635  0.079827   
-13  0.0  0.070794  0.071084  0.080839  0.080518  0.080048  0.079741  0.079043   
-14  0.0  0.070562  0.071051  0.080249  0.080070  0.079745  0.078880  0.078293   
+1   0.0  0.072254  0.074732  0.075050  0.075400  0.075727  0.068297  0.068215   
+2   0.0  0.074732  0.078265  0.076259  0.077107  0.077954  0.068418  0.068622   
+3   0.0  0.075050  0.076259  0.082604  0.082322  0.081988  0.078133  0.077663   
+4   0.0  0.075400  0.077107  0.082322  0.082315  0.082269  0.077387  0.077091   
+5   0.0  0.075727  0.077954  0.081988  0.082269  0.082525  0.076570  0.076454   
+6   0.0  0.068297  0.068418  0.078133  0.077387  0.076570  0.075952  0.075224   
+7   0.0  0.068215  0.068622  0.077663  0.077091  0.076454  0.075224  0.074613   
+8   0.0  0.068168  0.068876  0.077210  0.076818  0.076371  0.074495  0.074004   
+9   0.0  0.068163  0.069190  0.076775  0.076573  0.076324  0.073764  0.073398   
+10  0.0  0.060924  0.060401  0.071600  0.070601  0.069522  0.071006  0.070141   
+11  0.0  0.060711  0.060369  0.071122  0.070241  0.069283  0.070364  0.069583   
+12  0.0  0.060534  0.060381  0.070671  0.069911  0.069080  0.069738  0.069042   
+13  0.0  0.060394  0.060441  0.070245  0.069612  0.068912  0.069125  0.068517   
+14  0.0  0.060291  0.060550  0.069845  0.069343  0.068782  0.068524  0.068007   
 
           8         9         10        11        12        13        14  
 0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
-1   0.079499  0.079448  0.071751  0.071392  0.071071  0.070794  0.070562  
-2   0.080295  0.080548  0.071438  0.071284  0.071164  0.071084  0.071051  
-3   0.088502  0.088037  0.082839  0.082139  0.081471  0.080839  0.080249  
-4   0.088506  0.088238  0.082089  0.081533  0.081007  0.080518  0.080070  
-5   0.088348  0.088275  0.081194  0.080780  0.080395  0.080048  0.079745  
-6   0.085716  0.084906  0.082509  0.081559  0.080635  0.079741  0.078880  
-7   0.085210  0.084541  0.081481  0.080641  0.079827  0.079043  0.078293  
-8   0.084685  0.084157  0.080434  0.079704  0.078999  0.078326  0.077688  
-9   0.084157  0.083772  0.079379  0.078759  0.078165  0.077604  0.077079  
-10  0.080434  0.079379  0.079152  0.078033  0.076935  0.075863  0.074818  
-11  0.079704  0.078759  0.078033  0.077004  0.075996  0.075014  0.074061  
-12  0.078999  0.078165  0.076935  0.075996  0.075079  0.074187  0.073325  
-13  0.078326  0.077604  0.075863  0.075014  0.074187  0.073388  0.072618  
-14  0.077688  0.077079  0.074818  0.074061  0.073325  0.072618  0.071942  
+1   0.068168  0.068163  0.060924  0.060711  0.060534  0.060394  0.060291  
+2   0.068876  0.069190  0.060401  0.060369  0.060381  0.060441  0.060550  
+3   0.077210  0.076775  0.071600  0.071122  0.070671  0.070245  0.069845  
+4   0.076818  0.076573  0.070601  0.070241  0.069911  0.069612  0.069343  
+5   0.076371  0.076324  0.069522  0.069283  0.069080  0.068912  0.068782  
+6   0.074495  0.073764  0.071006  0.070364  0.069738  0.069125  0.068524  
+7   0.074004  0.073398  0.070141  0.069583  0.069042  0.068517  0.068007  
+8   0.073520  0.073044  0.069265  0.068792  0.068339  0.067905  0.067489  
+9   0.073044  0.072705  0.068375  0.067990  0.067628  0.067288  0.066969  
+10  0.069265  0.068375  0.067400  0.066672  0.065952  0.065237  0.064526  
+11  0.068792  0.067990  0.066672  0.066006  0.065350  0.064701  0.064057  
+12  0.068339  0.067628  0.065952  0.065350  0.064759  0.064176  0.063600  
+13  0.067905  0.067288  0.065237  0.064701  0.064176  0.063661  0.063155  
+14  0.067489  0.066969  0.064526  0.064057  0.063600  0.063155  0.062721
    diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html index 07507e971..907f755d4 100644 --- a/doc/LectureNotes/_build/html/chapter3.html +++ b/doc/LectureNotes/_build/html/chapter3.html @@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output" Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression +
  • + + Exercises week 36 + +
    • +
  • + + Week 36: Statistical interpretation of Linear Regression and Resampling techniques + +
    • @@ -787,10 +797,10 @@ number \(i\) is left out. Usin
    -
    Runtime: 0.0907831 sec
+
    Runtime: 0.0907788 sec
 Jackknife Statistics :
 original           bias      std. error
- 100.142        100.132        0.149864
+ 100.022        100.012        0.148734
    @@ -1009,7 +1019,7 @@ theorem.

    Bootstrap Statistics :
 original           bias      std. error
- 100.033  14.9292        100.032        0.149452
+ 99.7522  14.9594        99.7525        0.149991
    @@ -1591,9 +1601,9 @@ Mean squared error on training data: 0.00060705 Mean squared error on test data: 3250.17647619
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label='Test Error')
 
    
 
    
@@ -1827,7 +1837,7 @@ cross-validation (LOOCV).
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
 
    
 
    
@@ -2716,9 +2726,9 @@ linear system as an equation would reduce this down to
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
@@ -2862,9 +2872,9 @@ with the form utilized in linear regression, viz.
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
@@ -2904,9 +2914,9 @@ cost function is given by
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
@@ -2941,9 +2951,9 @@ cost function is given by
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
@@ -3005,34 +3015,34 @@ constant as opposed to ridge and OLS. We get a sparse solution with
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    100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:02<00:00,  7.30it/s]
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    100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:02<00:00,  7.32it/s]
 
    
 
    
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    100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:02<00:00,  4.68it/s]
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@@ -3179,9 +3189,9 @@ which polynomial fits the data best.
    
 
    
 
    
 
    
-
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3980313467.py:9: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3980313467.py:9: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
   ax = fig.gca(projection='3d')
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3980313467.py:37: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3980313467.py:37: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   fig.colorbar(surf, shrink=0.5, aspect=5)
 
    
 
    
diff --git a/doc/LectureNotes/_build/html/chapter4.html b/doc/LectureNotes/_build/html/chapter4.html
index bb3c8e5f9..0b527a199 100644
--- a/doc/LectureNotes/_build/html/chapter4.html
+++ b/doc/LectureNotes/_build/html/chapter4.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/chapter5.html b/doc/LectureNotes/_build/html/chapter5.html
index eecfa0f14..399f156ce 100644
--- a/doc/LectureNotes/_build/html/chapter5.html
+++ b/doc/LectureNotes/_build/html/chapter5.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/chapter6.html b/doc/LectureNotes/_build/html/chapter6.html
index 79fd9ac75..f7124abd2 100644
--- a/doc/LectureNotes/_build/html/chapter6.html
+++ b/doc/LectureNotes/_build/html/chapter6.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
@@ -715,9 +725,9 @@ predicting the target features of query instances is as follows:
    
 
    
 
    
 
    2nd degree coefficients:
-zero power:  -4.653578701904388
-first power:  0.17297886491529482
-second power:  -0.0007790285013223805
+zero power:  -6.548110376991839
+first power:  0.2232822462117919
+second power:  -0.0007480407244119591
 
    
 
    
 _images/chapter6_1_1.png
diff --git a/doc/LectureNotes/_build/html/chapter7.html b/doc/LectureNotes/_build/html/chapter7.html
index 735ae8dbe..eee3e2dd3 100644
--- a/doc/LectureNotes/_build/html/chapter7.html
+++ b/doc/LectureNotes/_build/html/chapter7.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/chapter8.html b/doc/LectureNotes/_build/html/chapter8.html
index c25b2f5eb..465d13b8e 100644
--- a/doc/LectureNotes/_build/html/chapter8.html
+++ b/doc/LectureNotes/_build/html/chapter8.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
@@ -669,10 +679,10 @@ covariance matrix through the np.linalg.eig() function.
    
 
    
 
    
 
    
-
    0.264540221699101
-4.673457751724773
-[[0.83632853 2.54078623]
- [2.54078623 8.44021223]]
+
    0.26662339374864535
+4.736115211426478
+[[ 1.20561803  3.63264564]
+ [ 3.63264564 12.10207647]]
 
    
 
    
 
    
@@ -712,10 +722,10 @@ a more brute force way. Here we scale the mean values for each column of the des
 
    
 
    
 
    
-
    0.08374032367704139
-1.7696835316227453
-[[1.         0.66443521]
- [0.66443521 1.        ]]
+
    0.09318696260700278
+1.9096305360355206
+[[1.         0.65907898]
+ [0.65907898 1.        ]]
 
    
 
    
 
    
@@ -744,30 +754,30 @@ this matrix we easily see that it is a positive definite matrix.
    
 
    
 
    
 
    
-
    [[ 1.20708879  4.33201844]
- [-0.3838783  -1.24125217]
- [ 0.74722409  1.60194224]
- [-0.04086326  0.37419664]
- [ 0.62422045  2.05587489]
- [ 1.75357263  5.63710438]
- [-2.53968309 -7.28219089]
- [-1.42054337 -4.90629812]
- [-0.13019959 -0.83320794]
- [ 0.18306165  0.26181254]]
+
    [[ 0.32360769  2.53264317]
+ [ 0.04531375 -0.70569833]
+ [ 0.17001895 -0.49570819]
+ [ 1.60938882  4.69896355]
+ [ 0.14052537  1.37535105]
+ [-0.12909917  1.25781559]
+ [-0.03016916  0.01780471]
+ [-0.38816656 -0.82894017]
+ [-0.34591885 -3.1893772 ]
+ [-1.39550083 -4.66285417]]
           0         1
-0  1.207089  4.332018
-1 -0.383878 -1.241252
-2  0.747224  1.601942
-3 -0.040863  0.374197
-4  0.624220  2.055875
-5  1.753573  5.637104
-6 -2.539683 -7.282191
-7 -1.420543 -4.906298
-8 -0.130200 -0.833208
-9  0.183062  0.261813
+0  0.323608  2.532643
+1  0.045314 -0.705698
+2  0.170019 -0.495708
+3  1.609389  4.698964
+4  0.140525  1.375351
+5 -0.129099  1.257816
+6 -0.030169  0.017805
+7 -0.388167 -0.828940
+8 -0.345919 -3.189377
+9 -1.395501 -4.662854
           0         1
-0  1.000000  0.992504
-1  0.992504  1.000000
+0  1.000000  0.899734
+1  0.899734  1.000000
 
    
 
    
 
    
@@ -824,37 +834,37 @@ this matrix we easily see that it is a positive definite matrix.
    
 
    
 
         0         1         2         3         4         5         6         7   \
 0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
-1   0.0  0.092290  0.091376  0.091772  0.091659  0.091579  0.083691  0.083321   
-2   0.0  0.091376  0.091209  0.090620  0.090923  0.091266  0.082166  0.082105   
-3   0.0  0.091772  0.090620  0.098069  0.097435  0.096845  0.093556  0.092724   
-4   0.0  0.091659  0.090923  0.097435  0.097111  0.096833  0.092478  0.091896   
-5   0.0  0.091579  0.091266  0.096845  0.096833  0.096872  0.091445  0.091113   
-6   0.0  0.083691  0.082166  0.093556  0.092478  0.091445  0.092007  0.090828   
-7   0.0  0.083321  0.082105  0.092724  0.091896  0.091113  0.090828  0.089857   
-8   0.0  0.083051  0.082145  0.091996  0.091418  0.090888  0.089744  0.088982   
-9   0.0  0.082877  0.082285  0.091369  0.091044  0.090768  0.088754  0.088201   
-10  0.0  0.075829  0.073985  0.087419  0.086021  0.084663  0.087860  0.086440   
-11  0.0  0.075277  0.073684  0.086460  0.085272  0.084124  0.086624  0.085384   
-12  0.0  0.074819  0.073477  0.085603  0.084624  0.083685  0.085484  0.084422   
-13  0.0  0.074453  0.073363  0.084846  0.084076  0.083348  0.084438  0.083555   
-14  0.0  0.074180  0.073344  0.084187  0.083627  0.083111  0.083485  0.082779   
+1   0.0  0.086358  0.084977  0.084028  0.085456  0.086705  0.073624  0.075225   
+2   0.0  0.084977  0.085778  0.080455  0.082791  0.085242  0.069320  0.071314   
+3   0.0  0.084028  0.080455  0.086647  0.086999  0.086904  0.078715  0.079856   
+4   0.0  0.085456  0.082791  0.086999  0.087848  0.088361  0.078426  0.079839   
+5   0.0  0.086705  0.085242  0.086904  0.088361  0.089641  0.077606  0.079333   
+6   0.0  0.073624  0.069320  0.078715  0.078426  0.077606  0.073298  0.074046   
+7   0.0  0.075225  0.071314  0.079856  0.079839  0.079333  0.074046  0.074971   
+8   0.0  0.076867  0.073465  0.080914  0.081223  0.081100  0.074653  0.075779   
+9   0.0  0.078521  0.075782  0.081827  0.082527  0.082877  0.075047  0.076403   
+10  0.0  0.063766  0.059453  0.069855  0.069291  0.068201  0.066209  0.066728   
+11  0.0  0.065175  0.061037  0.071105  0.070700  0.069782  0.067233  0.067873   
+12  0.0  0.066656  0.062742  0.072370  0.072150  0.071437  0.068239  0.069013   
+13  0.0  0.068206  0.064577  0.073634  0.073630  0.073163  0.069203  0.070127   
+14  0.0  0.069816  0.066556  0.074868  0.075121  0.074951  0.070092  0.071185   
 
           8         9         10        11        12        13        14  
 0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
-1   0.083051  0.082877  0.075829  0.075277  0.074819  0.074453  0.074180  
-2   0.082145  0.082285  0.073985  0.073684  0.073477  0.073363  0.073344  
-3   0.091996  0.091369  0.087419  0.086460  0.085603  0.084846  0.084187  
-4   0.091418  0.091044  0.086021  0.085272  0.084624  0.084076  0.083627  
-5   0.090888  0.090768  0.084663  0.084124  0.083685  0.083348  0.083111  
-6   0.089744  0.088754  0.087860  0.086624  0.085484  0.084438  0.083485  
-7   0.088982  0.088201  0.086440  0.085384  0.084422  0.083555  0.082779  
-8   0.088317  0.087746  0.085108  0.084230  0.083446  0.082756  0.082158  
-9   0.087746  0.087386  0.083859  0.083159  0.082553  0.082040  0.081620  
-10  0.085108  0.083859  0.085252  0.083832  0.082501  0.081258  0.080098  
-11  0.084230  0.083159  0.083832  0.082569  0.081395  0.080305  0.079298  
-12  0.083446  0.082553  0.082501  0.081395  0.080374  0.079437  0.078582  
-13  0.082756  0.082040  0.081258  0.080305  0.079437  0.078652  0.077949  
-14  0.082158  0.081620  0.080098  0.079298  0.078582  0.077949  0.077397  
+1   0.076867  0.078521  0.063766  0.065175  0.066656  0.068206  0.069816  
+2   0.073465  0.075782  0.059453  0.061037  0.062742  0.064577  0.066556  
+3   0.080914  0.081827  0.069855  0.071105  0.072370  0.073634  0.074868  
+4   0.081223  0.082527  0.069291  0.070700  0.072150  0.073630  0.075121  
+5   0.081100  0.082877  0.068201  0.069782  0.071437  0.073163  0.074951  
+6   0.074653  0.075047  0.066209  0.067233  0.068239  0.069203  0.070092  
+7   0.075779  0.076403  0.066728  0.067873  0.069013  0.070127  0.071185  
+8   0.076819  0.077713  0.067089  0.068365  0.069654  0.070936  0.072186  
+9   0.077713  0.078926  0.067224  0.068643  0.070096  0.071567  0.073038  
+10  0.067089  0.067224  0.060603  0.061468  0.062298  0.063070  0.063750  
+11  0.068365  0.068643  0.061468  0.062424  0.063353  0.064232  0.065030  
+12  0.069654  0.070096  0.062298  0.063353  0.064390  0.065389  0.066317  
+13  0.070936  0.071567  0.063070  0.064232  0.065389  0.066519  0.067593  
+14  0.072186  0.073038  0.063750  0.065030  0.066317  0.067593  0.068832  
 
    
 
    
 
    
@@ -1043,10 +1053,10 @@ We can write our own code or simply use either the functionaly of numpy<
 
    
 
    
 
              0         1
-0  3.956454  1.972286
-1  1.972286  1.977089
-[[3.95645365 1.97228638]
- [1.97228638 1.97708897]]
+0  4.059118  2.009163
+1  2.009163  2.004788
+[[4.05911793 2.00916336]
+ [2.00916336 2.00478786]]
 
    
 
    
 
    
@@ -1073,8 +1083,8 @@ Our own code here is not very elegant and asks for obvious improvements. It is t
 
    
 
    
 
    Centered covariance using own code
-[[3.95645365 1.97228638]
- [1.97228638 1.97708897]]
+[[4.05911793 2.00916336]
+ [2.00916336 2.00478786]]
 
    
 
    
 _images/chapter8_65_1.png
@@ -1134,16 +1144,16 @@ questions.
    
 
    
 
    
 
    Eigenvalues of Covariance matrix
-5.173439546289586
-0.7601030735620569
+5.288455813429108
+0.7754499790100834
 First eigenvector
-[0.85102768 0.52512084]
+[0.85299536 0.52191849]
 Second eigenvector
-[-0.52512084  0.85102768]
+[-0.52191849  0.85299536]
 
    
 
    
 
    Eigenvector of largest eigenvalue
-[-0.85102768 -0.52512084]
+[0.85299536 0.52191849]
 
    
 
    
 
    
diff --git a/doc/LectureNotes/_build/html/chapter9.html b/doc/LectureNotes/_build/html/chapter9.html
index 66dc4a971..4bca6f831 100644
--- a/doc/LectureNotes/_build/html/chapter9.html
+++ b/doc/LectureNotes/_build/html/chapter9.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/chapteroptimization.html b/doc/LectureNotes/_build/html/chapteroptimization.html
index 09ade64da..048ba187e 100644
--- a/doc/LectureNotes/_build/html/chapteroptimization.html
+++ b/doc/LectureNotes/_build/html/chapteroptimization.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
@@ -965,11 +975,11 @@ which equals
    
 
    
 
    
 
    
-
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20669/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16261/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
   ax = fig.gca(projection="3d")
 
    
 
    
-
    <mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x1336a6040>
+
    <mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x138098070>
 
    
 
    
 _images/chapteroptimization_61_2.png
@@ -1027,7 +1037,7 @@ which equals
    
 
    
 
    
 
    
-
    [<matplotlib.lines.Line2D at 0x133c2e1c0>]
+
    [<matplotlib.lines.Line2D at 0x1387541c0>]
 
    
 
    
 _images/chapteroptimization_69_1.png
@@ -1284,11 +1294,11 @@ when \(||\nabla_\beta C(\beta_k) || \
 
    
 
    
 
    
-
    [0.29972182 4.52744746]
-[[4.01247056]
- [2.97656972]]
-[[4.01247056]
- [2.97656972]]
+
    [0.28836053 4.52113415]
+[[4.19528375]
+ [2.90383424]]
+[[4.19528375]
+ [2.90383424]]
 
    
 
    
 _images/chapteroptimization_123_1.png
@@ -1317,9 +1327,9 @@ when \(||\nabla_\beta C(\beta_k) || \
 
    
 
    
 
    
-
    [[4.02158709]
- [2.93023603]]
-[4.00882596] [2.93522293]
+
    [[4.16575256]
+ [2.8620652 ]]
+[4.11520281] [2.85097049]
 
    
 
    
 
    
@@ -1390,10 +1400,10 @@ C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta|
 
    
 
    
 
    
-
    [[4.09781386]
- [2.97980332]]
-[[4.0527437 ]
- [3.01510929]]
+
    [[4.20793824]
+ [2.75460639]]
+[[4.106971  ]
+ [2.83724637]]
 
    
 
    
 _images/chapteroptimization_132_1.png
@@ -1643,15 +1653,15 @@ function.
    
 
    
 
    
 
    Own inversion
-[[3.87618586]
- [3.13847924]]
-Eigenvalues of Hessian Matrix:[0.3313155  4.62759057]
+[[4.31347523]
+ [2.69915639]]
+Eigenvalues of Hessian Matrix:[0.28457442 4.43693489]
 theta from own gd
-[[3.87618586]
- [3.13847924]]
+[[4.31347523]
+ [2.69915639]]
 theta from own sdg
-[[3.87379129]
- [3.15508406]]
+[[4.2891298 ]
+ [2.67783138]]
 
    
 
    
 _images/chapteroptimization_148_1.png
diff --git a/doc/LectureNotes/_build/html/clustering.html b/doc/LectureNotes/_build/html/clustering.html
index 5a39179c8..077fbf904 100644
--- a/doc/LectureNotes/_build/html/clustering.html
+++ b/doc/LectureNotes/_build/html/clustering.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/exercisesweek34.html b/doc/LectureNotes/_build/html/exercisesweek34.html
index febe54b5d..06c5ea5d0 100644
--- a/doc/LectureNotes/_build/html/exercisesweek34.html
+++ b/doc/LectureNotes/_build/html/exercisesweek34.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/exercisesweek35.html b/doc/LectureNotes/_build/html/exercisesweek35.html
index 31b481daa..b9dd5219b 100644
--- a/doc/LectureNotes/_build/html/exercisesweek35.html
+++ b/doc/LectureNotes/_build/html/exercisesweek35.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/exercisesweek36.html b/doc/LectureNotes/_build/html/exercisesweek36.html
new file mode 100644
index 000000000..cc93d03c9
--- /dev/null
+++ b/doc/LectureNotes/_build/html/exercisesweek36.html
@@ -0,0 +1,622 @@
+
+
+
+
+  
+    
+    
+    Exercises week 36 — Applied Data Analysis and Machine Learning
+    
+  
+  
+
+    
+  
+  
+  
+
+    
+      
+
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+  
+
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+    
+
+    
+    
+  
+  
+    
+    
+
+    
+
+    
    
+      
    
+          
+
+
+
+          
+
+
+          
+
    
+    
+    
    
+    
    
+        
    
+        
    
+            
+            
+            
+            
+
    
+    
+
+    
    
+        
+        
+        
+        
+        
+        
+    
    
+
    
+
+            
+
+            
+
+
+
+            
+
+        
    
+
+        
+        
    
+            
+            
    
+                 Contents
+            
    
+            
+        
    
+    
    
+
    
+    
    
+        
    
+            
+            
    
+                
    Exercises week 36
    
+                
+                
    
+                    
+                
    
+            
    
+            
+              
    
+                
+  
+
    
+
    Exercises week 36
    
+
    September 4-8, 2023
    
+
    Date: Deadline is Sunday September 10 at midnight
    
+
    
+
    Overarching aims of the exercises this week
    
+
    This set of exercises form an important part of the first project. The
+analytical exercises deal with the material covered last week on the
+mathematical interpretations of ordinary least squares and of Ridge
+regression. The numerical exercises can be seen as a continuation of
+exercise 3 from week 35, with the inclusion of Ridge regression. This
+material enters also the discussions of the first project.
    
+
    
+
    
+
    Exercise 1: Analytical exercises
    
+
    The aim here is to derive the expression for the optimal parameters
+using Ridge regression. Furthermore, using the singular value
+decomposition, we will analyze the difference between the ordinary
+least squares approach and Ridge regression.
    
+
    The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the
+optimization problem
    
+
    
+\[
+{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
+\]
    
+
    which we can also write as
    
+
    
+\[
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
+\]
    
+
    where we have used the definition of  a norm-2 vector, that is
    
+
    
+\[
+\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
+\]
    
+
    By minimizing the above equation with respect to the parameters
+\(\boldsymbol{\beta}\) we could then obtain an analytical expression for the
+parameters \(\boldsymbol{\beta}\).
    
+
    We can add a regularization parameter \(\lambda\) by
+defining a new cost function to be optimized, that is
    
+
    
+\[
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
+\]
    
+
    which leads to the Ridge regression minimization problem. One can require as part of the optimization problem
+that \(\vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t\), where \(t\) is
+a finite number larger than zero. We will not implement that here.
    
+
    
+
    a) Expression for Ridge regression
    
+
    Show that the optimal parameters
    
+
    
+\[
+\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+\]
    
+
    with \(\boldsymbol{I}\) being a \(p\times p\) identity matrix with the constraint that
    
+
    
+\[
+\sum_{i=0}^{p-1} \beta_i^2 \leq t,
+\]
    
+
    with \(t\) a finite positive number.
    
+
    The ordinary least squares result is
    
+
    
+\[
+\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+\]
    
+
    
+
    
+
    b) The singular value decomposition
    
+
    Use the singular value decomposition of an n\times p\( matrix \)\boldsymbol{X}$ (our design matrix)
    
+
    
+\[
+\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
+\]
    
+
    where \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are orthogonal matrices of dimensions
+\(n\times n\) and \(p\times p\), respectively, and \(\boldsymbol{\Sigma}\) is an
+\(n\times p\) matrix which contains the ingular values only. This material was discussed during the lectures of week 35.
    
+
    Show that you can write the
+OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix  \(\boldsymbol{U}\) as
    
+
    
+\[
+\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta}  = \sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}.
+\]
    
+
    For Ridge regression, show that the corresponding equation is
    
+
    
+\[
+\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
+\]
    
+
    with the vectors \(\boldsymbol{u}_j\) being the columns of \(\boldsymbol{U}\) from the SVD of the matrix \(\boldsymbol{X}\).
    
+
    Give an interpretation of the results.  Section 3.4 of Hastie et al’s textbook gives a good discussion of the above results.
    
+
    
+
    
+
    
+
    Exercise 2: Adding Ridge Regression
    
+
    This exercise is a continuation of exercise 3 from week 35, see https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html. We will use the same function to
+generate our data set, still staying with a simple function \(y(x)\)
+which we want to fit using linear regression, but now extending the
+analysis to include the Ridge regression method.
    
+
    In this exercise you need to include the same elements from last week, that is
    
+
      
+
    1. scale your data by subtracting the mean value from each column in the design matrix.
      2. 
+
    2. perform a split of the data in a training set and a test set.
      3. 
+
    
+
    The addition to the analysis this time is the introduction of the hyperparameter \(\lambda\) when introducing Ridge regression.
    
+
    Extend the code from exercise 3 from week 35 to include Ridge regression with the hyperparameter \(\lambda\). The optimal parameters \(\hat{\beta}\) for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations
    
+
    
+\[
+\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+\]
    
+
    The ordinary least squares result you encoded last week is given by
    
+
    
+\[
+\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+\]
    
+
    Use these results to compute the mean squared error for ordinary least
+squares and Ridge regression first for a polynomial of degree five
+with \(n=100\) data points and five selected values of
+\(\lambda=[0.0001,0.001, 0.01,0.1,1.0]\). Compute thereafter the mean
+squared error for the same values of \(\lambda\) for polynomials of degree ten
+and \(15\). Discuss your results for the training MSE and test MSE with
+Ridge regression and ordinary least squares.
    
+
    
+
    
+
+    
+    
+
+              
    
+              
+            
+                
+
     
+    
+        
+        
    
+            
    previous
    
+            
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
    
+        
    
+        
+    
+    
    
+        
    next
    
+        
    Week 36: Statistical interpretation of Linear Regression and Resampling techniques
    
+    
    
+    
+        
+
    
+            
+        
    
+    
    
+    
    
+  
    
+    
+      By Morten Hjorth-Jensen
    
+    
+        © Copyright 2021.
    
+  
    
+
    
+
    
+
+
+      
    
+    
    
+  
+  
+
+  
+
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/genindex.html b/doc/LectureNotes/_build/html/genindex.html
index 10e0e9656..34a146f90 100644
--- a/doc/LectureNotes/_build/html/genindex.html
+++ b/doc/LectureNotes/_build/html/genindex.html
@@ -264,6 +264,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/intro.html b/doc/LectureNotes/_build/html/intro.html
index b1f405215..a56260d38 100644
--- a/doc/LectureNotes/_build/html/intro.html
+++ b/doc/LectureNotes/_build/html/intro.html
@@ -265,6 +265,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/linalg.html b/doc/LectureNotes/_build/html/linalg.html
index c3a1e0757..eaabfff04 100644
--- a/doc/LectureNotes/_build/html/linalg.html
+++ b/doc/LectureNotes/_build/html/linalg.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
@@ -571,8 +581,8 @@ matrices and vectors.
    
 
    
 
    
 
    
-
    [-0.38763091 -2.70501534 -0.3581571  -0.96251494 -1.26223899  0.35309734
-  2.28186376 -1.85104809 -0.37114298 -1.20893188]
+
    [-0.87136737  1.4300745  -0.3322326  -0.66758934 -1.00283636 -0.27625974
+  1.89249454 -0.25006757  0.73565195  0.33697405]
 
    
 
    
 
    
@@ -793,26 +803,26 @@ as (recall that we user lowercase letters for vectors and uppercase letters for
 
    
 
    
 
    
-
    [[0.78459267 0.75453081 0.05363779 0.57350724 0.69764852 0.65795279
-  0.51507839 0.20461136 0.38788697 0.96496641]
- [0.25028968 0.96081861 0.18931988 0.51108791 0.30337713 0.43036842
-  0.52839842 0.15321987 0.78561443 0.09030825]
- [0.10097956 0.50584526 0.34989509 0.55626454 0.69154964 0.2895238
-  0.13393141 0.15503141 0.26015755 0.42902155]
- [0.25789255 0.9492866  0.90252116 0.904221   0.51933924 0.14432948
-  0.54445121 0.02699523 0.18657863 0.971688  ]
- [0.13392097 0.27801122 0.50931378 0.04234339 0.22442417 0.44065609
-  0.74943449 0.42451192 0.33736485 0.97952271]
- [0.95824108 0.59950055 0.91346044 0.58042237 0.13228567 0.31519573
-  0.12427889 0.64736858 0.60236782 0.18036103]
- [0.95911004 0.82027884 0.27547877 0.84317815 0.89842298 0.68322599
-  0.02377668 0.39943328 0.00162091 0.0525221 ]
- [0.94076632 0.88335933 0.75492292 0.7860324  0.41923956 0.86181269
-  0.45979894 0.44190304 0.07829878 0.00458014]
- [0.42915006 0.68127341 0.23875722 0.31988705 0.54956992 0.24801014
-  0.65335632 0.97713364 0.05635863 0.12160173]
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@@ -872,13 +882,13 @@ covariance matrix through the np.linalg.eig() function.
    
 
    
 
    
 
    
-
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diff --git a/doc/LectureNotes/_build/html/objects.inv b/doc/LectureNotes/_build/html/objects.inv
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diff --git a/doc/LectureNotes/_build/html/schedule.html b/doc/LectureNotes/_build/html/schedule.html
index 1d7fb3865..a8fe16c41 100644
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+ 
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+ 
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+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
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diff --git a/doc/LectureNotes/_build/html/search.html b/doc/LectureNotes/_build/html/search.html
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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations  with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","1. 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations  with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling 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\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/statistics.html b/doc/LectureNotes/_build/html/statistics.html
index ba447d4b5..a45121dc4 100644
--- a/doc/LectureNotes/_build/html/statistics.html
+++ b/doc/LectureNotes/_build/html/statistics.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
@@ -943,37 +953,27 @@ uncorrelated.
    
 
    
 
    
 
    
-
    3.0818000034712947
-[[1.82949718e-02 3.31658671e-01 3.68886089e-01 6.49666527e-01
-  1.49052317e-01 5.97305528e-01 4.57810898e-01 2.51578522e-01
-  2.89396164e-01 5.90342093e-01]
- [3.31658671e-01 6.01244295e+00 6.68731669e+00 1.17774184e+01
-  2.70208087e+00 1.08281969e+01 8.29938171e+00 4.56071752e+00
-  5.24629107e+00 1.07019610e+01]
- [3.68886089e-01 6.68731669e+00 7.43794243e+00 1.30993886e+01
-  3.00537912e+00 1.20436207e+01 9.23095558e+00 5.07264063e+00
-  5.83516719e+00 1.19032152e+01]
- [6.49666527e-01 1.17774184e+01 1.30993886e+01 2.30700873e+01
-  5.29294618e+00 2.12107137e+01 1.62571673e+01 8.93371943e+00
-  1.02766489e+01 2.09634375e+01]
- [1.49052317e-01 2.70208087e+00 3.00537912e+00 5.29294618e+00
-  1.21435514e+00 4.86635201e+00 3.72986500e+00 2.04965397e+00
-  2.35776087e+00 4.80961969e+00]
- [5.97305528e-01 1.08281969e+01 1.20436207e+01 2.12107137e+01
-  4.86635201e+00 1.95011994e+01 1.49468927e+01 8.21369083e+00
-  9.44838453e+00 1.92738529e+01]
- [4.57810898e-01 8.29938171e+00 9.23095558e+00 1.62571673e+01
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-  7.24181044e+00 1.47726407e+01]
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-  2.04965397e+00 8.21369083e+00 6.29546690e+00 3.45951629e+00
-  3.97955570e+00 8.11793498e+00]
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-  4.57776818e+00 9.33823452e+00]
- [5.90342093e-01 1.07019610e+01 1.19032152e+01 2.09634375e+01
-  4.80961969e+00 1.92738529e+01 1.47726407e+01 8.11793498e+00
-  9.33823452e+00 1.90491568e+01]]
+
    1.9474044318317307
+[[16.27404277  4.07430684  4.90189742  9.80462258  9.27412395  4.40264255
+   5.15060311 17.7265799  -0.62769765  9.00658717]
+ [ 4.07430684  1.02002781  1.22722021  2.45464765  2.32183405  1.10222868
+   1.28948521  4.43795846 -0.15714797  2.25485457]
+ [ 4.90189742  1.22722021  1.47649841  2.95324615  2.79345488  1.32611807
+   1.55141095  5.33941551 -0.18906854  2.71287025]
+ [ 9.80462258  2.45464765  2.95324615  5.90699099  5.58738148  2.65246006
+   3.10308387 10.67973263 -0.37816901  5.42619859]
+ [ 9.27412395  2.32183405  2.79345488  5.58738148  5.28506507  2.50894343
+   2.93518534 10.10188442 -0.35770742  5.13260331]
+ [ 4.40264255  1.10222868  1.32611807  2.65246006  2.50894343  1.19105386
+   1.39340081  4.79559972 -0.16981204  2.43656628]
+ [ 5.15060311  1.28948521  1.55141095  3.10308387  2.93518534  1.39340081
+   1.63012429  5.61031939 -0.19866124  2.85051211]
+ [17.7265799   4.43795846  5.33941551 10.67973263 10.10188442  4.79559972
+   5.61031939 19.308763   -0.68372271  9.8104687 ]
+ [-0.62769765 -0.15714797 -0.18906854 -0.37816901 -0.35770742 -0.16981204
+  -0.19866124 -0.68372271  0.0242106  -0.3473884 ]
+ [ 9.00658717  2.25485457  2.71287025  5.42619859  5.13260331  2.43656628
+   2.85051211  9.8104687  -0.3473884   4.98453971]]
 
    
 
    
 
    
@@ -1241,15 +1241,15 @@ more practically oriented methods like the blocking technique.
    
 
    
 
    
 
    
-
    -0.054244842835462305
-4.000854409696581
-0.13083543199018746
-0.8437169762110144 8.948675162607389 10.317825933186352
-2.601583341274718 2.1245596497124075 6.443538568902246
-[[ 0.84371698  2.60158334  2.12455965]
- [ 2.60158334  8.94867516  6.44353857]
- [ 2.12455965  6.44353857 10.31782593]]
-[16.80422999  0.07128434  3.23470374]
+
    0.08549632935144091
+4.438111004052204
+0.15485374225770068
+0.8225325933960339 7.887392472315955 7.062734874360835
+2.364244035362507 1.814860456269878 5.4421034379901165
+[[0.82253259 2.36424404 1.81486046]
+ [2.36424404 7.88739247 5.44210344]
+ [1.81486046 5.44210344 7.06273487]]
+[13.62123034  0.09662862  2.05480098]
 
    
 
    
 
    
@@ -1579,7 +1579,7 @@ assumption for approximating \(\sigma
 
    
 
    
 
    
-
    -0.048547423739546604 0.9959293935368551
+
    0.029574060388349064 0.9577775794806141
 
    
 
    
 _images/statistics_188_1.png
diff --git a/doc/LectureNotes/_build/html/teachers.html b/doc/LectureNotes/_build/html/teachers.html
index 361da019d..bebb743b2 100644
--- a/doc/LectureNotes/_build/html/teachers.html
+++ b/doc/LectureNotes/_build/html/teachers.html
@@ -266,6 +266,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/textbooks.html b/doc/LectureNotes/_build/html/textbooks.html
index 8176c6e2c..d41a9ad55 100644
--- a/doc/LectureNotes/_build/html/textbooks.html
+++ b/doc/LectureNotes/_build/html/textbooks.html
@@ -266,6 +266,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
diff --git a/doc/LectureNotes/_build/html/week34.html b/doc/LectureNotes/_build/html/week34.html
index b0278d538..69b9b0409 100644
--- a/doc/LectureNotes/_build/html/week34.html
+++ b/doc/LectureNotes/_build/html/week34.html
@@ -268,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
@@ -1733,8 +1743,8 @@ developed in the 1970s, namely EISPACK and LINPACK.  We describe them shortly he
 
    
 
    
 
    
-
    [-0.03483677 -1.06136773  0.56481272 -0.35947075 -1.64972151 -1.58590795
- -1.17428293  0.7225597  -1.14061742 -1.22893751]
+
    [-0.75408649 -0.05177092 -0.07339251  1.49372893 -1.21467644  0.24712854
+  0.67554302 -0.45143018  0.34212496  0.63262164]
 
    
 
    
 
    
@@ -1959,26 +1969,26 @@ lowercase letters for vectors and uppercase letters for matrices)
    
 
    
 
    
 
    
-
    [[0.24279008 0.63112036 0.80947943 0.97509292 0.19425617 0.3957482
-  0.1655226  0.83760781 0.07995375 0.76400155]
- [0.75621895 0.20893514 0.93082503 0.79419162 0.02783644 0.21296315
-  0.64298419 0.34578026 0.60975366 0.46369869]
- [0.77859437 0.23477043 0.35438626 0.63115792 0.2460037  0.35568525
-  0.0825971  0.94117118 0.14900336 0.30035718]
- [0.36692356 0.78972773 0.67655635 0.67160204 0.80108096 0.31507591
-  0.21328866 0.41340248 0.3005849  0.40672425]
- [0.21922061 0.88274486 0.86572911 0.06486061 0.07565581 0.26678445
-  0.03265139 0.22090974 0.33135331 0.66973261]
- [0.7221662  0.96941962 0.39707147 0.24929083 0.31531613 0.33079801
-  0.06538944 0.42352791 0.94227931 0.27809912]
- [0.07195822 0.31719317 0.47248297 0.18264218 0.64033527 0.51146442
-  0.49545491 0.91936525 0.81656508 0.78329097]
- [0.58666142 0.01646892 0.11029323 0.67442363 0.6914791  0.87902877
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- [0.25253892 0.57505712 0.24848907 0.9631064  0.46312791 0.96431281
-  0.28744729 0.09772449 0.17676228 0.51656406]
- [0.68664664 0.63281351 0.62806444 0.36809474 0.98129668 0.62914124
-  0.37885034 0.6577093  0.57947706 0.24688732]]
+
    [[0.11167579 0.88778091 0.24908246 0.32566092 0.37585988 0.61893735
+  0.58574143 0.60081665 0.81065392 0.00509527]
+ [0.02840884 0.39600607 0.96810393 0.28213741 0.19237495 0.54219245
+  0.85581869 0.17928538 0.43185254 0.37624303]
+ [0.82045496 0.22828694 0.52773891 0.54526769 0.84708672 0.59118929
+  0.18221254 0.94640767 0.25328924 0.03192893]
+ [0.98328759 0.68996179 0.81168086 0.08086066 0.10659682 0.66599246
+  0.78035464 0.57403561 0.96995249 0.78990036]
+ [0.9909606  0.39920034 0.59813773 0.85982436 0.66665241 0.60522226
+  0.08889302 0.74433499 0.85630009 0.2745333 ]
+ [0.60826013 0.94413158 0.32305813 0.27011455 0.38636467 0.02568505
+  0.0610602  0.92275042 0.09189927 0.67177934]
+ [0.72097126 0.35819312 0.98060869 0.46573851 0.70482138 0.51749473
+  0.17012048 0.53752634 0.66726905 0.08874264]
+ [0.79974359 0.05023564 0.84150451 0.05491301 0.90643349 0.75505597
+  0.7300372  0.57445493 0.39626502 0.22016672]
+ [0.44463741 0.01448144 0.01167264 0.85595734 0.09989861 0.81007682
+  0.30844205 0.68075712 0.9067362  0.96317459]
+ [0.87327422 0.12086283 0.50504743 0.60715594 0.9440887  0.54041662
+  0.7316348  0.42976761 0.63933182 0.22772039]]
 
    
 
    
 
    
@@ -2033,13 +2043,13 @@ covariance matrix through the np.linalg.eig() function.
    
 
    
 
    
 
    
-
    -0.19542686939553625
-3.3929303193461995
--0.3886211842840971
-[[ 0.96552318  2.67354673  2.55640013]
- [ 2.67354673  8.3318361   7.41646135]
- [ 2.55640013  7.41646135 11.19890453]]
-[18.10686259  0.09289204  2.29650918]
+
    -0.05279147850537411
+3.9698377974429513
+0.22403397554517984
+[[ 1.03173163  3.03859372  3.15084741]
+ [ 3.03859372  9.96514849  9.33589475]
+ [ 3.15084741  9.33589475 15.3141149 ]]
+[23.20313883  0.08502083  3.02283536]
 
    
 
    
 
    
@@ -2262,7 +2272,7 @@ Name: Aragorn, dtype: object
 
    
 
    
 
    
-
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20702/1326197715.py:6: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16293/1326197715.py:6: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
   data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
 
    
 
    
diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html
index 49b3cd2b5..28f234655 100644
--- a/doc/LectureNotes/_build/html/week35.html
+++ b/doc/LectureNotes/_build/html/week35.html
@@ -55,6 +55,7 @@ const thebe_selector_output = ".output, .cell_output"
     
     
     
+    
     
     
     
@@ -267,6 +268,16 @@ const thebe_selector_output = ".output, .cell_output"
    Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
   
  
+ 
  • 
+  
+   Exercises week 36
+  
+ 
    • 
+ 
  • 
+  
+   Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+  
+ 
    • 
 
 
     
    
@@ -1613,7 +1624,7 @@ Since we are not using Scikit-Learn here we can define our own
 
    
 
    
 
    
-
    0.9949347794612255
+
    0.9959033816551833
 
    
 
    
 
    
@@ -1630,7 +1641,7 @@ Since we are not using Scikit-Learn here we can define our own
 
    
 
    
 
    
-
    0.01028995427580139
+
    0.009290411029763584
 
    
 
    
 
    
@@ -1645,31 +1656,23 @@ Since we are not using Scikit-Learn here we can define our own
 
    
 
    
 
    
-
    [3.27961873e-03 1.75320666e-02 3.97840267e-02 1.41369795e-02
- 1.12890007e-02 3.55765644e-02 1.34914442e-03 9.92463154e-03
- 3.86416637e-02 8.40062254e-03 1.58912508e-02 7.41091395e-02
- 2.28266792e-02 1.29662593e-03 1.47977394e-02 1.48877744e-02
- 3.81874188e-02 7.92322465e-02 3.43012535e-02 1.47732343e-02
- 2.59567145e-02 1.77816763e-02 1.07826171e-02 3.45241645e-02
- 3.01752962e-02 4.99180856e-03 2.49838167e-04 4.38432672e-03
- 5.93034176e-03 2.54644646e-02 1.76834212e-02 1.85785878e-02
- 3.70827971e-02 9.47008799e-03 6.04008203e-02 2.45865699e-02
- 2.43113284e-03 2.65643756e-02 1.40768645e-01 4.45840226e-03
- 1.23365145e-02 3.49001938e-02 3.65693278e-02 2.08966600e-02
- 1.45112450e-02 1.27524966e-02 7.26193455e-02 3.00129543e-02
- 5.95594445e-03 1.06826023e-02 1.32609560e-02 1.42729886e-02
- 2.87131961e-02 6.39892216e-02 4.19650622e-02 2.78272242e-02
- 1.19023414e-02 2.32474290e-02 5.36904006e-02 7.48259891e-03
- 2.74997869e-02 3.08549564e-02 4.12211324e-02 1.06229862e-02
- 4.41816468e-02 5.87890188e-04 4.41486312e-02 1.41083418e-02
- 5.35759569e-03 2.89869600e-02 2.20517254e-02 2.65534935e-02
- 1.04481443e-02 2.47355725e-02 1.00634741e-02 2.73185363e-02
- 1.13375152e-02 4.40236659e-02 1.20669787e-02 7.63258368e-02
- 1.13208672e-04 1.15746031e-02 9.91094169e-03 4.09789363e-02
- 5.98604688e-02 1.47082726e-02 1.54957609e-02 3.71921767e-02
- 2.85688243e-03 7.15752533e-04 4.77553346e-02 2.33370315e-03
- 3.30638512e-05 7.19284552e-02 5.16049171e-02 7.79000942e-03
- 2.70465373e-02 5.84696315e-02 5.47099765e-02 6.36462687e-02]
+
    [0.02126934 0.07602623 0.0014937  0.01294617 0.05421908 0.02237669
+ 0.0026081  0.00564956 0.00399247 0.05763394 0.00040953 0.06861085
+ 0.0098629  0.01199845 0.0097577  0.03260201 0.020964   0.01933058
+ 0.02255062 0.01801774 0.04039087 0.00472303 0.01003763 0.01487742
+ 0.05554042 0.00886682 0.05110883 0.02944194 0.00806407 0.01028231
+ 0.03613949 0.03352185 0.0512238  0.01525206 0.00660801 0.01073938
+ 0.06353697 0.00700232 0.0391902  0.08741274 0.01227458 0.01049472
+ 0.04691549 0.00963223 0.0143088  0.05177527 0.00850988 0.01121347
+ 0.02768957 0.02259051 0.02233576 0.01322543 0.02143332 0.01400329
+ 0.00102864 0.01322099 0.00611932 0.01011376 0.13281267 0.00684221
+ 0.05358851 0.02232779 0.00695738 0.03054765 0.00554475 0.05748797
+ 0.03507211 0.00563446 0.03123832 0.00033779 0.01122997 0.1098906
+ 0.07003926 0.03718926 0.0695405  0.00605451 0.0456042  0.00477722
+ 0.01224109 0.01072866 0.04273116 0.01873409 0.02903947 0.01927709
+ 0.00819724 0.00628788 0.00086553 0.02341603 0.0525063  0.03546779
+ 0.03012368 0.05069808 0.00327082 0.00517074 0.00071305 0.01194406
+ 0.05454172 0.02480935 0.00577016 0.02925853]
 
    
 
    
 
    
@@ -1738,15 +1741,15 @@ but now splitting the data into a training set and a test set.
    
 
    
 
    
 
    
-
    [ 1.96667129  1.00622755  0.25455904  7.36570403 -3.55965719]
+
    [ 2.04860436 -0.39444293  5.97533203 -0.78980112  0.09575221]
 Training R2
-0.9967912002709148
+0.9960913291755783
 Training MSE
-0.007098683295819775
+0.008823125370709272
 Test R2
-0.9957763445154583
+0.9906601091977738
 Test MSE
-0.008754614401844201
+0.01969004537138309
 
    
 
    
 
    
@@ -4221,6 +4224,13 @@ C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymb
             
    Exercises week 35
    
         
    
     
+    
+    
    
+        
    next
    
+        
    Exercises week 36
    
+    
    
+    
+        
 
    
             
         
    
diff --git a/doc/LectureNotes/_build/html/week36.html b/doc/LectureNotes/_build/html/week36.html
new file mode 100644
index 000000000..b7a101b2a
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+++ b/doc/LectureNotes/_build/html/week36.html
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+
+
+
+
+  
+    
+    
+    Week 36: Statistical interpretation of Linear Regression and Resampling techniques — Applied Data Analysis and Machine Learning
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+                
    Week 36: Statistical interpretation of Linear Regression and Resampling techniques
    
+                
+                
    
+                    
    
+                        
+                        
    
+                            
     Contents 
    
+                        
    
+                        
+                    
    
+                
    
+            
    
+            
+              
    
+                
+  
+
    
+
    Week 36: Statistical interpretation of Linear Regression and Resampling techniques
    
+
    Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
+
    Date: September 4-8, 2023
    
+
    
+
    Plans for week 36
    
+
      
+
    • Material for the active learning sessions on Tuesday and Wednesday
      
+
+
      • 
+
    • Material for the lecture on Thursday September 7
      
+
        
+
      • Linear Regression and links with Statistics, Resampling methods
        • 
+
      • Recommended Reading: Goodfellow et al chapter 3 on probability theory, see URL:””
        • 
+
      • See also Murphy, sections 2.4 (Gaussian distributions) and 3.2 (Bayesian Statistics, basis)
        • 
+
      
+
      • 
+
    
+
    
+
    
+
    Material for the active learning sessions Tuesday and Wednesday
    
+
    The material here contains a summary from last Week and discussion of SVD, Ridge and Lasso regression with examples
    
+
    
+
    
+
    Linear Regression and  the SVD
    
+
    We used the SVD to analyse the matrix to invert in ordinary lineat regression
    
+
    
+\[
+\boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T.
+\]
    
+
    Since the matrices here have dimension \(p\times p\), with \(p\) corresponding to the singular values, we defined last week the matrix
    
+
    
+\[\begin{split}
+\boldsymbol{\Sigma}^T\boldsymbol{\Sigma} = \begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0}\\ \end{bmatrix}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \\ \boldsymbol{0}\end{bmatrix},
+\end{split}\]
    
+
    where the tilde-matrix \(\tilde{\boldsymbol{\Sigma}}\) is a matrix of dimension \(p\times p\) containing only the singular values \(\sigma_i\), that is
    
+
    
+\[\begin{split}
+\tilde{\boldsymbol{\Sigma}}=\begin{bmatrix} \sigma_0 & 0 & 0 & \dots & 0 & 0 \\
+                                    0 & \sigma_1 & 0 & \dots & 0 & 0 \\
+				    0 & 0 & \sigma_2 & \dots & 0 & 0 \\
+				    0 & 0 & 0 & \dots & \sigma_{p-2} & 0 \\
+				    0 & 0 & 0 & \dots & 0 & \sigma_{p-1} \\
+\end{bmatrix},
+\end{split}\]
    
+
    meaning we can write
    
+
    
+\[
+\boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^2\boldsymbol{V}^T.
+\]
    
+
    Multiplying from the right with \(\boldsymbol{V}\) (using the orthogonality of \(\boldsymbol{V}\)) we get
    
+
    
+\[
+\left(\boldsymbol{X}^T\boldsymbol{X}\right)\boldsymbol{V}=\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^2.
+\]
    
+
    
+
    
+
    What does it mean?
    
+
    This means the vectors \(\boldsymbol{v}_i\) of the orthogonal matrix \(\boldsymbol{V}\)
+are the eigenvectors of the matrix \(\boldsymbol{X}^T\boldsymbol{X}\) with eigenvalues
+given by the singular values squared, that is
    
+
    
+\[
+\left(\boldsymbol{X}^T\boldsymbol{X}\right)\boldsymbol{v}_i=\boldsymbol{v}_i\sigma_i^2.
+\]
    
+
    In other words, each non-zero singular value of \(\boldsymbol{X}\) is a positive
+square root of an eigenvalue of \(\boldsymbol{X}^T\boldsymbol{X}\).  It means also that
+the columns of \(\boldsymbol{V}\) are the eigenvectors of
+\(\boldsymbol{X}^T\boldsymbol{X}\). Since we have ordered the singular values of
+\(\boldsymbol{X}\) in a descending order, it means that the column vectors
+\(\boldsymbol{v}_i\) are hierarchically ordered by how much correlation they
+encode from the columns of \(\boldsymbol{X}\).
    
+
    Note that these are also the eigenvectors and eigenvalues of the
+Hessian matrix.
    
+
    If we now recall the definition of the covariance matrix (not using
+Bessel’s correction) we have
    
+
    
+\[
+\boldsymbol{C}[\boldsymbol{X}]=\frac{1}{n}\boldsymbol{X}^T\boldsymbol{X},
+\]
    
+
    meaning that every squared non-singular value of \(\boldsymbol{X}\) divided by \(n\) (
+the number of samples) are the eigenvalues of the covariance
+matrix. Every singular value of \(\boldsymbol{X}\) is thus a positive square
+root of an eigenvalue of \(\boldsymbol{X}^T\boldsymbol{X}\). If the matrix \(\boldsymbol{X}\) is
+self-adjoint, the singular values of \(\boldsymbol{X}\) are equal to the
+absolute value of the eigenvalues of \(\boldsymbol{X}\).
    
+
    
+
    
+
    And finally  \(\boldsymbol{X}\boldsymbol{X}^T\)
    
+
    For \(\boldsymbol{X}\boldsymbol{X}^T\) we found
    
+
    
+\[
+\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T=\boldsymbol{U}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{U}^T.
+\]
    
+
    Since the matrices here have dimension \(n\times n\), we have
    
+
    
+\[\begin{split}
+\boldsymbol{\Sigma}\boldsymbol{\Sigma}^T = \begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \\ \boldsymbol{0}\\ \end{bmatrix}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}}  \boldsymbol{0}\\ \end{bmatrix}=\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix},
+\end{split}\]
    
+
    leading to
    
+
    
+\[\begin{split}
+\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}\boldsymbol{U}^T.
+\end{split}\]
    
+
    Multiplying with \(\boldsymbol{U}\) from the right gives us the eigenvalue problem
    
+
    
+\[\begin{split}
+(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U}=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}.
+\end{split}\]
    
+
    It means that the eigenvalues of \(\boldsymbol{X}\boldsymbol{X}^T\) are again given by
+the non-zero singular values plus now a series of zeros.  The column
+vectors of \(\boldsymbol{U}\) are the eigenvectors of \(\boldsymbol{X}\boldsymbol{X}^T\) and
+measure how much correlations are contained in the rows of \(\boldsymbol{X}\).
    
+
    Since we will mainly be interested in the correlations among the features
+of our data (the columns of \(\boldsymbol{X}\), the quantity of interest for us are the non-zero singular
+values and the column vectors of \(\boldsymbol{V}\).
    
+
    
+
    
+
    Code for SVD and Inversion of Matrices
    
+
    How do we use the SVD to invert a matrix \(\boldsymbol{X}^\boldsymbol{X}\) which is singular or near singular?
+The simple answer is to use the linear algebra function for pseudoinvers, that is
    
+
    
+
    
+
    Ainv = np.linlag.pinv(A)
+
    
+
    
+
    
+
    
+
    ---------------------------------------------------------------------------
+NameError                                 Traceback (most recent call last)
+Input In [1], in <cell line: 1>()
+----> 1 Ainv = np.linlag.pinv(A)
+
+NameError: name 'np' is not defined
+
    
+
    
+
    
+
    
+
    Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD.
    
+
    
+
    
+
    import numpy as np
+# SVD inversion
+def SVDinv(A):
+    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
+    SVD is numerically more stable than the inversion algorithms provided by
+    numpy and scipy.linalg at the cost of being slower.
+    '''
+    U, s, VT = np.linalg.svd(A)
+    print('test U')
+    print( (np.transpose(U) @ U - U @np.transpose(U)))
+    print('test VT')
+    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
+
+
+    D = np.zeros((len(U),len(VT)))
+    D = np.diag(s)
+    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
+    return np.matmul(V,np.matmul(invD,UT))
+
+
+#X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
+# Non-singular square matrix
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+print(X)
+A = np.transpose(X) @ X
+# Brute force inversion
+B = np.linalg.inv(A)  # here we could use np.linalg.pinv(A)
+C = SVDinv(A)
+print(np.abs(B-C))
+
    
+
    
+
    
+
    
+
    
+
    
+
    Inverse of Rectangular Matrix
    
+
    Although our matrix to invert \(\boldsymbol{X}^T\boldsymbol{X}\) is a square matrix, our matrix may be singular.
    
+
    The pseudoinverse is the generalization of the matrix inverse for square matrices to
+rectangular matrices where the number of rows and columns are not equal.
    
+
    It is also called the the Moore-Penrose Inverse after two independent discoverers of the method or the Generalized Inverse.
+It is used for the calculation of the inverse for singular or near singular matrices and for rectangular matrices.
    
+
    Using the SVD we can obtain the pseudoinverse of a matrix \(\boldsymbol{A}\) (labeled here as \(\boldsymbol{A}_{\mathrm{PI}}\))
    
+
    
+\[
+\boldsymbol{A}_{\mathrm{PI}}= \boldsymbol{V}\boldsymbol{D}_{\mathrm{PI}}\boldsymbol{U}^T,
+\]
    
+
    where \(\boldsymbol{D}_{\mathrm{PI}}\) can be calculated by creating a diagonal matrix from \(\boldsymbol{\Sigma}\) where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD.
    
+
    
+
    
+
    import numpy as np
+# SVD inversion
+def SVDinv(A):
+    U, s, VT = np.linalg.svd(A)
+    # reciprocals of singular values of s
+    d = 1.0 / s
+    # create m x n D matrix
+    D = np.zeros(A.shape)
+    # populate D with n x n diagonal matrix
+    D[:A.shape[1], :A.shape[1]] = np.diag(d)
+    UT = np.transpose(U)
+    V = np.transpose(VT)
+    return np.matmul(V,np.matmul(D.T,UT))
+
+
+A = np.array([ [0.3, 0.4], [0.5, 0.6], [0.7, 0.8],[0.9, 1.0]])
+print(A)
+# Brute force inversion of super-collinear matrix
+B = np.linalg.pinv(A)
+print(B)
+# Compare our own algorithm with pinv
+C = SVDinv(A)
+print(np.abs(C-B))
+
    
+
    
+
    
+
    
+
    As you can see from this example, our own decomposition based on the SVD agrees with  the pseudoinverse algorithm provided by Numpy.
    
+
    
+
    
+
    Ridge and LASSO Regression
    
+
    Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is
+our optimization problem is
    
+
    
+\[
+{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
+\]
    
+
    or we can state it as
    
+
    
+\[
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
+\]
    
+
    where we have used the definition of  a norm-2 vector, that is
    
+
    
+\[
+\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
+\]
    
+
    
+
    
+
    From OLS to Ridge and Lasso
    
+
    By minimizing the above equation with respect to the parameters
+\(\boldsymbol{\beta}\) we could then obtain an analytical expression for the
+parameters \(\boldsymbol{\beta}\).  We can add a regularization parameter \(\lambda\) by
+defining a new cost function to be optimized, that is
    
+
    
+\[
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
+\]
    
+
    which leads to the Ridge regression minimization problem where we
+require that \(\vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t\), where \(t\) is
+a finite number larger than zero. We do not include such a constraints in the discussions here.
    
+
    By defining
    
+
    
+\[
+C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,
+\]
    
+
    we have a new optimization equation
    
+
    
+\[
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1
+\]
    
+
    which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.
    
+
    Here we have defined the norm-1 as
    
+
    
+\[
+\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert.
+\]
    
+
    
+
    
+
    Deriving the  Ridge Regression Equations
    
+
    Using the matrix-vector expression for Ridge regression and dropping the parameter \(1/n\) in front of the standard means squared error equation, we have
    
+
    
+\[
+C(\boldsymbol{X},\boldsymbol{\beta})=\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta},
+\]
    
+
    and
+taking the derivatives with respect to \(\boldsymbol{\beta}\) we obtain then
+a slightly modified matrix inversion problem which for finite values
+of \(\lambda\) does not suffer from singularity problems. We obtain
+the optimal parameters
    
+
    
+\[
+\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+\]
    
+
    with \(\boldsymbol{I}\) being a \(p\times p\) identity matrix with the constraint that
    
+
    
+\[
+\sum_{i=0}^{p-1} \beta_i^2 \leq t,
+\]
    
+
    with \(t\) a finite positive number.
    
+
    
+
    
+
    Note on Scikit-Learn
    
+
    Note well that a library like Scikit-Learn does not include the \(1/n\) factor in the expression for the mean-squared error. If you include it, the optimal parameter \(\beta\) becomes
    
+
    
+\[
+\hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+n\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+\]
    
+
    In our codes where we compare our own codes with Scikit-Learn, we do thus not include the \(1/n\) factor in the cost function.
    
+
    
+
    
+
    Comparison with OLS
    
+
    When we compare this with the ordinary least squares result we have
    
+
    
+\[
+\hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+\]
    
+
    which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix \(\boldsymbol{X}^T\boldsymbol{X}\).
    
+
    We see that Ridge regression is nothing but the standard OLS with a
+modified diagonal term added to \(\boldsymbol{X}^T\boldsymbol{X}\). The consequences, in
+particular for our discussion of the bias-variance tradeoff are rather
+interesting. We will see that for specific values of \(\lambda\), we may
+even reduce the variance of the optimal parameters \(\boldsymbol{\beta}\). These topics and other related ones, will be discussed after the more linear algebra oriented analysis here.
    
+
    
+
    
+
    SVD analysis
    
+
    Using our insights about the SVD of the design matrix \(\boldsymbol{X}\)
+We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \(\boldsymbol{U}\) as
    
+
    
+\[
+\tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta}  =\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}.
+\]
    
+
    For Ridge regression this becomes
    
+
    
+\[
+\tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
+\]
    
+
    with the vectors \(\boldsymbol{u}_j\) being the columns of \(\boldsymbol{U}\) from the SVD of the matrix \(\boldsymbol{X}\).
    
+
    
+
    
+
    Interpreting the Ridge results
    
+
    Since \(\lambda \geq 0\), it means that compared to OLS, we have
    
+
    
+\[
+\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1.
+\]
    
+
    Ridge regression finds the coordinates of \(\boldsymbol{y}\) with respect to the
+orthonormal basis \(\boldsymbol{U}\), it then shrinks the coordinates by
+\(\frac{\sigma_j^2}{\sigma_j^2+\lambda}\). Recall that the SVD has
+eigenvalues ordered in a descending way, that is \(\sigma_i \geq
+\sigma_{i+1}\).
    
+
    For small eigenvalues \(\sigma_i\) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.
    
+
    
+
    
+
    More interpretations
    
+
    For the sake of simplicity, let us assume that the design matrix is orthonormal, that is
    
+
    
+\[
+\boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}.
+\]
    
+
    In this case the standard OLS results in
    
+
    
+\[
+\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
+\]
    
+
    and
    
+
    
+\[
+\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}},
+\]
    
+
    that is the Ridge estimator scales the OLS estimator by the inverse of a factor \(1+\lambda\), and
+the Ridge estimator converges to zero when the hyperparameter goes to
+infinity.
    
+
    We will come back to more interpreations after we have gone through some of the statistical analysis part.
    
+
    For more discussions of Ridge and Lasso regression, Wessel van Wieringen’s article is highly recommended.
+Similarly, Mehta et al’s article is also recommended.
    
+
    
+
    
+
    Deriving the  Lasso Regression Equations
    
+
    Using the matrix-vector expression for Lasso regression, we have the following cost function
    
+
    
+\[
+C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1,
+\]
    
+
    Taking the derivative with respect to \(\boldsymbol{\beta}\) and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicity)
    
+
    
+\[\begin{split}
+\frac{d \vert \beta\vert}{d \beta}=\mathrm{sgn}(\beta)=\left\{\begin{array}{cc} 1 & \beta > 0 \\-1 & \beta < 0, \end{array}\right.
+\end{split}\]
    
+
    we have that the derivative of the cost function is
    
+
    
+\[
+\frac{\partial C(\boldsymbol{X},\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}=-\frac{2}{n}\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})+\lambda sgn(\boldsymbol{\beta})=0,
+\]
    
+
    and reordering we have
    
+
    
+\[
+\boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}+\lambda sgn(\boldsymbol{\beta})=\boldsymbol{X}^T\boldsymbol{y}.
+\]
    
+
    This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor \(2/n\) in a redefinition of the parameter \(\lambda\). We will solve this type of problems using libraries like scikit-learn.
    
+
    
+
    
+
    Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression
    
+
    Let us assume that our design matrix is given by unit (identity) matrix, that is a square diagonal matrix with ones only along the
+diagonal. In this case we have an equal number of rows and columns \(n=p\).
    
+
    Our model approximation is just \(\tilde{\boldsymbol{y}}=\boldsymbol{\beta}\) and the mean squared error and thereby the cost function for ordinary least sqquares (OLS) is then (we drop the term \(1/n\))
    
+
    
+\[
+C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2,
+\]
    
+
    and minimizing we have that
    
+
    
+\[
+\hat{\beta}_i^{\mathrm{OLS}} = y_i.
+\]
    
+
    
+
    
+
    Ridge Regression
    
+
    For Ridge regression our cost function is
    
+
    
+\[
+C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\beta_i^2,
+\]
    
+
    and minimizing we have that
    
+
    
+\[
+\hat{\beta}_i^{\mathrm{Ridge}} = \frac{y_i}{1+\lambda}.
+\]
    
+
    
+
    
+
    Lasso Regression
    
+
    For Lasso regression our cost function is
    
+
    
+\[
+C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\vert\beta_i\vert=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\sqrt{\beta_i^2},
+\]
    
+
    and minimizing we have that
    
+
    
+\[
+-2\sum_{i=0}^{p-1}(y_i-\beta_i)+\lambda \sum_{i=0}^{p-1}\frac{(\beta_i)}{\vert\beta_i\vert}=0,
+\]
    
+
    which leads to
    
+
    
+\[\begin{split}
+\hat{\boldsymbol{\beta}}_i^{\mathrm{Lasso}} = \left\{\begin{array}{ccc}y_i-\frac{\lambda}{2} &\mathrm{if} & y_i> \frac{\lambda}{2}\\
+                                                          y_i+\frac{\lambda}{2} &\mathrm{if} & y_i< -\frac{\lambda}{2}\\
+							  0 &\mathrm{if} & \vert y_i\vert\le  \frac{\lambda}{2}\end{array}\right.\\.
+\end{split}\]
    
+
    Plotting these results shows clearly that Lasso regression suppresses (sets to zero) values of \(\beta_i\) for specific values of \(\lambda\). Ridge regression reduces on the other hand the values of \(\beta_i\) as function of \(\lambda\).
    
+
    
+
    
+
    Yet another Example
    
+
    Let us assume we have a data set with outputs/targets given by the vector
    
+
    
+\[\begin{split}
+\boldsymbol{y}=\begin{bmatrix}4 \\ 2 \\3\end{bmatrix},
+\end{split}\]
    
+
    and our inputs as a \(3\times 2\) design matrix
    
+
    
+\[\begin{split}
+\boldsymbol{X}=\begin{bmatrix}2 & 0\\ 0 & 1 \\ 0 & 0\end{bmatrix},
+\end{split}\]
    
+
    meaning that we have two features and two unknown parameters \(\beta_0\) and \(\beta_1\) to be determined either by ordinary least squares, Ridge or Lasso regression.
    
+
    
+
    
+
    The OLS case
    
+
    For ordinary least squares (OLS) we know that the optimal solution is
    
+
    
+\[
+\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left( \boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+\]
    
+
    Inserting the above values we obtain that
    
+
    
+\[\begin{split}
+\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\begin{bmatrix}2 \\ 2\end{bmatrix},
+\end{split}\]
    
+
    The code which implements this simpler case is presented after the discussion of Ridge and Lasso.
    
+
    
+
    
+
    The Ridge case
    
+
    For Ridge regression we have
    
+
    
+\[
+\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}=\left( \boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+\]
    
+
    Inserting the above values we obtain that
    
+
    
+\[\begin{split}
+\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}=\begin{bmatrix}\frac{8}{4+\lambda} \\ \frac{2}{1+\lambda}\end{bmatrix},
+\end{split}\]
    
+
    There is normally a constraint on the value of \(\vert\vert \boldsymbol{\beta}\vert\vert_2\) via the parameter \(\lambda\).
+Let us for simplicity assume that \(\beta_0^2+\beta_1^2=1\) as constraint. This will allow us to find an expression for the optimal values of \(\beta\) and \(\lambda\).
    
+
    To see this, let us write the cost function for Ridge regression.
    
+
    
+
    
+
    Writing the Cost Function
    
+
    We define the MSE without the \(1/n\) factor and have then, using that
    
+
    
+\[\begin{split}
+\boldsymbol{X}\boldsymbol{\beta}=\begin{bmatrix} 2\beta_0 \\ \beta_1 \\0 \end{bmatrix},
+\end{split}\]
    
+
    
+\[
+C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\beta_0^2+\beta_1^2),
+\]
    
+
    and taking the derivative with respect to \(\beta_0\) we get
    
+
    
+\[
+\beta_0=\frac{8}{4+\lambda},
+\]
    
+
    and for \(\beta_1\) we obtain
    
+
    
+\[
+\beta_1=\frac{2}{1+\lambda},
+\]
    
+
    Using the constraint for \(\beta_0^2+\beta_1^2=1\) we can constrain \(\lambda\) by solving
    
+
    
+\[
+\left(\frac{8}{4+\lambda}\right)^2+\left(\frac{2}{1+\lambda}\right)^2=1,
+\]
    
+
    which gives \(\lambda=4.571\) and \(\beta_0=0.933\) and \(\beta_1=0.359\).
    
+
    
+
    
+
    Lasso case
    
+
    For Lasso we need now, keeping a  constraint on \(\vert\beta_0\vert+\vert\beta_1\vert=1\),  to take the derivative of the absolute values of \(\beta_0\)
+and \(\beta_1\). This gives us the following derivatives of the cost function
    
+
    
+\[
+C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\vert\beta_1\vert),
+\]
    
+
    
+\[
+\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_0}=-4(4-2\beta_0)+\lambda\mathrm{sgn}(\beta_0)=0,
+\]
    
+
    and
    
+
    
+\[
+\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_1}=-2(2-\beta_1)+\lambda\mathrm{sgn}(\beta_1)=0.
+\]
    
+
    We have now four cases to solve besides the trivial cases \(\beta_0\) and/or \(\beta_1\) are zero, namely
    
+
      
+
    1. \(\beta_0 > 0\) and \(\beta_1 > 0\),
      2. 
+
    2. \(\beta_0 > 0\) and \(\beta_1 < 0\),
      3. 
+
    3. \(\beta_0 < 0\) and \(\beta_1 > 0\),
      4. 
+
    4. \(\beta_0 < 0\) and \(\beta_1 < 0\).
      5. 
+
    
+
    
+
    
+
    The first Case
    
+
    If we consider the first case, we have then
    
+
    
+\[
+-4(4-2\beta_0)+\lambda=0,
+\]
    
+
    and
    
+
    
+\[
+-2(2-\beta_1)+\lambda=0.
+\]
    
+
    which yields
    
+
    
+\[
+\beta_0=\frac{16+\lambda}{8},
+\]
    
+
    and
    
+
    
+\[
+\beta_1=\frac{4+\lambda}{2}.
+\]
    
+
    Using the constraint on \(\beta_0\) and \(\beta_1\) we can then find the optimal value of \(\lambda\) for the different cases. We leave this as an exercise to you.
    
+
    
+
    
+
    Simple code for solving the above problem
    
+
    Here we set up the OLS, Ridge and Lasso functionality in order to study the above example. Note that here we have opted for a set of values of \(\lambda\), meaning that we need to perform a search in order to find the optimal values.
    
+
    First we study and compare the OLS and Ridge results.  The next code compares all three methods.
    
+
    
+
    
+
    %matplotlib inline
+
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+
+X = np.array( [ [ 2, 0], [0, 1], [0,0]])
+y = np.array( [4, 2, 3])
+
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y
+print(OLSbeta)
+# and then make the prediction
+ytildeOLS = X @ OLSbeta
+print("Training MSE for OLS")
+print(MSE(y,ytildeOLS))
+ypredictOLS = X @ OLSbeta
+
+# Repeat now for Ridge regression and various values of the regularization parameter
+I = np.eye(2,2)
+# Decide which values of lambda to use
+nlambdas = 100
+MSEPredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 4, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y
+#    print(Ridgebeta)
+    # and then make the prediction
+    ypredictRidge = X @ Ridgebeta
+    MSEPredict[i] = MSE(y,ypredictRidge)
+#    print(MSEPredict[i])
+    # Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Train')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
    
+
    
+
    
+
    
+
    We see here that we reach a plateau. What is actually happening?
    
+
    
+
    
+
    With Lasso Regression
    
+
    
+
    
+
    import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn import linear_model
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+
+X = np.array( [ [ 2, 0], [0, 1], [0,0]])
+y = np.array( [4, 2, 3])
+
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y
+print(OLSbeta)
+# and then make the prediction
+ytildeOLS = X @ OLSbeta
+print("Training MSE for OLS")
+print(MSE(y,ytildeOLS))
+ypredictOLS = X @ OLSbeta
+
+# Repeat now for Ridge regression and various values of the regularization parameter
+I = np.eye(2,2)
+# Decide which values of lambda to use
+nlambdas = 100
+MSERidgePredict = np.zeros(nlambdas)
+MSELassoPredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 4, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y
+    print(Ridgebeta)
+    # and then make the prediction
+    ypredictRidge = X @ Ridgebeta
+    MSERidgePredict[i] = MSE(y,ypredictRidge)
+    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
+    RegLasso.fit(X,y)
+    ypredictLasso = RegLasso.predict(X)
+    print(RegLasso.coef_)
+    MSELassoPredict[i] = MSE(y,ypredictLasso)
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSERidgePredict, 'r--', label = 'MSE Ridge Train')
+plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Train')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
    
+
    
+
    
+
    
+
    
+
    
+
    Another Example, now with a polynomial fit
    
+
    
+
    
+
    import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+
+# number of features p (here degree of polynomial
+p = 3
+#  The design matrix now as function of a given polynomial
+X = np.zeros((len(x),p))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x*x
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
+print(OLSbeta)
+# and then make the prediction
+ytildeOLS = X_train @ OLSbeta
+print("Training MSE for OLS")
+print(MSE(y_train,ytildeOLS))
+ypredictOLS = X_test @ OLSbeta
+print("Test MSE OLS")
+print(MSE(y_test,ypredictOLS))
+
+# Repeat now for Lasso and Ridge regression and various values of the regularization parameter
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 100
+MSEPredict = np.zeros(nlambdas)
+MSETrain = np.zeros(nlambdas)
+MSELassoPredict = np.zeros(nlambdas)
+MSELassoTrain = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 4, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    Ridgebeta = np.linalg.inv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
+    # include lasso using Scikit-Learn
+    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
+    RegLasso.fit(X_train,y_train)
+    # and then make the prediction
+    ytildeRidge = X_train @ Ridgebeta
+    ypredictRidge = X_test @ Ridgebeta
+    ytildeLasso = RegLasso.predict(X_train)
+    ypredictLasso = RegLasso.predict(X_test)
+    MSEPredict[i] = MSE(y_test,ypredictRidge)
+    MSETrain[i] = MSE(y_train,ytildeRidge)
+    MSELassoPredict[i] = MSE(y_test,ypredictLasso)
+    MSELassoTrain[i] = MSE(y_train,ytildeLasso)
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSETrain, label = 'MSE Ridge train')
+plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Test')
+plt.plot(np.log10(lambdas), MSELassoTrain, label = 'MSE Lasso train')
+plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
    
+
    
+
    
+
    
+
    
+
    
+
    Material for lecture Thursday September 7
    
+
    
+
    
+
    Linking the regression analysis with a statistical interpretation
    
+
    We will now couple the discussions of ordinary least squares, Ridge
+and Lasso regression with a statistical interpretation, that is we
+move from a linear algebra analysis to a statistical analysis. In
+particular, we will focus on what the regularization terms can result
+in.  We will amongst other things show that the regularization
+parameter can reduce considerably the variance of the parameters
+\(\beta\).
    
+
    The
+advantage of doing linear regression is that we actually end up with
+analytical expressions for several statistical quantities.
    
+Standard least squares and Ridge regression  allow us to
+derive quantities like the variance and other expectation values in a
+rather straightforward way.
    
+
    It is assumed that \(\varepsilon_i
+\sim \mathcal{N}(0, \sigma^2)\) and the \(\varepsilon_{i}\) are
+independent, i.e.:
    
+
    
+\[\begin{split}
+\begin{align*} 
+\mbox{Cov}(\varepsilon_{i_1},
+\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
+& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2.  \end{array} \right.
+\end{align*}
+\end{split}\]
    
+
    The randomness of \(\varepsilon_i\) implies that
+\(\mathbf{y}_i\) is also a random variable. In particular,
+\(\mathbf{y}_i\) is normally distributed, because \(\varepsilon_i \sim
+\mathcal{N}(0, \sigma^2)\) and \(\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}\) is a
+non-random scalar. To specify the parameters of the distribution of
+\(\mathbf{y}_i\) we need to calculate its first two moments.
    
+
    Recall that \(\boldsymbol{X}\) is a matrix of dimensionality \(n\times p\). The
+notation above \(\mathbf{X}_{i,\ast}\) means that we are looking at the
+row number \(i\) and perform a sum over all values \(p\).
    
+
    
+
    
+
    Assumptions made
    
+
    The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)
+that there exists a function \(f(\boldsymbol{x})\) and  a normal distributed error \(\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)\)
+which describe our data
    
+
    
+\[
+\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
+\]
    
+
    We approximate this function with our model from the solution of the linear regression equations, that is our
+function \(f\) is approximated by \(\boldsymbol{\tilde{y}}\) where we want to minimize \((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\), our MSE, with
    
+
    
+\[
+\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
+\]
    
+
    
+
    
+
    Expectation value and variance
    
+
    We can calculate the expectation value of \(\boldsymbol{y}\) for a given element \(i\)
    
+
    
+\[
+\begin{align*} 
+\mathbb{E}(y_i) & =
+\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
+\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, 
+\end{align*}
+\]
    
+
    while
+its variance is
    
+
    
+\[\begin{split}
+\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i
+- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
+[\mathbb{E}(y_i)]^2  \\  & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
+\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &
+= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
+\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
+\ast} \, \beta)^2 \\  & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
+\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
+\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 
+\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
+\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.  
+\end{align*}
+\end{split}\]
    
+
    Hence, \(y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\), that is \(\boldsymbol{y}\) follows a normal distribution with
+mean value \(\boldsymbol{X}\boldsymbol{\beta}\) and variance \(\sigma^2\) (not be confused with the singular values of the SVD).
    
+
    
+
    
+
    Expectation value and variance for \(\boldsymbol{\beta}\)
    
+
    With the OLS expressions for the optimal parameters \(\boldsymbol{\hat{\beta}}\) we can evaluate the expectation value
    
+
    
+\[
+\mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
+\]
    
+
    This means that the estimator of the regression parameters is unbiased.
    
+
    We can also calculate the variance
    
+
    The variance of the optimal value \(\boldsymbol{\hat{\beta}}\) is
    
+
    
+\[\begin{split}
+\begin{eqnarray*}
+\mbox{Var}(\boldsymbol{\hat{\beta}}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
+\\
+& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \}
+\\
+% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1}  \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+\\
+& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \,  \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \,  \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
+% \\
+% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T  \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
+\\
+& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}  + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
+\end{eqnarray*}
+\end{split}\]
    
+
    where we have used  that \(\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) =
+\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} +
+\sigma^2 \, \mathbf{I}_{nn}\). From \(\mbox{Var}(\boldsymbol{\beta}) = \sigma^2
+\, (\mathbf{X}^{T} \mathbf{X})^{-1}\), one obtains an estimate of the
+variance of the estimate of the \(j\)-th regression coefficient:
+\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \). This may be used to
+construct a confidence interval for the estimates.
    
+
    In a similar way, we can obtain analytical expressions for say the
+expectation values of the parameters \(\boldsymbol{\beta}\) and their variance
+when we employ Ridge regression, allowing us again to define a confidence interval.
    
+
    It is rather straightforward to show that
    
+
    
+\[
+\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}.
+\]
    
+
    We see clearly that
+\(\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}\) for any \(\lambda > 0\). We say then that the ridge estimator is biased.
    
+
    We can also compute the variance as
    
+
    
+\[
+\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
+\]
    
+
    and it is easy to see that if the parameter \(\lambda\) goes to infinity then the variance of Ridge parameters \(\boldsymbol{\beta}\) goes to zero.
    
+
    With this, we can compute the difference
    
+
    
+\[
+\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
+\]
    
+
    The difference is non-negative definite since each component of the
+matrix product is non-negative definite.
+This means the variance we obtain with the standard OLS will always for \(\lambda > 0\) be larger than the variance of \(\boldsymbol{\beta}\) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.
    
+
    
+
    
+
    Deriving OLS from a probability distribution
    
+
    Our basic assumption when we derived the OLS equations was to assume
+that our output is determined by a given continuous function
+\(f(\boldsymbol{x})\) and a random noise \(\boldsymbol{\epsilon}\) given by the normal
+distribution with zero mean value and an undetermined variance
+\(\sigma^2\).
    
+
    We found above that the outputs \(\boldsymbol{y}\) have a mean value given by
+\(\boldsymbol{X}\hat{\boldsymbol{\beta}}\) and variance \(\sigma^2\). Since the entries to
+the design matrix are not stochastic variables, we can assume that the
+probability distribution of our targets is also a normal distribution
+but now with mean value \(\boldsymbol{X}\hat{\boldsymbol{\beta}}\). This means that a
+single output \(y_i\) is given by the Gaussian distribution
    
+
    
+\[
+y_i\sim \mathcal{N}(\boldsymbol{X}_{i,*}\boldsymbol{\beta}, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
+\]
    
+
    
+
    
+
    Independent and Identically Distrubuted (iid)
    
+
    We assume now that the various \(y_i\) values are stochastically distributed according to the above Gaussian distribution.
+We define this distribution as
    
+
    
+\[
+p(y_i, \boldsymbol{X}\vert\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]},
+\]
    
+
    which reads as finding the likelihood of an event \(y_i\) with the input variables \(\boldsymbol{X}\) given the parameters (to be determined) \(\boldsymbol{\beta}\).
    
+
    Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event \(\boldsymbol{y}\) as the product of the single events, that is we have
    
+
    
+\[
+p(\boldsymbol{y},\boldsymbol{X}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}=\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta}).
+\]
    
+
    We will write this in a more compact form reserving \(\boldsymbol{D}\) for the domain of events, including the ouputs (targets) and the inputs. That is
+in case we have a simple one-dimensional input and output case
    
+
    
+\[
+\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})].
+\]
    
+
    In the more general case the various inputs should be replaced by the possible features represented by the input data set \(\boldsymbol{X}\).
+We can now rewrite the above probability as
    
+
    
+\[
+p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
+\]
    
+
    It is a conditional probability (see below) and reads as the likelihood of a domain of events \(\boldsymbol{D}\) given a set of parameters \(\boldsymbol{\beta}\).
    
+
    
+
    
+
    Maximum Likelihood Estimation (MLE)
    
+
    In statistics, maximum likelihood estimation (MLE) is a method of
+estimating the parameters of an assumed probability distribution,
+given some observed data. This is achieved by maximizing a likelihood
+function so that, under the assumed statistical model, the observed
+data is the most probable.
    
+
    We will assume here that our events are given by the above Gaussian
+distribution and we will determine the optimal parameters \(\beta\) by
+maximizing the above PDF. However, computing the derivatives of a
+product function is cumbersome and can easily lead to overflow and/or
+underflowproblems, with potentials for loss of numerical precision.
    
+
    In practice, it is more convenient to maximize the logarithm of the
+PDF because it is a monotonically increasing function of the argument.
+Alternatively, and this will be our option, we will minimize the
+negative of the logarithm since this is a monotonically decreasing
+function.
    
+
    Note also that maximization/minimization of the logarithm of the PDF
+is equivalent to the maximization/minimization of the function itself.
    
+
    
+
    
+
    A new Cost Function
    
+
    We could now define a new cost function to minimize, namely the negative logarithm of the above PDF
    
+
    
+\[
+C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})},
+\]
    
+
    which becomes
    
+
    
+\[
+C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}.
+\]
    
+
    Taking the derivative of the new cost function with respect to the parameters \(\beta\) we recognize our familiar OLS equation, namely
    
+
    
+\[
+\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0,
+\]
    
+
    which leads to the well-known OLS equation for the optimal paramters \(\beta\)
    
+
    
+\[
+\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}!
+\]
    
+
    Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics.
    
+
    
+
    
+
    More basic Statistics and Bayes’ theorem
    
+
    A central theorem in statistics is Bayes’ theorem. This theorem plays a similar role as the good old Pythagoras’ theorem in geometry.
+Bayes’ theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.
    
+
    Assume we have two domains of events \(X=[x_0,x_1,\dots,x_{n-1}]\) and \(Y=[y_0,y_1,\dots,y_{n-1}]\).
    
+
    We define also the likelihood for \(X\) and \(Y\) as \(p(X)\) and \(p(Y)\) respectively.
+The likelihood of a specific event \(x_i\) (or \(y_i\)) is then written as \(p(X=x_i)\) or just \(p(x_i)=p_i\).
    
+
    Union of events is given by.
    
+
    
+\[
+p(X \cup Y)= p(X)+p(Y)-p(X \cap Y).
+\]
    
+
    The product rule (aka joint probability) is given by.
    
+
    
+\[
+p(X \cup Y)= p(X,Y)= p(X\vert Y)p(Y)=p(Y\vert X)p(X),
+\]
    
+
    where we read \(p(X\vert Y)\) as the likelihood of obtaining \(X\) given \(Y\).
    
+
    If we have independent events then \(p(X,Y)=p(X)p(Y)\).
    
+
    
+
    
+
    Marginal Probability
    
+
    The marginal probability is defined in terms of only one of the set of variables \(X,Y\). For a discrete probability we have
    
+
    
+\[
+p(X)=\sum_{i=0}^{n-1}p(X,Y=y_i)=\sum_{i=0}^{n-1}p(X\vert Y=y_i)p(Y=y_i)=\sum_{i=0}^{n-1}p(X\vert y_i)p(y_i).
+\]
    
+
    
+
    
+
    Conditional  Probability
    
+
    The conditional  probability, if \(p(Y) > 0\), is
    
+
    
+\[
+p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}.
+\]
    
+
    
+
    
+
    Bayes’ Theorem
    
+
    If we combine the conditional probability with the marginal probability and the standard product rule, we have
    
+
    
+\[
+p(X\vert Y)= \frac{p(X,Y)}{p(Y)},
+\]
    
+
    which we can rewrite as
    
+
    
+\[
+p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)},
+\]
    
+
    which is Bayes’ theorem. It allows us to evaluate the uncertainty in in \(X\) after we have observed \(Y\). We can easily interchange \(X\) with \(Y\).
    
+
    
+
    
+
    Interpretations of Bayes’ Theorem
    
+
    The quantity \(p(Y\vert X)\) on the right-hand side of the theorem is
+evaluated for the observed data \(Y\) and can be viewed as a function of
+the parameter space represented by \(X\). This function is not
+necesseraly normalized and is normally called the likelihood function.
    
+
    The function \(p(X)\) on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.
    
+
    Let us try to illustrate Bayes’ theorem through an example.
    
+
    
+
    
+
    Example of Usage of Bayes’ theorem
    
+
    Let us suppose that you are undergoing a series of mammography scans in
+order to rule out possible breast cancer cases.  We define the
+sensitivity for a positive event by the variable \(X\). It takes binary
+values with \(X=1\) representing a positive event and \(X=0\) being a
+negative event. We reserve \(Y\) as a classification parameter for
+either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a  positive thing).
    
+
    We let \(Y=1\) represent the the case of having breast cancer and \(Y=0\) as not.
    
+
    Let us assume that if you have breast cancer, the test will be positive with a probability of \(0.8\), that is we have
    
+
    
+\[
+p(X=1\vert Y=1) =0.8.
+\]
    
+
    This obviously sounds  scary since many would conclude that if the test is positive, there is a likelihood of \(80\%\) for having cancer.
+It is however not correct, as the following Bayesian analysis shows.
    
+
    
+
    
+
    Doing it correctly
    
+
    If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number.
+Let us assume that the prior probability in the population as a whole is
    
+
    
+\[
+p(Y=1) =0.004.
+\]
    
+
    We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have
    
+
    
+\[
+p(X=1\vert Y=0) =0.1.
+\]
    
+
    Using Bayes’ theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute
    
+
    
+\[
+p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=0)p(Y=0)}=\frac{0.8\times 0.004}{0.8\times 0.004+0.1\times 0.996}=0.031.
+\]
    
+
    That is, in case of a positive test, there is only a \(3\%\) chance of having breast cancer!
    
+
    
+
    
+
    Bayes’ Theorem and Ridge and Lasso Regression
    
+
    Hitherto we have discussed Ridge and Lasso regression in terms of a
+linear analysis. This may to many of you feel rather technical and
+perhaps not that intuitive. The question is whether we can develop a
+more intuitive way of understanding what Ridge and Lasso express.
    
+
    Before we proceed let us perform a Ridge, Lasso  and OLS analysis of a polynomial fit.
    
+
    
+
    
+
    Test Function for what happens with OLS, Ridge and Lasso
    
+
    We will play around with a study of the values for the optimal
+parameters \(\boldsymbol{\beta}\) using OLS, Ridge and Lasso regression.  For
+OLS, you will notice as function of the noise and polynomial degree,
+that the parameters \(\beta\) will fluctuate from order to order in the
+polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS.
    
+
    For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one.
    
+
    
+
    
+
    import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+# Make data set.
+n = 10000
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)
+
+Maxpolydegree = 5
+X = np.zeros((len(x),Maxpolydegree))
+X[:,0] = 1.0
+
+for polydegree in range(1, Maxpolydegree):
+    for degree in range(polydegree):
+        X[:,degree] = x**(degree)
+
+
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
+print(OLSbeta)
+ypredictOLS = X_test @ OLSbeta
+print("Test MSE OLS")
+print(MSE(y_test,ypredictOLS))
+# Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn
+# Decide which values of lambda to use
+nlambdas = 4
+MSERidgePredict = np.zeros(nlambdas)
+MSELassoPredict = np.zeros(nlambdas)
+lambdas = np.logspace(-3, 1, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    # Make the fit using Ridge and Lasso
+    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
+    RegRidge.fit(X_train,y_train)
+    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
+    RegLasso.fit(X_train,y_train)
+    # and then make the prediction
+    ypredictRidge = RegRidge.predict(X_test)
+    ypredictLasso = RegLasso.predict(X_test)
+    # Compute the MSE and print it
+    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+    MSELassoPredict[i] = MSE(y_test,ypredictLasso)
+    print(lmb,RegRidge.coef_)
+    print(lmb,RegLasso.coef_)
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSERidgePredict, 'b', label = 'MSE Ridge Test')
+plt.plot(np.log10(lambdas), MSELassoPredict, 'r', label = 'MSE Lasso Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
    
+
    
+
    
+
    
+
    How can we understand this?
    
+
    
+
    
+
    Invoking Bayes’ theorem
    
+
    Using Bayes’ theorem we can gain a better intuition about Ridge and Lasso regression.
    
+
    For ordinary least squares we postulated that the maximum likelihood for the doamin of events \(\boldsymbol{D}\) (one-dimensional case)
    
+
    
+\[
+\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})],
+\]
    
+
    is given by
    
+
    
+\[
+p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
+\]
    
+
    In Bayes’ theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set \(\boldsymbol{\beta}\) given a domain of events \(\boldsymbol{D}\)?  That is, how can we define the posterior probability
    
+
    
+\[
+p(\boldsymbol{\beta}\vert\boldsymbol{D}).
+\]
    
+
    Bayes’ theorem comes to our rescue here since (omitting the normalization constant)
    
+
    
+\[
+p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
+\]
    
+
    We have a model for \(p(\boldsymbol{D}\vert\boldsymbol{\beta})\) but need one for the prior \(p(\boldsymbol{\beta}\)!
    
+
    
+
    
+
    Ridge and Bayes
    
+
    With the posterior probability defined by a likelihood which we have
+already modeled and an unknown prior, we are now ready to make
+additional models for the prior.
    
+
    We can, based on our discussions of the variance of \(\boldsymbol{\beta}\) and the mean value, assume that the prior for the values \(\boldsymbol{\beta}\) is given by a Gaussian with mean value zero and variance \(\tau^2\), that is
    
+
    
+\[
+p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
+\]
    
+
    Our posterior probability becomes then (omitting the normalization factor which is just a constant)
    
+
    
+\[
+p(\boldsymbol{\beta\vert\boldsymbol{D})}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
+\]
    
+
    We can now optimize this quantity with respect to \(\boldsymbol{\beta}\). As we
+did for OLS, this is most conveniently done by taking the negative
+logarithm of the posterior probability. Doing so and leaving out the
+constants terms that do not depend on \(\beta\), we have
    
+
    
+\[
+C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
+\]
    
+
    and replacing \(1/2\tau^2\) with \(\lambda\) we have
    
+
    
+\[
+C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
+\]
    
+
    which is our Ridge cost function!  Nice, isn’t it?
    
+
    
+
    
+
    Lasso and Bayes
    
+
    To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution (Laplace in this case) with zero mean value,  that is
    
+
    
+\[
+p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
+\]
    
+
    Our posterior probability becomes then (omitting the normalization factor which is just a constant)
    
+
    
+\[
+p(\boldsymbol{\beta}\vert\boldsymbol{D})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
+\]
    
+
    Taking the negative
+logarithm of the posterior probability and leaving out the
+constants terms that do not depend on \(\beta\), we have
    
+
    
+\[
+C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1,
+\]
    
+
    and replacing \(1/\tau\) with \(\lambda\) we have
    
+
    
+\[
+C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1,
+\]
    
+
    which is our Lasso cost function!
    
+
    
+
    
+
+    
+    
+
+              
    
+              
+            
+                
+
     
+    
+        
+        
    
+            
    previous
    
+            
    Exercises week 36
    
+        
    
+        
+
    
+            
+        
    
+    
    
+    
    
+  
    
+    
+      By Morten Hjorth-Jensen
    
+    
+        © Copyright 2021.
    
+  
    
+
    
+
    
+
+
+      
    
+    
    
+  
+  
+
+  
+
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb
index 9b2dd9a85..5af78146a 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb
@@ -343,7 +343,7 @@
    "outputs": [
     {
      "data": {
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\n",
+      "image/png": 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\n",
       "text/plain": [
        "
    "
       ]
@@ -515,7 +515,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "
    "
       ]
@@ -583,18 +583,18 @@
      "output_type": "stream",
      "text": [
       "The intercept alpha: \n",
-      " [2.11366595]\n",
+      " [2.05177695]\n",
       "Coefficient beta : \n",
-      " [[4.95136998]]\n",
+      " [[5.05971574]]\n",
       "Mean squared error: 0.28\n",
-      "Variance score: 0.89\n",
+      "Variance score: 0.88\n",
       "Mean squared log error: 0.01\n",
-      "Mean absolute error: 0.41\n"
+      "Mean absolute error: 0.45\n"
      ]
     },
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "
    "
       ]
@@ -822,7 +822,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
+      "image/png": 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\n",
       "text/plain": [
        "
    "
       ]
@@ -838,7 +838,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.004999999999999996\n"
+      "0.004999999999999997\n"
      ]
     }
    ],
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb
index ff15fcc41..bc64986ba 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb
@@ -1077,7 +1077,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1655,7 +1655,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1673,7 +1673,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1691,7 +1691,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1709,7 +1709,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1727,7 +1727,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1745,7 +1745,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1763,7 +1763,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1781,11 +1781,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1803,11 +1803,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1825,11 +1825,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1847,11 +1847,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1869,11 +1869,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1891,7 +1891,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -1909,11 +1909,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1931,11 +1931,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1953,11 +1953,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1975,11 +1975,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -1997,11 +1997,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -2019,11 +2019,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -2041,11 +2041,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -2063,11 +2063,11 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:43: RuntimeWarning: overflow encountered in exp\n",
       "  exp_term = np.exp(self.z_o)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide\n",
       "  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n"
      ]
     },
@@ -2128,15 +2128,15 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20528/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16145/953065564.py:4: RuntimeWarning: overflow encountered in exp\n",
       "  return 1/(1 + np.exp(-x))\n"
      ]
     },
@@ -2669,34 +2669,24 @@
       "Learning rate  =  0.01\n",
       "Lambda =  0.1\n",
       "Accuracy score on test set:  0.9888888888888889\n",
-      "\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "\n",
       "Learning rate  =  0.01\n",
       "Lambda =  1.0\n",
       "Accuracy score on test set:  0.9722222222222222\n",
-      "\n",
+      "\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
       "Learning rate  =  0.01\n",
       "Lambda =  10.0\n",
       "Accuracy score on test set:  0.9527777777777777\n",
-      "\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "\n",
       "Learning rate  =  0.1\n",
       "Lambda =  1e-05\n",
       "Accuracy score on test set:  0.9027777777777778\n",
-      "\n",
-      "Learning rate  =  0.1\n",
-      "Lambda =  0.0001\n",
-      "Accuracy score on test set:  0.8583333333333333\n",
       "\n"
      ]
     },
@@ -2704,6 +2694,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Learning rate  =  0.1\n",
+      "Lambda =  0.0001\n",
+      "Accuracy score on test set:  0.8583333333333333\n",
+      "\n",
       "Learning rate  =  0.1\n",
       "Lambda =  0.001\n",
       "Accuracy score on test set:  0.8722222222222222\n",
@@ -2721,20 +2715,10 @@
       "Learning rate  =  0.1\n",
       "Lambda =  0.1\n",
       "Accuracy score on test set:  0.8805555555555555\n",
-      "\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "\n",
       "Learning rate  =  0.1\n",
       "Lambda =  1.0\n",
       "Accuracy score on test set:  0.8722222222222222\n",
-      "\n",
-      "Learning rate  =  0.1\n",
-      "Lambda =  10.0\n",
-      "Accuracy score on test set:  0.8666666666666667\n",
       "\n"
      ]
     },
@@ -2742,6 +2726,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Learning rate  =  0.1\n",
+      "Lambda =  10.0\n",
+      "Accuracy score on test set:  0.8666666666666667\n",
+      "\n",
       "Learning rate  =  1.0\n",
       "Lambda =  1e-05\n",
       "Accuracy score on test set:  0.08611111111111111\n",
@@ -2749,10 +2737,6 @@
       "Learning rate  =  1.0\n",
       "Lambda =  0.0001\n",
       "Accuracy score on test set:  0.10555555555555556\n",
-      "\n",
-      "Learning rate  =  1.0\n",
-      "Lambda =  0.001\n",
-      "Accuracy score on test set:  0.10555555555555556\n",
       "\n"
      ]
     },
@@ -2760,6 +2744,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Learning rate  =  1.0\n",
+      "Lambda =  0.001\n",
+      "Accuracy score on test set:  0.10555555555555556\n",
+      "\n",
       "Learning rate  =  1.0\n",
       "Lambda =  0.01\n",
       "Accuracy score on test set:  0.17777777777777778\n",
@@ -2785,6 +2773,10 @@
       "Learning rate  =  10.0\n",
       "Lambda =  1e-05\n",
       "Accuracy score on test set:  0.17222222222222222\n",
+      "\n",
+      "Learning rate  =  10.0\n",
+      "Lambda =  0.0001\n",
+      "Accuracy score on test set:  0.11666666666666667\n",
       "\n"
      ]
     },
@@ -2792,10 +2784,6 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Learning rate  =  10.0\n",
-      "Lambda =  0.0001\n",
-      "Accuracy score on test set:  0.11666666666666667\n",
-      "\n",
       "Learning rate  =  10.0\n",
       "Lambda =  0.001\n",
       "Accuracy score on test set:  0.10555555555555556\n",
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb
index d921b1909..0f5b522f8 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb
@@ -3029,24 +3029,18 @@
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py:20\u001b[0m, in \u001b[0;36munary_to_nary..nary_operator..nary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     18\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m     19\u001b[0m     x \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mtuple\u001b[39m(args[i] \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m argnum)\n\u001b[0;32m---> 20\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43munary_operator\u001b[49m\u001b[43m(\u001b[49m\u001b[43munary_f\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mnary_op_args\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mnary_op_kwargs\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py:78\u001b[0m, in \u001b[0;36mhessian\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m     75\u001b[0m \u001b[38;5;129m@unary_to_nary\u001b[39m\n\u001b[1;32m     76\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mhessian\u001b[39m(fun, x):\n\u001b[1;32m     77\u001b[0m     \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mReturns a function that computes the exact Hessian.\u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[0;32m---> 78\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mjacobian\u001b[49m\u001b[43m(\u001b[49m\u001b[43mjacobian\u001b[49m\u001b[43m(\u001b[49m\u001b[43mfun\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\u001b[43m(\u001b[49m\u001b[43mx\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py:20\u001b[0m, in \u001b[0;36munary_to_nary..nary_operator..nary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     18\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m     19\u001b[0m     x \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mtuple\u001b[39m(args[i] \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m argnum)\n\u001b[0;32m---> 20\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43munary_operator\u001b[49m\u001b[43m(\u001b[49m\u001b[43munary_f\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mnary_op_args\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mnary_op_kwargs\u001b[49m\u001b[43m)\u001b[49m\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py:57\u001b[0m, in \u001b[0;36mjacobian\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m     47\u001b[0m \u001b[38;5;129m@unary_to_nary\u001b[39m\n\u001b[1;32m     48\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mjacobian\u001b[39m(fun, x):\n\u001b[1;32m     49\u001b[0m     \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m     50\u001b[0m \u001b[38;5;124;03m    Returns a function which computes the Jacobian of `fun` with respect to\u001b[39;00m\n\u001b[1;32m     51\u001b[0m \u001b[38;5;124;03m    positional argument number `argnum`, which must be a scalar or array. Unlike\u001b[39;00m\n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m     55\u001b[0m \u001b[38;5;124;03m    (out1, out2, ...) then the Jacobian has shape (out1, out2, ..., in1, in2, ...).\u001b[39;00m\n\u001b[1;32m     56\u001b[0m \u001b[38;5;124;03m    \"\"\"\u001b[39;00m\n\u001b[0;32m---> 57\u001b[0m     vjp, ans \u001b[38;5;241m=\u001b[39m \u001b[43m_make_vjp\u001b[49m\u001b[43m(\u001b[49m\u001b[43mfun\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mx\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     58\u001b[0m     ans_vspace \u001b[38;5;241m=\u001b[39m vspace(ans)\n\u001b[1;32m     59\u001b[0m     jacobian_shape \u001b[38;5;241m=\u001b[39m ans_vspace\u001b[38;5;241m.\u001b[39mshape \u001b[38;5;241m+\u001b[39m vspace(x)\u001b[38;5;241m.\u001b[39mshape\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:10\u001b[0m, in \u001b[0;36mmake_vjp\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mmake_vjp\u001b[39m(fun, x):\n\u001b[1;32m      9\u001b[0m     start_node \u001b[38;5;241m=\u001b[39m VJPNode\u001b[38;5;241m.\u001b[39mnew_root()\n\u001b[0;32m---> 10\u001b[0m     end_value, end_node \u001b[38;5;241m=\u001b[39m  \u001b[43mtrace\u001b[49m\u001b[43m(\u001b[49m\u001b[43mstart_node\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mfun\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mx\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     11\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m end_node \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[1;32m     12\u001b[0m         \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mvjp\u001b[39m(g): \u001b[38;5;28;01mreturn\u001b[39;00m vspace(x)\u001b[38;5;241m.\u001b[39mzeros()\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py:10\u001b[0m, in \u001b[0;36mtrace\u001b[0;34m(start_node, fun, x)\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m trace_stack\u001b[38;5;241m.\u001b[39mnew_trace() \u001b[38;5;28;01mas\u001b[39;00m t:\n\u001b[1;32m      9\u001b[0m     start_box \u001b[38;5;241m=\u001b[39m new_box(x, t, start_node)\n\u001b[0;32m---> 10\u001b[0m     end_box \u001b[38;5;241m=\u001b[39m \u001b[43mfun\u001b[49m\u001b[43m(\u001b[49m\u001b[43mstart_box\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     11\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m isbox(end_box) \u001b[38;5;129;01mand\u001b[39;00m end_box\u001b[38;5;241m.\u001b[39m_trace \u001b[38;5;241m==\u001b[39m start_box\u001b[38;5;241m.\u001b[39m_trace:\n\u001b[1;32m     12\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m end_box\u001b[38;5;241m.\u001b[39m_value, end_box\u001b[38;5;241m.\u001b[39m_node\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py:15\u001b[0m, in \u001b[0;36munary_to_nary..nary_operator..nary_f..unary_f\u001b[0;34m(x)\u001b[0m\n\u001b[1;32m     13\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m     14\u001b[0m     subargs \u001b[38;5;241m=\u001b[39m subvals(args, \u001b[38;5;28mzip\u001b[39m(argnum, x))\n\u001b[0;32m---> 15\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mfun\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43msubargs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py:20\u001b[0m, in \u001b[0;36munary_to_nary..nary_operator..nary_f\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     18\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m     19\u001b[0m     x \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mtuple\u001b[39m(args[i] \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m argnum)\n\u001b[0;32m---> 20\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43munary_operator\u001b[49m\u001b[43m(\u001b[49m\u001b[43munary_f\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mnary_op_args\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mnary_op_kwargs\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py:61\u001b[0m, in \u001b[0;36mjacobian\u001b[0;34m(fun, x)\u001b[0m\n\u001b[1;32m     59\u001b[0m jacobian_shape \u001b[38;5;241m=\u001b[39m ans_vspace\u001b[38;5;241m.\u001b[39mshape \u001b[38;5;241m+\u001b[39m vspace(x)\u001b[38;5;241m.\u001b[39mshape\n\u001b[1;32m     60\u001b[0m grads \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mmap\u001b[39m(vjp, ans_vspace\u001b[38;5;241m.\u001b[39mstandard_basis())\n\u001b[0;32m---> 61\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m np\u001b[38;5;241m.\u001b[39mreshape(\u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mstack\u001b[49m\u001b[43m(\u001b[49m\u001b[43mgrads\u001b[49m\u001b[43m)\u001b[49m, jacobian_shape)\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_wrapper.py:88\u001b[0m, in \u001b[0;36mstack\u001b[0;34m(arrays, axis)\u001b[0m\n\u001b[1;32m     83\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mstack\u001b[39m(arrays, axis\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m0\u001b[39m):\n\u001b[1;32m     84\u001b[0m     \u001b[38;5;66;03m# this code is basically copied from numpy/core/shape_base.py's stack\u001b[39;00m\n\u001b[1;32m     85\u001b[0m     \u001b[38;5;66;03m# we need it here because we want to re-implement stack in terms of the\u001b[39;00m\n\u001b[1;32m     86\u001b[0m     \u001b[38;5;66;03m# primitives defined in this file\u001b[39;00m\n\u001b[0;32m---> 88\u001b[0m     arrays \u001b[38;5;241m=\u001b[39m [array(arr) \u001b[38;5;28;01mfor\u001b[39;00m arr \u001b[38;5;129;01min\u001b[39;00m arrays]\n\u001b[1;32m     89\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m arrays:\n\u001b[1;32m     90\u001b[0m         \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mneed at least one array to stack\u001b[39m\u001b[38;5;124m'\u001b[39m)\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_wrapper.py:88\u001b[0m, in \u001b[0;36m\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m     83\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mstack\u001b[39m(arrays, axis\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m0\u001b[39m):\n\u001b[1;32m     84\u001b[0m     \u001b[38;5;66;03m# this code is basically copied from numpy/core/shape_base.py's stack\u001b[39;00m\n\u001b[1;32m     85\u001b[0m     \u001b[38;5;66;03m# we need it here because we want to re-implement stack in terms of the\u001b[39;00m\n\u001b[1;32m     86\u001b[0m     \u001b[38;5;66;03m# primitives defined in this file\u001b[39;00m\n\u001b[0;32m---> 88\u001b[0m     arrays \u001b[38;5;241m=\u001b[39m [array(arr) \u001b[38;5;28;01mfor\u001b[39;00m arr \u001b[38;5;129;01min\u001b[39;00m arrays]\n\u001b[1;32m     89\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m arrays:\n\u001b[1;32m     90\u001b[0m         \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mneed at least one array to stack\u001b[39m\u001b[38;5;124m'\u001b[39m)\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:14\u001b[0m, in \u001b[0;36mmake_vjp..vjp\u001b[0;34m(g)\u001b[0m\n\u001b[0;32m---> 14\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mvjp\u001b[39m(g): \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mbackward_pass\u001b[49m\u001b[43m(\u001b[49m\u001b[43mg\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mend_node\u001b[49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:21\u001b[0m, in \u001b[0;36mbackward_pass\u001b[0;34m(g, end_node)\u001b[0m\n\u001b[1;32m     19\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m node \u001b[38;5;129;01min\u001b[39;00m toposort(end_node):\n\u001b[1;32m     20\u001b[0m     outgrad \u001b[38;5;241m=\u001b[39m outgrads\u001b[38;5;241m.\u001b[39mpop(node)\n\u001b[0;32m---> 21\u001b[0m     ingrads \u001b[38;5;241m=\u001b[39m \u001b[43mnode\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mvjp\u001b[49m\u001b[43m(\u001b[49m\u001b[43moutgrad\u001b[49m\u001b[43m[\u001b[49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     22\u001b[0m     \u001b[38;5;28;01mfor\u001b[39;00m parent, ingrad \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mzip\u001b[39m(node\u001b[38;5;241m.\u001b[39mparents, ingrads):\n\u001b[1;32m     23\u001b[0m         outgrads[parent] \u001b[38;5;241m=\u001b[39m add_outgrads(outgrads\u001b[38;5;241m.\u001b[39mget(parent), ingrad)\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:67\u001b[0m, in \u001b[0;36mdefvjp..vjp_argnums..\u001b[0;34m(g)\u001b[0m\n\u001b[1;32m     64\u001b[0m         \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mNotImplementedError\u001b[39;00m(\n\u001b[1;32m     65\u001b[0m             \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mVJP of \u001b[39m\u001b[38;5;132;01m{}\u001b[39;00m\u001b[38;5;124m wrt argnum 0 not defined\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;241m.\u001b[39mformat(fun\u001b[38;5;241m.\u001b[39m\u001b[38;5;18m__name__\u001b[39m))\n\u001b[1;32m     66\u001b[0m     vjp \u001b[38;5;241m=\u001b[39m vjpfun(ans, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[0;32m---> 67\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mlambda\u001b[39;00m g: (\u001b[43mvjp\u001b[49m\u001b[43m(\u001b[49m\u001b[43mg\u001b[49m\u001b[43m)\u001b[49m,)\n\u001b[1;32m     68\u001b[0m \u001b[38;5;28;01melif\u001b[39;00m L \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m2\u001b[39m:\n\u001b[1;32m     69\u001b[0m     argnum_0, argnum_1 \u001b[38;5;241m=\u001b[39m argnums\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:82\u001b[0m, in \u001b[0;36m\u001b[0;34m(g)\u001b[0m\n\u001b[1;32m     80\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mlog10,  \u001b[38;5;28;01mlambda\u001b[39;00m ans, x : \u001b[38;5;28;01mlambda\u001b[39;00m g: g \u001b[38;5;241m/\u001b[39m x \u001b[38;5;241m/\u001b[39m anp\u001b[38;5;241m.\u001b[39mlog(\u001b[38;5;241m10\u001b[39m))\n\u001b[1;32m     81\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mlog1p,  \u001b[38;5;28;01mlambda\u001b[39;00m ans, x : \u001b[38;5;28;01mlambda\u001b[39;00m g: g \u001b[38;5;241m/\u001b[39m (x \u001b[38;5;241m+\u001b[39m \u001b[38;5;241m1\u001b[39m))\n\u001b[0;32m---> 82\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39msin,    \u001b[38;5;28;01mlambda\u001b[39;00m ans, x : \u001b[38;5;28;01mlambda\u001b[39;00m g: \u001b[43mg\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43m \u001b[49m\u001b[43manp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mcos\u001b[49m\u001b[43m(\u001b[49m\u001b[43mx\u001b[49m\u001b[43m)\u001b[49m)\n\u001b[1;32m     83\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mcos,    \u001b[38;5;28;01mlambda\u001b[39;00m ans, x : \u001b[38;5;28;01mlambda\u001b[39;00m g: \u001b[38;5;241m-\u001b[39m g \u001b[38;5;241m*\u001b[39m anp\u001b[38;5;241m.\u001b[39msin(x))\n\u001b[1;32m     84\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mtan,    \u001b[38;5;28;01mlambda\u001b[39;00m ans, x : \u001b[38;5;28;01mlambda\u001b[39;00m g: g \u001b[38;5;241m/\u001b[39m anp\u001b[38;5;241m.\u001b[39mcos(x) \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m2\u001b[39m)\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_boxes.py:27\u001b[0m, in \u001b[0;36mArrayBox.__mul__\u001b[0;34m(self, other)\u001b[0m\n\u001b[0;32m---> 27\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m__mul__\u001b[39m(\u001b[38;5;28mself\u001b[39m, other): \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43manp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmultiply\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mother\u001b[49m\u001b[43m)\u001b[49m\n",
+      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:78\u001b[0m, in \u001b[0;36mdefvjp..vjp_argnums..\u001b[0;34m(g)\u001b[0m\n\u001b[1;32m     76\u001b[0m     vjp_0 \u001b[38;5;241m=\u001b[39m vjp_0_fun(ans, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m     77\u001b[0m     vjp_1 \u001b[38;5;241m=\u001b[39m vjp_1_fun(ans, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[0;32m---> 78\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mlambda\u001b[39;00m g: (vjp_0(g), \u001b[43mvjp_1\u001b[49m\u001b[43m(\u001b[49m\u001b[43mg\u001b[49m\u001b[43m)\u001b[49m)\n\u001b[1;32m     79\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m     80\u001b[0m     vjps \u001b[38;5;241m=\u001b[39m [vjps_dict[argnum](ans, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs) \u001b[38;5;28;01mfor\u001b[39;00m argnum \u001b[38;5;129;01min\u001b[39;00m argnums]\n",
+      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:660\u001b[0m, in \u001b[0;36munbroadcast_f..\u001b[0;34m(g)\u001b[0m\n\u001b[1;32m    658\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21munbroadcast_f\u001b[39m(target, f):\n\u001b[1;32m    659\u001b[0m     target_meta \u001b[38;5;241m=\u001b[39m anp\u001b[38;5;241m.\u001b[39mmetadata(target)\n\u001b[0;32m--> 660\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mlambda\u001b[39;00m g: unbroadcast(\u001b[43mf\u001b[49m\u001b[43m(\u001b[49m\u001b[43mg\u001b[49m\u001b[43m)\u001b[49m, target_meta)\n",
+      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:35\u001b[0m, in \u001b[0;36m\u001b[0;34m(g)\u001b[0m\n\u001b[1;32m     30\u001b[0m \u001b[38;5;66;03m# ----- Binary ufuncs -----\u001b[39;00m\n\u001b[1;32m     32\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39madd,         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(x, \u001b[38;5;28;01mlambda\u001b[39;00m g: g),\n\u001b[1;32m     33\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: g))\n\u001b[1;32m     34\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mmultiply,    \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(x, \u001b[38;5;28;01mlambda\u001b[39;00m g: y \u001b[38;5;241m*\u001b[39m g),\n\u001b[0;32m---> 35\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: \u001b[43mx\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43m \u001b[49m\u001b[43mg\u001b[49m))\n\u001b[1;32m     36\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39msubtract,    \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(x, \u001b[38;5;28;01mlambda\u001b[39;00m g: g),\n\u001b[1;32m     37\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: \u001b[38;5;241m-\u001b[39mg))\n\u001b[1;32m     38\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mdivide,      \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(x, \u001b[38;5;28;01mlambda\u001b[39;00m g:   g \u001b[38;5;241m/\u001b[39m y),\n\u001b[1;32m     39\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: \u001b[38;5;241m-\u001b[39m g \u001b[38;5;241m*\u001b[39m x \u001b[38;5;241m/\u001b[39m y\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m2\u001b[39m))\n",
+      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_boxes.py:35\u001b[0m, in \u001b[0;36mArrayBox.__rmul__\u001b[0;34m(self, other)\u001b[0m\n\u001b[0;32m---> 35\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m__rmul__\u001b[39m(\u001b[38;5;28mself\u001b[39m, other): \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43manp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmultiply\u001b[49m\u001b[43m(\u001b[49m\u001b[43mother\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m)\u001b[49m\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py:45\u001b[0m, in \u001b[0;36mprimitive..f_wrapped\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     43\u001b[0m     argnums \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mtuple\u001b[39m(argnum    \u001b[38;5;28;01mfor\u001b[39;00m argnum, _   \u001b[38;5;129;01min\u001b[39;00m boxed_args)\n\u001b[1;32m     44\u001b[0m     ans \u001b[38;5;241m=\u001b[39m f_wrapped(\u001b[38;5;241m*\u001b[39margvals, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[0;32m---> 45\u001b[0m     node \u001b[38;5;241m=\u001b[39m \u001b[43mnode_constructor\u001b[49m\u001b[43m(\u001b[49m\u001b[43mans\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mf_wrapped\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43margvals\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mkwargs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43margnums\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mparents\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     46\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m new_box(ans, trace, node)\n\u001b[1;32m     47\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n",
       "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:36\u001b[0m, in \u001b[0;36mVJPNode.__init__\u001b[0;34m(self, value, fun, args, kwargs, parent_argnums, parents)\u001b[0m\n\u001b[1;32m     33\u001b[0m     fun_name \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mgetattr\u001b[39m(fun, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124m__name__\u001b[39m\u001b[38;5;124m'\u001b[39m, fun)\n\u001b[1;32m     34\u001b[0m     \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mNotImplementedError\u001b[39;00m(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mVJP of \u001b[39m\u001b[38;5;132;01m{}\u001b[39;00m\u001b[38;5;124m wrt argnums \u001b[39m\u001b[38;5;132;01m{}\u001b[39;00m\u001b[38;5;124m not defined\u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m     35\u001b[0m                               \u001b[38;5;241m.\u001b[39mformat(fun_name, parent_argnums))\n\u001b[0;32m---> 36\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mvjp \u001b[38;5;241m=\u001b[39m \u001b[43mvjpmaker\u001b[49m\u001b[43m(\u001b[49m\u001b[43mparent_argnums\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mvalue\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:76\u001b[0m, in \u001b[0;36mdefvjp..vjp_argnums\u001b[0;34m(argnums, ans, args, kwargs)\u001b[0m\n\u001b[1;32m     73\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mKeyError\u001b[39;00m:\n\u001b[1;32m     74\u001b[0m     \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mNotImplementedError\u001b[39;00m(\n\u001b[1;32m     75\u001b[0m         \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mVJP of \u001b[39m\u001b[38;5;132;01m{}\u001b[39;00m\u001b[38;5;124m wrt argnums 0, 1 not defined\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;241m.\u001b[39mformat(fun\u001b[38;5;241m.\u001b[39m\u001b[38;5;18m__name__\u001b[39m))\n\u001b[0;32m---> 76\u001b[0m vjp_0 \u001b[38;5;241m=\u001b[39m \u001b[43mvjp_0_fun\u001b[49m\u001b[43m(\u001b[49m\u001b[43mans\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     77\u001b[0m vjp_1 \u001b[38;5;241m=\u001b[39m vjp_1_fun(ans, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m     78\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mlambda\u001b[39;00m g: (vjp_0(g), vjp_1(g))\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:34\u001b[0m, in \u001b[0;36m\u001b[0;34m(ans, x, y)\u001b[0m\n\u001b[1;32m     30\u001b[0m \u001b[38;5;66;03m# ----- Binary ufuncs -----\u001b[39;00m\n\u001b[1;32m     32\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39madd,         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(x, \u001b[38;5;28;01mlambda\u001b[39;00m g: g),\n\u001b[1;32m     33\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: g))\n\u001b[0;32m---> 34\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mmultiply,    \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : \u001b[43munbroadcast_f\u001b[49m\u001b[43m(\u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mlambda\u001b[39;49;00m\u001b[43m \u001b[49m\u001b[43mg\u001b[49m\u001b[43m:\u001b[49m\u001b[43m \u001b[49m\u001b[43my\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43m \u001b[49m\u001b[43mg\u001b[49m\u001b[43m)\u001b[49m,\n\u001b[1;32m     35\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: x \u001b[38;5;241m*\u001b[39m g))\n\u001b[1;32m     36\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39msubtract,    \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(x, \u001b[38;5;28;01mlambda\u001b[39;00m g: g),\n\u001b[1;32m     37\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: \u001b[38;5;241m-\u001b[39mg))\n\u001b[1;32m     38\u001b[0m defvjp(anp\u001b[38;5;241m.\u001b[39mdivide,      \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(x, \u001b[38;5;28;01mlambda\u001b[39;00m g:   g \u001b[38;5;241m/\u001b[39m y),\n\u001b[1;32m     39\u001b[0m                         \u001b[38;5;28;01mlambda\u001b[39;00m ans, x, y : unbroadcast_f(y, \u001b[38;5;28;01mlambda\u001b[39;00m g: \u001b[38;5;241m-\u001b[39m g \u001b[38;5;241m*\u001b[39m x \u001b[38;5;241m/\u001b[39m y\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m2\u001b[39m))\n",
-      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/numpy/numpy_vjps.py:659\u001b[0m, in \u001b[0;36munbroadcast_f\u001b[0;34m(target, f)\u001b[0m\n\u001b[1;32m    658\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21munbroadcast_f\u001b[39m(target, f):\n\u001b[0;32m--> 659\u001b[0m     target_meta \u001b[38;5;241m=\u001b[39m \u001b[43manp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmetadata\u001b[49m\u001b[43m(\u001b[49m\u001b[43mtarget\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    660\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mlambda\u001b[39;00m g: unbroadcast(f(g), target_meta)\n",
+      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py:56\u001b[0m, in \u001b[0;36mdefvjp..vjp_argnums\u001b[0;34m(argnums, ans, args, kwargs)\u001b[0m\n\u001b[1;32m     53\u001b[0m argnums \u001b[38;5;241m=\u001b[39m kwargs\u001b[38;5;241m.\u001b[39mget(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124margnums\u001b[39m\u001b[38;5;124m'\u001b[39m, count())\n\u001b[1;32m     54\u001b[0m vjps_dict \u001b[38;5;241m=\u001b[39m {argnum : translate_vjp(vjpmaker, fun, argnum)\n\u001b[1;32m     55\u001b[0m              \u001b[38;5;28;01mfor\u001b[39;00m argnum, vjpmaker \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mzip\u001b[39m(argnums, vjpmakers)}\n\u001b[0;32m---> 56\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mvjp_argnums\u001b[39m(argnums, ans, args, kwargs):\n\u001b[1;32m     57\u001b[0m     L \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mlen\u001b[39m(argnums)\n\u001b[1;32m     58\u001b[0m     \u001b[38;5;66;03m# These first two cases are just optimizations\u001b[39;00m\n",
       "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
      ]
     }
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diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter1_9_0.png b/doc/LectureNotes/_build/jupyter_execute/chapter1_9_0.png
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diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb
index acc85f976..f2170c67a 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb
@@ -1798,10 +1798,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.008947823579448178\n",
-      "4.14822021080294\n",
-      "[[0.73947737 2.16006961]\n",
-      " [2.16006961 7.24782786]]\n"
+      "0.01354598394614281\n",
+      "4.037503978471253\n",
+      "[[0.95927495 2.85834701]\n",
+      " [2.85834701 9.70233292]]\n"
      ]
     }
    ],
@@ -1845,10 +1845,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.08657894597048958\n",
-      "2.219222942590059\n",
-      "[[1.        0.6343356]\n",
-      " [0.6343356 1.       ]]\n"
+      "0.09309965420717024\n",
+      "1.6546329613199204\n",
+      "[[1.         0.57867898]\n",
+      " [0.57867898 1.        ]]\n"
      ]
     }
    ],
@@ -1905,30 +1905,30 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[ 0.29724806  0.26804287]\n",
-      " [-0.15626984 -1.36853738]\n",
-      " [-0.77070756 -2.13536532]\n",
-      " [-0.45372697 -3.1582408 ]\n",
-      " [ 0.52580392  2.72567956]\n",
-      " [-0.86515815 -1.35704388]\n",
-      " [-0.73738602 -2.12933164]\n",
-      " [-0.10486183  1.06292011]\n",
-      " [ 1.75670484  5.27381733]\n",
-      " [ 0.50835355  0.81805914]]\n",
+      "[[-0.49352496 -2.38394242]\n",
+      " [ 0.18849928  0.73454039]\n",
+      " [-1.58104393 -5.16350368]\n",
+      " [ 0.34695383  0.23472346]\n",
+      " [ 0.95953339  2.95819409]\n",
+      " [ 1.31331481  3.59914165]\n",
+      " [ 0.14846308  1.1180677 ]\n",
+      " [ 0.26022531  0.23496851]\n",
+      " [-0.12178678 -0.11868087]\n",
+      " [-1.02063403 -1.21350883]]\n",
       "          0         1\n",
-      "0  0.297248  0.268043\n",
-      "1 -0.156270 -1.368537\n",
-      "2 -0.770708 -2.135365\n",
-      "3 -0.453727 -3.158241\n",
-      "4  0.525804  2.725680\n",
-      "5 -0.865158 -1.357044\n",
-      "6 -0.737386 -2.129332\n",
-      "7 -0.104862  1.062920\n",
-      "8  1.756705  5.273817\n",
-      "9  0.508354  0.818059\n",
+      "0 -0.493525 -2.383942\n",
+      "1  0.188499  0.734540\n",
+      "2 -1.581044 -5.163504\n",
+      "3  0.346954  0.234723\n",
+      "4  0.959533  2.958194\n",
+      "5  1.313315  3.599142\n",
+      "6  0.148463  1.118068\n",
+      "7  0.260225  0.234969\n",
+      "8 -0.121787 -0.118681\n",
+      "9 -1.020634 -1.213509\n",
       "          0         1\n",
-      "0  1.000000  0.915549\n",
-      "1  0.915549  1.000000\n"
+      "0  1.000000  0.949087\n",
+      "1  0.949087  1.000000\n"
      ]
     }
    ],
@@ -1974,37 +1974,37 @@
      "text": [
       "     0         1         2         3         4         5         6         7   \\\n",
       "0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   \n",
-      "1   0.0  0.086925  0.087377  0.087520  0.087878  0.088164  0.079680  0.079580   \n",
-      "2   0.0  0.087377  0.089240  0.088074  0.089063  0.089924  0.079827  0.080060   \n",
-      "3   0.0  0.087520  0.088074  0.094227  0.094252  0.094139  0.089441  0.088974   \n",
-      "4   0.0  0.087878  0.089063  0.094252  0.094610  0.094812  0.089040  0.088778   \n",
-      "5   0.0  0.088164  0.089924  0.094139  0.094812  0.095315  0.088488  0.088425   \n",
-      "6   0.0  0.079680  0.079827  0.089441  0.089040  0.088488  0.087315  0.086524   \n",
-      "7   0.0  0.079580  0.080060  0.088974  0.088778  0.088425  0.086524  0.085876   \n",
-      "8   0.0  0.079499  0.080295  0.088502  0.088506  0.088348  0.085716  0.085210   \n",
-      "9   0.0  0.079448  0.080548  0.088037  0.088238  0.088275  0.084906  0.084541   \n",
-      "10  0.0  0.071751  0.071438  0.082839  0.082089  0.081194  0.082509  0.081481   \n",
-      "11  0.0  0.071392  0.071284  0.082139  0.081533  0.080780  0.081559  0.080641   \n",
-      "12  0.0  0.071071  0.071164  0.081471  0.081007  0.080395  0.080635  0.079827   \n",
-      "13  0.0  0.070794  0.071084  0.080839  0.080518  0.080048  0.079741  0.079043   \n",
-      "14  0.0  0.070562  0.071051  0.080249  0.080070  0.079745  0.078880  0.078293   \n",
+      "1   0.0  0.072254  0.074732  0.075050  0.075400  0.075727  0.068297  0.068215   \n",
+      "2   0.0  0.074732  0.078265  0.076259  0.077107  0.077954  0.068418  0.068622   \n",
+      "3   0.0  0.075050  0.076259  0.082604  0.082322  0.081988  0.078133  0.077663   \n",
+      "4   0.0  0.075400  0.077107  0.082322  0.082315  0.082269  0.077387  0.077091   \n",
+      "5   0.0  0.075727  0.077954  0.081988  0.082269  0.082525  0.076570  0.076454   \n",
+      "6   0.0  0.068297  0.068418  0.078133  0.077387  0.076570  0.075952  0.075224   \n",
+      "7   0.0  0.068215  0.068622  0.077663  0.077091  0.076454  0.075224  0.074613   \n",
+      "8   0.0  0.068168  0.068876  0.077210  0.076818  0.076371  0.074495  0.074004   \n",
+      "9   0.0  0.068163  0.069190  0.076775  0.076573  0.076324  0.073764  0.073398   \n",
+      "10  0.0  0.060924  0.060401  0.071600  0.070601  0.069522  0.071006  0.070141   \n",
+      "11  0.0  0.060711  0.060369  0.071122  0.070241  0.069283  0.070364  0.069583   \n",
+      "12  0.0  0.060534  0.060381  0.070671  0.069911  0.069080  0.069738  0.069042   \n",
+      "13  0.0  0.060394  0.060441  0.070245  0.069612  0.068912  0.069125  0.068517   \n",
+      "14  0.0  0.060291  0.060550  0.069845  0.069343  0.068782  0.068524  0.068007   \n",
       "\n",
       "          8         9         10        11        12        13        14  \n",
       "0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  \n",
-      "1   0.079499  0.079448  0.071751  0.071392  0.071071  0.070794  0.070562  \n",
-      "2   0.080295  0.080548  0.071438  0.071284  0.071164  0.071084  0.071051  \n",
-      "3   0.088502  0.088037  0.082839  0.082139  0.081471  0.080839  0.080249  \n",
-      "4   0.088506  0.088238  0.082089  0.081533  0.081007  0.080518  0.080070  \n",
-      "5   0.088348  0.088275  0.081194  0.080780  0.080395  0.080048  0.079745  \n",
-      "6   0.085716  0.084906  0.082509  0.081559  0.080635  0.079741  0.078880  \n",
-      "7   0.085210  0.084541  0.081481  0.080641  0.079827  0.079043  0.078293  \n",
-      "8   0.084685  0.084157  0.080434  0.079704  0.078999  0.078326  0.077688  \n",
-      "9   0.084157  0.083772  0.079379  0.078759  0.078165  0.077604  0.077079  \n",
-      "10  0.080434  0.079379  0.079152  0.078033  0.076935  0.075863  0.074818  \n",
-      "11  0.079704  0.078759  0.078033  0.077004  0.075996  0.075014  0.074061  \n",
-      "12  0.078999  0.078165  0.076935  0.075996  0.075079  0.074187  0.073325  \n",
-      "13  0.078326  0.077604  0.075863  0.075014  0.074187  0.073388  0.072618  \n",
-      "14  0.077688  0.077079  0.074818  0.074061  0.073325  0.072618  0.071942  \n"
+      "1   0.068168  0.068163  0.060924  0.060711  0.060534  0.060394  0.060291  \n",
+      "2   0.068876  0.069190  0.060401  0.060369  0.060381  0.060441  0.060550  \n",
+      "3   0.077210  0.076775  0.071600  0.071122  0.070671  0.070245  0.069845  \n",
+      "4   0.076818  0.076573  0.070601  0.070241  0.069911  0.069612  0.069343  \n",
+      "5   0.076371  0.076324  0.069522  0.069283  0.069080  0.068912  0.068782  \n",
+      "6   0.074495  0.073764  0.071006  0.070364  0.069738  0.069125  0.068524  \n",
+      "7   0.074004  0.073398  0.070141  0.069583  0.069042  0.068517  0.068007  \n",
+      "8   0.073520  0.073044  0.069265  0.068792  0.068339  0.067905  0.067489  \n",
+      "9   0.073044  0.072705  0.068375  0.067990  0.067628  0.067288  0.066969  \n",
+      "10  0.069265  0.068375  0.067400  0.066672  0.065952  0.065237  0.064526  \n",
+      "11  0.068792  0.067990  0.066672  0.066006  0.065350  0.064701  0.064057  \n",
+      "12  0.068339  0.067628  0.065952  0.065350  0.064759  0.064176  0.063600  \n",
+      "13  0.067905  0.067288  0.065237  0.064701  0.064176  0.063661  0.063155  \n",
+      "14  0.067489  0.066969  0.064526  0.064057  0.063600  0.063155  0.062721  \n"
      ]
     }
    ],
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
index b7e0134a0..f183f152e 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
@@ -489,10 +489,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Runtime: 0.0907831 sec\n",
+      "Runtime: 0.0907788 sec\n",
       "Jackknife Statistics :\n",
       "original           bias      std. error\n",
-      " 100.142        100.132        0.149864\n"
+      " 100.022        100.012        0.148734\n"
      ]
     }
    ],
@@ -917,7 +917,7 @@
      "text": [
       "Bootstrap Statistics :\n",
       "original           bias      std. error\n",
-      " 100.033  14.9292        100.032        0.149452\n"
+      " 99.7522  14.9594        99.7525        0.149991\n"
      ]
     }
    ],
@@ -975,7 +975,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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dQGEhS9KDDPHcC5Vkh6hb7UVHIaIHYEkiIoPasgV45BGgSRPRScxXoP11PFbrLCeWJDJzLElEZDAaDfD779yLpI+hnnsQldEeRaUK0VGI6D5YkojIYA4f1l7+//jjopOYv6Fee5FX6oidWSGioxDRfbAkEZHBbNkCeHgAHTuKTmL+mjteQmOHK7zhLZEZY0kiIoPZsgUYOBCQy0UnMX8ymXZiyS23OqG0VHQaItKFJYmIDOLCBeCvv3g+UmUM8dyLG2o37N8vOgkR6cKSREQG8fvvgJ0dEBEhOonlaO9yGn52N7Fpk+gkRKQLSxIRGcSWLUCvXkCtWqKTWA4bmYTHPfdh0yZAkkSnIaJ7sSQRUbVlZgLx8TzUVhVDPffi0iXgxAnRSYjoXixJRFRt27cDpaXak7apcrrXPg5HR+0YEpF5YUkiomr77TcgJASoW1d0EsujtFGjZ0+WJCJzxJJERNWiUgHbtvFQW3X06wfs3Qvk5opOQkR3Y0kiomrZvRvIyWFJqo6+fYGSEuCPP0QnIaK7sSQRUbVs2QLUqwe0bi06ieVq0ABo3Fi7R46IzIfQkjRv3jy0bdsWzs7O8Pb2xpAhQ5CUlPTQ58XHxyMkJAT29vZo0KABlixZYoK0RHQvSdKWpMGDtTNIU9X17as9L4lTARCZD6ElKT4+HpMnT8aBAwcQGxuLkpISREREID8//77PuXjxIvr374+uXbvi2LFjeOutt/Dyyy9jw4YNJkxORABw8iRw+TIPtRlC377asdTj70QiMhFbkW++/Z7LOX744Qd4e3sjISEB3bp10/mcJUuWoF69eli4cCEAoFmzZjhy5Ajmz5+PYcOGVVi/uLgYxcXFZZ/n5OQAANRqNdRqtYG2xLRu57bU/KbCcdJPdcZpyxYb1JIXo9MnT0I9v+T+K9rZVTWeWVErFOU+Gu6F1ejcGVAqbREZqUHDhhrDvr6J8WdPPxwn/YgcJ5kkmc/O3X/++QeNGzfGyZMn0bJlS53rdOvWDcHBwfjiiy/Klm3atAlPPfUUCgoKoLjnl9ecOXPw3nvvVXidNWvWwNHR0bAbQFTDvP12Zzg5qfHWW4dER7EKc+Z0hEwmYfbsA6KjEJmdgoICjBw5EtnZ2XBxcTHJe5pNSZIkCY8//jgyMzOxZ8+e+673yCOPYOzYsXjrrbfKlu3fvx+dO3fGtWvX4OfnV259XXuSAgICcPPmTZMNsqGp1WrExsYiPDy8QimkOzhO+qnqOOXmAj4+tljQ6GtMrLfViAnNh1qhQOyECQhfuhQKQ/5Vu1U7fl98YYN33rHB9eslcHAw3MubGn/29MNx0s/tcWrfvj38/PxMWpKEHm6725QpU3DixAns3bv3oevK7jlD9HbPu3c5ACiVSiiVygrLFQqFxX9TWsM2mALHST+VHae9e7WXrfdzOQCFSmXEZOZHoVYbdpv/HfcBA4DXXwf271egb1/Dvbwo/NnTD8dJPyLGyCymAHjppZewZcsW7Nq1C3UfMmWvr68v0tLSyi1LT0+Hra0tPDw8jBmTiO4SHa29dL2R4zXRUaxG06ba6RQ4+zaReRBakiRJwpQpU7Bx40b88ccfCAoKeuhzOnbsiNjY2HLLYmJiEBoayiZOZELR0UCfPqJTWBeZ7M5UAEQkntCSNHnyZKxevRpr1qyBs7Mz0tLSkJaWhsLCwrJ1Zs6cidGjR5d9PnHiRFy+fBnTpk3D6dOnsXz5cixbtgzTp08XsQlENdL589oHS5Lh9e2rnQbg4kXRSYhIaElavHgxsrOz0aNHD/j5+ZU9fv7557J1UlNTkZycXPZ5UFAQoqKiEBcXhzZt2mDu3Ln48ssvdV7+T0TGER0N2NoCYWGik1ifXr20Y8u9SUTiCT1xW58L61asWFFhWffu3XH06FEjJCIifURHA506ARZ6gahZc3EBOnfWlqRJk0SnIarZzOLEbSKyHCqV9kasPNRmPH37Ajt3aseaiMRhSSKiSjlwAMjLY0kypr59gfx8YN8+0UmIajaWJCKqlOhowMsLCA4WncR6tW4N+PryvCQi0ViSiKhSoqOB8HDAhr89jEYm0+6pY0kiEstsZtwmIvN34wZw9Cjw8suik1gZHZcJ9r0ehh9Pv4trnf4Df+Ut7cJdu0wcjKhm49+CRKS32FhAkoCICNFJrF+4ewJsUIrtGe1ERyGqsViSiEhv0dF3zpch4/JQ5KCdyxmWJCKBWJKISC+SBMTEcC+SKfV1P4TYzBCUaPirmkgE/uQRkV5OnADS0njpvyn1dT+MrBJnHMptJjoKUY3EkkREeomOBhwdgS5dRCepOUKdk+Bum81DbkSCsCQRkV6io4EePQClUnSSmkMu0yDC/Qi2Z7QVHYWoRmJJIqKHys8H9u7loTYR+rofxpHcJrihchUdhajGYUkiooeKi9PeR4wlyfT6uB2CBBvEZoaKjkJU47AkEdFDRUcDgYHAI4+ITlLz+CozEVzrLM9LIhKAJYmIHio6WrsXSSYTnaRm6ut+GNsz2kKjEZ2EqGZhSSKiB7p0CTh7lofaROrrfgg31G44dkx0EqKahSWJiB4oOhqQy4FevUQnqbk6uPwNJ5tCxMaKTkJUs7AkEdEDRUcDHToArry4Shg7mxL0qJ3IkkRkYixJRHRfJSXAzp081GYOwt0TsHcvUFAgOglRzcGSRET3dfAgkJPD+7WZg3C3I1CpgD17RCchqjlYkojovqKjAXd3IJRT9AjXzPEy/P2BHTtEJyGqOViSiOi+oqOB3r21J26TWDIZEB4OnpdEZEIsSUSk061bwOHDPB/JnISHA8ePA9evi05CVDOwJBGRTjt2AJLE85HMSe/e2o87d4rNQVRTsCQRkU6xsUDz5kDduqKT0G0+PkCrVjzkRmQqLElEVIEkaf9HHB4uOgnd6/Z5SZIkOgmR9WNJIqIK/vkHSE6+c3iHzEd4OJCSApw5IzoJkfWzFR2AiMxPbCxgawt0/19/4LNC0XHoLl27AnZ22v9GzZqJTkNk3ViSiGqqsDDtRzs74MUXgYEDAZUKABD71/vo6OQCZ1sWJHPj6Ah07qwtSS+/LDoNkXXj4TYiKqdEY4M/MoMR7pYgOgrdR3g4EBcHqNWikxBZN5YkIirnSG4T5JTWQm+WJLMVHg7k5QEHDohOQmTdWJKIqJzYzFC4yvPQ1plnBpur4GDt7WI4FQCRcVWpJK1YsQIFvBU1kVWKzQxBmNsx2NpoREeh+5DLgV69WJKIjK1KJWnmzJnw9fXFc889h/379xs6ExEJklvigD9zWvB8JAsQHg4cOgRkZ4tOQmS9qnR129WrVxEZGYkVK1YgLCwMQUFBGDduHMaMGQNfX19DZyQiE9md3Rolki3PRzJXt69IBBBe6AONZh12dZ6FIV777qyza5eAYETWqUp7kuRyOQYPHoyNGzfiypUreP755/HTTz+hXr16GDx4MH777TdoNNxVT2RpYjNCUE+ZhsYOV0VHoYeo73AdjRyuIjYzVHQUIqtV7RO3vb290blzZ3Ts2BE2NjY4efIkxo4di4YNGyIuLs4AEYnIVGIzQxHulgCZTHQS0ke4WwJiM0NExyCyWlUuSdevX8f8+fPRokUL9OjRAzk5Odi6dSsuXryIa9eu4YknnsCYMWMMmZWIjCilyAN/F9RHuDsPtVmK3m4JOFcYgMtFPqKjEFmlKpWkQYMGISAgACtWrMCECROQkpKCtWvXove/N3pycHDAa6+9hitXrhg0LBEZzx8ZwQCAnrWPCk5C+upZ+yhsUIrYDO5NIjKGKp247e3tjfj4eHTs2PG+6/j5+eHixYtVDkZEprXzVjCCa52Flx0vl7IUtRX5aOuchNjMUIz3jxIdh8jqVGlPUvfu3fHYY49VWK5SqbBy5UoAgEwmQ2BgYPXSEZFJSBLwxy3eisQShbsfwc7Mx6CReCIZkaFVqSSNGzcO2Tom58jNzcW4ceOqHYqITCs52RlpKg+Eux0RHYUqKdwtAbdKXHEsr7HoKERWp0olSZIkyHRc/nL16lW4urpWOxQRmdbx415Q2qjQ2fUv0VGokjq4/A0nm0Kel0RkBJU6Jyk4OBgymQwymQy9evWCre2dp5eWluLixYvo27evwUMSkXElJnqjS+2/4CBXiY5ClWRnU4IetRMRmxmKNwPXio5DZFUqVZKGDBkCAEhMTESfPn1Qq1atsq/Z2dmhfv36GDZsmEEDEpFxFWsUOHXKA7Pr/S46ClVRuHsCZpx/HgWlSjiKDkNkRSpVkmbPng0AqF+/Pp5++mnY29sbJRQRmc7BrKYoLrZFLw9e+m+pwt2OQCVNwZ7sVugjOgyRFanSOUljxoxhQSKyEjtuPQYXl2K0dr4gOgpVUTPHy/C3u4EdmRWvOiaiqtN7T5K7uzvOnj0LT09PuLm56Txx+7aMjAyDhCMi49t56zG0anUDNjJJdBSqIpns31uUZISWuwnuffEmuER60bskff7553B2di7794NKEhFZhkx1LSTkNMak1ieAc6LTUHX0dkvAj9f74rrKDT52maLjEFkFvUvS3fdhGzt2rDGyEJGJ7coKhgZytGlzgyXJwt2+597OzMcw0men4DRE1kHvkpSTk6P3i7q4uFQpDBGZVmxmCBo5XoWXV6HoKFRNPnaZaOV0HjEZoSxJRAaid0mqXbv2Qw+x3Z5ksrS0tNrBiMj4YjNCEe7FW5FYiwj3w/jpem9IkvY8JSKqHr1L0i6e6EdkVS4W+uJ8UR187LEUQLDoOGQAEW5HMP/KcPxdUB8tnC6JjkNk8fQuSd27dzf4m+/evRuffvopEhISkJqaik2bNpVNWKlLXFwcwnRcuXH69Gk0bdrU4PmIrNmOzBDYoBQ93I5jH0uSVejiehJKmQoxGaEsSUQGoHdJOnHiBFq2bAkbGxucOHHigeu2atVKr9fMz89H69atMW7cuErN1J2UlFTuvCcvLy+9n0tEWrGZoWjncgauigLRUchAHOQqdKt9HDGZoXg1YL3oOEQWT++S1KZNG6SlpcHb2xtt2rSBTCaDJFWcV6Uy5yT169cP/fr10z/tv7y9vVG7du1KP4+ItEpLgZ2ZwZhc5zfRUcjAwt0SMPvSWBRrFFDaqEXHIbJoepekixcvlu2xuXjxotEC6SM4OBhFRUVo3rw5Zs2apfMQ3G3FxcUoLi4u+/z2VXpqtRpqtWX+Armd21LzmwrH6f6OHgUySlwR5n0caoUCAMo+0v1ZwliF+RzHjAv2iM9vgzCP47pXMvLPBH/29MNx0o/IcZJJunYHCSCTyR56TlJSUhJ2796NkJAQFBcXY9WqVViyZAni4uLQrVs3nc+ZM2cO3nvvvQrL16xZA0dH3gqSaqb16xtj/fpHsGpVFBQKs/gVQAai0QDPPtsHPXtewejRf4uOQ2QwBQUFGDlyJLKzs0021VCVS1JSUhK++uornD59GjKZDE2bNsVLL72EJk2aVC2IHiVJl0GDBkEmk2HLli06v65rT1JAQABu3rxpsfM5qdVqxMbGIjw8HAoz/otWNI7T/YWHy1Hr74PYFDwHaoUCsRMmIHzpUij4F+0DWcpYjTk5A2fy6uFgxym6V9i61ajvz589/XCc9HN7nNq3bw8/Pz+TliS9D7fdbf369RgxYgRCQ0PRsWNHAMCBAwfQsmVLrFmzBk8++aRBQz5Ihw4dsHr16vt+XalUQqlUVliuUCgs/pvSGrbBFDhO5eXlAfv3AwsCD0GhUpUtV6jV5T6n+zP3serreghrU3shK88BXnbZFVcw0c8Df/b0w3HSj4gxqlJJmjFjBmbOnIn333+/3PLZs2fjjTfeMGlJOnbsGPz8/Ez2fkSWLj5ee0pKH/fDoqOQkfR2u3OLkuE+nOOOqKpsqvKktLQ0jB49usLyZ555BmlpaXq/Tl5eHhITE5GYmAhAe0J4YmIikpOTAQAzZ84s9z4LFy7E5s2bce7cOZw6dQozZ87Ehg0bMGXKfXYpE1EF0dFA/fpAI4cU0VHISPyVt9DS6QJiMtuKjkJk0aq0J6lHjx7Ys2cPGjVqVG753r170bVrV71f58iRI+WuTJs2bRoA7c10V6xYgdTU1LLCBAAqlQrTp09HSkoKHBwc0KJFC0RGRqJ///5V2QyiGikmBoiIAGRnRSchY4pwO4Kf08N4ixKiatC7JN19YvTgwYPxxhtvICEhAR06dACgPSfp119/1Xkl2f306NFD51xLt61YsaLc5zNmzMCMGTP0fn0iKu/yZSApCfjoIwAsSVYt3O0IFlx9CmcK6qGZU/LDn0BEFehdknRddbZo0SIsWrSo3LLJkydj4sSJ1Q5GRIYXEwPY2AA9ewL4SnQaMqZutU/ATqZCTGZbliSiKtL7nCSNRqPXQ9/ZtonI9GJigPbtAU5Yb/0c5cXo6noSMRmhoqMQWawqnbhNRJantBTYsUN7PhLVDOHuRxCX1RrFGl5eTlQVVTpxG9DenDY+Ph7JyclQ3TNfyMsvv1ztYERkWIcPA1lZQJ8+opOQqUS4HcGbF17An9nN0cPtPrcoIaL7qlJJOnbsGPr374+CggLk5+fD3d0dN2/ehKOjI7y9vVmSiMxQTAzg6gq05VXhNUbrWufhpchETGZbliSiKqjS4bZXX30VgwYNQkZGBhwcHHDgwAFcvnwZISEhmD9/vqEzEpEBxMQAvXoBtlXef0yWxkYmobfbUcRmhoiOQmSRqlSSEhMT8dprr0Eul0Mul6O4uBgBAQH45JNP8NZbbxk6IxFVU3Y2cOAAD7XVRBFuh5GQ+whuqS3zXpVEIlWpJCkUCsj+nZ3Mx8enbMJHV1fXcpM/EpF5+OMP7Ynb4eGik5CphbsnQIINdmY+JjoKkcWpUkkKDg7GkSNHAABhYWF499138dNPP2Hq1Kl49NFHDRqQiKovJgZo3BgIChKdhEytjvImmjte4lQARFVQpZL00Ucfld1Udu7cufDw8MCkSZOQnp6O7777zqABiah6JEl7vzYeaqu5wt2OICYzFA+4wQER6VClUzhDQ+/8ReLl5YWoqCiDBSIiwzp/Hrh4kfMj1WQR7kfwRcp/cLYwAE0cr4iOQ2QxqnWdS3p6OpKSkiCTydCkSRN4eXkZKhcRGUhMjPaKth49RCchUbrXPg6FTI2YjFCWJKJKqNLhtpycHIwaNQp16tRB9+7d0a1bN/j7++OZZ55Bdna2oTMSUTVERwOdOwPOzqKTkChO8iJ0dv0LMZk8L4moMqpUksaPH4+DBw9i69atyMrKQnZ2NrZu3YojR45gwoQJhs5IRFWkVmuvbOOhNopwO4K4rDZQaThRFpG+qlSSIiMjsXz5cvTp0wcuLi5wdnZGnz59sHTpUkRGRho6IxFV0YEDQF4eSxJpz0vKK3XEgZzmoqMQWYwqlSQPDw+4urpWWO7q6go3N7dqhyIiw4iJATw8gMc4RU6NF1zrHDxsszkVAFElVKkkzZo1C9OmTUNqamrZsrS0NLz++ut45513DBaOiKonOlo7gaRNlX7SyZpob1GSgFiel0SkN70PTgcHB5fNsg0A586dQ2BgIOrVqwcASE5OhlKpxI0bN/DCCy8YPikRVcqtW8CRI8CkSaKTkLkId0/AhKTXkJEBuLuLTkNk/vQuSUOGDDFiDCIyqLAw7EzvAUmajfDvngRW3hSdiMxAhNthSLBBbCzw9NOi0xCZP71L0uzZs42Zg4gMLDqjLVo4XkRdexYk0gqwv4GWThewbVsDliQiPVTrWtCEhAScPn0aMpkMzZs3R3BwsKFyEVE1SBIQkxmKJ73iRUchM9Pf/SBWbGsAjYbnqhE9TJVKUnp6OoYPH464uDjUrl0bkiQhOzsbYWFhWLduHWfeJhLsTEE9XC32RoTbYdFRyMz09ziITxJH4NgxICREdBoi81alkvTSSy8hJycHp06dQrNmzQAAf//9N8aMGYOXX34Za9euNWhIIqqc6Iy2UMpU6Fb7hOgoZGY6ufwFF3keov7zC0Lqr7r/irt2mS4UkZmq0s7W7du3Y/HixWUFCQCaN2+Ob775Btu2bTNYOCKqmpjMUHStfQKO8mLRUcjMKGxKEe6WgKiM9qKjEJm9KpUkjUYDhUJRYblCoYBGo6l2KCKquuJiIC6rDSLcjoiOQmaqv8dBHMxphpsqF9FRiMxalUpSz5498corr+DatWtly1JSUvDqq6+iV69eBgtHRJW3dy9QqLFHH3eej0S69XU/BAk2iMlsKzoKkVmrUkn6+uuvkZubi/r166Nhw4Zo1KgRgoKCkJubi6+++srQGYmoErZtA3ztbuFRpwuio5CZ8lfeQpta5xB1i4fciB6kSiduBwQE4OjRo4iNjcWZM2cgSRKaN2+O3r17GzofEVVSZCQwwP0A7pogn6iC/u4H8e21QSiVbCCX8TQJIl0qXZJKSkpgb2+PxMREhIeHIzw83Bi5iKgKzp8HzpwBPmpxQHQUMnP9PQ7io+RncCS3Cdq7nBYdh8gsVfpwm62tLQIDA1FaWmqMPERUDZGRgEIB9HZLEB2FzFx757/hZpvDQ25ED1Clc5JmzZqFmTNnIiMjw9B5iKgaIiOBHj0AZ9tC0VHIzNnaaBDhdoRTARA9QJXOSfryyy/xzz//wN/fH4GBgXBycir39aNHjxokHBHpLy8PiIsDPvkEwGbBYcgi9Pc4iDFnZuK6yg0+dpmi4xCZnSqVpCFDhkAmk0GSJEPnIaIq2rEDUKmAAQPAkkR66et+CIB2hvbRvjGC0xCZn0qVpIKCArz++uvYvHkz1Go1evXqha+++gqenp7GykdEeoqMBJo0ARo1Ep2ELIW3XRbaOp9G1K32LElEOlTqnKTZs2djxYoVGDBgAEaMGIEdO3Zg0qRJxspGRHqSpH8v/R8gOglZmn7uhxCd2RYlmiqdokpk1Sq1J2njxo1YtmwZhg8fDgD473//i86dO6O0tBRyudwoAYno4Y4dA1JTWZKo8vp7HMT7l8fgQE5zdKn9l+g4RGalUn86XLlyBV27di37vF27drC1tS13exIiMr3ISMDZGejSRXQSsjShzknwVGRhG69yI6qgUiWptLQUdnZ25ZbZ2tqipKTEoKGIqHIiI4E+fYB7fjyJHkou06Cv+yFOBUCkQ6UOt0mShLFjx0KpVJYtKyoqwsSJE8tNA7Bx40bDJSSiB0pPBw4dAiZOFJ2ELFV/94NYfT0CKcWeqKO8KToOkdmoVEkaM2ZMhWXPPPOMwcIQUeVt26Y9cbtfP9FJyFJFuB+BDUqxPaMdnvOLEh2HyGxUqiT98MMPxspBRFUUGQm0awf4+IhOQpbKQ5GD9i6nse0WSxLR3XjNJ5EFU6uB6Ghe1UbV19/9IGIyQ6HW8EplottYkogs2N69QE4OSxJVX3+Pg8gtdcK+7JaioxCZDZYkIgsWGQn4+QHBwaKTkKVrU+sf+CgyOBUA0V1YkogsWGQk0L8/YMOfZKomG5mEfh4HORUA0V2qdINbIhIoLAwAcL7QH2fO/ISP5O8AYXsFhyJr0N/9IFak9UNykTfqiQ5DZAb49yeRhYq81QF2MhV6uyWIjkJWItztCOQo5SE3on+xJBFZqMhbHdC99nE42xaKjkJWorYiH51c/8K2W+1ERyEyCyxJRBYor8QecVmtMcDjoOgoZGX6ux/EjswQFBeLTkIkHksSkQXakRkClWSHAe5/io5CVqa/x0HkaxwQHy86CZF4LElEFigyowOaOCSjkeM10VHIyjzqdAH17VOxebPoJETisSQRWRhJ0p6PNMDjgOgoZIVkMuAJzz3YvBnQaESnIRKLJYnIwhzLa4xUlSdLEhnNUM89SE0FDvKUN6rhhJak3bt3Y9CgQfD394dMJsNmPfbvxsfHIyQkBPb29mjQoAGWLFli/KBEZiTyVge4yPPQxfWk6ChkpTq6/g0fH2DjRtFJiMQSWpLy8/PRunVrfP3113qtf/HiRfTv3x9du3bFsWPH8NZbb+Hll1/Ghg0bjJyUyHxsvdUREe5HYGdTIjoKWSm5TIPHH9eWJEkSnYZIHKEzbvfr1w/9+vXTe/0lS5agXr16WLhwIQCgWbNmOHLkCObPn49hw4YZKSWR+UhPBw7nNsEk/99ERyEr98QTwHffASdPAq1aiU5DJIZF3Zbkzz//RERERLllffr0wbJly6BWq6FQKCo8p7i4GMV3TfiRk5MDAFCr1VCr1cYNbCS3c1tqflOxxnH6/XcZABv09j0GtZ2dQV5T/e/PjVrHzw+VV5PGqksXNVxdbfHrrxo0a1a5M7it8WfPGDhO+hE5ThZVktLS0uDj41NumY+PD0pKSnDz5k34+flVeM68efPw3nvvVVgeExMDR0dHo2U1hdjYWNERLII1jdO337bDI02USHh1pMFfO3bCBIO/prWqEWO1Iwpt2jyG1atdEBoaV6WXsKafPWPiOOln165dJn9PiypJACCTycp9Lv17wPze5bfNnDkT06ZNK/s8JycHAQEBiIiIgIuLi/GCGpFarUZsbCzCw8N17j0jLWsbp5wc4PhxW3wY9C36L9pksNdVKxSInTAB4UuXQsG/aB+oRo3V1q0oKpJh+HBbNGnSHw0b6v9Ua/vZMxaOk35uj1PYvzf3NiWLKkm+vr5IS0srtyw9PR22trbw8PDQ+RylUgmlUllhuUKhsPhvSmvYBlOwlnGKjgZUKuApt11QqFQGf32FWm2U17VGNWKsFAoMHAjY2wNbtyowfXpVXsI6fvaMjeOkHxFjZFHzJHXs2LHCbsmYmBiEhobyG4ys3vr1QLt2QD37dNFRqIZwcgL69OFUAFRzCS1JeXl5SExMRGJiIgDtJf6JiYlITk4GoD1UNnr06LL1J06ciMuXL2PatGk4ffo0li9fjmXLlmF6Vf7EIbIgeXnAtm3Af/4jOgnVNE88Afz5J5CaKjoJkekJLUlHjhxBcHAwgoODAQDTpk1DcHAw3n33XQBAampqWWECgKCgIERFRSEuLg5t2rTB3Llz8eWXX/Lyf7J6UVFAURFLEpnewIGAXA7ey41qJKHnJPXo0aPsxGtdVqxYUWFZ9+7dcfToUSOmIjI/69cDISFAUJDoJFTTuLsDPXoAmzYBkyaJTkNkWhZ1ThJRTVRQAERGci8SifPEE8CuXUBmpugkRKbFkkRk5rZv1xYlHlUmUR5/HCgpAbZuFZ2EyLRYkojM3Pr1QOvWQOPGopNQTVWnDtChA69yo5qHJYnIjBUVAb//zkNtJN7Qodq5uvLzRSchMh2WJCIzFhOjvfyfJYlEGzoUKCzUFiWimoIliciMrV8PtGgBNG0qOgnVdI0bAy1baq9yI6opWJKIzFRxMbBlC/cikfl44gnt4V9rvyML0W0Wde82Iqt31w0cd95qj+zsj/GfyHFA/CVxmYj+NXQo8P77QFwcEBEhOg2R8XFPEpGZWn+jO5o4JKOF0yXRUYgAaK+yDAriVW5Uc7AkEZkhtUaOzTc74z9e8ZDJRKch0pLJtHuTNm8GSktFpyEyPpYkIjO0KysYmSUu+I9XvOgoROU88QRw/Tpw4IDoJETGx3OSiMzQ+hvd0dA+Ba1rnRcdhWqqu86Pu1tHSQYfxXpsGh6LzleeNnEoItPiniQiM1OiscGmm114qI3Mko1MwhDPvdh4sysecH9yIqvAkkRkZnZnt8ZNdW0eaiOzNdRrLy4W+eP4cdFJiIyLJYnIzKy/0R2ByjSEOJ8VHYVIp561j8LDNhs//SQ6CZFxsSQRmZFSyQYbb3TloTYyawqbUozw2YmffuJVbmTdWJKIzMi+7Ja4rnbnoTYye6N9YpCaCuzcKToJkfGwJBGZkV9vdEddZTrauZwRHYXogUKdk9C0KbBypegkRMbDkkRkJjQaYMONbhjmuRs2Ml42ROZNJgNGjdLOvp2bKzoNkXGwJBGZiT//BFJVnjzURhbjmWeAwkLepoSsF0sSkZlYswbwt7uBTq6nREch0ku9eto5J3nIjawVSxKRGSgq0pak0b4xPNRGFmXUKGDXLuDKFdFJiAyPJYnIDGzeDGRlAeN8t4uOQlQpw4YB9vbgnElklViSiMzADz8AnTsDjzheFR2FqFJcXIChQ7WH3HibErI2LElEgiUnA7GxwLPPik5CVDWjRgGnTwMJCaKTEBkWSxKRYCtXAg4OwJNPik5CVDW9ewO+vsCqVaKTEBkWSxKRQBqN9lDbU08Bzs6i0xBVja0t8N//AmvXAmq16DREhsOSRCTQnj3AhQvAuHGikxBVz6hRwI0bQHS06CREhsOSRCTQ8uVAo0ZA166ikxBVT+vWQKtWnDOJrAtLEpEgOTnA+vXA2LHaWzwQWZywsHKP0fmLsGW9Cll9ntZ+feBAsfmIqokliUiQX37R3tJhzBjRSYgMY6T3TqglOTZc7yY6CpFBsCQRCfLDD0BEBFC3rugkRIbhp8xAuFsCVl/rLToKkUGwJBEJcOYMsH8/50Yi6zPaNwb7sloiLc1RdBSiamNJIhJgxQrAzQ0YPFh0EiLDGuK5F7XkBYiP5y5SsnwsSUQmVlIC/PgjMHKk9p5XRNbEUV6MJ3z2YteuAN6mhCyeregARDVGWBgAIPpWB6SlzcOzB54Hws4JDkVkeM/478DKIxE4UK85eAo3WTLuSSIyseWp/dDK6TyCa7EgkXXq5nYCnp4F+OlaL9FRiKqFJYnIhG6oXPH7rY541m8b50Yiq2Ujk9C9+1X8ktYdxcWi0xBVHUsSkQn9dF17afR/vWMFJyEyrrCwK8gqccamTaKTEFUdSxKRiUgSsDytHwZ77IenXY7oOERGVbduHnq4JeLLL0UnIao6liQiEzma9whO5jfEOL/toqMQmcRLgZvw55/AwYOikxBVDUsSkYn8kNoXfnY30cftkOgoRCbR3+sQGjQAvvhCdBKiqmFJIjKBoiLgp/TeGO0TA1sbjeg4RCYhl2nw8svAr78CKSmi0xBVHksSkQls3gxklTjzUBvVOOPGAQ4OwKJFopMQVR5LEpGRSRLw+edAd9dENHG8IjoOkUm5uGjvUfjtt0Bhoeg0RJXDkkRkZHv2AIcOAa/X+1l0FCIhXnoJyMgAfvpJdBKiymFJIjKyTz8FmjcH+rnzEh+qmRo21N7MeeFC8H5uZFFYkoiM6O+/ga1bgddf185CTFRTvfIKcOoUsHOn6CRE+mNJIjKizz4D/P2BkSNFJyESq0cPoFUrTgdAloUlichIUlOB1au1f0Hb2YlOQySWTAZMnards3qO93YmC8GSRGQkX34JKJXACy+ITkJkHkaMALy8wFuVkMVgSSIygtxcYPFi4PnnAVdX0WmIzIO9PTBxIvDDD0BWlug0RA9nKzoAkVUICyv36dIrTyI/53lM3TcCCLspKBSR+Zk0Cfj4Y2D5cmDaNNFpiB6Me5KIDEytkWPh1WEY6b0Tde1ZkIju5ucHDB8OfPUVUFoqOg3RgwkvSYsWLUJQUBDs7e0REhKCPXv23HfduLg4yGSyCo8zZ86YMDHRg/2cHoYrxT6YHsDJI4l0eeUV4NIlYMsW0UmIHkxoSfr5558xdepUvP322zh27Bi6du2Kfv36ITk5+YHPS0pKQmpqatmjcePGJkpM9GCSBHx6ZTj6uh/Eo7Uuio5DZJZCQoAuXbSTSxKZM6HnJC1YsADPPfccxo8fDwBYuHAhoqOjsXjxYsybN+++z/P29kbt2rX1eo/i4mIUFxeXfZ6TkwMAUKvVUKvVVQ8v0O3clprfVEw6Tv9e4x978zGcyG+IT5t+C7WFXPevVijKfaT741jpp9w43efnb8oUGYYPt8WhQ2oEB5synfng73L9iBwnmSSJmSRepVLB0dERv/76K4YOHVq2/JVXXkFiYiLi4+MrPCcuLg5hYWGoX78+ioqK0Lx5c8yaNQth95w0e7c5c+bgvffeq7B8zZo1cHR0NMzGEP3r3Xc7IT9fgfnz4yGTiU5DZL5KS2WYOLE3Wra8iVdeOSY6DlmAgoICjBw5EtnZ2XBxcTHJewrbk3Tz5k2UlpbCx8en3HIfHx+kpaXpfI6fnx++++47hISEoLi4GKtWrUKvXr0QFxeHbt266XzOzJkzMe2uSyhycnIQEBCAiIgIkw2yoanVasTGxiI8PBwK/kV7XyYdp4EDcSynEU6ceByrW32EAYsrlnxzpVYoEDthAsKXLoWCf9E+EMdKP/qO0zm363g3bgx+/LE/fH1NGNBM8He5fm6P04N2iBiL8CkAZPf8uS1JUoVltzVp0gRNmjQp+7xjx464cuUK5s+ff9+SpFQqoVQqKyxXKBQW/01pDdtgCiYZJ5UKCy88gfr2qXi69k7YqjTGfT8jUKjVUKhUomNYBI6Vfh42Ti94/YaPzo/AggW18PnnJgxmZvi7XD8ixkjYiduenp6Qy+UV9hqlp6dX2Lv0IB06dMA5znFPgl0u8sEv6T0wre6vsLWxvIJEJEJtRT5m1FuHRYu0V7sRmRthJcnOzg4hISGIjY0ttzw2NhadOnXS+3WOHTsGPz8/Q8cjqpTPr/wHrrZ5eNZvm+goRBZlat0NcHcH3n1XdBKiioQebps2bRpGjRqF0NBQdOzYEd999x2Sk5MxceJEANrziVJSUrBy5UoA2qvf6tevjxYtWkClUmH16tXYsGEDNmzYIHIzqIbLzAS+Tx2AaQG/wkleJDoOkUVxkhdh9mzgxReB114DWrcWnYjoDqEl6emnn8atW7fw/vvvIzU1FS1btkRUVBQCAwMBAKmpqeXmTFKpVJg+fTpSUlLg4OCAFi1aIDIyEv379xe1CUT44gugRJJjSp1NoqMQWaTnngMWLABmzgSiokSnIbpD+InbL774Il588UWdX1uxYkW5z2fMmIEZM2aYIBWRfq5eBT79FHil7gZ422WJjkNkkRQK4KOPgCefBHbtqnArRCJhhN+WhMiSvfUWUKsW8HbgatFRiCzasGFAu3bAG29oZ64nMgcsSURVdPgwsGoV8MEHgIttgeg4RBZNJgP+9z/tzxVPMyVzIfxwG5ElkiRg6lSgVSvg2WcBrBGdiMiC/Xt8rQeAvu4f4+3Rfnj8q2ehsCm9s86uXUKiUc3GPUlEVfDLL8D+/cDnnwNyueg0RNbj4wZLca6wLpan8YIcEo8liaiSCguBGTOAxx8HevYUnYbIurSudR7/9dmBOZfGIL/UXnQcquFYkogqacECIDVVe1UbERne3PrLkaF2xsKrw0RHoRqOJYmoEq5dA+bNA15+GWjcWHQaIutU3+E6Jvlvwf+SR+CmyjJvRE7WgSduEz3MXZO2vH1mBhxUHTHrwDNAWL7AUETW7e3A1Vie1g8fJj+DzxstEh2HaijuSSLSU0LuI/gxrQ/er/8DaitYkIiMycsuGzMC1mFRyuO4VKj/Tc+JDIkliUgPkgS8+s+LaO54GRP8toqOQ1QjvBqwHm62eXj30rOio1ANxZJEpIcNN7phT3ZrfN7oG9jaaETHIaoRnORFmFN/BVZf742jR0WnoZqIJYnoIYpKFXj9wkQM9NiPcPcE0XGIapTn/KLQ0ukixo0DiotFp6GahiWJ6CEWXv0PrhZ7YX7DJaKjENU4CptSrGo2D6dPA3PmiE5DNQ1LEtEDpKUBHyY/g8n+m9HE8YroOEQ1Uuta5/Hee8AnnwD79olOQzUJSxLRfUgS8NJLgJ1MjXfrrxQdh6hGe/11oH17YMwYIC9PdBqqKViSiO5j6VJg/Xrg20cWwF2RKzoOUY1mawv8+KN2tvvXXxedhmoKliQiHf76C3jlFWDiROA/3rtFxyEiaGe5nz8fWLIEiI4WnYZqAs64TTXbXbNp31ZQqsTwhMVoJJew4NQkQC4gFxHpNHEi8NtvwLPPAidPAu7uohORNeOeJKJ7vPrPZFwo8sO65nPhIFeJjkNEd5HJgGXLgIICYMoU0WnI2rEkEd3l1/Tu+C51EL5o9DVaOF0SHYeIdKhTB/jmG2DtWuCXX0SnIWvGkkT0r4uFvpiQNB1Pee3CeL9I0XGI6AFGjACefBKYNEl7MjeRMbAkEQFQa+QY8fc7cFPk4rsmn0EmE52IiB5EJgMWLQLs7IDx47VTdhAZGksSEYB3L41DQt4jWNd8Llxt80XHISI9eHoC338PREVpPxIZGksS1XixGSH4OPm/+DBoGdq7nBYdh4gqYcAAYMIE4NVXgXPnRKcha8OSRDXadZUbRp1+CxFuhzE94GfRcYioCj77DKhbF+jbF7h+XXQasiacJ4lqLI0GGH16JgBgZbN5sJHxpAYis6VjTrPbnAFsr+2DTse+Rr9GGYhr8ypcbAt0r7xrl3HykVXiniSqsd57D4jJbItVzT6Cj12m6DhEVA31Ha5je6s3cKHQH0/89T6KNQrRkcgKsCRRjfTJJ8D77wPzgr5DuHuC6DhEZACtal3Ab4/Owt7sRzH69ExoJF6mStXDkkQ1zsKFwBtvAO++C7wZuFZ0HCIyoO61j2NN8w+w/kY3TP1nCqcGoGphSaIaZdEi7VUwb7wBzJkjOg0RGcMTXnuw6JEv8FXKE/g4eaToOGTBeOI21Rjffw9MngxMnQrMmwdOGElkxV7w/x1pKje8dXECvO2y8JxflOhIZIFYkqhGWLkSeP554MUXgQULWJCIaoJ3A1ciTeWO55OmwUuRhcGe+0VHIgvDw21kvQYOBAD8Evopxo0pxXO+W/HVqZ6Q9QzTXk78gEuKicjyyWTA142/xBDPfXj673exL7ul6EhkYViSyKr9+acfxpx8A8/47MC3jyzgXEhENYxcpsFPzT5Ae+fTGHTyQyTwYlaqBJYkslqRN9rjs89CMcxnN5Y3/YQFiaiGspersbnlLDR2SEHXrsC6daITkaVgSSKr9NtvwNOJsxAamoYfWn4KuUwjOhIRCVRbkY+4NlMxbBgwYgQwcyZQWio6FZk7liSyKmo18PrrwJAhwACvg3jttSNQ2PA3IREBDnIVVq4E5s/XTig7eDCQnS06FZkzliSyGsnJQPfu2skiP/sMWNf6AygUPMRGRHfIZMBrrwFRUcD+/UC7dsCZM6JTkbliSSKrsHUr0KYNkJIC7NkDTJvGy/yJ6P769AEOHQLkcqB9eyAyUnQiMkcsSWTRbh9eGzQI6NoVOHYM6NBBdCoiMlthd6YAafx8GA54DEB3xT4MGqjBxw2+g9SDU4PQHZxMkixTWBiSi7wx/O93cDi3KT5r+B1ezf4VsmF3rWNnJyweEVkGF9sCbG75DmZfGouZF59HYl4jfJ8H1KolOhmZA+5JIou09WYHBB/5DinFntjT5mVMC/iVh9eIqEpsZBLmBv2AX5rPwe+3OqJJE2D1akDDi2JrPJYksigpKcCECcCgv+ahi+tfOBb6PDq4nhYdi4iswJPe8TjVdhw6dgRGjQI6dwYOHxadikTi4TayCBkZwP/+B3z5JeDkBHzd+Au86L+Ze4+IyKDqO1zH+lth2NW6DV45OQXt2jXEWN9tmBe0FL7KzDsr7tolLiSZDEsSmZ+77qmWX2qPL64OwyfJw1EiyTEj4Be8FvALXGwLBAYkImsX5paIoyHP4/vUAZh18Tmsv9Ed7wSuwit1N0BpoxYdj0yEJYnMkkpji+9TB+D9S6ORUeKMSf5b8HbganjbZYmORkQ1hK2NBhPr/I6nvOPw3qUxeOvCeCxNHYAFDRdhoMRpRmoCliQyrbAHX16rkWRYl94L71wch4tFfhjlE4v36v+A+g7XTRSQiKg8d0Uuvmj8NZ7334qp/0zG4L8+QpvHgBdeAEaOBFxcRCckY2FJIrNwtqAuVl0Px+rr4bhU5IfBHvvwW8tZaFnrkuhoREQAgBZOlxDT6nXEZLbF4iuDMXlSR0yfrMIIn514we93hLqcLf8Enrdk8ViSSJibKhf8fKMnVqWF42Buc7jI8/CUdxzG+0WhvQuvWCMi8yOTAX3cD6OP+2GkFHtiWWp/fJ/aH9+nDsRjtc7ief+tGOm9A862haKjkgGwJJFJFWsU2HqrI1amRSAqoz0kSYZ+Hgfxc/P3MMhjPxzkKtERiYj0Ukd5E+/WX4m3A1dje0Y7fHttIF48+wpe+2cSRvrsxH9itHcCcHAQnZSqiiWJjEqSgKQkYPduID4eiNq/AVklzgh1PoPPGi7GcO8/eDI2EVk0uUyDAR4HMMDjAK4UeWF5Wj8sS+2PpX0ApUyFLq4nEe5+BOFuCWhT6x/YyP698badHfDii2LD0wOxJJFBaTTAX39pC9Hu3dpHejpgYwM89hjwov9veMYnFs2ckkVHJSIyuAD7G5hdfyXeDVyJvwvqIzYjBLGZoZh7aTTevPACPBVZ6FX7KMLdE9DD+7jouPQQLElUZUVFwLlzwJkz2sfhL/ZhT3YrZJU4QyFTo53zGYyvfRzdWp1AJ5e/tMfoeT8kIqoBZDLtid4tnC5hasAGqDS2+DOnRVlpej5pGjRJctQ+WYQQm5No5XgejzpdwKNOF9Dc6TIc5cV3XowngAsjvCQtWrQIn376KVJTU9GiRQssXLgQXbt2ve/68fHxmDZtGk6dOgV/f3/MmDEDEydONGHimqWkRLsn6OJFbRE6ffpOKbp48c69jTw9gValDni17np0cz2O9i6neX4REdG/7GxK0L32cXSvfRwfYDky1bWwIy8U65sMR1FcIbbc7ISFV4dBgg1k0KCRQwoedbqIR50uoMFKICAAqFtX++A5TqYjtCT9/PPPmDp1KhYtWoTOnTvj22+/Rb9+/fD333+jXr16Fda/ePEi+vfvjwkTJmD16tXYt28fXnzxRXh5eWHYsGE63oHuJUlAfj6QnQ1kZWk/3rgBpKYC165pP9797/T0O0VIJgOCgoBmzYAhQ4CmTe88PD0BhL0mcMuIiCyHmyIPQ3z2w25kG/TPWgSFSoX8Unucyq+Pk/kNcDIvCCfzG2DxtcFIH1P+uR4ed0rT7Y/e3oCbG+Dufueju7v2Nk6c9LLqhJakBQsW4LnnnsP48eMBAAsXLkR0dDQWL16MefPmVVh/yZIlqFevHhYuXAgAaNasGY4cOYL58+ebRUlSqbR7XiRJWyw0Gt3/Li19+KOkBFCrtQ+Vqvy/CwtlSEgIwNWZi6FSyVGoUaJQY4eCUvuyfxeWKlGoUSK3VedyhSgrS/v695LLAR8fwN8f8PMD2rXTfvTz0y6r995zeMThCuzlaiAfQMK/DyIiMggneRHauZxBO5cz5ZYXRO5CSgpw9Spw5Ur5j/v3az9mZGj/H3MvW9s7xcnZWVuaHB21j9v/vvujUql92Nnd/6FQaP+fYWt7/48uLkDt2qYZN2MSVpJUKhUSEhLw5ptvllseERGB/fv363zOn3/+iYiIiHLL+vTpg2XLlkGtVkOhUFR4TnFxMYqL7xzbzc7OBgBkZGRArTbs/Xcef1yOfftsDPqa99cIMgTB0aYY9jYqOMhVUNqo4CAvhoNcBQcbFZSyItSunYXAQAnOzoCrq/Yb19VV+vcj4Owswf3N5+FplwO57N9dRvkAzv37uM1Fuzgfptq+6lMDKCgowC0AChvLyW1qHCf9caz0w3HSj97jNKgX3AG4A2h179c8tI9SyQY5JY7IVNdCtroWMktqIUvthEx1LWSVOiOroBbyc+1RUKpEfqk9MkrtcEVjj8JSJfJLlSgq1X6t2K4WVCqgtLR6u58mTizFBx9oqvUat6nVahQUFCAjIwMAIOlqg8YiCZKSkiIBkPbt21du+Ycffig98sgjOp/TuHFj6cMPPyy3bN++fRIA6dq1azqfM3v2bAkAH3zwwQcffPBhBY8rV64YpojoQfiJ27J7DpZKklRh2cPW17X8tpkzZ2LatGlln2s0GmRkZMDDw+OB72POcnJyEBAQgCtXrsCFNw26L46TfjhO+uNY6YfjpB+Ok35uj1NycjJkMhn8/f1N9t7CSpKnpyfkcjnS0tLKLU9PT4ePj4/O5/j6+upc39bWFh4eHjqfo1QqoVQqyy2rbQ0HSgG4uLjwB0sPHCf9cJz0x7HSD8dJPxwn/bi6upp8nIQdLLazs0NISAhiY2PLLY+NjUWnTp10Pqdjx44V1o+JiUFoaKjO85GIiIiIqkroGXXTpk3D999/j+XLl+P06dN49dVXkZycXDbv0cyZMzF69Oiy9SdOnIjLly9j2rRpOH36NJYvX45ly5Zh+vTpojaBiIiIrJTQc5Kefvpp3Lp1C++//z5SU1PRsmVLREVFITAwEACQmpqK5OQ7t68ICgpCVFQUXn31VXzzzTfw9/fHl19+aRaX/5uSUqnE7NmzKxxGpPI4TvrhOOmPY6UfjpN+OE76ETlOMkky5bV0RERERJaBE1gQERER6cCSRERERKQDSxIRERGRDixJRERERDqwJAmWm5uLqVOnIjAwEA4ODujUqRMOHz5c9vXr169j7Nix8Pf3h6OjI/r27Ytz58494BWBpUuXomvXrnBzc4Obmxt69+6NQ4cOGXtTjM4YY3W3devWQSaTYciQIUZIbzrGGqesrCxMnjwZfn5+sLe3R7NmzRAVFWXMTTEqY43TwoUL0aRJEzg4OCAgIACvvvoqioqKjLkpBrN7924MGjQI/v7+kMlk2Lx5c7mvS5KEOXPmwN/fHw4ODujRowdOnTpVbp3i4mK89NJL8PT0hJOTEwYPHoyrV68+9L0XLVqEoKAg2NvbIyQkBHv27DHkphmcqLGaN28e2rZtC2dnZ3h7e2PIkCFISkoy9OYZjMjvqdvmzZsHmUyGqVOnVjo/S5Jg48ePR2xsLFatWoWTJ08iIiICvXv3RkpKCiRJwpAhQ3DhwgX89ttvOHbsGAIDA9G7d2/k5+ff9zXj4uIwYsQI7Nq1C3/++Sfq1auHiIgIpKSkmHDLDM8YY3Xb5cuXMX36dHTt2tUEW2JcxhgnlUqF8PBwXLp0CevXr0dSUhKWLl2KOnXqmHDLDMsY4/TTTz/hzTffxOzZs3H69GksW7YMP//8M2bOnGnCLau6/Px8tG7dGl9//bXOr3/yySdYsGABvv76axw+fBi+vr4IDw9Hbm5u2TpTp07Fpk2bsG7dOuzduxd5eXkYOHAgSktL7/u+P//8M6ZOnYq3334bx44dQ9euXdGvX79yU8CYG1FjFR8fj8mTJ+PAgQOIjY1FSUkJIiIi9Po9J4Kocbrt8OHD+O6779CqVYVbA+vHZHeJowoKCgokuVwubd26tdzy1q1bS2+//baUlJQkAZD++uuvsq+VlJRI7u7u0tKlS/V+n5KSEsnZ2Vn68ccfDZbd1Iw5ViUlJVLnzp2l77//XhozZoz0+OOPG2MTTMJY47R48WKpQYMGkkqlMlp2UzLWOE2ePFnq2bNnuWXTpk2TunTpYtgNMAEA0qZNm8o+12g0kq+vr/Txxx+XLSsqKpJcXV2lJUuWSJIkSVlZWZJCoZDWrVtXtk5KSopkY2Mjbd++/b7v1a5dO2nixInlljVt2lR68803DbQ1xmXKsbpXenq6BECKj4+v/oYYmanHKTc3V2rcuLEUGxsrde/eXXrllVcqnZl7kgQqKSlBaWkp7O3tyy13cHDA3r17UVxcDADlvi6Xy2FnZ4e9e/fq/T4FBQVQq9Vwd3c3THABjDlW77//Pry8vPDcc88ZPriJGWuctmzZgo4dO2Ly5Mnw8fFBy5Yt8dFHH+n1l5w5MtY4denSBQkJCWWHty9cuICoqCgMGDDACFthWhcvXkRaWhoiIiLKlimVSnTv3h379+8HACQkJECtVpdbx9/fHy1btixb514qlQoJCQnlngMAERER932OuTPWWOmSnZ0NABb5+93Y4zR58mQMGDAAvXv3rnJGliSBnJ2d0bFjR8ydOxfXrl1DaWkpVq9ejYMHDyI1NRVNmzZFYGAgZs6ciczMTKhUKnz88cdIS0tDamqq3u/z5ptvok6dOtX6RhHNWGO1b98+LFu2DEuXLjXh1hiPscbpwoULWL9+PUpLSxEVFYVZs2bhs88+w4cffmjCrTMcY43T8OHDMXfuXHTp0gUKhQINGzZEWFgY3nzzTRNunXHcvrn4vTcg9/HxKftaWloa7Ozs4Obmdt917nXz5k2UlpY+8HUtjbHG6l6SJGHatGno0qULWrZsaYDkpmXMcVq3bh2OHj2KefPmVSsjS5Jgq1atgiRJqFOnDpRKJb788kuMHDkScrkcCoUCGzZswNmzZ+Hu7g5HR0fExcWhX79+kMvler3+J598grVr12Ljxo0V/mq2NIYeq9zcXDzzzDNYunQpPD09Tbw1xmOM7ymNRgNvb2989913CAkJwfDhw/H2229j8eLFJtwywzLGOMXFxeHDDz/EokWLcPToUWzcuBFbt27F3LlzTbhlxiWTycp9LklShWX30medqryuuTPWWN02ZcoUnDhxAmvXrq1yRnNg6HG6cuUKXnnlFaxevbra/99jSRKsYcOGiI+PR15eHq5cuYJDhw5BrVYjKCgIABASEoLExERkZWUhNTUV27dvx61bt8q+/iDz58/HRx99hJiYmKqftGZGDD1W58+fx6VLlzBo0CDY2trC1tYWK1euxJYtW2Bra4vz58+bcvMMxhjfU35+fnjkkUfKFYRmzZohLS0NKpXK6NtkDMYYp3feeQejRo3C+PHj8eijj2Lo0KH46KOPMG/ePGg0GlNtmlH4+voCQIW/3tPT08v2BPj6+kKlUiEzM/O+69zL09MTcrn8ga9raYw1Vnd76aWXsGXLFuzatQt169Y1UHLTMtY4JSQkID09HSEhIWW/2+Pj4/Hll1/C1ta2UqcJsCSZCScnJ/j5+SEzMxPR0dF4/PHHy33d1dUVXl5eOHfuHI4cOVLh6/f69NNPMXfuXGzfvh2hoaHGjG5yhhqrpk2b4uTJk0hMTCx7DB48GGFhYUhMTERAQIApNsdoDPk91blzZ/zzzz/l/kd/9uxZ+Pn5wc7OzmjbYAqGHKeCggLY2JT/tSqXyyFJEiQLv01mUFAQfH19ERsbW7ZMpVIhPj4enTp1AqAtlgqFotw6qamp+Ouvv8rWuZednR1CQkLKPQcAYmNj7/scc2essQK0e1CmTJmCjRs34o8//tDrD2ZzZaxx6tWrV4Xf7aGhofjvf/+LxMREvY/EAODVbaJt375d2rZtm3ThwgUpJiZGat26tdSuXbuyq4h++eUXadeuXdL58+elzZs3S4GBgdITTzxR7jVGjRpV7iqQ//3vf5KdnZ20fv16KTU1teyRm5tr0m0zNGOM1b0s/eo2STLOOCUnJ0u1atWSpkyZIiUlJUlbt26VvL29pQ8++MCk22ZIxhin2bNnS87OztLatWvLXrdhw4bSU089ZdJtq6rc3Fzp2LFj0rFjxyQA0oIFC6Rjx45Jly9fliRJkj7++GPJ1dVV2rhxo3Ty5ElpxIgRkp+fn5STk1P2GhMnTpTq1q0r7dixQzp69KjUs2dPqXXr1lJJSUnZOj179pS++uqrss/XrVsnKRQKadmyZdLff/8tTZ06VXJycpIuXbpkuo2vJFFjNWnSJMnV1VWKi4sr9/u9oKDAdBtfCaLG6V5VvbqNJUmwn3/+WWrQoIFkZ2cn+fr6SpMnT5aysrLKvv7FF19IdevWlRQKhVSvXj1p1qxZUnFxcbnX6N69uzRmzJiyzwMDAyUAFR6zZ8820VYZhzHG6l7WUJKMNU779++X2rdvLymVSqlBgwbShx9+WO6XlKUxxjip1Wppzpw5UsOGDSV7e3spICBAevHFF6XMzEwTbVX17Nq1S+fvjtvbqNFopNmzZ0u+vr6SUqmUunXrJp08ebLcaxQWFkpTpkyR3N3dJQcHB2ngwIFScnJyuXUCAwMr/D765ptvpMDAQMnOzk567LHHzP6SdlFjpes9AUg//PCDkbe4akR+T92tqiVJJkkWvg+YiIiIyAh4ThIRERGRDixJRERERDqwJBERERHpwJJEREREpANLEhEREZEOLElEREREOrAkEREREenAkkRERESkA0sSERERkQ4sSUREREQ6sCQRERER6cCSREQW78aNG/D19cVHH31UtuzgwYOws7NDTEyMwGREZMl4g1sisgpRUVEYMmQI9u/fj6ZNmyI4OBgDBgzAwoULRUcjIgvFkkREVmPy5MnYsWMH2rZti+PHj+Pw4cOwt7cXHYuILBRLEhFZjcLCQrRs2RJXrlzBkSNH0KpVK9GRiMiC8ZwkIrIaFy5cwLVr16DRaHD58mXRcYjIwnFPEhFZBZVKhXbt2qFNmzZo2rQpFixYgJMnT8LHx0d0NCKyUCxJRGQVXn/9daxfvx7Hjx9HrVq1EBYWBmdnZ2zdulV0NCKyUDzcRkQWLy4uDgsXLsSqVavg4uICGxsbrFq1Cnv37sXixYtFxyMiC8U9SUREREQ6cE8SERERkQ4sSUREREQ6sCQRERER6cCSRERERKQDSxIRERGRDixJRERERDqwJBERERHpwJJEREREpANLEhEREZEOLElEREREOrAkEREREenwf/nuTiTl/ihiAAAAAElFTkSuQmCC\n",
       "text/plain": [
        "
    "
       ]
@@ -1707,9 +1707,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/626635268.py:73: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/626635268.py:73: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(trainingerror), label='Training Error')\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/626635268.py:74: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/626635268.py:74: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(testerror), label='Test Error')\n"
      ]
     },
@@ -2051,7 +2051,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n"
      ]
     },
@@ -3725,9 +3725,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -4037,9 +4037,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -4116,9 +4116,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -4203,9 +4203,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -4284,7 +4284,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 20%|█████████████████████████████▍                                                                                                                     | 2/10 [00:00<00:03,  2.15it/s]"
+      " 20%|█████████████████████████████▍                                                                                                                     | 2/10 [00:00<00:03,  2.12it/s]"
      ]
     },
     {
@@ -4292,7 +4292,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 30%|████████████████████████████████████████████                                                                                                       | 3/10 [00:01<00:02,  3.06it/s]"
+      " 30%|████████████████████████████████████████████                                                                                                       | 3/10 [00:01<00:02,  3.10it/s]"
      ]
     },
     {
@@ -4300,7 +4300,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 40%|██████████████████████████████████████████████████████████▊                                                                                        | 4/10 [00:01<00:01,  3.41it/s]"
+      " 40%|██████████████████████████████████████████████████████████▊                                                                                        | 4/10 [00:01<00:01,  3.80it/s]"
      ]
     },
     {
@@ -4308,7 +4308,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 50%|█████████████████████████████████████████████████████████████████████████▌                                                                         | 5/10 [00:01<00:01,  4.28it/s]"
+      " 50%|█████████████████████████████████████████████████████████████████████████▌                                                                         | 5/10 [00:01<00:01,  4.70it/s]"
      ]
     },
     {
@@ -4316,7 +4316,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 60%|████████████████████████████████████████████████████████████████████████████████████████▏                                                          | 6/10 [00:01<00:00,  5.00it/s]"
+      " 60%|████████████████████████████████████████████████████████████████████████████████████████▏                                                          | 6/10 [00:01<00:00,  5.50it/s]"
      ]
     },
     {
@@ -4324,7 +4324,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 70%|██████████████████████████████████████████████████████████████████████████████████████████████████████▉                                            | 7/10 [00:01<00:00,  5.70it/s]"
+      " 70%|██████████████████████████████████████████████████████████████████████████████████████████████████████▉                                            | 7/10 [00:01<00:00,  6.18it/s]"
      ]
     },
     {
@@ -4332,7 +4332,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 80%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████▌                             | 8/10 [00:01<00:00,  6.29it/s]"
+      " 80%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████▌                             | 8/10 [00:01<00:00,  6.77it/s]"
      ]
     },
     {
@@ -4340,7 +4340,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      " 90%|████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████▎              | 9/10 [00:02<00:00,  6.85it/s]"
+      " 90%|████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████▎              | 9/10 [00:01<00:00,  7.22it/s]"
      ]
     },
     {
@@ -4348,7 +4348,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      "100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:02<00:00,  7.30it/s]"
+      "100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:02<00:00,  7.32it/s]"
      ]
     },
     {
@@ -4356,7 +4356,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      "100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:02<00:00,  4.68it/s]"
+      "100%|██████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:02<00:00,  4.88it/s]"
      ]
     },
     {
@@ -4592,9 +4592,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3980313467.py:9: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3980313467.py:9: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n",
       "  ax = fig.gca(projection='3d')\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20614/3980313467.py:37: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16213/3980313467.py:37: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  fig.colorbar(surf, shrink=0.5, aspect=5)\n"
      ]
     },
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3_51_0.png b/doc/LectureNotes/_build/jupyter_execute/chapter3_51_0.png
index 4f3ade9f3..09ea68850 100644
Binary files a/doc/LectureNotes/_build/jupyter_execute/chapter3_51_0.png and b/doc/LectureNotes/_build/jupyter_execute/chapter3_51_0.png differ
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter6.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter6.ipynb
index 9a93ed3d7..bbeb7e0fa 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter6.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter6.ipynb
@@ -86,14 +86,14 @@
      "output_type": "stream",
      "text": [
       "2nd degree coefficients:\n",
-      "zero power:  -4.653578701904388\n",
-      "first power:  0.17297886491529482\n",
-      "second power:  -0.0007790285013223805\n"
+      "zero power:  -6.548110376991839\n",
+      "first power:  0.2232822462117919\n",
+      "second power:  -0.0007480407244119591\n"
      ]
     },
     {
      "data": {
-      "image/png": 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QBEEQnMqytGfPHvLz8/X1Q4cO0b9/f0aOHFlsu2PHjllZhEJDQytNRkEQBEEQahdOpSzZKjnvvPMOTZs2pXfv3sW2CwsLK1VVb0EQBEEQrjycahrOkpycHBYvXsyDDz5Y4lxkp06diIyMpF+/fmzbtq3EvrOzs0lNTbX6CIIgCILgnDitsrRy5UqSk5MZM2ZMkftERkYyd+5cli9fzooVK2jRogX9+vXjl19+Kbbvt99+G39/f/0jaQMEQRAEwXlx2kK6AwcOxM3NjdWrV5ep3dChQzEYDKxatarIfbKzs8nOztbXtdDDlJQUiYYTBEEQhFpCamoq/v7+JT6/ncpnSePMmTNs3ryZFStWlLlt9+7dWbx4cbH7uLu74+7uXl7xBEEQBEGoRTjlNNyCBQsICwtjyJAhZW67f/9+IiMjK0EqQRAEQRBqI05nWTKbzSxYsID7778fFxfrw5syZQoXLlxg0aJFAMycOZNGjRrRpk0b3SF8+fLlLF++vDpEFwRBEAShBuJ0ytLmzZs5e/YsDz74YKHvoqKiOHv2rL6ek5PDpEmTuHDhAp6enrRp04a1a9cyePDgqhRZEARBEIQajNM6eFclpXUQEwRBEASh5lDa57dT+iwJgiAIgiA4CqebhhOEKxmzYuZcyrlC2z1dPQnzDqsSGRIzE7mUfQlfd1+CPIOqZMzaxKXsSyRmJhZ7TTJyM/B08SwxoW5tIzM3E3cXd4wGeU8XaheiLAmCEzHk2yGsP7He7ndzb57Lw1c/XKnjb/5vMwMXD8SsmDEZTGy4ZwP9mvSr1DFrEycST9Dus3Zk5WUB8H83/x+PXP2I1T6nkk7RZk4bRrcbzRfDvqgOMSuF5KxkrvrkKrrU6cLPo3+ubnEEoUyIei8IToKiKGw9tRUAd5M7Hi4eeLh44GJU34m2n9le6TL8cuYXzIoZgHwln1/P/lrpY9Ymdp/frStKADvO7Ci0zx8X/iAzL5Ntp0suvVSbOBR7iPiMeLad2oa4ygq1DVGWBMFJSMlOISc/B4Ck55PIfDGTzBczWXDLAgBi0mIqXQbbMapizNpEac6Pti0m3bnOnXZc2fnZpGZLPU2hdiHKkiA4CdFp0QD4ufvh6eqpbw/3Dgeq5uEbna7K0D68vdW6oKJdow7hHazW7e2TlpNGek561QlXyVj+/dk7bkGoyYiyJAhOgvbmHuETYbVdW6+KB5Qmg6YMiGXJGk1h6BDRwWrd3j5FfV9bsfz7c6bjEq4MRFkSBCdBexhpliSNcB91PSEjgdz83CqRoTjLyZWM7fmJz4gvdE0sz5kznT9LxdmZjku4MhBlSRCcBO1t3dayFOwZjMlgQkEhLiOu0sZXFMWu5USceQvQzk/r0NaYDCaAQtfEyrLkRJY5yylZZzou4cpAlCVBcBKKsiyZjCZCvUOByn1IpWan6pFems9SRm4GaTlplTZmbUO7RpE+kXqOJVsri1iWBKHmIcqSIDgJRfksWW6rzIeUZhHxdfMlzDsMb1dvq+1XOvnmfOIz4gH1emjXxFKJMCtmYtNj9XVnOnfisyTUZkRZEgQnQZvm0HyULKmKiDjdsnV5fO1/sSKoxGXEYVbMGA1GQrxC7J6fxMxE8sx5+rqznDvLKVpwnuMSrhxEWRIEJ6HaLUs249uznFzJaOch1CsUk9FUcH6K8VFyFguM5RQtOM9xCVcOoiwJgpOgPYBsfZYst1Wm4mLrM1WVKQtqA/r10Sxv3oUtS7bnylkUTVvlSP4mhNqGKEuC4AQoilKsZUmf8qnEJJG20XhVmQyzNqApCIUsb3byKnm6eFq1qe1ox6EdV0yaREkKtQtRlgTBCUjKSiLXrObrsVfJviqmxLS+xbJkH9vzY8/apy23C2+nrjtJ6gXtuNqGtQUg15xLUlZSdYokCGVClCVBcAI0hSTQIxB3F/dC39ub8nG4DOnWlhOxLFlTlGXJ3jSclrTSWVIvaMfVwL8BAR4BgPNMMQpXBqIsCYIToFst7ETCgf0pn8qWQSxL1tj6lGnnyd40XJPAJvi4+RT6vrZiOUUrfxdCbUSUJUFwAmytFrZoD+bEzERy8nOqRAZdGRALAlA4tYJ2niyviaWTfFVYA6sKe8flDEqgcOUgypIgOAHFRcIBBHkG4WJ0AbBKeugoLPPo2PNZcga/m4pi6wAf6BGIq9EVKLgm9iwwzqBsimVJqO2IsiQITkBJliWjwag7flfGwzc5K1m3jtiGxmfnZ5OanerwMWsbtqkVDAZDocSUltYnZ0rqaTlFWxVpLATB0bhUtwCCIFSckixLoCpSFy9dZM6eOTQPbg5Ar4a96FG/R7nGPBJ3hKNxR7mt9W36+P7u/ni4eADg6eqJn7sfqdmpvPXrWwR5Bult3V3cubPtnUUqdzWNfHM+Sw8t5boG19EooFGZ2+fm55KQkQBYK7Th3uGcTz1PTFoM+eZ84tLj9H0ivB3rZxabHsv6E+sZ2Xoknq6eDumzKBIzE/nm4Ddk5GYAcDLpJGBtWdpyagtz9szh3vb34uvuW6nyCEJFEWVJEJyAkixLAPX96rMvah9f/v2lvs3P3Y/EyYmYjKYyjzl6xWj+jv6bA+MOkJiZaHf8+n71ORx3mHd/f7dQ+0Oxh5g3bF6Zx60ONpzcwD0/3sPNzW9m9V2ry9w+LiMOBQWTwUSwV7C+3XJKKiEzgXwlH1CzfDvasjRt2zQ+/+tzsvKyeOTqRxzSZ1HM+G2G3Wte17cu9f3rA7Dn4h72XNxDSlYKU3pOqVR5BKGiiLIkCE5ASdFwAK/1fY0Inwh9umzB3wtIzU4lITPBbm6mkjiReAKAk4knC03BacwePJvFBxdjVsz6ttPJp9l2eptubagNWB5redCuT5h3GEZDgfeDpbOztk+IVwiuJleHRzCeSKrYMZRnrGvrX0uL4BYAdK3Tlbp+dRnRagSHYg+x5dQW9l7cW6v+DoQrF1GWBMEJKI1lqX14ez6/+XN9ffW/q4nPiCcmLabMylJ6Trqe/ycmPYbsvGy74/dp1Ic+jfpYbdt6aivbTm+rVT4rmqzlVVxsI+E0LC1Ltj5Njo6Gq+gxlGesZ7o/w+2tb7f6zsvVi3dufIcv/vqCvRf3SlScUCsQB29BqOWYFbMeTVWcz5ItFXkY21aQL43PlCPGrS40WcubesE2Ek7DMteS7T6OjoazdSKvTGwVP3s4kwO74PyIsiQItZyEjAJfl7JYiCoyzWNboqM0li3bcZOyknSLVE3H8hyVJ/VCUcqDXcuSj3XSSkekXsgz5xGfEQ9UkWWpCOXQEmdKjSA4P6IsCUItR3swBXsG42pyLXW7irzZW5XoSC+bZSnQM7BScz5VBvZKkpSFooocW4bR6/t4W5eLcUTqhbh01cEcKt+SYzlFW5wPnaW/luThEmo6oiwJQi2nLFYdS/TQ9HK82VuV6CijZcloMNa6LM62x1tWtLp5xVqW0q0tS1rqBe37imApf1x6HPnm/Ar1V5qxPF088XUrOiWAdpw5+TkkZyVXmjyC4AhEWRKEWk5pIuHsYa82WWmxtbSUVYba5K+iKIqVguRQy9Ll85CSncKZ5DOF9nFURJyl/PlKPgmZCRXqrzgspxMNBkOR+3m4eODv7q/KV0uUZuHKRZQlQajllNuyVIGyE7bKQ2l8VOyNXRv8VZKyksg15+rrFVEubZVJf3d/3E3uAByMOajuY2F9cpQzvG37yjzvRSmG9pDSJ0JtQZQlQajllMVfyJKKTIVpU0YAmXmZ5JnzgNI7mNemiDhbGSsSPWirQFiWPEnJTim0j6OUSttrXJnnvTSRcBpSbFmoLYiyJAi1nOq2LGkEegTiZnIr09i1YfrF9ljLKnNOfo6e4dyeAmG7zdL6VGmWpUo872WxMoplSagtiLIkCLWccluWLj+U4zPiy+zwa+/hVhZl7UqyLGkRf65GVwI9Awt9b3nejAYjoV6hhb6rsM9STbUs1TJHf+HKRZQlQajllNeyFOoVitFgxKyYicuIK3U7RVHsWg/K4mBeqyxLNsda1ikj7frYljrRsFQqQrxCrOr0OcoR3vZvpFJ9lsSyJDghoiwJQi2nvNFwJqOJEK8Qqz5KQ1pOml5NvkN4B317mSxLtSgaTpNRO9ayylySw7M9HyXbdUdFw+nHkF4FlqVS/D2KZUmoLYiyJAi1mHxzvm4VKqtlybJNWRQA7cHm7epN08Cm+vayTAPWpmg47Xg1RSMlO4WsvKxSty9JebDno2S77ijLknYMEg0nCGVDlCVBqMXEZ8RjVswYMOhWorJQnjd7y4e/5YO+PD5LZVU8qgPteFuEtNAd2MuibOjTUt5ltyxZRouVN8t1bn6unlepQ0T5rGOlRVEUiYYTnBKX6hbAkUyfPp1XX33Valt4eDjR0UXfGHbs2MHEiRM5fPgwderUYfLkyYwbN66yRRUEh6A9mEK9Q/USImWhPNNhlpYDezmBSkOARwCuBldyE3NZs2kNucm5nD9/nqSkJDIzM8nKyiI7Oxs3Nze8vb3x8vLCz8+POnXqUKdOHerWrUvDhg1xd3cv9ZjlxfZ4z6WeIzotmoYBDUvVvkTLUjHnUFvPNeeSlJVEkGdQmeXXHMxNBhOtQ1sDlTftlZaTRmZeJlD2aTizYrbr0yUINQGnUpYA2rRpw+bNm/V1k8lU5L6nTp1i8ODBPPzwwyxevJidO3fy+OOPExoaym233VYV4gpChShvJJxGeUqeWFoOirOK2BIbG8umTZvYs2cP+/btI+/PPMiGkbNGlkNyMBqNNGnShJYtW9K6dWu6dOlC9+7dqVevXrGZo8uK7fGeSz1XJmWjpGtU3Dl0d3EnwCOA5KxkYtJiyqUs6eP7hBPpEwmoJU/yzHnlUrCLQztX3q7e+Lj5lLi/lpcrz5xHUmYSwV7BDpVHEByF0ylLLi4uRESUbjrg888/p0GDBsycOROAVq1asXfvXt5//31RloRaQXkj4TR0y1IZHH4to52s/G3sWBL+/vtvVqxYwc8//8xff/1VeCrJBBF1ImjRpAX16tUjJCQET09PPDw8cHNzIzc3l/T0dDIyMkhOTubChQtcvHiR8+fPk56ezokTJzhx4gRr1qzRu4yMjKRbt25069aNnj17cs011+DqWvoCw5aYFbNumbE83rJY4kq6RiWdwwifCJKzkolOi6ZVaKtSj2s7frh3OCFeIXoEZHxGfLn/boqirJnc3V3cCfQIJCkriei0aFGWhBqL0ylLx48fp06dOri7u9OtWzfeeustmjRpYnffXbt2MWDAAKttAwcOZP78+eTm5hZ5g83OziY7O1tfT02tWEVwQbDH39F/88LmF3ir31t0juxsd5/yRsJpaA+11cdW0/HzjoAavr5w+ELq+dUrfswiLEtxcXF88803LFy4kAMHDli17dixI7169eLqq69mfvR8fkn7hVdveZVHrn6kVPK+vPVlVv+7mmY047b6t3Gd53Ws+30dy7YtwzfOl3+P/EtUVBQrV65k5cqVAPj4+NCnTx/69evHjTfeSJs2bUpteUrMTCRfUXNQhXmH6Za46dunM2fPHKt9b2t1Gy/3frlQHyVdI183XzxdPMnMy7SrZIR7h/NP/D/FWrM2/7eZF7e+SHZedqHvkrKSAPX6mIwmQr1CiUlXix8XpdQs/HshH//xMWbFTM8GPflk8CecTz3PmJVjiM+Ix9/Dn/+7+f9oGdKyTMdqjwifCJKykhi+bDi9GvRi3rB5xGfEc8+P9xCTFoOPmw+zB8+mY0THUvcpCI7GqZSlbt26sWjRIpo3b05MTAxvvPEG1157LYcPHyY4uPAbS3R0NOHhNj4C4eHk5eURHx9PZGSk3XHefvvtQr5RguBoFv69kA0nN3BV0FVFKku61aII5+GS0KKjLuVc4kBMgWKz/Mhynur+lP0x0wssJXV866i5gQwmEs8l8tL7L7F48WJyc9Vaam5ubtx8880MHTqUgQMHWv2mfl31K7/s/6VgClBRIC4Ozp1TP9HRkJSkf/IT47n24Ep654NbPnhymK4h7Wmcepbh2fFEBjckvE8fPvl3Cwfz4GgWHE+DtLQ01qxZo1ufIiIiGDBgADfddBP9+/cnJKRox3jt/AZ7BuNqctUdpKPSoohKi7La93DcYV7s9WIhv5uSLEsGg4EOER3Ye3EvrUIKW45KEzH26Z5P+fPCn0V+D9A+vD2gKjIx6THFTr2+89s7HEs4BsCBmANM7TmV5UeWs+XUFn2fb//3La/1fc2qXXksnR0iOnA0/ignEk9wIvEEk6+bzO/nfmfjyY36Pgv/XsjMm2aWuk9BcDROpSwNGjRIX27Xrh09evSgadOmfPXVV0ycONFuG9s3TG2aoLg3zylTplj1l5qaSv369SsiuiAUQnsYFzdFZumPUh7ahbfjyONHOJ96HoDP//qcFUdXFPtgtrQeuJncWNxtMR9/8DHtJ7fXfz9dunThgQce4M477yQoyMbPJi4Ojh7l5m0XaL0bBq76AlK+gvPnIbuwZUTDBAyy2pIHZ/fRGGgMcO4McIYpFnuYgQPAFmAz8AvqS9KiRYtYtGgRBqBr/foMvPZabrrtNq4ZNgwXC6dxW0vJ410fp0udLlzKvlQwhmLmpm9uIs+cR0JGAqHeBRm4s/Ky9JpvxfmVrR+9noTMBOr61S30ne4EXYxyE3VJ/Vt5tc+r9KjXo9D37i7u+vYInwgOxhws9hrbKoLRadGlymReHh+6L4d9ycOdH+aBnx7gbMrZUo8lCFWJUylLtnh7e9OuXTuOHz9u9/uIiIhCkXKxsbG4uLjYtURpuLu7V0kUjnBloz0ci3tIVtRnCaBVaCvdF2bPxT2sOLqi2CkfbUzjJSP33nsvixcv1r+75ZZbeOGFF+jevbu6ISEBNmyAPXtg7171/4sX1X31VucKOjcYIDwc6teHyEgICoLAQAgM5LQhlel738fHJ5C43BSyjGa+vG0Rr+14lXOxJ+kW1pE7rrqVtzZNoy6+eKSm459u5t76Q+iUksGk8+fJPnOGnTk5bADWAweBP8+d489ly3h92TICgBuDg7mpY0cGDhtGeuNcUArOr9FgpHu97oXOSbBnMAmZCcSkx1gpS9q1czO5EeARUOQ59ffwx9/D3+53umWpFEpz/yb96VG/sLJkSUnpIjJzM0nNVl0LGgU04nTSac4mnOXUhVOQAPUC6nE+/Txno86SmZmJp6en3rY8f4+erp7c0PgGGvo35GzKWX2KEKBxQGNOJZ+SpJVCtePUylJ2djZHjx6lZ8+edr/v0aMHq1evttq2ceNGunTpUm6HUEFwFNoDo1grTwWj4WwpacpHURSik6JhB9wx4w6yMtUcSaNHj2bq1Km0Dg2F7dvh669h2zY4erRwJwYDNGxIVP1Avs3bT27zprzw4JfQoAHUqQNu9ovxHjy2iq/y36drnWacTz1PVFoUZ65rww9nM7kYAmcijbTv25UvkqBTRDPiM+I5l3qOzmNf4Zq61wDgrijcEBfHDWfPMuPUKS7++Scbf/mF9f/+y8bkZJKAHxIS+GHLFtiyhTbAeBO0anaQrLgP8Bg4ENq0UY/B5rwlZCYQnRZN27C2+nbL61PeCL2SchFZ5jYqbyLIrKwsjhw5woEDB9i1bxdsA0OqQbUwpcDwV4fr+55HtUJuYANej3oRFhZGo0aNaNSoEQfyD4AH+PX2K/NxWsqlJwKN6MCp5FNiWRKqHadSliZNmsTQoUNp0KABsbGxvPHGG6SmpnL//fcD6vTZhQsXWLRoEQDjxo1j9uzZTJw4kYcffphdu3Yxf/58lixZUp2HIQhAwYO2NFYeR0U1lWR1WLd5HdmfZEMSZJHF9dddx0f33UeXEyfgjjvg0KHCja66Crp2hS5d1P87dgQfH06c+ZVJC3vRLMjAC716lSib5ZRYrjlX9Ru6FKVHq8WkFVgkwn3CMRlNapi/pZJhMEBYmPrp0oU6I0cyBhgD5OfksPfHH1m/bBnrd+3iz+hoDgOH84Fj8Tw3aRJ9J03iJh8fBvbqxVV33IFhyBAIDibcJ5zDcYcLKTSOuD4lKbCXci7piT1Lmo5VFAWPDA84DluPb+Xur+7mwIEDHDt2jPx862LKCgrZ2EyLuoKryZXcnFx1jhPVGh8bG8uffxb4TE1aNomV163k9ttvZ+TIkYV8Q+1hOd1omXF85T8rJWmlUO04lbJ0/vx57rrrLuLj4wkNDaV79+7s3r2bhg3V5HFRUVGcPXtW379x48asW7eOZ555hk8//ZQ6derw8ccfS9oAodrJyssiOSsZgNTsVDJzM/F09bTaJzc/l4QMNTNzeX2WbCnqwZyWlsaUKVOYPXs2AF6e8FXn67jt0CEMjz5q3Un79tCnD/TtCz17QhFT2mUNw7d0Zs/NVx3Ij8YfJc+cB6gKnuZrE+4drucQKm3/Jjc3ut1xB93uuINpQGJiIre9fguH1v2Gct6dhIxs1gHr0tJg3Toar1vHTUC/li25tbWRc5GFx6potKJ2LNrx2UMb09fNFy9XL327Zi06ePAgBw4c0P9PSFD/Zg5c/qcRHBxMhw4d8KrrxZq4NTRt1JRr217L1ye/5rHrHmPlqZVEpUexdNRSbvvuNtwMbkRNUO+pp0+f5r///uOVJa+QfjKd/KR8duzYwY4dO3jmmWcYMWIETz31FNdee22Rx2nPsqQ5pSdlJZGdl427i7g/CNWDUylLS5cuLfb7hQsXFtrWu3dv9u3bV0kSCUL50KwlGjHpMTQKaGS1LS4jDgUFk8FEsKdj8tNoD/XY9Fg9o/Kff/7JnXfcwanTpwF4wAAfZYL/zp1qo+BgGDwYhgyBfv2gmOgyS7SHY1pOGuk56Xi7eRe7v6Uze65ZVZYsI/jyzHkcjT+q960pS+X1dwkKCsLtai/iA2DBsM/pbOrMhrVrWf/DD/z699+cMpv5DPjsn3/gH4gANiyYSsZ1G+h6zz20veEG3fG6vNGK2rGAqnjZy3J9IfkCxIF3pjevvvoqhw8f5vDhw3atRQBGkxFzkBn/hv68cNsLdOjQgQ4dOhAZGYnBYGDuX3NZs2YNrZu3pmOjjnwd/zWJSiKxGerfpKbA5Cg5mLxMdOzYkY4dO6IoClMzpkI+7LhlB/u272PJkiX8+eeffPfdd3z33Xf069ePN998k27duhWSS59utPBZahnSElejK7nmXGLTY6nvL4E0QvXgVMqSIDgL9qKBbJUlzWoR6h2KyVh0pvqyYJlROSEtjiVTXmfSZ5+RazbTAJgH9FfgTIQn/mOegptvhu7doZhM+UXh6+aLh4sHWXlZxKTH0MTNfj40DcspLc2ydCDaOo+Ttl4ey5I99FInvhG0b9ae9u3b89yUKaSlpbF9+3bWf/89v23ZwqELF4gGojNy2LRpE2zaBICnqwlTCOxqvYspu6ZQv3596tevT1hYGIGBgQQFBeHv74+Li4uVT5OiKOTm5pKZmUl6XDpchPyMfD6b9xkJUQmcOXOG06dPq/+fOQ15EE0005luJX9QUJCuDLVv354OHTqghCh0WdAFF08XXpj8QpHn2TKP1pG4I3q+qYb+DfFz9yM1O5XotGjdMT01O5XsfHXarmubrvTq2Iunn36agwcP8vHHH7No0SK2bNnCli1buP/++5kxY4bV9Jw21pmUM7pVVUsEej71PNFp0aIsCdWGKEuCUAMpyvfF3jZHZmF2M7nROjeAgb8m8+i7jfkxQ63zdRswv359TvRpQUevzVx1w2C+H/V2hcYyGAxE+ERwOvk0MWkxNAksnbIU7h1uNQ1niSMtS5Zj2p5jHx8fbr75Zm6++WYA5u6ay6TZj9LvRDhepxX+io3lBJCZmw9RcDjqMIe3HC52LBcXF1xdXTGbzVZJby2ZsHiC/cauENggkFt63kLbtm1p06YNbdu2pW7duoUcy+PS4wBIyEwgNz8XV5N1MIu92n/aedXyTUX4ROjKUouQFlbnys/dz2rKuH379sybN4+XX36Z6dOns3DhQr766ivWrFnD559/zu233w4UTDf+E/+PekhGVwI9AonwidCVJUGoLkRZEoQaSFG+L1bbHBkJpyiqNeTTT/lhdTLDFfiXTFyBD66/ngkzZmDo0YMft73MgV83c70DHcpPJ58u1YPQspSGNg2n+StpaOvhPhW3LOWb84nLiNPlLI6GoQ251Bz+6xnOgXEH4MIFcr75hq0fTCM9NouTqAkSzvn4cC4ggHggKSWFS5cK8jXl5eWRl5dXqG8PDw/yPPLIc8ujU7NOdG7VmYYNG9KoUSMaNmzI9xe+Z/Y/s7m7293MHjy7xOMK9grGZDCRr+QTmx5bKLeTlqIg3KfAsmR5XrXz8W/Cv1aKaEl/jw0bNmTBggWMGzeORx99lAMHDjBy5EgeeughZs+ebXcsg8FQos+WIFQFoiwJQg3E9sFQaZalzEz45huYORMOH2YHMAJIBML9fVi1ai3XWESqOdqapfvjlOJBaBnppilLxfWrW5bKGUkVnxGPWTFjwGCVO8kehZzV69bFbfJknvCYS/DBk6y8NJiINdshLU39+PrCgw+S9/jjpIaEkJubq3+MRiMeHh64u7vj4eGBh4cH/Rb1Y+uprTx767OMbj/aauyvV30NxtIrzUaDkTDvMKLSoohJjymkLFlZlmwc07XrZS8QoLR/G926dePPP//ktdde4+2332b+/Pns27eP75Z/V+qxBKGqMZa8iyAIVY2e+PGyM689ZcKyRluZiYmBV15Rcxs9/DAcPsxCd3f6G40kAtSFcYuftVKULOVwVF4nrZ+SHoTpOemk5aQB1tNDGrZOz5b+NuV9yGrHGuIVoiteRaGNFZ8RT765wKk6Jj2WP+pD6pyPICoKPvsMWreGS5dg1ixcWrUiaOxYwk+coF69ejRu3JiGDRsSHh5OQEAAHh4e+vFYymSJZfmZ0lLcubGc7gzyDLI6dk0Oe1nFyxL55+bmxhtvvMGGDRsICQlh//799L6+N76JvqUaSxCqGlGWBKEGoj0Umwc3B4p4qJXjIcnFi/D009C4Mbz+OsTHo9Svz2sDBvBAdja5ZjNX9bwKxkCme2bhMSvLslTCg1A7Hx4uHvi6+RYaXztPcNnXxTNQf8heyrlERm5GmWUry7GGeIVgwIBZMetTdxm5GVzKuVTQh58fjBun5qJavx4GDACzGX78Ea6/Xk23sGWLOiVqQ3HKTXnSExSX6NJyulOzQtnKUaxlqQyRfzfeeCN79+6lVatWXLx4kfS56XC2iLGKyWAuCJWNKEuCUAPRHmJaodtiLUuleUiePQvjx6tK0qxZ6vRb166Yly5l4q23Mm2jWrR06tSpjHljDLjafzg5Im+QJfr0VQkPQsupIYPBQKBnoJXFQztPoEb0GQ1G/Nz98HDxsGpfFspyrC5GF32qzrZMjabg6RgMMHCgWgbm0CF46CFwdYUdO+DGG1XFaf16K6WpWMuShSWotBSlfFla8LTjtlQWdWuPT2F5ylunsGHDhuzcuZNevXphzjLD18BpO2OJZUmoRkRZEoQaiGUGY8t1e/sU+5CMilKtGc2awZw5kJOjPow3bCBv507GbtjAzI8/BmDmzJm8+eab1PGrAxR+OCmKUmnlVUp6ENoeq9FgtJLBUlnS+qyoc3BZlRDbKUVLy1SRpU7atIF58+DkSZgwAdzd4fffYdAg6NYNNm+2OibbvwPLa1IWa19R50Vb93Tx1BU8y+N3hM+SPQIDA/n5558JbxcOucA3kH0mu8ixBKGqEWVJEGoglrWxoOTpkkKkpMCLL0LTpvB//we5uWrCyO3b4ddfye3blzvvuosFCxZgNBpZuHAhTz31FFC0H1FyVjI5+TnqPo6yLJXSZ8nesVrKoJ0n2+0VedCWVQmxdVYvk2JZvz588gmcOgUTJ4Knp1p0uH9/GDCAq86o03m2fwflvSZFnRdLa5qm4Nk75/aUrYoq0l5eXtz++u3QGMiFuc/O5ejRoxINJ9QIRFkShBqGZdV3zWKSnlswPQKQk59DYmYiYPOQzMqCDz+EJk3grbfU6bZrr4VfflGtFL17k5uby1133cXy5ctxc3Pjhx9+0OsnQtERatqD1d/dX5/eqiiWYyl2fHVsx7Zn5TBgoF1Yu4Lt3oUf7uWZwimzZcmnaMtSqYmMhA8+gNOn4amn1Om5TZu4dvgTfPMDuJ8+b7W7do3Kek2KUkDsyVycZSkmreC6OcKfrW5QXbgTqAOXki8xePBgXDLV6Vat7I8gVAeiLAlCDcPSmbmObx293pflA18rh+JidCHIM0j1b1m+HFq2hGefhcREaNUKfvoJfvtNrdGGmsvn3nvv1RWln376iVtvvdVqfMuSJ9aRXWWf7ikJbayM3AwrZdAWS58lve3lh3iIVwiRvpEYMFj1CQWKU5VYlrytpxQrFK0YFqamczh2DEaPRjEYuPsQ/Pp+AuZnJ0KqqkxbplMoC0ValuxYh+ydc83pO9ecS1JWkjod6AB/tgifCHAHRkODRg04ffo0Y+4eg5viZiWfIFQ1oiwJQg3iUvYl/rygVm8P97ZOyvfnhT85GneUo3FH+eP8H8BlZ+YjR9XpmttvhzNnoG5dmD8fDh6EYcNUh2IgPz+fMWPGsGzZMlxdXVm+fDk33XRTIRlCvUL1yK6ETLXoamJmIvuj9qtyOWgKDsDHzQdvV7UmnPYgTM1OtVLSTief5mTSyUJjWzoAuxhdCPYKttpuuX9ZHrKKonA84ThnU84WGrM4tP2OJx7naNxRjiUcK1N7uzRuDIsXk7/3T9Y3Bbd8MH74EUrz5rBwITGpl2vPlVGBtWdxi7oUxeFYNcu4vXNomW/K3cWdAI8AAHad28WfF/7Uc19ZRs+VFf1cecN3P36Hn58fv/32G26bVGXpj/N/cCn7UjE9CELlIEkpBaGGcCn7Eo1mNSo0vRbhE8Gp5FPcveJuq/39M+G9X3JhcgfIz1edgydPhuefB2/rorSKovDII4/wzTff4OLiwnfffaeX6rDF1eRKsFcw8RnxRKdFk56TTstPWxb4xjjIuVsj3Cec/5L+IzotGl83X5p+3JQ+jfqw5u41zNw9k2c2PFOwrx2Lh7YtwieC+Iz4CvssPfHzE3y651O7YxaHNtbqf1ez+t/VZW5fHC6du3Dfo6FcfSCOmeuhRUwMPPAAvdo0pMu1EN66fJalpKwksvOy+e3sb/T/uj8K6pSavXMY7BVsFYEY4RNBclYyNy8p+Duq6BStNpa7yZ1rOlzD0qVLGTJkCGm/p0EE3MmdBHkGceqpU/i5+5V7HEEoK2JZEoQawrGEYyRmJmLAQJh3GA92fBCABzo+QJh3GEGeQerHI5BH/vHm+GwDd2+NUxWlW2+Fo0fhtdcKKUoAU6ZM4csvv8RoNLJkyRKGDx9erCyWPil/Rf1FTn4OLkYX6vjW4Z729zj0uC3HOhBzgPTcdH4/9zsAO8/tBMDL1YsO4R3o06iP3m7wVYNpF9aO+zuo/lZjO42lU0Qn+jXup+9THudgbUxfN19uaHwDbcLalKrdjU1upG1Y24Lr5BlEy5CWDL5qcKnHLo5xXcaxp30w7R6D5/qD4uNN5OEz7PkCnlpwBJKTS91XoEcgrka1Jlxseiy7zu9CQcHd5E6TwCbc2rJgavaautfQo14Pxl09zqqPsZ3GEuoVqh9rsGcwj3V5rELH2C6sHX0b9eWxLo9hMBgYNGgQL730EgCGNQaIU62c/yb8W6FxBKHMKEKFSUlJUQAlJSWlukURajGrj61WmI7SZW6Xonc6dUpRBg5UFNVLSVFatVKUjRuL7ff9999XAAVQ5s+fXypZ+n3VT2E6yqK/Fymf/PGJwnSU25bdVoajKT23Lr1VYTrK7D9mK1/9/ZXCdBSmo2TlZinXf3m9wnSU7w59V66+fzvzm8J0lCazmpS6TcT7EQrTUfZd3FeuMSuboBlBCtNR/jm4Tfmtb9OCv4WICEX5/ntFMZtL1U+9D+spTEf58/yfyvi14xWmo7y45cVKlr7s5OXlKf369VMAxauBl8JLKKuPra5usQQnobTPb7EsCUINoViH4Lw8NcqtTRs1maG7O7z5Jvz9t+qvVARfffUVkyZNAmDGjBk8+OCDpZLFMkqtQo7K5RwLVItHRZ2GyxoNl2/OJy49rkJjVjbadbjgY+atsS3oNQaSG4ZDdDSMHAnDh8P588X2YdlPTHqMw/NnORKTycRXX31FYGAgGWczYIckqBSqHlGWBKGGUGTo9cGD0L27GuWWkQG9e6vbpk4FN7ci+9uwYQMPPfQQAM8++yzPPfdcqWWxzH/k6BInpRnLUWNr7WxTLxRFQmYC+Uq+6szsVXzx3OrC0g8rOi2aXxvB76vnwMsvq6kGVq1S68999pnd0ikalqkOKvsaV5S6devyf//3f+rKb7B3z97qFUi44hBlSRBqCIXe7vPz4Z13oEsX+OsvCAiAL76ArVuhefOiOwIOHTrEqFGjyM/P55577uHdd98tOou0HaysPeUsY1GRsQBOJZ/Sa6uV1+Lh4+ZjN/VCUWj7BHsF42pyLdeYlY2ltUyTNyy4geqvtn8/9OihFup9/HG1rEoRVibLVAeOLmNTGYwcOZLWN7QGBb5/53tycnKqWyThCkKUJUGoIVi93Z84Ab16wZQpavbtYcNUB+6xY8FY/M82JiaGm2++mdTUVHr16sX8+fMxltDGlqq0OtgbC+BgzEFAzTdVkcinskTE1XQLCxQoOVFpUYUV7DZt1Lxas2apWcA3bYK2bWHx4kJWptpkWdIYPXk0eEHC6QTefffd6hZHuIIQZUkQaggx6TGgwLVrD0KHDmqNMF9fWLAAVq6EiJIfZJmZmQwfPpwzZ87QrFkzVqxYgVsxU3VFYRmhVtn+LPbGAjgQc0AftyxWMVvKEhFXk313NDQl52j8UfLMeYBNbiOjEZ58UrUyXXONWvrm3ntVf6a4OH037bw7woJXVTSt2xQupwZ74403OHnyZPUKJFwxiLIkCDWEnKjzrP0Grn59nuqb1Lcv/O9/MGaMnliyOBRF4cEHH2T37t0EBgaydu1agoODyyVLTfBZOhB9wCHjOp1l6bJs2vkJ9AjE3cW98I4tWsDOnfD66+DiomZ4b9dODRCg4Lw7yoJXFUT4REA78GrhRXZ2tl7PUBAqG1GWBKEmsHUrP759isEnwOzhrpa62LwZGjYsdRfTp09n6dKluLi4sGLFCpqX4NdUHNoDOS4jzuHFc23R+s3OzyY+I17ffi71nEPG1S1LZfBZqskWFk027fwUq9i5uMBLL8Eff6hO3zExcNNN8MILRLqHWPVTUQteVRDuEw4GMA424urqytq1a1m9enXJDQWhgoiyJAjVSV4evPwyyo03EnFJ4VAopO/crhZRLYOf0TfffMNrr70GwNy5c+nTp0+FxArxCsFoKBjfkcVzbfFy9cLXzbfI7yuquJTJspReeyxLGqVSJjt3hr17YdzlxJIzZtD17mdpmFR0vzURTcY0/zSefPpJACZOnEhubm51iiVcAYiyJAjVxfnzcMMN8MYbGBSFuZ2h5zg3fDp1K1M3+/btY+zYsQA8//zzPPDAAxUWzWQ0WYXOV3aUVHEP6oo+xMtSH642RIXZylbq8+PpqaYT+O478PfHc+8B/v4cRhyx329NxN/dHzeT6oP3wBMPEBoayokTJ5g/f341SyY4O6IsCUJ1sH696sT966/g68vx2a/z6DDwC4oo01RIfHw8I0aMICsriyFDhvDWW285TER79cEqC8uxtFB//buqtCzVAp8lrdCxRpnPz8iRsH8/SrduBGTD8u/g43VQ161m5pWyxGAwFFiXDGm8/PLLALz66qukp6dXp2iCkyPKkiBUJWYzvPEGDB4MiYlw9dWwbx9Hb2wPlO0hnZeXx1133aVHvi1evLjMKQKKw1KWyvbhsRyrfXj7Ir8rD84WDacVOtYo1/lp3BjDr7/y2Q3q9OcTf8Lz0zZCVJSjxKw0LK/no48+SuPGjYmOjmbWrFnVLJngzIiyJAhVRUqKWvD25ZfVnDfjxqnRSs2a6RaNsjykX3rpJTZv3oy3tzc//vgjAQEBDhXXUpZKtyxZjNUiuAXupoLoropOD1lalpRiMlrnmfP0Uic12bIEDlJkXV1ZcGdLht0JKe7Q8NA51bdp504HSVk5WF5PNzc33njjDUAt55OQkFCdoglOjChLglAVHD4MXbuqpSjc3WH+fNV/xF1VCjRfmdI+pH/44QdmzJgBwJdffknbtm0dLrKlLJWtPNiO5cgpQK2vrLwsPZ+QPeIz4lFQMBqMhHiFVGjMysZR1ybCJ4LVLaHrw5DSrL5aX65PH5g9u9hSKdWJbXTjnXfeSceOHUlNTXXoNLQgWCLKkiBUNj/8AN26wfHjUL++6qdkU9C2LJalw4cPM2bMGAAmTZrEqFGjHC6yrSyVPS1lO5YjpwAto+2K81vSHr6hXqGYjKYKjVnZWJ2vCljetH6Oh8CRVfNh1Cg1QvOJJ+D++yErq8KyOhpbHzSj0cg777wDwOzZszl79my1ySY4L6IsCUJloShqva6RIyE9XY18++sv1cJkg+YrU5KVIDU1lVtvvZX09HT69evH22+/XSmi28pS5Zalyw9xL1cvfNx8Kty/ZT21oqgNzt0ajrQsaYSFN4GlS+H998Fkgq+/VhOjxpTs61WV2ItuHDBgAH379iUnJ0fKoAiVgihLglAZZGXB6NEwbZq6/swzaubkUPsRR7plqRgrgaIojBs3juPHj9OgQQM9AWVlYSlLZYeV246lPcQdlSixNBFxlV0w2JFYWpYsUzyUuR/ba2wwwLPPqn+rAQGwe7daMuXgwYqI61DsXUuDwaBHxs2fP5+YGqbgCbUfUZZqOXHpcXy25zNSslKqWxRBIyZGfSNfskTNoDx3Lnz4Id8f+5E/zv9hv0kpLEsLFy5kyZIlmEwmli5dSkhI5frVVKdlSVt31LjFRcTl5ucyZ88cvvnfNw4dszLRZAzxCsHV5Frhfrxdva0teP36qVm/r7oKzp6Fa6+FGpIp2/ZaJmUm8f7v77NN2Uajto3IysqSyDjB4YiyVMt57/f3eHzd48zZM6e6RRFAreV2zTXqG3lgIGzcCA8/zKmkU4z6YRS3f3+73WYl+SwdPXqUCRMmAGoB0R49elSO/BbU9a2L0WDEzeRmXai1Egj3DsfN5IbRYKSObx0a+DcA0P+vKMVZltb8u4bx68az8eRGdUw/x4xZmTjq/BTbT/Pm6t/xDTeo08i33KJO0VWz47fttfzkz094btNzvP7r65xuexqATz/9lJQUeYEUHEfl2fCFKuG/pP8AtXK4UM38/LPqIJuWpr6Rr1mjPnAouD7nU8+Tm59rZQ1Iz0knLScNsG/VyMzM5M477yQjI4Mbb7yRyZMnV8HBQKBnIF/f+jUeLh561uTKwt3FnSW3LSErL4sAjwDubHsnsemx3N7avnJZVoqrD6f9hloEt2B4y+GMv2a8Q8asTHo27Mn7/d/nugbXVaifrnW6MnPgTK6uc7X9HYKC1ASqEyaoFtLnnoN//4U5c1SraTWgTR2m5aSRnpOuXz8AmkN4o3BiTsfw2Wef8cILL1SLjILzIZalWo5mii5Nwj2hElm4EIYOVRWlvn3VN3KLQraWD+nY9Firptq183TxtOvMPGnSJA4ePEhYWBhff/21QxNPlsTd7e5mRKsRVTLWiFYjuLvd3QD4uPkwtedUmgeXvxiwJbo1Ir2wZUk7/4OvGsw7N75TK6bhjAYjz177LN3rda9QPwaDgae6P8X1Da4veidXV/j8c7W4s8EAX3wBI0ZARkaFxi4vvm6+ep3CmPQY/foFegSCETrf3hmAjz76iMzMzGqRUXA+RFmq5Wim6NKUchAqAUWBN9+EBx6A/Hy47z71TTwoyGo3y+tje60scyzZOjOvWLGCOXPUKdavv/6aiIia/yCviRQXDVeehKBXHAaDWtx5+XLw8FD9l264AeLjq0GUgpInMWkx+vXrENEBgMBrAmnYsCGxsbEsWLCgyuUTnBNRlmo52s2/uJBooZLIz1enJ156SV1/4QXVwuRWeMrK0vJnawUsKhLuzJkzPPTQQwBMnjyZAQMGOFD4K4vifJZKm7ZBQM1Av3mz6o/3xx+q4/epqncB0BTb6LRo/d7XIVxVluKy4pg0aRIA7733Hvn5+VUun+B8iLJUi8nIzdAzEpdUykFwMJmZqn/SnDnqW/cnn8Dbb6vLdijWsmTnYZ2bm8tdd91FcnIy3bp100s6COXDMoLK9ndSmrQNggXXXaeWRGnQQE202qMH7NtXpSJov5WotCh9WltTlqLTonnooYcICgri9OnTrF27tkplE5wTp1KW3n77bbp27Yqvry9hYWEMHz6cY8eOFdtm+/btGAyGQp9//vmniqQuP5bWpOz8bFKzU6tRmiuI5GQYMABWrFCtSN99p1qYisHKspRWhGXJYhpo+vTp7Nq1Cz8/P5YsWYKra/nDw4UCRSgnP4fkrGSr78paakYAWrWCXbugfXs1VUbv3rBtW5UNr/1WjsQdIV9RLUftwtsB6m/N09OTsWPHAmpWb0GoKE6lLO3YsYPx48eze/duNm3aRF5eHgMGDCA9Pb3EtseOHSMqKkr/XHXVVVUgccWwtVCI31IVEBenOnD/9hv4+6upAW4vOWKrtD5LAJs3b9Yzc8+bN4/GjRs7SvorFg8XD/zd/QFrxTXfnE9chlo8V3yWykidOvDLL6rvUloaDBqkRoBWAdpv5UDMAUDNN1XPrx6g5p7LM+fx2GOPYTAY2LRpU614+RVqNk6lLK1fv54xY8bQpk0bOnTowIIFCzh79ix//fVXiW3DwsKIiIjQPyZTza4NBYV9XyQirpK5cAF69YK//4awMNixQ32jLgWW1qRCPkvpBZalmJgY7r33XhRF4ZFHHmHkyJEOE/9KR7MuWSqr8RnxmBUzBgyEepc/E/YVi78/rF2r5mDKzlZ9mpYtq/RhtWt5IFpVlsK9wwn1CsVoMKKgEJ8RT6NGjRg6dCiAHiQhCOXFqZQlW7SkZEE2kUn26NSpE5GRkfTr149tJZiTs7OzSU1NtfpUB2JZqkJOnYKePeGff6BePfWNukOHUjU1K2ardAFFXbcwrzDuv/9+oqOjadOmDR999JHj5BesIqg0tHMf6h2Ki1HSzpULDw/4/nu4+261CO9dd8G8eZU6pHYtNZ/NCJ8ITEYTIV5qVnvtumqJXBcuXMilS5cqVSbBuXFaZUlRFCZOnMj1119P27Zti9wvMjKSuXPnsnz5clasWEGLFi3o168fv/zyS5Ft3n77bfz9/fVP/fr1K+MQSsTW90Ui4iqJo0fh+utVhalpU/j1V2jRotTNEzISdL8KsGMRvHzdtn67lQ0bNuDp6cmyZcvw8vJyjPwCYB1BpaHXg5MpuIrh6qoW3n30UTWdxsMPQyUq+7bXS7M02SrE/fr1o0WLFly6dImvv/660uQRnB+nVZYmTJjAwYMHWbJkSbH7tWjRgocffpjOnTvTo0cP5syZw5AhQ3j//feLbDNlyhRSUlL0z7lz5xwtfqkoKgRdcCD796tTbxcvQuvWqqLUqFGZuijOAqgoirp+Hua+OxeAWbNm0aZNmwqLLlijP0jTC1uWxLnbARiN8NlnapZvgIkT4bXXKmUo2+sV4V1QeBkKrqvRaGT8eDUj++zZsyViWCg3TqksPfHEE6xatYpt27ZRr169Mrfv3r07x48fL/J7d3d3/Pz8rD7VgXZDiPSJBMRnyeH89VdB4r3OnVUfpcjIMnejXRftOiVnJZOdlw2oJRsyL2XCD5CXl8eoUaP0KB7Bsdi1LF22QEjaAAdhMMCMGaClupg2DaZPd/gwtterkGXJ4l54//334+Pjw9GjR0t0sRCEonAqZUlRFCZMmMCKFSvYunVruaOI9u/fT2Q5HopVjXZD0DLXimXJgfz1F9x4o5omoEcP2LoVQkLK1ZV2XVqGtMTVqKYA0K5d1KUoWA0kQ6NGjZg7d26hLN6CYyjWsuQtliWHYTDAiy+qRXcBXn1VVZocaNXxcfPB29VbX9eurT2F2M/Pj/vuuw+Azz77zGEyCFcWTqUsjR8/nsWLF/Ptt9/i6+tLdHQ00dHRVvWBpkyZov9wAGbOnMnKlSs5fvw4hw8fZsqUKSxfvlx3DKzJ6Gn+LydjE8uSg9i3D/r3VxWla69Vy5f4+5e7O8vUALZlN+bNnwdHACMsXboU/wqMIxSPvWg43WdJLEuO59ln4YMP1OXXXlMtTA5UmCyvmaYk2VOIAR5++GEAVq1aRUJCgsNkEK4cnEpZ+uyzz0hJSaFPnz5ERkbqn2UWoaxRUVGcPXtWX8/JyWHSpEm0b9+enj178ttvv7F27VpGjKia4qEVwTbNv1iWHMC+fapFKSlJtSj9/DNUcJrV0i/GsuzG4cOHmTltJgANb2tIt27dKjSOUDz2Sp6Iz1IlM3GitcLkQAuT5TXTLUt2FGKAjh070qlTJ3Jycvj2228dMr5wZeFUsbKlcd5buHCh1frkyZOZPHlyJUlUeaTlpJGeqybb1KbhYtLUUg4yjVNO9u+3VpTWr6+wogTWEVfaG/DZ+LNMeXQKudm50BQ6j+hc4XGE4tHOfWx6LGbFjNFglGi4qmDiRHVqbuJEeP11ddurrxZZGqi0WF6zoqLhLHnggQfYv38/X375JU888USFxhauPJzKsnQlod0MvF29aRLYBIBccy5JWUnVKVbt5e+/K0VRAvuWpYUzFnL48GF8gnzgVojwFctGZRPmHQZAnjmPxMxEQCxLVcYzz8CHH6rLr7+uKksVRLtmRoORUC81oag9nyWNu+++Gzc3N/7++2/2799f4fGFKwtRlmoplsU/rUo5SK6lsnPkiKooJSZC9+4OVZTA2i8m3DscDsHe1XsxGAz0eroX+MjDuipwd3En0CMQUH8nufm5JGSo/ivis1QFWCpMr75a4ABeTjTFKMQrBJNRrbig/Y4SMhPIzc+12j84OJjhw4cDsGDBggqNLVx5iLJUS7GtVG/PH0MoBf/9pzpzJyRA166wYUOFFaWdZ3fSa0Ev9kepb6+W1gvXVFc1+g012MDUTL3JyzRQ1WD5O4nLiENBwWQwEewZXM2SXSE88wy89Za6/Nxz8Pnn5e7K9t4HEOwVjMmg/qZi02NJzEzkxkU3sujAIkCdigP45ptvyM7OttvvT//8RNs5bWn+SXOGLhlKTn5OuWUUnAdRlmoptpXqi4oCEYrhwgXVonTxIrRt6zCL0sK/F/Lr2V/59n/fkm/OJz4jHoAgtyC+mfYNZINfMz+mT59eSOkVKhc9GjE9RrfChnqH6pYJoQqYMkX9ADz+uJr5uxx0jlT9/DpFdNK3GQ1Gfbo1Jj2GDSc2sOXUFmb9MQuA/v37U7duXRITE1m1apXdfufsncPhuMMcTzzOmn/XsOfCnnLJJzgXoizVUmwr1RcVBSIUQVycalE6dQqaNYNNm6AUNQRLg1YYNzpdtV5ohVo/eecTTvzvBHhA0D1BuLq6Wk2nCpWPpWVJ/JWqkTffhCeeUCPjxoyBFSvK3EXXul059dQpvhj6hdV2y3uhdo21/00mE/fffz9Q9FSc1NwU7CHKUi2lkGXJu+goEMGGlBQYOFCt+VavHmzeDBGOe2Bq1yAmrcB64XfOj/ffu+yjMQwS3RNRFKWQ0itULtrvJSYtRiLhqhODAWbOVBUlsxnuvFO17JaRRgGNcDW5Wm2zjIjTrrEWAQkFU3EbNmzgwoULhfrUfpONA9SkxmKtF0CUpVqL7fSN/jaVLm9BxZKeDkOGqGkCwsJgyxZo2NChQ1i+zUanRcMlSFuWBsBDjz4ErSE1O5XotGiy81W/CXlgVw26ZSldLEvVjtEI8+bByJGQmwu33qrWXqwglhFx2jW2jIBs1qwZ1113HWaz2SoHH0C+OZ+4jDhAKiMI1oiyVEuxnb4pLr+IcJncXLjtNti5EwICYONGaN7coUMoiqIrsjHpMWo5kxWQn5ZP+/bt+eSjT3A3uQNwMOYgAH7ufni6ejpUDsE+VpalNLEsVTsmEyxerL7AZGXBsGFw6FCFurT037S0ClneG++++26AQoXW4zPi9WnztqFtC7UTrlxEWaqlFLIsFZNfRED1jXjoITXazctLzczdoYPDh0nOStajZ+LS41jy+RI4BSZ3E0uXLsXT01NXcA/EHADkYV2VWPkspYtlqUbg5gbffaeWFkpOhptuAosqC2XFnmVJW9cYOXIkJpOJvXv3WhVN1/YJ8Qqhrl9ddZtY6wVEWaqVKIoi0XBlZcoUNerGZIIfflDzKVUCludfOauwad4mAG4cfyOtWrUCCq6VpizJw7rqsBcNJ871NQAvL1i9Glq3VqNUBw5U03mUAyvLkoVVyPK3GRoayo033ghYW5csX0LFWi9YIspSLeRSziWy8rKAghu9ZYFWzZFRuMzHH8OMGeryvHkwaFClDaW/vWYAy0ExK9AO+o3op++jKbgHoi9bluRhXWVoD8DY9FguXrpotU2oZoKCVCfvevXgn3/g5ptVH8Myov2eLl66SGx6rL7dVunRpuK+/fZbvVSWpXuDWOsFS0RZqoVoP15fN1+8XL2AglIO+Uq+7sgooJr3n35aXX7rLTX6phKJSYsBBfgJSAGCgCH2i37+E/+Puu4tD+uqQiuLYVbMnEg8Acg0aI2ifn1VYQoIgN274Y47VF/DMqD9vk4mniRfyde32yo9w4cPx8PDg2PHjvH3338D1ilZLC1Upak7Kjg3oizVQuyFm7uZ3AjyVPMEyZvQZbZtg3vvVf2VJkyAF16o9CGj06JhN3AMMAEjAQ/ra6U9nLUbuViWqg5XkyshXiFAwfkXy1INo00bWLMGPDxg7Vp45BH1N1xKbH9fGrYuCn5+fgwZMgQomIqzdG/QfpdZeVmkZqeW71gEp0GUpVpIUYkMZY7dggMHYPhwyMmB229Xc7pUsMp5adi3dx9surxyExCpLlpeK9uHszysqxZLS5KL0YVAz8BqlEawy3XXwbJlanqBhQvLVHg30DMQV6Nroe1FFdcFWLp0KWaz2cpnycvVC183X0B8QQVRlmolRZXIkDn2y1y4oIYip6ZCr14Fjt2VTFJSEiteWwFmoA3QpeA7K8uSjZIr00BVi62Vz2iQ22CNZNiwgtpxr74KixaVqpllyRNtHewrPIMHD8bPz49z586xc+fOQoEzUhlB0JC7RC3E9getIRFxQFoaDB2qKkytWsFPP6nm/EpGURQefPBB0uLSIBBaPNACLhuyjAajVaFWsSxVL5bKqkyB1nAefhief15dHjsWduwoVTPL69oypCVgX+Hx8PDg1ltvBdSpuKIKlIu1XhBlqRZSVImMK96ylJ8Po0er2blDQ1V/h4CAKhn6448/ZuXKlRhcDDASOjfqrH8X5h1mVajVVsmVB3bVYulQL4pqLeCtt9SpdC3L97FjJTaxvK4dwtV8anHpceSb8wvte9dddwGwfPlyolKjAIso4yv9niroiLJUC9GSpIllyYbnnoNVq8DdXbUoNW5cJcPu2bOH5557DgC/oX5Qp+AGDUVfJw3LKQOh8rGyLMkUaM3HaFSn4Lp1g6QkGDxYLYRdDJbXtW1YWwwYyFfyScgsnLvphhtuIDAwkNjYWBL+Ub8vZFm6Uu+pgo4oS7WQIi1LV/L8+mefwUcfqctffQU9elTJsMnJyYwaNYrc3FxG3DaCtI5qDTitrhQUvk4+bj54uqjlTQI8AvBwqfxpQqEAe2kchBqOp6f6ItS4Mfz3nxq8kZVV5O6W17WeXz09AtLedJqrqyvDhg1TV45aT5uLZUnQEGWpFiLRcDasXw9PPKEuv/GGmpulClAUhYceeojTp0/TuHFj3v/kfT1cuX14e30/2+tkMBgKmfmFqsPynMv5r0WEhRVMrf/+u5ozzWw/Aa/tNS7pRXLEiBHqwlEI9QzVp83FsiRoiLJUy7As1Co+S8D//gejRqn+SvffD1OnVtnQ77//PitWrMDV1ZVly5aRYcoAIMgziAifCEyGyzdcO0knbc38QtUhlqVaTKtWsGIFuLioqQXeeMPubrbXuCSlZ8CAAXh6e0Iq+Mb76tuvaGu9YIUoS7WI9Jx0fj37q16o1dbXRbshxGXYd2S0JTEzkX1R+/TPvwn/lpip9uKli/o+WXlZetuM3IzyHFLFiItTI98uXYLevWHu3CrJpQSwdetWXric5PLjjz+ma9euVkqsZfiyPQdu29BkoeqQaLhaTt++8H//py5Pm6YqTzbYXmPt97Yvah9nUwoX6fXw8KDdde0AyDucp28vylqvKApRl6L09dTsVNJy0sp5QEJtQJSlWoKiKHT5ogu9F/YGwN/dv5CvS6h3KAYMmBUz8RnxxfaXkpVC41mNuXru1fqnxewWfLHviyLbLPx7IXU/rMucPXMA6Leon972mi+uqeARlpHcXBg5Es6cgaZN1Rumm1uVDH327FnuuOMOzGYzY8aM4dFHHwUKp3TQbrT2pnqK+06oXEK9QvXcO3L+aykPPlhQxujee9UktBZovy8DBkK8QvT1j3Z/RMOZDfnt7G+Fumx2fTMAEv5K0F8Itb8P25InL259kTof1mHjyY3k5OfQcnZL2n/WXupyOjGiLNUS0nLS9Fpi9fzq8XT3pwvt42J0KXBkLGGO/VjCMVKzUzEZTNTzq4e/uz8Aey7sKbLNnxf+VPe5uAezYrba93Dc4aq1Lj3zjJpzxcdHdfwMCqqSYbOysrj99tuJj4+nc+fOzJkzB8Nla5at4/1jXR7j2vrXMqDpgEL9jG43mq51unJ3u7urRG6hAJPRxNPdnmZ4y+G0CGlR3eII5eW996B/f8jIgFtusYqQax7cnFtb3srT3Z/GxejCHW3uoHVoa72W5t6Lewt1F9A2AFzgUvQlDh48CBRYqHLyc0jOStb31e6Fey/u5ULqBaLSojiVfIqEjMLRdoJzIMpSLUGzWvi4+XDumXNM7zPd7n6lnWPXvu8c2Zlzz5zjvf7vqdvTi26ntYlOiyYpM4lcs1rgUs+QW1WO5V98AZ9+qk65ffMNtG5dNeMCTz75JHv27CEoKIjly5fj6empf2drWXr46ofZ+eBOQr1DC/XTs2FP/nz4T7rX6141ggtWfDDwA36840fJ3l2b0fyWmjVTLcy3366WN0K9J624YwUfDvwQgK51u3L48cM80vkRwP79MdGcCE3V5eXLlwPg4eKhv0havoBa3gst+xLfJudF7hS1BO2HWtK0QWkj4rTv9aisy/8X106TISY9Rl8O9Aikvl99q+8rld9+g/Hj1eXXX1dLIlQR8+bN44svvsBgMLBkyRIaNWpk9X1RjveCIFQSgYGqZdnXF375BZ58stjd9fucnXtVTFoMXH7vWmHhB6X9ni0VIXv3wqL6FZwDUZZqCdoPtaQHcWkj4vT+vK2jsoprZ+9tyjLSpNLfqs6dg9tuK/BXqsLItz/++IPxl5W0N954gwEDCk+tFZXSQRCESqRVK1iyRLU0/9//qTnXiqC4e1V0WjQ0BxdXFw4fPsyxy5nCbV8kc/NzdZ9QsSxdOYiyVEuwtQQVRWnzguiWKpt8P7aOjPZkiEuP4+Kli3r70lilKkxGhpqILjYWOnSABQuqLPLt3LlzDB8+nJycHIYPH65HwdkiliVBqCaGDIF33lGXn3iiyBpy+n3Ozr0qJj0GPKF7T3Vq/McffwQKK1hxGQW+UTFpMVZ9XXE57q4gRFmqJdhagoqizJYlLSqrCEdGjbScNNJz0wHIV/I5EndEH0+TqdLeqhRFLai5bx+EhMDKleDtXTlj2ZCWlsawYcOIjo6mXbt2LFq0CKPR/s+mqALHgiBUAc89p9aGzM9Xc69duFBol6IsSzn5OSRmJgJwyy23ALB69WrA+kXStq1Ylq4cRFmqJdhagoqitJYl2wd7UY6M+vg2b0wHYg7o41V64rZPPoFvv1UdOn/4AWx8hSoLs9nMvffey99//01YWBirV6/G19fX/r6Kmbh09Y1TLEuCUA0YDGqutfbtVQv0yJG6w7eGdq+KTY+1CvOPTY8F1IjiUbeOAmDXrl3ExsYWUrAs73Mp2SmcSTmjr4vPkvMiylItodQ+S6VUXOxNGZU4n2/BgWhVWQr3Dq/ckgC//w7PPqsuv/eemnyyinjxxRdZuXIlbm5urFy5koYNGxa5b0JGAvlKPgYMdqPfBEGoAry81JxrAQGwa1fBveMyoV6hBUV1LcL8tftbmHcYDeo3oHPnziiKwrp16wpZlmxfHA/GHCzUj+B8iLJUS3B0NJw9Z+TifI9sFaELly7o41VamZWYGPXtMC9Prff21FOO7b8YFi1axDuXfSC+/PJLepRQmFc79hCvEFyMLpUunyAIRdC0KSxerC7Pnl2wDLiaXAn2UovkWkWx2eRIGzp0KACrVq0q1rIEBfdC2z4F50KUpVpCWaPh4jPiyTPn2d0nPSddT81fXsuSPp5PJVmW8vLgzjvh4kU12mXevCpz6N65cycPP/wwoFqXRo8eXWKb0k6TCoJQBQwZAq+8oi4/8ohVhm979zlbt4Rhl1OSbNy4kQCXAKBAoSruPieWJedFlKVagKIopY6GC/EKwWgwoqDoPjS2aD92DxcPfN0sikZ6l5CDxA6V5rP04ouwfbuaoXvFCvX/KuDYsWPccsst5OTkMGLECF577bVStRPnbkGoYbzyCtx0E2RmwogRkJQE2I+Is3VL6NSpE3Xr1iU9PZ3/9v+n72NWzMXe5+Iz4ktVl1OofYiyVAtIyU4hOz8bKPlhbDKaCPVSfWaKegOyNDkbLKw1pbEs+bhZKy2WPksZuRmOKSb544/w7rvq8oIF0LJlxfssBVFRUdx0000kJCTQtWvXYiPfbLE14wuCUM2YTGqG/0aN4L//4L77wGwulWXJYDDoU3G/bvoVgDxzHkmZSfp91fJe6OniidFgVAM9Muy/pAq1G1GWagHag9jP3Q9PV88S9i45wWRRVpBiLUuXt7UPb2+1Pcw7DB83H73mUoWtS//+C/ffry5PnKiWMKgCUlNTGTx4MKdPn6ZZs2asXbsW7zKkJxDLkiDUQIKCVMu0hwesWQNvv233Pmcv4EVTltatWUegRyBgnSrA8l4Y6RtZ8JIquZacElGWagGl9VfSKClJZFHJE0tjWeoQ3kHfFuwZjKvJ1apthW4U6elqhu5Ll6Bnz4Ikc5VMTk4Ot912m54iYP369YSGli2iTRJSCkINpVOngqzer7xCl+Nqvji7liULN4cbbrgBLy8vLly4gH9SQVoV7R5neS+skhQqQrUiylItoLSRcBrltiyVIhrO9gaht3VERNyTT8KhQxARoRbIdHUtf1+lJC8vj9GjR7N582a8vb1Zu3YtTZs2LXM/UupEEGowY8aoFmuzmVteW0ZoWvHRcAAeHh4FZY3UyiecTTlLUpbq+2R5L6z0FCpCteOUytKcOXNo3LgxHh4eXH311fz666/F7r9jxw6uvvpqPDw8aNKkCZ9//nkVSVo6ymxZKmY6DYr2r7H8sVsmbFMUpcCyFGFxg7BQDCp8o1i8GL78EoxGtc5TZGT5+ikDZrOZsWPH8sMPP+Dq6sqKFSvo0qVLufoSy5Ig1HA+/RRatcIrLpmvf4SY1Cj9q6JeILWouJSDKUBBTiVXoystQwp8KSs1hYpQI3A6ZWnZsmU8/fTTvPjii+zfv5+ePXsyaNAgzp49a3f/U6dOMXjwYHr27Mn+/fuZOnUqTz75JMuXL69iyYtGj4RzlGUp3b4VJMw7DChwZNS4lHOJrLwsANqGtcVoMFqNYylbuW4Ux47BuHHq8iuvQJ8+Ze+jjCiKwhNPPMFXX32FyWRi2bJldovjlhbxWRKEGo63N3z/PWYPDwaehFGrTwGQlZdFSraqDNm+7AwZMgSDwUDSf0mQWlC5INwnnEjfghc6K8uS+Cw5JU6nLH344Yc89NBDjB07llatWjFz5kzq16/PZ0VUov78889p0KABM2fOpFWrVowdO5YHH3yQ999/v4olL5qqsiy5mdwI8gyyGtNy2dfNFx83H92R0VIxKPeNIjNTreOUng59+8JLL5WtfTlQFIUXXniBOXPmYDAY+Oqrr7j11lvL3V++OV+vQi6WJUGowbRpw6UP3wbg+Q1p5O/Yrt+z3ExuBHgEWO0eFhZWYG0+YV25wPb+p78wpotlyRlxKmUpJyeHv/76q5CFYMCAAfz+++922+zatavQ/gMHDmTv3r3k5ubabZOdnU1qaqrVpzIpa8LDoixLR+KOMGXzFP6J/8dqP0vsRorYKFeaHPYSWm4/s51p26YRdanAxA3w/eHv2XBiQ2Fhn30WDh6E0FA1zNdkKtUxlhdFUXj++ed593Jqgs8//7xUSSeLIy4jDrNixmgwEuIV4ggxBUGoJHwemcCiDmBSgLvuIuGsej8M9w63SqWiMXjwYHXhBCRkqiVSInwi8HP3w8PFQ23rI5YlZ8eplKX4+Hjy8/MJD7dxXA4PJzq6iCmp6Gi7++fl5REfH2+3zdtvv42/v7/+qV+/vmMOoAgcFQ331PqneGfnO7qDYgP/BoXaFpuD5HK/jQIaAdDQv6BWmrbt34R/ee2X13jv9/f07+Iz4rlz+Z2M+G6EdcK2778viFJZvLjS/ZQUReGZZ57hvfdU2T7++GMeeeSRCvernedQr1BMxspV9gRBqBgmkwvTRoVyJARMUdHUGz8Fg7nol9FBgwapCyeBy7evhv4NMRgM+n2vUUAjiYZzcpxKWdKwfTtQFMXuG0Nx+9vbrjFlyhRSUlL0z7lz5yoocfE82OlBnrzmSSuHwuLQFJ6EzARy8wusY6eTTwMwut1ovhnxDfX86hVqa0/RsnVefq//e8wcOJPhLYfr+9zY5EY+GvgRA5oOsBoL4FzKOcyKmYzcDH26iv/+g7Fj1eUpU6AC/kKlwWw288QTTzBr1iwAPvvsM5544gmH9C2RcIJQu/ALimTUSMj3cCfst/1M3ln0y2iXLl0ICQmBbLjT/05e6fUKU3tOBWDhLQv5YugXdAjvINFwTo5TVfwMCQnBZDIVsiLFxsYWsh5pRERE2N3fxcWF4OBgu23c3d1xd3d3jNCl4PGuj5dp/yDPIEwGE/lKPrHpsdT1qwsUKECv9H6F5sHN7baN8C45u23z4OaF2puMJp7u/jT1/eqz8eRGuwnftOVw1wC1MG5qKlx3HZSypEh5ycvL45FHHmHBggUYDAbmzZvHgw8+6LD+JRJOEGoX4d7hbAqHP164h2unz+f1bfDuTfZtByaTiYEDB/LNN9/QMK4hr/Z9Vf+uW71udKvXTe8TVEt6bn6unoNOcA6cyrLk5ubG1VdfzaZNm6y2b9q0iWuvvdZumx49ehTaf+PGjXTp0gXXKsjzUxkYDcZCJuGM3Awu5VwCio/Y0i1LJeQgKbG9hWXKUvGKSYtR677t3atm112yBFwqT2fPyMhg+PDhLFiwAKPRyMKFCx2qKIFEwglCbUO7l+3s15y/ejXD1QzjPvpVTYhrB81v6eeffy6yz2CvYEwGdRpeSp44HxVWlrKyshwhh8OYOHEi8+bN48svv+To0aM888wznD17lnGXQ9OnTJnCfffdp+8/btw4zpw5w8SJEzl69Chffvkl8+fPZ9KkSdV1CA7B1lFbU148XDzwc/crsp1dn6X00isD9tpb+U5t3gwffKAuL1gAlejvlZCQQL9+/Vi7di0eHh6sXLnS6to7CqkLJwi1i4LItRhmjWnFGX8IvpgERUzNDxgwAIPBwMGDBzl//rzdfYwGo55+RfyWnI9yKUtms5nXX3+dunXr4uPjw3//qVWZX375ZebPn+9QAcvKHXfcwcyZM3nttdfo2LEjv/zyC+vWraNhQ9UZOSoqyirnUuPGjVm3bh3bt2+nY8eOvP7663z88cfcdttt1XUIDsFWabHMAl6c/5a9efcyWZYu34TSc9P1orqaDEEZ0OPFywk/H3sMLid8qwxOnjzJ9ddfz+7duwkMDGTLli16rSdHUxZlUhCE6sfyPneKJEaPAMVogK++Uq3dNoSEhNCtmzrdtn79+pL7lYg4p6NcytIbb7zBwoULeffdd3Fzc9O3t2vXjnnz5jlMuPLy+OOPc/r0abKzs/nrr7/o1auX/t3ChQvZvn271f69e/dm3759ZGdnc+rUKd0KVZuxnQ4rbUSdveSSZXFgtiyqq40dkx4DCnyxCnwSUqFlS6jEPFbbtm3jmmuu4Z9//qFevXr89ttvRU7DOgKxLAlC7cLSTSE6LZqdDeHshMtW53Hj4PTpQm20qLjipuIkIs55KZeytGjRIubOncvo0aMxWeTFad++Pf/884/DhBPKj62jtp4FvASFR3vgx6XHkW/OR1GUMjkwGwyGQlOA0WnRPLQPRvwDeS5G+PZb8PIqx1GVzGeffcaAAQNITEzkmmuu4Y8//qB169aVMpaGRMMJQu3C0rKk3Ruzp06GHj3UwJN77oG8PKs2mt/Spk2byMnJKbFfwbkol7J04cIFmjVrVmi72WwuMpGjULXYOmrrliXv4hWeUO9QDBjIV/JJyEwgOSuZnHz1xqDNx5eE7RSgx39nmXXZcj3/9qZqFXAHk5WVxaOPPsrjjz+uF8fdvn07derUcfhYtkg0nCDULrQXutPJpwsCX/zrqolxfX1h5054802rNp07dyYsLIxLly4VmeRY6sM5L+VSltq0aWO3OO33339Pp0p4EAplp0ifpRKsHy5GFz0LdUxajN4uwCNAz1ZbElZTgDk5vDX/NN65sKUxzLne8SkXjh07Rrdu3Zg7dy4Gg4F33nmHr7/+Gk9PT4ePZUtufq6eO0p8lgShdqDdHzW/Sj3wpXHjgkS5r72mKk2XMRqN3HTTTQCsW7eu2H7FsuR8lEtZmjZtGhMmTGDGjBmYzWZWrFjBww8/zFtvvcUrr7ziaBmFcmBvKgzKFv6vzedb9lcaLKcA8195mY4X8kn0gPtuhagMx95EFi9ezNVXX83BgwcJDQ1l/fr1PP/888U6sTsSLUTYZDAR7GU/L5cgCDULyzB/sAl8GT1anYYzm9Vli3JWmrK0ceNGu/2KZcl5KZeyNHToUJYtW8a6deswGAy88sorHD16lNWrV9O/f39HyyiUg+Ki4Urb1nI+vyxTTJqy5b97P8Z31dIiDw+Di35qwrY8c15xzUtFfHw8d911F/feey/p6en07duXAwcOFKrzV9lo5zfMOwyjwanSlgmC02IZ5g927m+ffgqNGsGZM/DMM/rmG2+8EYADBw4QE1P4xU+i4ZyXct/dBw4cyI4dO0hLSyMjI4Pffvutyh9UQtFoCktyVjLZedllckK2fDsqj/NyuHc4fllw7webMSgK8zrBzq7hGA1GFBTi0iuWsG3FihW0adOGpUuXYjKZmDZtGps2bSKykmvL2UMi4QShdmJ5Tyt0f/PzU9MIGAzw5ZewahUAoaGhdO7cGYDNmzcX2adYlpyPcilLe/bs4Y8//ii0/Y8//mDv3r0VFkqoOIEegbga1QzkZbUQWb4d6c7LJTiG27afuR5CEzJJrx/B0zdBXb+6hHqF6vKUh/Pnz3PHHXdw2223ERsbS5s2bdi9ezfTp0+3isqsSiQSThBqJ5b3Qrv3t1694Nln1eWHH4Y49SVPMwrYm4rTXjSTspLIzst2sMRCdVIuZWn8+PF2i8deuHCB8ePHV1gooeIYDAb9AX4y8STpuelA6abhLP2dSptywJKWv//LA3+D2QDbp48h3V3ts7xvXVlZWbz11lu0aNGC7777DpPJxNSpU/nrr7/o0qVLmfpyNBIJJwi1E8t7YZH3t9dfhzZtIDYWHn0UFMVKWdKKrmsEeha8pMamx1aO4EK1UC5l6ciRI7op0pJOnTpx5MiRCgslOAbtAX4g5gAAXq5e+Lj5lLpddFq0np261MpAfDxXTVH9lGZe58KBq3z19mWdzzebzXz//fe0bduWF198kYyMDK677jr27NnDm2++WaXFjItC6sIJQu3EyrJU1P3NwwO+/lqtX/njj7B4Mddeey1eXl5ER0dz6NAhq90tfaEkIs65KJey5O7ubte5LSoqCpdKLIoqlA3tAa4pSyWVOtHb+dixLJVGGVAUeOwxXOISOBQKU3vncTzxuN6+tJEiiqLw008/0blzZ0aNGsXJkyeJjIxk8eLF/PrrrzUqPYVYlgShdmJlWSru/tapE0yfri5PmIB7bCy9e/cG7E/F2auPKdR+yqUs9e/fnylTppCSkqJvS05OZurUqRINV4PQfrQHYw5arZe2naWDd6naLl0KP/wALi6MG+VJtqv12CXlIMnNzWXp0qV07dqV4cOHc+DAAfz8/Jg2bRrHjh1j9OjRVZYSoLSIZUkQaielsixpPP88dOumphF44AEGXH7O2fVbsik1JTgH5TIDffDBB/Tq1YuGDRvqb/l///034eHhfP311w4VUCg/umUp+rJlqZR+R1q72PRYPRy+xLYXL4Lmr/byy0QHLIKkk1ZjaykDbJWlhIQEvvjiC2bPns2FCxcA8Pb25sknn2TSpEkEBQWVSu7qQKLhBKF2Umw0nC0uLrBoEXTsCFu2MOCaawD45ZdfyMrKwsOjIGGvbakpwTkol7JUt25dDh48yDfffMOBAwfw9PTkgQce4K677sLV1dXRMgrlRHuA5yv56nopI9pCvEIwGoyYFTNmxQyUUOpEUWDsWEhKgquvhilTiPh6IyeTThaM7RNBvlldXn5kOX+c/YP7PO/jwMYDrF69Wi+TEx4ezuOPP85jjz1GaGhouY67tGTmZjLk2yHc2ORGpvacWq4+JBpOEGonZbIsATRvDu+9BxMm0OrDD6kbHs6FmBh+++03Pf8SFC41JTgH5XYw8vb25pFHHnGkLIKD6Vq3q9X6NXWvKVU7k9FElzpd+PPCnwB0jOiIm8mt6Abz5sHPP4O7u/r25epKt7rd2HlOLRXg5epFm9A2+Bv9MfxrIPtoNiePnWRaxjS9i06dOvHUU09x5513Vpnj9u7zu9l2ehtH4o6US1nKzssmKSsJEMuSINQ2Ggc0JsQrhFCv0FIFvgDw2GPw008YNm1igNnMAtSpOEtlSXyWnJNyK0v//vsv27dvJzY2FrPZbPWdlDypGXSv152zT58lLiMOXzdfrgq+qtRtd4zZwZE4NbKxVUironc8dQomTlSX33wTWrcG4L0B73Ff+/s4dPAQ/+3/jwfvfJAtW7agZBaE2rr5u/HE2Ce47777aN++fdkPsIJoN7O4jDjyzfmYjGXL1aSFBrsaXQn0CHS4fIIgVB6erp4cf+J48S+CthiN6sth27b0j4vTlaV3331X38W21JTgHJRLWfriiy947LHHCAkJISIiwsrpVit/ItQM6vvXp75//TK383DxoHNk4fQQVmjTb2lp5F1/PaeGDGH/d9+xb98+/vrrL/bs2WMVBADQoEEDOvbuyCpW0aJbC94f/36ZZXMU2s3MrJhJyEwofqqxmPbhPqWLMhQEoWYR4BFQ9kYNGsB773HjuHGAWvokOjqaiAjVoiSWJeekXMrSG2+8wZtvvsnzzz/vaHmEGkp2djZRUVFcuHCh4PPzz/y3dSvHDAZO/PEHOa0KW6D8/Pzo1asXffr0oV+/fnTo0IEDMQdY9X+riMms3jcvy5tZdFp0mZUliYQThCuURx4h9Lvv6Lx1K/uAzRs3cs999wESDeeslEtZSkpKYuTIkY6WRSiCH3/8kdjYWBRFqfAH1DxGeXl5ZGVlkZ2dTVZWltVyRkYGycnJVp+MjIyiBVQUyM3F3d2ddu3acfXVV+ufDh06FCpFor15xWfEl2v6y1FYmslj0mKgjDqPRMIJwhWKwQDz5jGgRQv25eaycfZsXVnS7gcp2Slk5WXh4eJRXE9CLaFcytLIkSPZuHEj4y6bIYXK5a233qoRNffc3NyoW7eu+jl5krpRUTRo2JCWc+bQonVrGjRogNFYcuquUK9QPdouLiOu2pQNyze/8pjMxbIkCFcwjRsz4LHHeOfjj9m0dy/Kf/9haNIEf3d/3Exu5OTnEJMWQ8OAhtUtqeAAyqUsNWvWjJdffpndu3fTrl27QukCnnzySYcIJ6j06dOHunXrYjAYHPZxcXHBw8MDd3d3PDw8rJY9PT0JCAggICCAwMBAfTkgIED1zfnmG7jnHnBzg7Vr1dpJZcBkNBHiFUJseiwxaTHVpixZKkjlccaU7N2CcGVz7Tvv4DV7NtFmM4fuuot2u3djMBiI8IngbMpZotOiRVlyEsqlLM2dOxcfHx927NjBjh07rL4zGAyiLDmY9957r7pFKCAmBrTr+/LLZVaUNCJ8IohNjyU6LZoOdHCggKXHUkGqkGVJciwJwhWJu6cnvXv25OcdO9j455+0mzsXHn2UcO9wzqaclYg4J6JcytKpU6ccLYdQW3jiCUhMVDPZVsDBv7rDa82K2WoaTixLgiCUhwHDh6vKEvDsc8/BoEFlLhou1HzKVRtOuEJZsQK+/x5MJvjyS6hAtvbqDq9NzEzUs4uXVw7xWRIEYcCAAQD8YjSSdekSPPII4V5qZK2kD3Aeyp2U8vz586xatYqzZ8+Sk5Nj9d2HH35YYcGEGkZiIjz+uLr8/PNqJe4KoFuWqunNy/YmVh45JBpOEIRWrVoRGRlJVFQUv7u6csOGDQzsdAvzPCQxpTNRLmVpy5YtDBs2jMaNG3Ps2DHatm3L6dOnURSFzp1LSGQo1E4mTlT9lVq2VH2VKohuWUqvnjcvTdExGUzkK/llfgPMyssiJVtNuCk+S4Jw5WIwGOjXrx+LFy9my7XXcsOOHQz5fAvBD4tlyZko1zTclClTePbZZzl06BAeHh4sX76cc+fO0bt3b8m/5Ixs2gRffaXmFvnyS/CoeN6Q6k7cpt3EWoWqiTTjM+LJM+eVur0mt7vJHX93f8cLKAhCraFfv34AbMnMhLZt8UxO4/2NYllyJsqlLB09epT7778fABcXFzIzM/Hx8eG1115jxowZDhVQqGYyM9XikQATJkCPHg7ptrp9lrSbWJvQNhgNRhQU4tLjSt3eMhJOSp0IwpWNpizt2buXlI8+QjEYGHMAmvz1XzVLJjiKcilL3t7eZGdnA1CnTh1OnjypfxcfH+8YyYSaweuvw8mTULcuvPGGw7qt7mg4Tdmp41uHUK/QMssikXCCIGjUr1+fq666CrPZzI6MDFIeGg3A9CVRUFz1A6HWUC5lqXv37uzcuROAIUOG8Oyzz/Lmm2/y4IMP0r17d4cKKFQjhw6BluNp9mzw83NY15YlT3Lzcx3Wb2nRi+B6h5fLyiWRcIIgWKJPxW3ZgvGttznvC40TFXKnS2F5Z6BcytKHH35It27dAJg+fTr9+/dn2bJlNGzYkPnz5ztUQKGaMJvh0UchLw+GD1c/DiTYKxiTQa0JF5dR+ukvR6EpOxE+EeXyn5JIOEEQLLFUlnxD6jJxmBsALh/OhIMHq1EywRGUKxquSZMm+rKXlxdz5sxxmEBCDWHuXPj9d/DxgU8+cXj3RoORMO8wotKiiE6Lpo5vHYePURyWyo5YlgRBqCh9+/bFYDBw+PBhoqOj2dO1Dj/sO83tR/Nh7FjYtUvNUSfUSsplWWrSpAkJCQmFticnJ1spUkItJSoKXnhBXX7zTahXr1KG0Sw6209v51zKObv75JvzOZl40u535SErL4vfz/3OhUsXdBlK8p86n3qe387+xh/n/9Aj5sRnSRAES4KDg+nYsSMAW7duJcIngicHQa6PF+zZA59+Wr0CChWiXMrS6dOnyc/PL7Q9OzubCxcuVFgooZp56ilISYGuXWH8+EobRlM0nt34LE0/bsqF1MJ/O1O3TKXZJ81Y8+8ah4w56vtRXPfldcRnxOsyFGdZiroURZNZTei5oCfd53dn0sZJVvtKjiVBEDQsp+LCvcOJ8oNdE25Rv5w6Fc6erUbphIpQpmm4VatW6csbNmzA378gv0x+fj5btmyhUaNGDhNOqAbWri0oaTJ3bqWajR/v8jjnU89zPOE42fnZHI47TF2/ulb77IveB8D+qP3c3PzmCo+5L0rtr6F/QwY0HUC4d/GWpcNxh8k1Fzig74/eb7WvWJYEQdDo168f77//Plu2bOGmW28CYFv/q+j163Wwc6dahHzlyuoVUigXZVKWhl928jUYDHqeJQ1XV1caNWrEBx984DDhhComPb2gpMkzz6jFciuRoS2GMrTFUAZ8PYBN/22ya9nRtjkiH5NZMetKzs4Hd+qKWXGWJW2bq9GVXHNuIXnEZ0kQBI2ePXvi6urK2bNncUtRHbyjM2Lh//5PvZ/+9BOsXg1Dh1avoEKZKdM0nNlsxmw206BBA2JjY/V1s9lMdnY2x44d4+abK/72L1QT06apZuKGDWH69CobtrhoNG2bI/IxJWUm6T5HYd5hZRq/Q0QHfT09J520nDRALEuCIBTg7e2tp89JOKz69cakx0CbNmrJKFCtS5J7qdZRLp+lU6dOERISYrUtOTnZEfII1cX//gczZ6rLc+aAt3eVDa1ZZ2wtO3nmPN23yBGWJa2PIM8gXE2u+nZN4UnITCiU80lr0z6sPQAp2SmcSTkDgKeLJz5uPhWWSxAE5+HGG28E4PT+04DFvevll6F+fTh9Gt56q3qEE8pNuZSlGTNmsGzZMn195MiRBAUFUbduXQ4cOOAw4YQqQlFUR+78fBgxAgYPrtLhNWXF1noUlx6HgmL3u/JQlJ9RkGeQnvMpNj3WbpsWIS1wN7kDcCD6gN6PlDoRBMESzcn7yJ9HwGxx7/LxgVmz1OV334Vjx6pJQqE8lEtZ+r//+z/q168PwKZNm9i8eTPr169n0KBBPPfccw4VsLScPn2ahx56iMaNG+Pp6UnTpk2ZNm0aOTk5xbYbM2YMBoPB6nPFZSFfvBh+/RW8vOCjj6p8+KIsS5brjrQs2foZGQ1GfSquKBksk1ceiFGVJYmEEwTBlmuuuQYfHx9SklIgxuaeMny4+jKam6u+oCpKtckplI1yKUtRUVG6srRmzRpGjRrFgAEDmDx5Mnv27HGogKXln3/+wWw283//938cPnyYjz76iM8//5ypU6eW2Pamm24iKipK/6xbt64KJK4hJCfDJDUcnpdfhgYNqlyEoixLlutpOWmk56RXaJzism4XFRFnryyKpiyJv5IgCLa4urrSq1cvdeUUZORm6D6OGAxqkl8PD9iyBSxmaISaTbmUpcDAQM6dU5MIrl+/Xp+jVRTFbv6lquCmm25iwYIFDBgwgCZNmjBs2DAmTZrEihUrSmzr7u5ORESE/gkKCqoCiWsIr7wCsbHQsmWBA2IVU5JVR6OiU3HFRbAVFRFnZVm63E6bhpNIOEEQ7KFNxZlOq9P7VveVJk3gxRfV5WeeUXPaCTWecilLI0aM4O6776Z///4kJCQwaNAgAP7++2+aNWvmUAErQkpKSqkUn+3btxMWFkbz5s15+OGHiY2NLbGNU7B/f0FW2dmzwc2tWsTQHawzEvRoNSgcnVaW2m32KC43kr2IuHxzvu5gHu5TYFmKSosqsh9BEARNWTKfNkOenXvXc8/BVVdBdLT6wirUeMqlLH300UdMmDCB1q1bs2nTJnx81IigqKgoHtfy9FQzJ0+e5JNPPmHcuHHF7jdo0CC++eYbtm7dygcffMCePXu44YYbyM7OLrJNdnY2qampVp9ah9ms5lQym+GOO+Dyj7s6CPYMxmgwoqAQl15QVLckS1NZKS7rdoR3YctSXEYcZsWM0WAk1Cu0kCVJLEuCINijXbt2hISEoOQocMHOvcvd3fpFdf/+qhdSKBPlUpZcXV2ZNGkSs2bNolOnTvr2p59+mrFjxzpMOIDp06cXcsC2/ezdu9eqzcWLF7npppsYOXJkifLccccdDBkyhLZt2zJ06FB+/vln/v33X9auXVtkm7fffht/f3/9o/lv1SoWLoTdu9UIjWpOJGoymvS8R1ZO3emVMw1XrGXJYgztbTDEKwST0VSonViWBEGwh9Fo5IYbblBXThVx7+rfX31RNZvhscfU/4UaS6kzeK9atYpBgwbh6upqVfbEHsOGDauwYBoTJkzgzjvvLHYfyxIrFy9epG/fvvTo0YO5c+eWebzIyEgaNmzI8ePHi9xnypQpTLTw70lNTa1dClNiIjz/vLo8fTrUrVvs7lVBuHc40WnRdpUVXzdfLuVcqrBlydJZ2xZ7Pku2ypWtRUqi4QRBKIp+/frx3XffwX/FWMU//BDWrYM//oAFC+Chh6pWSKHUlFpZGj58ONHR0YSFhellT+xhMBgc6uQdEhJSKAFmUVy4cIG+ffty9dVXs2DBAozGshvOEhISOHfuHJGRkUXu4+7ujru7e5n7rjFMnQrx8WpW2SefrG5pAFUhORBzwK6y0iGiA7+d/a1CPkv55nw9h1Jpo+FslSuxLAmCUFo0vyXOw/n48/Z3qlNHfWF99lmYMgVuuw0CAqpKRKEMlFqbMJvNhIWF6ctFfaorGu7ixYv06dOH+vXr8/777xMXF0d0dDTR0dYafcuWLfnxxx8BSEtLY9KkSezatYvTp0+zfft2hg4dSkhICLfeemt1HEbls2ePWiAX1Ezdrq7F719F2HOw1pSVDuFqqRHbabmykJCZgFkxY8BAqHdooe9LZVkSnyVBEEpJkyZNCIoMAjMc/eto0Ts+8QS0agVxcfDqq1UnoFAmymx6MZvNfPnll9x88820bduWdu3accstt7Bo0SKUakywtXHjRk6cOMHWrVupV68ekZGR+seSY8eOkXI5VNNkMvG///2PW265hebNm3P//ffTvHlzdu3aha+vb3UcRuViNsOECWoitHvuAS0XSA3A1sE6Jz+HxMxEoEBZqohlSes3xCsEF2Nhg6qmrCVnJZOdl201nj3Lko+bD95uVVcSRhCE2oXBYKBTD9Wn98z+M0Xv6OpakNn7k0/g8OEqkE4oK2VSlhRFYdiwYYwdO5YLFy7Qrl072rRpw+nTpxkzZky1WmPGjBmDoih2P5YoisKYMWMA8PT0ZMOGDcTGxpKTk8OZM2dYuHBh7fI/Kgtffw1//qk6db/7bnVLY4Wtg7U2ZeZidKFVaCugYtFwuuJThJ9RoEcgrkZXKxk0S5amJPm4+eDp4qn2I1YlQRBKoGefnkBBUd0i6d9fze6dnw9PPSWZvWsgZVKWFi5cyC+//MKWLVvYv38/S5YsYenSpRw4cIDNmzezdetWFi1aVFmyChXh0iV44QV1+eWXoRifrOrAdhrMMoFkpI8qa0x6TLmtl8VFwoH6Fmg7FWirYBkMhiKdvQVBEGwZ1F/NQZhzMYf4+Pjid/7wQzWlwJYtcNlVRKg5lElZWrJkCVOnTqVv376Fvrvhhht44YUX+OabbxwmnOBA3nhDTYDWrJn65lLDsHWwtlRUNMXEqmxAGSkuEk6jKIXNUsHSZBHnbkEQSqJt07YQCiiwbmMJZbQaN4bJk9XliRMhI6PS5RNKT6mj4QAOHjzIu8VM3wwaNIiPP/64wkIJDub48YICudrbSw1DU0JOJZ1i/NrxHEtQK3KHe4er/kGu3qTnphOTHoOve/H+ZN8d/o4dp3dYbdt9YbfeX5Ey2CpsdhQsbVmm4QRBKAkvVy9cm7mSG5fLuo3ruO/u+4pv8MILag68M2fgvfdg2rQqkVMomTIpS4mJiYSHF/OwCQ8nKSmpwkIJDubZZ9Uq1wMHws03V7c0dqnnVw8XowuZeZnM2TtH394ksAmgKlP/Jf1HdFo0zYKKLqmTnpPO6BWjrcqmWKL1Zw9Ly1Jufq5e6sTSiqS1L64fQRAEjZC2IUTtiuK3Hb+VvLOXl5okeNQoeOcduP9+sMgjKFQfZVKW8vPzcXEpuonJZCIvz/5DSqgmNmyA1avBxQVmzlSrXtdAAjwC+OnOn/jzwp/6Ng8XD8Z0HAOoCst/Sf+VGBEXnRZNnjkPDxcPnr/u+UJjaP3ZQ7cspcUQl6GWXTEZTAR7Bev7vHD9CzQPbs7d7e4uy+EJgnCF0rBDQ6IMUVw4fYGzZ8/SoEGD4hvcfjv06QPbt8OkSfDDD1UhplACZVKWtEiyohIyFldPTagGcnPh6afV5SeegJYtq1Wckhh81WAGXzXY7neaIlNSRJz2fV3fukzvM71M42tTgdHp0Xo/Yd5hGA0Frn1h3mGM61J8vUFBEASNemH1oA5wAbZs2cIDDzxQfAODAT7+GDp1guXLVYfvaqzdKaiUycH7/vvvJywszKoumuUnLCyM++4rYU5WqDpmz4Z//oHQ0Fpf2VqbCiupPlxxxXJLPUZaTImpBgRBEEpDuHc4XJ6137JlS+katWunFjoHtcqCzNhUO2WyLC1YsKCy5BAcTWysmkYf4K23an0K/dJaljRlqjzRapZjlJRqQBAEoTRE+ERAY+BXVVlSFAVDadwhXn0Vvv0WjhxRqy5oypNQLZS9eJpQO3jxRUhNhc6doSSzby2gzJalckSrWY5RmlQDgiAIJRHuHQ71wehqJDo6mqNHiyl9YklgYEH5k1degeTkSpNRKBlRlpyRfftg/nx1+eOPwWSqXnkcgL3acfbQvi+XZenyGKnZqZxKOlXufgRBEDQifCLAFXya+QCwefPm0jd+9FG1blxCgporT6g2RFlyNhQFnnlG/f+uu+C666pbIodgr9CtPbQSJeWxCPm7++NuUoMXDsYeLHc/giAIGnoFgKbq1FuZlCUXFzWVAKgvvidOOFo8oZSIsuRs/PQT/PILeHjAjBnVLY3DsEwYWVzJk4pYlixLnhyMOVjufgRBEDS0e0h6/XQAtm/fTm5ubuk7GDRIzZGXmwvPP1/y/kKlIMqSM5GTY50u34kKAmtKTFZeFqnZqUXuV5FoOCi4sWXkZlSoH0EQBFDTjQDkheYRGBTIpUuX2LNnT9k6+eADMBphxQrYsaPk/QWHI8qSM/HZZ2ppk7CwgqK5ToKXqxe+bmqZk6KcvBVFqbBjtm07sSwJglARPFw8CPAIACNcc/01QBmn4gDatIFHHlGXJ04Es9mxQgolIsqSs5CUBK+9pi6//jr4Fl8/rTZSkt9SanYqWXlZQMUtS0WtC4IglBXtJaxNtzZAOZQlUO/vfn5qAM+iRY4UTygFoiw5C2+8AYmJ0LYtPPhgdUtTKZQUEadZlXzdfPFy9SrfGBaWJVejK4EegeXqRxAEQUN76WrQSS11smvXLtLS0srWSWgovPSSujx1KpS1vVAhRFlyBk6cgE8+UZfff1+NoHBCSrIsOSKRpGXbcJ/w0iWPEwRBKAbd0h0IjRs3Ji8vj19++aXsHT35JDRpAlFR8N57jhVSKBZRlpyBF15QIyUGDlQ/ToplRJw9HFGixLKtpA0QBMERRHgXvOjdeOONQDmn4tzd4d131eX33oPz5x0lolACoizVdn79VS22aDSqViUnpqotS+KvJAiCI9BdCNJjKqYsAYwYAb16QWYmTJniKBGFEhBlqTZjNsOzz6rLY8eq/kpOTImWJQeUKLFsK5YlQRAcgeWL3g033ADA//73P6Kji0+yaxeDAT78UF1evFh1+BYqHVGWajNLl8KePeDjUxAJ58SIZUkQhNqI5YteSEgInTp1AmDr1q3l6/Dqq2H0aHV58mS1YoNQqYiyVFvJzCzIpTRlCoQ7vxWktNFwFbEI+bj54OniaTWeIAhCRbB90avwVByoEdBubrBlC2zcWGEZheIRZam2MmsWnDunZul+5pnqlqZK0G4451LPETQjqNDn5+M/W+1XHgwGg95eLEuCIDgC7cUrNj0Ws2LWlaXVP6+m9aetORB9oOydNmoEEyaoy5MnQ36+g6QV7CHKUm0kIQHeeUddfvNN8PSsXnmqiDq+dWgc0BiApKykQp98JR8PFw86RXaq0DjXN7geN5MbV0de7QixBUG4wtFLnpjzSMxM5Prrr8fNzY346HiO/nOUn479VL6OX3wRAgLg4EHVf0moNJwzIY+z89ZbkJICHToUzFtfAbgYXTj8+GHOpJwpcp8Inwi1tEAFWDh8IZ8M+gR/D/8K9SMIggDgZnIjyDOIxMxEYtJiaBPWhuuuu45t27bBf0W7FpRIUJCaoHLyZDVh5ahRV8zLc1UjlqXaxpkzMHu2ujxjhpoy4ArC09WTliEti/xUVFECMBqMoigJguBQNF9KW78l/is6wrdUPPEENGig5lzSkhMLDufKetI6A6+8Ajk5cMMNMGBAdUsjCIIglALNB1JTjHRl6RREpUaVv2MPD7UeKKizDgkJFRFTKAJRlmoTBw7A11+ry++8o+bbEARBEGo8mpO3Zllq0a4FuAPZcPafsxXrfPRo1S0jJUX1YxUcjihLtYkpU9R8GqNGQdeu1S2NIAiCUEq0kieaf1J8Zjyo8SrE/i+2Yp2bTAVlUGbPhlOnKtafUAhRlmoL27bBzz+rRXLlzUEQBKFWoVuW0lXLUkx6DDRRv8v5N4f0nPSKDTBgAPTvr9YJfemlivUlFEKUpdqAosDzz6vLjz4KzZpVrzyCIAhCmdB9li5blqLTonVliXNwOu50xQeZMUP9/9tv4a+/Kt6foCPKUm3ghx/Usibe3vDyy9UtjSAIglBGbKPhYtJiIBjwA/Jh0/ZNFR+kUye45x51efLkivcn6IiyVNPJzVXzaABMmnRFlDURBEFwNmyj4aLTosGAbl3asXWHYwbSyqBs3QoVKaciWCHKUk1n3jw4cQLCwuDZZ6tbGkEQBKEcaD5Lcelx5JvzC3IrXVaW/trpoGmzhg1h3Dh1eepUKbLrIERZqsmkpcGrr6rLr7wCvr7VK48gCIJQLkK9QjFgIF/JJyEzQZ+OMzUzAXDu2Dni4+MdM9iLL6puG3v2wI8/OqbPKxxRlmoyH34IMTHQtCk8/HB1SyMIgiCUE1eTK8FewYDqr6RZllo3ag1q6Tg2O2raLCysoMD6Sy9JkV0HIMpSTSY2Vk08+eab6hy0IAiCUGvR/Jai06J1y1KHiA5wOcB5w4YNjhts0iS1dtzRo1Jk1wGIslSTmT0bDh+GkSOrWxJBEAShglhGxGkpBDqEd4Cm6vcbNmxAcZSPkb8/vPCCujxtGmRnO6bfKxSnUpYaNWqEwWCw+ryg/bEUgaIoTJ8+nTp16uDp6UmfPn04fPhwFUlcClq1uuKK5QqCIDgjmmXpROIJMvMygcvKUgMwuBmIiorif//7n+MGHD8eIiPVAuxz5zqu3ysQp3sKv/baa0RFRemfl0rIZPruu+/y4YcfMnv2bPbs2UNERAT9+/fn0qVLVSSxIAiCcCWgWZYOxBwAwMfNhyaBTcAVDI3UWp/r16933IBeXmpwEKgpBdLSHNf3FYbTKUu+vr5EREToHx8fnyL3VRSFmTNn8uKLLzJixAjatm3LV199RUZGBt9++20VSi0IgiA4O5plSVOWInwi9JQC5qZmwMHKEsBDD6lBQrGxMGuWY/u+gnA6ZWnGjBkEBwfTsWNH3nzzTXJycorc99SpU0RHRzNgwAB9m7u7O7179+b333+vCnEFQRCEKwRNMTqdfFpd9w7Hx80Hb1dv3cn7t99+Iy0tDUVR+F/M/8jJL/oZVipcXeG119Tl996DxMSK9XeF4lTK0lNPPcXSpUvZtm0bEyZMYObMmTz++ONF7h8drUYjhNtkxQ4PD9e/s0d2djapqalWH0EQBEEojkifSKt1zdIU6RsJQUAA5ObmsnHzRlYdW0X7z9vzwubi/W5LxZ13Qvv2kJJSUD9OKBM1XlmaPn16Iadt28/evXsBeOaZZ+jduzft27dn7NixfP7558yfP5+EhIRixzAYDFbriqIU2mbJ22+/jb+/v/6pX79+xQ9UEARBcGp6NuzJrS1vpWNER7rV7cb4ruMBmHL9FDrX6QxXqfutXLtSn6o7GHOw4gMbjWoKGoBPPoGLFyve5xWGQXFYnGLlEB8fX2JW00aNGuHh4VFo+4ULF6hXrx67d++mW7duhb7/77//aNq0Kfv27aNTp0769ltuuYWAgAC++uoru+NlZ2eTbRGGmZqaSv369UlJScHPz6+0hyYIgiAIOmEPhRH3ZRx1G9Zl2KfD+GzvZ7QJbcOhxw9VvHNFgeuug1274LHHYM6civfpBKSmpuLv71/i89ulCmUqFyEhIYSEhJSr7f79+wGIjIy0+33jxo2JiIhg06ZNurKUk5PDjh07mFGMqdLd3R13d/dyySQIgiAI9mjYqSFxxjgunLnAyZMngYLCuxXGYIC334Y+feCLL2DyZGjUyDF9XwHU+Gm40rJr1y4++ugj/v77b06dOsV3333Ho48+yrBhw2jQoIG+X8uWLfnxcq0cg8HA008/zVtvvcWPP/7IoUOHGDNmDF5eXtx9993VdSiCIAjCFUidkDpw+XF1/M/jAMRnxJObn+uYAXr3hv79IS9PTSUglBqnUZbc3d1ZtmwZffr0oXXr1rzyyis8/PDDLFmyxGq/Y8eOkZKSoq9PnjyZp59+mscff5wuXbpw4cIFNm7ciK8UrRUEQRCqkAjvCD2bd8z+AotSXEac4wbRirMvXAgnTjiuXyenxvss1QZKO+cpCIIgCEXxyrZXeH3Z6/B/gCswWf3/r0f+onNkZ8cNNHgw/Pwz3HcfFOGbe6VQ2ue301iWBEEQBKE2E+ETARHgHugOucBpdbtWR85haNalxYvh2DHH9u2kiLIkCIIgCDWAcO9wMICxxeVH87/qf9FpRef9Kxddu8LQoWA2FySsFIpFlCVBEARBqAFoSSozG6tFdvkXUBwYEWeJZl1asgSOHHF8/06GKEuCIAiCUAPQyqHQGDWxTwoQWwmWJYBOnWDECDX/0vTpju/fyRBlSRAEQRBqAJplCTdUhQngWCVZlqBASfr+ezjogEzhTowoS4IgCIJQA/Bx88HL1UtdaXF547+VZFkCaNcORo1Sl8W6VCyiLAmCIAhCDUG3Ll2uE8d5uBB1ofIGnD5dze7944+wb1/ljVPLEWVJEARBEGoI4d6X/Zb8IaSJWurrwr5KVJZatQKtYoVYl4pElCVBEARBqCHoliWge9/uAGQcziA7L7uoJhXnlVfAaITVq+HPPytvnFqMKEuCIAiCUEPQLUvAsGHD1IWTcD7pfOUN2rw53HuvujxtWuWNU4sRZUkQBEEQagh6+gCg73V9MfoYIRs2bttYuQO//DKYTLB+PezeXblj1UJEWRIEQRCEGoLlNFykbyQB7QIAWLdmXeUO3LSpWisO4PXXK3esWogoS4IgCIJQQ9Cm4XzcfPB286bxtWrCpZ0bdmI2mx06lqIofPD7B6z5d426YepU1Xdp3TrYu9dq38zcTF7e+jJ7L+6105PzI8qSIAiCINQQmgU1A6BpYFMAWndrDW6QFJvEnj17HDrWP/H/MGnTJMauGnt58GYwerS6bGNdWnVsFW/8+gZTtkxxqAy1BVGWBEEQBKGG0C68HStGreDb274FoF5QPT3n0vLlyx061rnUc4CaITwnP0fd+OKLat6lVatg//5C+55P/f/27js8imp94Ph308smAQJpEEiACyH00APXEJAmIE1BEAVBlN70B1cRDCJVKXIV5CICKgoiRUA6JKFJiyAlkZbQU6gJ6W1+f6w7ZEmyCZDu+3meedidOXPmzMmSvPvOmTOFONC8BJNgSQghhChBetXphXclb+Dvy3K6l2zYsAFFUQrsOFlnBo9JiNG9qF0bXntN9/rTT7OVLbTZxEs4CZaEEEKIEspF6wI1wcTchPDwcP78888Cqzs6PjrH12p2aeNGOHtWt/3v59M9TH5YuHM+lVASLAkhhBAllLPWGSzBpo7umXEbN24ssLqzZokMMkZ168Irr+he/51dyrq90B7sW4JJsCSEEEKUUPqpBDJqZwAFO24pa9CTLQD66CPdv+vXQ2ho7lmofwgJloQQQogSSj+VQFL1JMzNzQkNDeWvv/4qkLpzzSwBNGgAvXqBosDMmcbL/gNIsCSEEEKUUOWsymFhagHW4PuCL1Bw2SWDzFJO2aKpUwFQ1q6l/PU7Oe73TyHBkhBCCFFCaTQaNbvUulNroOCCJYNsUUIO2aLGjaF7dzSZmXxwMOf9/ikkWBJCCCFKMP24Ja82XpiYmHDq1CkuXbr0XHWmZaRxL/Ge+j7XcUh/Z5cGnoHq9/MoW4ZJsCSEEEKUYPqH6yZbJNOhQwcA1qxZ81x13km8g8LjOZtyzRY1a8Ydv6aYKajZpRyzUGWcBEtCCCFECeZiq8ssRcVH8cYbbwDwww8/PNcElU9mh4yNQzrxVmcABv0J1R5IZkkIIYQQJYw+sxSdEE3Pnj2xtbXlypUr/P77789cpz6TVM2hGqCbbDI5PTnHsmdraNldHcwzYdJhGeAthBBCiBJGP2YpKj4KW1tbevfuDcD333//zHXqA57aFWvr7rYjyyNPcig78wXd6yGnIDPy9jMft7SSYEkIIYQowfR3w+kDHP2luHXr1pGamvpMdeozSy5aF7X+3MYtRcVHcaAaXPeuglUGvB0cT1Ja0jMdt7SSYEkIIYQowbJmlgDatWuHm5sbDx48YPv27c9Up37ckYuti1p/bmORohOiQQNXhvcFYOQJuHPr4jMdt7SSYEkIIYQowdQxS38HM6ampgwYMAB49ktx+jvanLXOav3GMksAGS91JtTNHLtU0Hy55JmOW1pJsCSEEEKUYPrMz6PURySmJQIwcOBAALZt28aDBw+euk41s6R1Ue+2y23gtlrWzpUfu1YFwGnFT5CQ8NTHLa0kWBJCCCFKMDsLO6zMrIDHgUvDhg2pX78+qamprF+//qnr1GeLnG2NZ5bSMtK4l3RPLXvuBS8ulwfLh49g+fJnOp/SSIIlIYQQogTTaDRqdqn2l7Xp8H0HMpVMdaD3t99+m22fxLREfJb5MHbHWAC2XtiKxyIPDlw7ADzOIrlos4xZyiGzpL9DzlRjiqONI04Obsxto9t28+MJ2AdYYvWplbrYzrJlwe8LCvDsSwYJloQQQogSrp1HOwDSMtPYG76Xm3E3efPNNzEzM+PYsWP8+eefBuX/iPyDU1GnWP3nagA2hG3gWuw1tlzYQmpGKveTdM8ucdY6G70bTh9AOdk6YaIxoa1HW75rCLfsoEoc9P0jlZSMFHVJTEvkx7M/Flo/FBcJloQQQogS7puXv+HGhBu42bkBustxzs7O9OrVC4Bly5YZlNdfrotLiSMpLUkNeqITotVskZmJGRWsKxi9Gy7rFAMAA+oP4NYHd7D5j+6ZcUvPe3BtTDjXxl9ja/+t6jHKGgmWhBBCiBJOo9FQxb6KGizpg5jhw4cDusefxMfHq+WzZomiE6LV91HxUeprfbbI2JglfQClLwNQ0aYi5cdOAkdHzMOvUnXPcao6VKWhc0N1n+d5FEtJZFbcDfgnycjIIC0trbibIUSBsLCwwMREvm8JUZSenKDS39+ff/3rX1y6dIk1a9bw7rvvGmwHXfCiD3qyvtZni/R16u+2szG3Ufd9MrOk0mph7Fj4+GOYNQv69sXJ1gnQXSp8kPyACtYVCvTci5MES0VAURSioqJ4+PBhcTdFiAJjYmKCp6cnFhYWxd0UIf4xnpygUqPRMGLECCZOnMjixYt555130Gg0Blmi249uq5fesmaW9EGSvaU9VmZWJKcnEx0fjWd5T3VffdClL2tg9Gj47DM4cwa2b8eya1fKW5XnQfIDouKjJFgST0cfKDk5OWFjY4NGoynuJgnxXDIzM7l9+zaRkZFUrVpVPtNCFBE1s5RlfNGQIUOYNm0aoaGh7N69m06dOhlklkLvhJKhZABwN/Eutx7dAh4HXhqNBmdbZ67FXiM6wTBYyjWzBFChAowYoQuYZs2Cl17CWevMg+QHRMdH413Ju2BPvhiVmWApKCgIf3//HLcdP36cZs2a5bht8ODBrF692mBdixYtOHr0aIG0KyMjQw2UHB0dC6ROIUqCSpUqcfv2bdLT0zE3Ny/u5gjxj6BmlhIeZ44cHBwYOnQoX3zxBYsWLaJTp04GmaU/ox/fKaegcC7mHGCYLXLRunAt9lq2cUtGM0sAEybA4sVw5AgcPIiL1oW/7v6V62zgpVWZGXDg6+tLZGSkwfL222/j4eFB06ZNje7buXNng/2e9Vk7OdGPUbKxscmjpBCli/7yW0ZGRjG3RIh/jicffaI3duxYNBoNO3fu5Pz58wbbswZLWd9nzRblVq/RzBKAqyu89Zbu9axZ2cZUlRVlJliysLDAxcVFXRwdHdmyZQtDhgzJ8xKBpaWlwb4VKhT8dVa5TCHKGvlMC1H0nhyzpFe9enV1GoHZs2cbbL9075JBWf37rHe46R95ki2zlMPdcNn83/+BiQns2kXjO6bZ6rl79y5bt25l8eLF/PTTT6Vy/G6ZCZaetGXLFu7evcvgwYPzLBsUFISTkxO1atVi2LBhxMTEGC2fkpJCXFycwSKEEEIUNmOZmylTpgDw008/kRKToq5XMLyNX/8+x8xSlnpT0lN4kPwgW9lsqleHV18FoOvmULWe6Ohoxo8fT9WqVXn55ZcZN24cAwYMoFKlSkycOLFUZaXLbLC0YsUKOnXqhLu7u9FyXbp0Yc2aNezfv5/58+dz4sQJ2rVrR0pKSq77zJ49GwcHB3XJ6xji6Xl4eLBo0aLibkaBuHr1KhqNhtOnTwO64Fyj0ZTKb1dCiOKlD1r0k01m5ePjw0svvURmZiYcyruuJ8csgWFGSH8HnbmJOeWtyhuv7P/+DwDvfWeo+hDO/X6O+vXr88UXX5CUlISXlxe9e/fG29ub9PR0Fi5cSJ8+fUhMTMy7oSVAiQ+WAgIC0Gg0RpeTJ08a7HPz5k127drF0KFD86y/X79+dO3alXr16tG9e3d27NjBxYsX+e2333Ld54MPPiA2NlZdbty48dznWdIMHjxY7V8zMzOqVq3KiBEjnunp1qXFzp07dbfcRhmmoV1cXLIFxDdv3kSj0bB79+4iaVvWn4e5uTnOzs506NCBb7/9VveL8SmsWrWKcuXKFU5DhRCFyt7SHktTSyDn7NJHH32ke/EnmD0wvIfL3MTwRgyDzFIOGSt1igGtc96X3Zs0gfbtMcnIpMFGODnvJHfu3KF+/frs2bOH0NBQNmzYwPnz51m/fj2Wlpb8+uuvjBo1Kn8nXsxKfLA0evRowsLCjC716tUz2GflypU4Ojry8ssvP/XxXF1dqVatGpcuXcq1jKWlJfb29gZLWaQf+H716lW++eYbtm7dysiRI4u7WYWmTZs2mJmZERQUpK4LCwsjOTmZuLg4Ll++rK4PDAzE3Nyc1q1bF1n7sv48duzYgb+/P+PGjaNbt26kp6cXWTuEEMUn60N1c7rjrFWrVjRq0wgywfagrcG2rLfyW5haUM6qnPo+pzrzvBPuCcr77zMJ2HYdUOCdd97h2LFjvPjiiwbB1iuvvMJvv/2GRqNh1apV7N+/P1/1F6cSHyxVrFgRLy8vo4uVlZVaXlEUVq5cyZtvvvlMtzPfu3ePGzdu4OrqWpCnoVIUhYTUhGJZnnb6ef3A9ypVqtCxY0f69etnkEnJyMhg6NCheHp6Ym1tTe3atfniiy8M6hg8eDA9e/bk888/x9XVFUdHR0aNGmUwk3lMTAzdu3fH2toaT09P1qxZk60t169fp0ePHmi1Wuzt7enbty/R0Y+/AQUEBNCoUSO+/fZbqlatilarZcSIEWRkZDBv3jxcXFxwcnJi5syZuZ6vVqulWbNmBsFSUFAQbdq0oU2bNtnWN2/eHFtbW3bu3EmbNm0oV64cjo6OdOvWjStXruS7n5OSkujatSstW7bk/v37uZbT/zwqV66Mj48PH374Ib/++is7duxg1apVarkFCxZQv359bG1tcXd3Z+TIkepjEIKCgnjrrbeIjY1VM1UBAQGA7nEJTZs2xc7ODhcXFwYMGJDn+D0hRNHL7c41vU7vdgIg9o9YLCMt1fUNXRo+rsPWMFuUU5153gmXhaIoTNixg8/+ft+uJiz9einW1tY5lm/fvj0jRowAdI9sSU5OzvMYxanMzLOkt3//fiIiInK9BOfl5cXs2bPp1asX8fHxBAQE0KdPH1xdXbl69SoffvghFStWVO8qKGiJaYloZ2sLpe68xH8Qj62Fbd4FcxAeHs7OnTsNAtDMzEyqVKnCzz//TMWKFTly5AjvvPMOrq6u9O3bVy0XGBiIq6srgYGBXL58mX79+tGoUSOGDRsG6AKqGzdusH//fiwsLBg7dqzBH2lFUejZsye2trYEBweTnp7OyJEj6devn0EAc+XKFXbs2MHOnTu5cuUKr7zyChEREdSqVYvg4GCOHDnCkCFDaN++PS1btszxPP39/fnll18M2t62bVsyMzMJDAzk7bffVte//vrrACQkJDBx4kTq169PQkIC06ZNo1evXpw+fTrPx4HExsbSrVs3rKys2LdvH7a2T/fzadeuHQ0bNmTjxo1q20xMTFi8eDEeHh5EREQwcuRIJk2axJIlS/D19WXRokVMmzaNCxcuALogESA1NZUZM2ZQu3ZtYmJimDBhAoMHDy7QqTSEEM/PWGYJwMzNDBoBp4HdwBuACeqz2yD73W36OhPSEohPjUdroX18J1wemSVFUfjwww/5YvFiAJYD3SPh3v1bVHLMfUzvrFmz2Lx5M5cuXWL58uWMGTPG6HGKU5kLllasWIGvry916tTJcfuFCxeIjY0FwNTUlLNnz/Ldd9/x8OFDXF1d8ff3Z926ddjZ2RVls0ukbdu2odVqycjIUKP+BQsWqNvNzc2ZPn26+t7T05MjR47w888/GwRL5cuX58svv8TU1BQvLy+6du3Kvn37GDZsGBcvXmTHjh0cPXqUFi1aALqfYdaf3969ezlz5gwRERHq2KHvv/+eunXrcuLECXXC0czMTL799lvs7Ozw9vbG39+fCxcusH37dkxMTKhduzZz584lKCgo12Cpbdu2zJo1i8jISFxdXQkODub//u//yMzMVLNmN27cICIiQp0EtU+fPgZ1rFixAicnJ0JDQ7NdIs4qOjqafv36UaNGDX766adnfmyIl5cXZ86cUd+PHz9efe3p6cmMGTMYMWIES5YswcLCAgcHB10q38Xw2+KQIUPU19WrV2fx4sU0b96c+Ph4NaASQhS/vOYyioqPAn+wuGBBSkQKnAKaGAZLT2aLtBZabMxtSExLJDo+Gm0Fbb4zS59++ilz5swB4KvFi+k0bTzODzO5tfIbeH96rvs5ODgwZcoURo0axdKlSxk9enSJnZKkzAVLP/74o9HtWS9FWVtbs2vXrsJukgEbcxviP4jPu2AhHftp+Pv7s3TpUhITE/nmm2+4ePFitsj/66+/5ptvvuHatWskJSWRmppKo0aNDMrUrVsXU1NT9b2rqytnz54FdGOCzMzMDCYO9fLyMhiAHBYWhru7u8Ega29vb8qVK0dYWJgaLHl4eBgEuc7Ozpiamhpkd5ydnY1eWmrdujUWFhYEBQXRsGFDkpKS8PHxQVEU4uLiuHTpEr///juWlpb4+voCuozW1KlTOXr0KHfv3lUHXF+/ft1osPTiiy/SrFkzfv75Z4P+eVqKohj8ggkMDGTWrFmEhoYSFxdHeno6ycnJJCQkGM1cnTp1ioCAAE6fPs39+/cNzsPbu+w8tkCI0i6vzFJ0QjQ4QI/hPVi/cD3sAY2XhnpOj38f5ZQtctG6EP4gnKj4KGpUqPF4zJKROZY+//xzpk2bBsD8+fMZOWYMcw7M4j+/RFHuqxUwYRoY+f02cOBAJk+eTFhYGMHBwbRt2zbP8y8OJX7MUlmj0WiwtbAtluVpI3ZbW1tq1qxJgwYNWLx4MSkpKQaZpJ9//pkJEyYwZMgQdu/ezenTp3nrrbdITU01qOfJsWMajUb9Q6wPXo217clgILf1OR3H2LFzYmNjQ/PmzQkMDCQwMJA2bdpgamqKmZkZvr6+6vpWrVqpY+W6d+/OvXv3WL58OceOHePYsWMA2frhSV27duXgwYOEhoYaLZeXsLAwPD11z3K6du0aL730EvXq1WPDhg2EhITw1VdfARiME3tSQkICHTt2RKvV8sMPP3DixAk2bdqUr/MQQhStfGWWgAFDB1CxRkVIBovtFlS0qYipRhe45JQterLevDJLCxYs4P/+njJgxowZTJw4EYCDHWpx3wpsr96CX381ei729vYMHDgQgCVLlhgtW5wkWBL59vHHH/P5559z+/ZtAA4ePIivry8jR46kcePG1KxZ86kGNgPUqVOH9PR0g+kfLly4YDAHkbe3N9evXzeYoiE0NJTY2NhcL7c+D39/f4KCgggKCjL4luPn56eu11+Cu3fvHmFhYXz00Ue0b9+eOnXq5Ht6hTlz5jBo0CDat2//zAHT/v37OXv2rHop8OTJk6SnpzN//nxatmxJrVq11J+XnoWFRbbJ4P766y/u3r3LnDlz+Pe//42Xl5cM7haihHoys5SSnsKuy7vY/NdmNv+1meux1wFwK+dGr0m9wARSzqfw9dKv1SxRbpmlrPUauxtu4cKFvPfeewBMnTr18ZQFQLmKVViifxzr3Llg5OaihNQE6napC8DGTRtZdWgVN+NuApCRmcH+iP3qeV19eDXvzikkEiyJfGvbti1169Zl1qxZANSsWZOTJ0+ya9cuLl68yNSpUzlx4sRT1Vm7dm06d+7MsGHDOHbsGCEhIbz99tsGd1C8+OKLNGjQgNdff50//viD48eP8+abb+Ln55fnc/+ehb+/P5cuXWLnzp34+fmp6/38/Ni2bRtXr15Vg6Xy5cvj6OjI//73Py5fvsz+/fvVb1f58fnnn/P666/Trl07/vrrL6NlU1JSiIqK4tatW/zxxx/MmjWLHj160K1bN958800AatSoQXp6Ov/9738JDw/n+++/5+uvvzaox8PDg/j4ePbt28fdu3dJTEykatWqWFhYqPtt2bKFGTNm5Ps8hBBFRx/U6Adgzzo4i85rOtNrXS96reulTibpqnXFx8cHOur2mzhxIvZ3dVPduNplv+NbzSzFG88sLVy4UP09N3XqVIMrDqB7dMp/W0CauSkcPw4HD+Z6LuN2jmPMqTHgBhnpGbz12Vu0/KYliqLw7alvaf9de/W8dl8pmnntciLBkngqEydOZPny5dy4cYPhw4fTu3dv+vXrR4sWLbh3794zzcO0cuVK3N3d8fPzo3fv3rzzzjs4OTmp2zUaDZs3b6Z8+fK88MILvPjii1SvXp1169YV5KmpWrVqhaWl7nbbJk2aqOubNWtGRkYG1tbW6mB0ExMT1q5dS0hICPXq1WPChAl89tlnOdabm4ULF9K3b1/atWvHxYsXcy23c+dOXF1d8fDwoHPnzgQGBrJ48WJ+/fVXdcxTo0aNWLBgAXPnzqVevXqsWbOG2bNnG9Tj6+vL8OHD6devH5UqVWLevHlUqlSJVatWsX79ery9vZkzZw6ff/75U52HEKJo6LND+mDm3J1zANQoXwNfd1983X15r9V7uDu407tOb1556xV8X/QlNTWV28tu09WxK51rds5Wb9bMUlJaEnEpcQbHUxSFmTNnZguUnhwm4ax1JkYLB/2r61bMm5fruZyL0bW9YqOKuhUX4dajW8SmxKrbqthXwdfdN9/zPRUKRTy32NhYBVBiY2OzbUtKSlJCQ0OVpKSkYmiZEIVHPttCFI9HKY8UAlAIQHmU8kjxXeGrEICy/vz6XPeJjY1VfHx8FEDx8PBQwsPDs5VZemKpQgBKj596KBEPIhQCUCxnWCqZmZlKcnKy8tZbbymAAihTp05VMjMzczzW6tOrFQJQBn/WWlE0GkUBRTl3Lsey1RZWUwhAWbF1ha5uCxQ+Qgm7E6b0Xd9XIQBl4e8Ln6mf8sPY3++sJLMkhBBClCL62/xBd8lMf9nM2C3+9vb2bN++nZo1a3L16lWaN2/O4cOHDcpkzSypcyxpnQkLC6N58+asXLkSExMTlixZwieffJLrjTn6DNAf2kfQu7duZQ6ZakVR1HFRfq38cHZ2hlTgev7Pq6hIsCSEEEKUMuq4pYTox89wy+MylbOzM0FBQfj4+HD37l38/PyYMGGCekNN1rvhohOiIR5StqXQqFEjzpw5Q6VKldi+fbs683aebYuPVh+wy5o1cOuWQbm4lDiS03Vz+Lnau9KlSxfdhou6gC2/51UUJFgSQgghShl9AHHl/hUS0hKA/GVgKleuzIEDB+jfvz8ZGRksWrQIV1dXXn31VTav2AwhcHPrTaYMmQILIHpfNGlpaXTt2pUzZ87QqVOnvNv29xinO4l3yGjWFF54AdLS4L//NSinzyrZWdhhY25D165ddRsuZQnY8nlehU2CJSGEEKKU0QcQZ6J1s/dbm1mjtcjfTPu2trb8+OOP7Nq1i3r16pGcnMwvv/zCvOnzYCuk70vn3JFzkAmValVi9+7dbNu2Ldus/7mpZFMJE40JmUomdxPvgv4O4WXLIP7xpMxZL/UBdOjQATTAPQiLCONh8kOD7cVJgiUhhBCilNFnlv6M/hPQBU9PO/Fwx44dOXPmDCEhIUybNo033ngD09qm0Bj+1e9fMBreXfquLoh5CqYmplS00d3dFhUfBd27Q82a8PAhZHno95NTEzg4OODsoTuv34/+DoC5iTnlrco/1fELgwRLQgghRCmjDzD0wdKzZl80Gg0+Pj5Mnz6d7777Ds9RntADYpvGQsVnvwSWdUwVJiYwYYJuw6JF8PekuDlNelmrYS0ALp25pNumdS4Rz4uTYEkIIYQoZfTBkX4CyoIa16OvR1/vswZh+gBIfX7doEFQvjxcuQJbtxpsy9r2hk11D/tNjEjMtq04SbAkhBBClDJPBhEFdcfYk/U8d2bp73FJ2NrC8OG61wsWGGzLeszWLVvrXtwCMkvGnXAgwZIoYlevXkWj0XD69OnibkqJ4+HhwaJFi4q7GUKIUqCggpq86n3WYCVbZglg9GgwN9c9/uTECaISsmeWWjZuCRZAGhAjmSVRws2ePZtmzZphZ2eHk5MTPXv25MKFC0Vy7LZt26LRaNBoNFhaWlK5cmW6d+/Oxo0bi+T4hSnruWVd0tPTOXHiBO+8845aVv+YFyGEeFJhZZaerLdAxizpublB//661wsWZLsbDnTzLVH57zc3JbMkSrjg4GBGjRrF0aNH2bNnD+np6XTs2JGEhIQiOf6wYcOIjIzk8uXLbNiwAW9vb1577TWDYKKwpKamFmr9+nPLupiZmVGpUiVsbGwK9dhCiLLhybFEBZZZylLv00xHkFs9BpkleDzQe/16TG7cBAzbbmlmiZWHle7NLcksiRJu586dDB48mLp169KwYUNWrlzJ9evXCQkJUct4eHgwa9YshgwZgp2dHVWrVuV///ufQT3Hjx+ncePGWFlZ0bRpU06dOpWv49vY2ODi4oK7uzstW7Zk7ty5LFu2jOXLl7N371613K1bt+jXrx/ly5fH0dGRHj16cPXqVXV7eno6Y8eOpVy5cjg6OjJ58mQGDRpEz5491TJt27Zl9OjRTJw4kYoVK6q3yYaGhvLSSy+h1WpxdnbmjTfe4O7du+p+iqIwb948qlevjrW1NQ0bNuSXX37J97llXfT9qb8M5+HhAUCvXr3QaDTqeyGEALAxt8HOwk59X1BzEWUNTp5lOoIn6zHILAE0agTt2kFGBn33Zx+zBOD4L0fdi1slY44lkGCp6CkKJCQUz6Ioz9zs2NhYACpUqGCwfv78+WoQNHLkSEaMGMFff/0FQEJCAt26daN27dqEhIQQEBDA+++//8xtGDRoEOXLl1cvxyUmJuLv749Wq+XAgQMcOnQIrVZL586d1ezQ3LlzWbNmDStXruTw4cPExcXleGlr9erVmJmZcfjwYZYtW0ZkZCR+fn40atSIkydPsnPnTqKjo+nbt6+6z0cffcTKlStZunQp58+fZ8KECQwcOJDg4OBnPke9EydOALBy5UoiIyPV90IIoZc1kCiMAd7PE6jkOGZJ7+9JKoeezESbkv04lWv+fR3uHjhaOj5zGwpUoT3K9x/E2FOLsz2ZPT5e9wTm4lji45/p/DIzM5Xu3bsrbdq0MVhfrVo1ZeDAgQblnJyclKVLlyqKoijLli1TKlSooCQkJKhlli5dqgDKqVOncj2en5+fMm7cuBy3tWjRQunSpYuiKIqyYsUKpXbt2gZPvk5JSVGsra2VXbt2KYqiKM7Ozspnn32mbk9PT1eqVq2q9OjRw+B4jRo1MjjO1KlTlY4dOxqsu3HjhgIoFy5cUOLj4xUrKyvlyJEjBmWGDh2q9O/f3+i5mZubK7a2tuoyceJERVF0/blw4UK1LKBs2rQp17qKW7bPthCiSLX5to1CAAoBKI9SHhVInVcfXFXr7Lm25zPXExMfoxCAognQKKnpqYYbMzKU5JqeigLK5G5W2fbtu66vgjkKoGw/sv2Z25Afxv5+Z2VWnIGaKB1Gjx7NmTNnOHToULZtDRo0UF9rNBpcXFyIidHNzxEWFkbDhg0NxuG0atXqudqiKIqaFg4JCeHy5cvY2dkZlElOTubKlSvExsYSHR1N8+bN1W2mpqY0adKEzMxMg32aNm1q8D4kJITAwEC02uzX6/V1JycnZ5vZNjU1lcaNGxs9h9dff50pU6ao78uVK2e0vBBC5ESfvbE1t33msUXZ6iygbJWjjSOmGlMylAzuJN7Bzc7t8UYTE66+1ZPaUxYy+kg6pKeD2eNwxMXOBSoBt+HO1TvwfH82CoQES0XNxsbg2ThFfuynNGbMGLZs2cKBAweoUqVKtu3m5uYG7zUajRqIKM9x2S8nGRkZXLp0iWbNmgGQmZlJkyZNWLNmTbaylSpVMmhTVjm1y9bW1uB9ZmYm3bt3Z+7cudnKurq6cu7cOQB+++03KleubLDd0tLS6Hk4ODhQs2ZNo2WEECIv+nFBBTkI2srMCgdLB2JTYp+rXhONCU62TkTGRxIdH20YLAFnOzbG8VOocj8dNm+GV15Rt7loHwdLERcjnrkNBUmCpaKm0egm5yrhFEVhzJgxbNq0iaCgIDw9PZ+6Dm9vb77//nuSkpKwtrYG4OjRo8/cptWrV/PgwQP69OkDgI+PD+vWrcPJyQl7e/sc93F2dub48eP8+9//BnQB16lTp2jUqJHRY/n4+LBhwwY8PDwwM8v+38Tb2xtLS0uuX7+On5/fM5+TMebm5mT8/VgAIYR4kj7zU9CDoJ21zsSmxD73OChnrTOR8ZFM2DWB9p7t+eiFj0jNSOU/e//DwesHCW0G0w6gm6QyS7DkrHUGJ93r0NDQ52pDQZEB3iJHo0aN4ocffuDHH3/Ezs6OqKgooqKiSEpKyncdAwYMwMTEhKFDhxIaGsr27dv5/PPP87VvYmIiUVFR3Lx5k2PHjjF58mSGDx/OiBEj8Pf3B3SXsypWrEiPHj04ePAgERERBAcHM27cOG7e1N2SOmbMGGbPns2vv/7KhQsXGDduHA8ePMjzDo9Ro0Zx//59+vfvz/HjxwkPD2f37t0MGTKEjIwM7OzseP/995kwYQKrV6/mypUrnDp1iq+++orVq1fnu4+M8fDwYN++fURFRfHgwYMCqVMIUXbUctQ9R+1fFf5VOPU6Pl+9+nqCrwUzLWgaYXfD2Hl5J4uOLSIkMoQlzSDd3BR+/123ZN3v74sD58+ff642FBQJlkSOli5dSmxsLG3btsXV1VVd1q1bl+86tFotW7duJTQ0lMaNGzNlypQcL2vlZPny5bi6ulKjRg169epFaGgo69atY8mSJWoZGxsbDhw4QNWqVenduzd16tRhyJAhJCUlqZmmyZMn079/f958801atWqFVqulU6dOWFlZGT2+m5sbhw8fJiMjg06dOlGvXj3GjRuHg4MDJia6/zYzZsxg2rRpzJ49mzp16tCpUye2bt36TFm4nMyfP589e/bg7u6e5zgoIcQ/T+86vdnQdwOfdfisQOv9uuvXbO63mfae7Z+rni86f8HXXb+mir1uCMetuFvcjNN9kW3s0phZr68gUz9J5cKF6n6t3VvzzdBvALh48SJpaWnP1Y6CoFEKemDJP1BcXBwODg7ExsZmuxyUnJxMREQEnp6eef6BFoUvMzOTOnXq0LdvX2bMmFHczSnV5LMthMiPjt93ZE/4Hr7r+R0X713k04OfMrLpSL7q+hWcPQsNGoCpKUREgLs7oBsKYm9vT3x8POfPn8fb27tQ2mbs73dWklkSZdq1a9dYvnw5Fy9e5OzZs4wYMYKIiAgGDBhQ3E0TQoh/hKyzeesnqVTHWdWvD/7+kJEBX32l7qPRaNQAqSRcipNgSZRpJiYmrFq1imbNmtG6dWvOnj3L3r17qVOnTnE3TQgh/hH0A8WjE6LVSSoN7rQbN0737/LlkJiorq5bty5QMoIluRtOlGnu7u4cPny4uJshhBD/WPrAyCCzlPVOu27dwNNTdxluzRoYNgxA/VJ78eLFom1wDiSzJIQQQohCk2dmydQURo/WvV68WH00V40aNQAIDw8vusbmQoIlIYQQQhQag8xS/BNjlvSGDNHNQXjuHAQGAlC9enVAgiUhhBBClHH6wOjy/cukZKTo1j054WW5cjBokO71F18AqNOw3Llzh0ePHhVJW3MjwZIQQgghCo0+s5ScngyAvaU91ubW2QuOGaP7d+tWCA/HwcEBR0dHACIiivexJxIsCSGEEKLQOFo7YqJ5HG7k+sw5Ly/o1Ek3ZunLL4GScylOgiUhhBBCFBpTE1OcbJ3U90afOaefRmDFCoiPl2BJ/HMFBATk+SBbobNq1SrKlStX3M0QQojnkjVAyjWzBLrMUq1aEBcHq1dLsCRKtgMHDtC9e3fc3NzQaDRs3rw5WxlFUQgICMDNzQ1ra2vatm2bbfKw3PZ9WkFBQWg0GjQaDSYmJjg4ONC4cWMmTZpEZGTkc9dfnLKeW9blo48+ol+/fgZzjEigKYQojbIGSEYzSyYmj8cuLV5MdQ8PAK5cuVKIrcubBEsiRwkJCTRs2JAv/75unJN58+axYMECvvzyS06cOIGLiwsdOnQo1LsWLly4wO3btzlx4gSTJ09m79691KtXj7NnzxbaMUEXGKanpxfqMS5cuEBkZKS6/Oc//8Ha2honJ6e8dxZCiBIs61QBRjNLoLsrzt4eLl6k+v37gGSWRAnVpUsXPv30U3r37p3jdkVRWLRoEVOmTKF3797Uq1eP1atXk5iYyI8//giAx9/fCHr16oVGo1Hf633//fd4eHjg4ODAa6+9lq8gy8nJCRcXF2rVqsVrr73G4cOHqVSpEiNGjDAot3LlSurUqYOVlRVeXl4sWbLEYPuRI0do1KgRVlZWNG3alM2bN6PRaDh9+jTwONuza9cumjZtiqWlJQcPHkRRFObNm0f16tWxtramYcOG/PLLLwZ1h4aG8tJLL6HVanF2duaNN97g7t27+T43/aLVag0uw61atYrp06fz559/qtmnVatW5VmvEEIUNxfbLJmlJ+dYepKdnW7eJaDGzp0AXL16lYyMjEJrX15KTbA0c+ZMfH19sbGxyXUMx/Xr1+nevTu2trZUrFiRsWPHkpqaarTelJQUxowZQ8WKFbG1teXll1/m5s2bhXAGOoqikJCQUCyL8vesqAUhIiKCqKgoOnbsqK6ztLTEz8+PI0eOAHDixAlAF7hERkaq70GXUt28eTPbtm1j27ZtBAcHM2fOnKduh7W1NcOHD+fw4cPExMQAsHz5cqZMmcLMmTMJCwtj1qxZTJ06ldWrVwPw6NEjunfvTv369fnjjz+YMWMGkydPzrH+SZMmMXv2bMLCwmjQoAEfffQRK1euZOnSpZw/f54JEyYwcOBAgoODAYiMjMTPz49GjRpx8uRJdu7cSXR0NH379n3qc3tSv379eO+996hbt66aferXr99z1yuEEIXtqTJLoLsUp9FQJTgYMzMzUlNTuX37diG20LhS82y41NRUXn31VVq1asWKFSuybc/IyKBr165UqlSJQ4cOce/ePQYNGoSiKPz3v//Ntd7x48ezdetW1q5di6OjI++99x7dunUjJCQEU1PTAj+PxMREtFptgdebH/Hx8dja2hZIXVFRuinrnZ0NvyE4Oztz7do1ACpVqgRAuXLlcHEx/M+RmZnJqlWrsLOzA+CNN95g3759zJw586nb4uXlBei+eTg5OTFjxgzmz5+vZsU8PT0JDQ1l2bJlDBo0iDVr1qDRaFi+fDlWVlZ4e3tz69Ythv39PKKsPvnkEzp06ADoLk0uWLCA/fv306pVK0B3W+uhQ4dYtmwZfn5+LF26FB8fH2bNmqXW8e233+Lu7s7FixepVatWrudRpUoVg/f6ftSztrZGq9ViZmaWrT+FEKIkyzpOyeiYJb3q1aF7d0y3bMHD1pbLsbGEh4fj7u5eiK3MXakJlqZPnw6Q62WH3bt3Exoayo0bN3BzcwNg/vz5DB48mJkzZ2Jvb59tn9jYWFasWMH333/Piy++CMAPP/yAu7s7e/fupVOnToVzMmWIRqMxeK8oSrZ1OfHw8FADJQBXV1c1M/S09BkzjUbDnTt3uHHjBkOHDjUIftLT03FwcAB0Y4MaNGiAlZWVur158+Y51t20aVP1dWhoKMnJyWrwpJeamkrjxo0BCAkJITAwMMeA+MqVK0aDpYMHDxr0Sfny5XMtK4QQpUnWbFK+MksAY8fCli1Uj4/nMrpxS35+foXTwDyUmmApL7///jv16tVTAyWATp06kZKSQkhICP7+/tn2CQkJIS0tzeBSkpubG/Xq1ePIkSO5BkspKSmkpKSo7+Pi4vLdThsbG+Lj4/NdviDZ2NgUWF36zEZUVBSurq7q+piYmGzZppyYm5sbvNdoNGRmZj5TW8LCwgBdAKavY/ny5bRo0cKgnD5TmFNAl9slyqyZOH3dv/32G5UrVzYoZ2lpqZbp3r07c+fOzVZX1n7Kiaenp0wTIIQok7Jehss655JR7dpB3bq0PH+ejH/9q1i/QJaZYCkqKirbH+ny5ctjYWGhXjLKaR8LC4tsPwBnZ+dc9wGYPXu2mul6WhqNpsAuhRUnT09PXFxc2LNnj5pVSU1NJTg42CBQMDc3L9RBeUlJSfzvf//jhRdeUC/7Va5cmfDwcF5//fUc9/Hy8mLNmjWkpKSoQc7JkyfzPJa3tzeWlpZcv3491283Pj4+bNiwAQ8PD8zMCv6/l4WFRbEOchRCiGdRs0JNapSvQVWHqliaWeZvJ40Gxo5l+rvvQmYmvPxy4TbSiGId4B0QEJDj/DJZl/z8EdPL6fJPfi8LPc0+H3zwAbGxsepy48aNp6q/NIiPj+f06dPq3WERERGcPn2a69evA7q+Hj9+PLNmzWLTpk2cO3eOwYMHY2Njw4ABA9R6PDw82LdvH1FRUTx48OC52xUTE0NUVBSXLl1i7dq1tG7dmrt377J06VK1TEBAALNnz+aLL77g4sWLnD17lpUrV7JgwQIABgwYQGZmJu+88w5hYWHs2rWLzz//XD2v3NjZ2fH+++8zYcIEVq9ezZUrVzh16hRfffWVOnh81KhR3L9/n/79+3P8+HHCw8PZvXs3Q4YMKZAgx8PDQ/1Z3L171yDDKYQQJZWVmRV/jf6LfW/ue7odX39d95DdK1dgx45CaVt+FGtmafTo0bz22mtGyzx5u3luXFxcOHbsmMG6Bw8ekJaWlutlIRcXF1JTU3nw4IFBdikmJgZfX99cj2VpaalmJMqqkydPGly6nDhxIgCDBg1Sx41NmjSJpKQkRo4cyYMHD2jRogW7d+82GHczf/58Jk6cyPLly6lcuTJXr159rnbVrl0bjUaDVqulevXqdOzYkYkTJxoMeH777bexsbHhs88+Y9KkSdja2lK/fn3Gjx8PgL29PVu3bmXEiBE0atSI+vXrM23aNAYMGGAwjiknM2bMwMnJidmzZxMeHk65cuXw8fHhww8/BHSXcQ8fPszkyZPVy8DVqlWjc+fOmJg8/3eTPn36sHHjRvz9/Xn48CErV65k8ODBz12vEEIUNjOTZwg5bG110wh88w0U491wGqUg7ycvAqtWrWL8+PE8fPjQYP2OHTvo1q0bN2/eVMeGrFu3jkGDBhETE5PrAO9KlSrxww8/qLd2R0ZGUqVKFbZv357vAd5xcXE4ODgQGxub7TjJyclERETg6emZ5x9iUXzWrFnDW2+9RWxsLNbWOTwNW2Qjn20hRJG4fx8sLKAQ7iQ39vc7q1IzZun69evcv3+f69evk5GRoV4eqlmzJlqtlo4dO+Lt7c0bb7zBZ599xv3793n//fcZNmyY2gG3bt2iffv2fPfddzRv3hwHBweGDh3Ke++9h6OjIxUqVOD999+nfv366t1xomz67rvvqF69OpUrV+bPP/9k8uTJ9O3bVwIlIYQoaSpUKO4WlJ5gadq0aeq4EEAdVBwYGEjbtm0xNTXlt99+Y+TIkbRu3Rpra2sGDBigjkUBSEtL48KFCyQmJqrrFi5ciJmZGX379iUpKYn27duzatWqQpljSZQcUVFRTJs2Tb2b79VXX32mOZ6EEEKUfaXuMlxJJJfhxD+RfLaFEKVdfi/DlZrHnQghhBBCFAcJloQQQgghjJBgqYjI1U5R1shnWgjxTyHBUiHTP9Yj66ByIcqC1NRUALkZQghR5pWau+FKK1NTU8qVK6c+JNbGxuapZxQXoqTJzMzkzp072NjYFMpjXYQQoiSR33JFQD+7tD5gEqIsMDExoWrVqhL8CyHKPAmWioBGo8HV1RUnJyfS0tKKuzlCFAgLC4sCeYSLEEKUdBIsFSFTU1MZ3yGEEEKUMvK1UAghhBDCCAmWhBBCCCGMkGBJCCGEEMIIGbNUAPST88XFxRVzS4QQQgiRX/q/23lNsivBUgF49OgRAO7u7sXcEiGEEEI8rUePHuHg4JDrdo0izyx4bpmZmdy+fRs7O7sCnXMmLi4Od3d3bty4YfRpyOL5SV8XDennoiH9XDSkn4tGYfazoig8evQINzc3o1OhSGapAJiYmFClSpVCq9/e3l7+IxYR6euiIf1cNKSfi4b0c9EorH42llHSkwHeQgghhBBGSLAkhBBCCGGEBEslmKWlJR9//DGWlpbF3ZQyT/q6aEg/Fw3p56Ih/Vw0SkI/ywBvIYQQQggjJLMkhBBCCGGEBEtCCCGEEEZIsCSEEEIIYYQES0IIIYQQRkiwVIItWbIET09PrKysaNKkCQcPHizuJpVqAQEBaDQag8XFxUXdrigKAQEBuLm5YW1tTdu2bTl//nwxtrh0OHDgAN27d8fNzQ2NRsPmzZsNtuenX1NSUhgzZgwVK1bE1taWl19+mZs3bxbhWZR8efXz4MGDs32+W7ZsaVBG+jlvs2fPplmzZtjZ2eHk5ETPnj25cOGCQRn5TD+//PRzSfpMS7BUQq1bt47x48czZcoUTp06xb///W+6dOnC9evXi7tppVrdunWJjIxUl7Nnz6rb5s2bx4IFC/jyyy85ceIELi4udOjQQX32n8hZQkICDRs25Msvv8xxe376dfz48WzatIm1a9dy6NAh4uPj6datGxkZGUV1GiVeXv0M0LlzZ4PP9/bt2w22Sz/nLTg4mFGjRnH06FH27NlDeno6HTt2JCEhQS0jn+nnl59+hhL0mVZEidS8eXNl+PDhBuu8vLyU//znP8XUotLv448/Vho2bJjjtszMTMXFxUWZM2eOui45OVlxcHBQvv766yJqYekHKJs2bVLf56dfHz58qJibmytr165Vy9y6dUsxMTFRdu7cWWRtL02e7GdFUZRBgwYpPXr0yHUf6ednExMTowBKcHCwoijymS4sT/azopSsz7Rklkqg1NRUQkJC6Nixo8H6jh07cuTIkWJqVdlw6dIl3Nzc8PT05LXXXiM8PByAiIgIoqKiDPrc0tISPz8/6fPnkJ9+DQkJIS0tzaCMm5sb9erVk75/SkFBQTg5OVGrVi2GDRtGTEyMuk36+dnExsYCUKFCBUA+04XlyX7WKymfaQmWSqC7d++SkZGBs7OzwXpnZ2eioqKKqVWlX4sWLfjuu+/YtWsXy5cvJyoqCl9fX+7du6f2q/R5wcpPv0ZFRWFhYUH58uVzLSPy1qVLF9asWcP+/fuZP38+J06coF27dqSkpADSz89CURQmTpxImzZtqFevHiCf6cKQUz9DyfpMmxVobaJAaTQag/eKomRbJ/KvS5cu6uv69evTqlUratSowerVq9VBg9LnheNZ+lX6/un069dPfV2vXj2aNm1KtWrV+O233+jdu3eu+0k/52706NGcOXOGQ4cOZdsmn+mCk1s/l6TPtGSWSqCKFStiamqaLTKOiYnJ9m1GPDtbW1vq16/PpUuX1LvipM8LVn761cXFhdTUVB48eJBrGfH0XF1dqVatGpcuXQKkn5/WmDFj2LJlC4GBgVSpUkVdL5/pgpVbP+ekOD/TEiyVQBYWFjRp0oQ9e/YYrN+zZw++vr7F1KqyJyUlhbCwMFxdXfH09MTFxcWgz1NTUwkODpY+fw756dcmTZpgbm5uUCYyMpJz585J3z+He/fucePGDVxdXQHp5/xSFIXRo0ezceNG9u/fj6enp8F2+UwXjLz6OSfF+pku0OHiosCsXbtWMTc3V1asWKGEhoYq48ePV2xtbZWrV68Wd9NKrffee08JCgpSwsPDlaNHjyrdunVT7Ozs1D6dM2eO4uDgoGzcuFE5e/as0r9/f8XV1VWJi4sr5paXbI8ePVJOnTqlnDp1SgGUBQsWKKdOnVKuXbumKEr++nX48OFKlSpVlL179yp//PGH0q5dO6Vhw4ZKenp6cZ1WiWOsnx89eqS89957ypEjR5SIiAglMDBQadWqlVK5cmXp56c0YsQIxcHBQQkKClIiIyPVJTExUS0jn+nnl1c/l7TPtARLJdhXX32lVKtWTbGwsFB8fHwMbqkUT69fv36Kq6urYm5urri5uSm9e/dWzp8/r27PzMxUPv74Y8XFxUWxtLRUXnjhBeXs2bPF2OLSITAwUAGyLYMGDVIUJX/9mpSUpIwePVqpUKGCYm1trXTr1k25fv16MZxNyWWsnxMTE5WOHTsqlSpVUszNzZWqVasqgwYNytaH0s95y6mPAWXlypVqGflMP7+8+rmkfaY1fzdaCCGEEELkQMYsCSGEEEIYIcGSEEIIIYQREiwJIYQQQhghwZIQQgghhBESLAkhhBBCGCHBkhBCCCGEERIsCSGEEEIYIcGSEEIIIYQREiwJIcqsmJgY3n33XapWrYqlpSUuLi506tSJ33//HdA9OX7z5s3F20ghRIlnVtwNEEKIwtKnTx/S0tJYvXo11atXJzo6mn379nH//v3ibpoQohSRzJIQokx6+PAhhw4dYu7cufj7+1OtWjWaN2/OBx98QNeuXfHw8ACgV69eaDQa9T3A1q1badKkCVZWVlSvXp3p06eTnp6ubtdoNCxdupQuXbpgbW2Np6cn69evV7enpqYyevRoXF1dsbKywsPDg9mzZxfVqQshCpgES0KIMkmr1aLVatm8eTMpKSnZtp84cQKAlStXEhkZqb7ftWsXAwcOZOzYsYSGhrJs2TJWrVrFzJkzDfafOnUqffr04c8//2TgwIH079+fsLAwABYvXsyWLVv4+eefuXDhAj/88INBMCaEKF3kQbpCiDJrw4YNDBs2jKSkJHx8fPDz8+O1116jQYMGgC5DtGnTJnr27Knu88ILL9ClSxc++OADdd0PP/zApEmTuH37trrf8OHDWbp0qVqmZcuW+Pj4sGTJEsaOHcv58+fZu3cvGo2maE5WCFFoJLMkhCiz+vTpw+3bt9myZQudOnUiKCgIHx8fVq1ales+ISEhfPLJJ2pmSqvVMmzYMCIjI0lMTFTLtWrVymC/Vq1aqZmlwYMHc/r0aWrXrs3YsWPZvXt3oZyfEKJoSLAkhCjTrKys6NChA9OmTePIkSMMHjyYjz/+ONfymZmZTJ8+ndOnT6vL2bNnuXTpElZWVkaPpc8i+fj4EBERwYwZM0hKSqJv37688sorBXpeQoiiI8GSEOIfxdvbm4SEBADMzc3JyMgw2O7j48OFCxeoWbNmtsXE5PGvzKNHjxrsd/ToUby8vNT39vb29OvXj+XLl7Nu3To2bNggd+EJUUrJ1AFCiDLp3r17vPrqqwwZMoQGDRpgZ2fHyZMnmTdvHj169ADAw8ODffv20bp1aywtLSlfvjzTpk2jW7duuLu78+qrr2JiYsKZM2c4e/Ysn376qVr/+vXradq0KW3atGHNmjUcP36cFStWALBw4UJcXV1p1KgRJiYmrF+/HhcXF8qVK1ccXSGEeE4SLAkhyiStVkuLFi1YuHAhV65cIS0tDXd3d4YNG8aHH34IwPz585k4cSLLly+ncuXKXL16lU6dOrFt2zY++eQT5s2bh7m5OV5eXrz99tsG9U+fPp21a9cycuRIXFxcWLNmDd7e3uqx586dy6VLlzA1NaVZs2Zs377dIDMlhCg95G44IYR4SjndRSeEKLvka44QQgghhBESLAkhhBBCGCFjloQQ4inJ6AUh/lkksySEEEIIYYQES0IIIYQQRkiwJIQQQghhhARLQgghhBBGSLAkhBBCCGGEBEtCCCGEEEZIsCSEEEIIYYQES0IIIYQQRkiwJIQQQghhxP8DY42xV8DKGxMAAAAASUVORK5CYII=\n",
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\n",
       "text/plain": [
        "
    "
       ]
@@ -107,7 +107,7 @@
     },
     {
      "data": {
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\n",
+      "image/png": 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\n",
       "text/plain": [
        "
    "
       ]
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter6_1_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter6_1_1.png
index 0a10641b4..2db05379b 100644
Binary files a/doc/LectureNotes/_build/jupyter_execute/chapter6_1_1.png and b/doc/LectureNotes/_build/jupyter_execute/chapter6_1_1.png differ
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter6_1_2.png b/doc/LectureNotes/_build/jupyter_execute/chapter6_1_2.png
index 2221d7900..cd4f3218e 100644
Binary files a/doc/LectureNotes/_build/jupyter_execute/chapter6_1_2.png and b/doc/LectureNotes/_build/jupyter_execute/chapter6_1_2.png differ
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter8.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter8.ipynb
index ca0c6655a..815adfb3c 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter8.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter8.ipynb
@@ -295,10 +295,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.264540221699101\n",
-      "4.673457751724773\n",
-      "[[0.83632853 2.54078623]\n",
-      " [2.54078623 8.44021223]]\n"
+      "0.26662339374864535\n",
+      "4.736115211426478\n",
+      "[[ 1.20561803  3.63264564]\n",
+      " [ 3.63264564 12.10207647]]\n"
      ]
     }
    ],
@@ -340,10 +340,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.08374032367704139\n",
-      "1.7696835316227453\n",
-      "[[1.         0.66443521]\n",
-      " [0.66443521 1.        ]]\n"
+      "0.09318696260700278\n",
+      "1.9096305360355206\n",
+      "[[1.         0.65907898]\n",
+      " [0.65907898 1.        ]]\n"
      ]
     }
    ],
@@ -397,30 +397,30 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[ 1.20708879  4.33201844]\n",
-      " [-0.3838783  -1.24125217]\n",
-      " [ 0.74722409  1.60194224]\n",
-      " [-0.04086326  0.37419664]\n",
-      " [ 0.62422045  2.05587489]\n",
-      " [ 1.75357263  5.63710438]\n",
-      " [-2.53968309 -7.28219089]\n",
-      " [-1.42054337 -4.90629812]\n",
-      " [-0.13019959 -0.83320794]\n",
-      " [ 0.18306165  0.26181254]]\n",
+      "[[ 0.32360769  2.53264317]\n",
+      " [ 0.04531375 -0.70569833]\n",
+      " [ 0.17001895 -0.49570819]\n",
+      " [ 1.60938882  4.69896355]\n",
+      " [ 0.14052537  1.37535105]\n",
+      " [-0.12909917  1.25781559]\n",
+      " [-0.03016916  0.01780471]\n",
+      " [-0.38816656 -0.82894017]\n",
+      " [-0.34591885 -3.1893772 ]\n",
+      " [-1.39550083 -4.66285417]]\n",
       "          0         1\n",
-      "0  1.207089  4.332018\n",
-      "1 -0.383878 -1.241252\n",
-      "2  0.747224  1.601942\n",
-      "3 -0.040863  0.374197\n",
-      "4  0.624220  2.055875\n",
-      "5  1.753573  5.637104\n",
-      "6 -2.539683 -7.282191\n",
-      "7 -1.420543 -4.906298\n",
-      "8 -0.130200 -0.833208\n",
-      "9  0.183062  0.261813\n",
+      "0  0.323608  2.532643\n",
+      "1  0.045314 -0.705698\n",
+      "2  0.170019 -0.495708\n",
+      "3  1.609389  4.698964\n",
+      "4  0.140525  1.375351\n",
+      "5 -0.129099  1.257816\n",
+      "6 -0.030169  0.017805\n",
+      "7 -0.388167 -0.828940\n",
+      "8 -0.345919 -3.189377\n",
+      "9 -1.395501 -4.662854\n",
       "          0         1\n",
-      "0  1.000000  0.992504\n",
-      "1  0.992504  1.000000\n"
+      "0  1.000000  0.899734\n",
+      "1  0.899734  1.000000\n"
      ]
     }
    ],
@@ -461,37 +461,37 @@
      "text": [
       "     0         1         2         3         4         5         6         7   \\\n",
       "0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   \n",
-      "1   0.0  0.092290  0.091376  0.091772  0.091659  0.091579  0.083691  0.083321   \n",
-      "2   0.0  0.091376  0.091209  0.090620  0.090923  0.091266  0.082166  0.082105   \n",
-      "3   0.0  0.091772  0.090620  0.098069  0.097435  0.096845  0.093556  0.092724   \n",
-      "4   0.0  0.091659  0.090923  0.097435  0.097111  0.096833  0.092478  0.091896   \n",
-      "5   0.0  0.091579  0.091266  0.096845  0.096833  0.096872  0.091445  0.091113   \n",
-      "6   0.0  0.083691  0.082166  0.093556  0.092478  0.091445  0.092007  0.090828   \n",
-      "7   0.0  0.083321  0.082105  0.092724  0.091896  0.091113  0.090828  0.089857   \n",
-      "8   0.0  0.083051  0.082145  0.091996  0.091418  0.090888  0.089744  0.088982   \n",
-      "9   0.0  0.082877  0.082285  0.091369  0.091044  0.090768  0.088754  0.088201   \n",
-      "10  0.0  0.075829  0.073985  0.087419  0.086021  0.084663  0.087860  0.086440   \n",
-      "11  0.0  0.075277  0.073684  0.086460  0.085272  0.084124  0.086624  0.085384   \n",
-      "12  0.0  0.074819  0.073477  0.085603  0.084624  0.083685  0.085484  0.084422   \n",
-      "13  0.0  0.074453  0.073363  0.084846  0.084076  0.083348  0.084438  0.083555   \n",
-      "14  0.0  0.074180  0.073344  0.084187  0.083627  0.083111  0.083485  0.082779   \n",
+      "1   0.0  0.086358  0.084977  0.084028  0.085456  0.086705  0.073624  0.075225   \n",
+      "2   0.0  0.084977  0.085778  0.080455  0.082791  0.085242  0.069320  0.071314   \n",
+      "3   0.0  0.084028  0.080455  0.086647  0.086999  0.086904  0.078715  0.079856   \n",
+      "4   0.0  0.085456  0.082791  0.086999  0.087848  0.088361  0.078426  0.079839   \n",
+      "5   0.0  0.086705  0.085242  0.086904  0.088361  0.089641  0.077606  0.079333   \n",
+      "6   0.0  0.073624  0.069320  0.078715  0.078426  0.077606  0.073298  0.074046   \n",
+      "7   0.0  0.075225  0.071314  0.079856  0.079839  0.079333  0.074046  0.074971   \n",
+      "8   0.0  0.076867  0.073465  0.080914  0.081223  0.081100  0.074653  0.075779   \n",
+      "9   0.0  0.078521  0.075782  0.081827  0.082527  0.082877  0.075047  0.076403   \n",
+      "10  0.0  0.063766  0.059453  0.069855  0.069291  0.068201  0.066209  0.066728   \n",
+      "11  0.0  0.065175  0.061037  0.071105  0.070700  0.069782  0.067233  0.067873   \n",
+      "12  0.0  0.066656  0.062742  0.072370  0.072150  0.071437  0.068239  0.069013   \n",
+      "13  0.0  0.068206  0.064577  0.073634  0.073630  0.073163  0.069203  0.070127   \n",
+      "14  0.0  0.069816  0.066556  0.074868  0.075121  0.074951  0.070092  0.071185   \n",
       "\n",
       "          8         9         10        11        12        13        14  \n",
       "0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  \n",
-      "1   0.083051  0.082877  0.075829  0.075277  0.074819  0.074453  0.074180  \n",
-      "2   0.082145  0.082285  0.073985  0.073684  0.073477  0.073363  0.073344  \n",
-      "3   0.091996  0.091369  0.087419  0.086460  0.085603  0.084846  0.084187  \n",
-      "4   0.091418  0.091044  0.086021  0.085272  0.084624  0.084076  0.083627  \n",
-      "5   0.090888  0.090768  0.084663  0.084124  0.083685  0.083348  0.083111  \n",
-      "6   0.089744  0.088754  0.087860  0.086624  0.085484  0.084438  0.083485  \n",
-      "7   0.088982  0.088201  0.086440  0.085384  0.084422  0.083555  0.082779  \n",
-      "8   0.088317  0.087746  0.085108  0.084230  0.083446  0.082756  0.082158  \n",
-      "9   0.087746  0.087386  0.083859  0.083159  0.082553  0.082040  0.081620  \n",
-      "10  0.085108  0.083859  0.085252  0.083832  0.082501  0.081258  0.080098  \n",
-      "11  0.084230  0.083159  0.083832  0.082569  0.081395  0.080305  0.079298  \n",
-      "12  0.083446  0.082553  0.082501  0.081395  0.080374  0.079437  0.078582  \n",
-      "13  0.082756  0.082040  0.081258  0.080305  0.079437  0.078652  0.077949  \n",
-      "14  0.082158  0.081620  0.080098  0.079298  0.078582  0.077949  0.077397  \n"
+      "1   0.076867  0.078521  0.063766  0.065175  0.066656  0.068206  0.069816  \n",
+      "2   0.073465  0.075782  0.059453  0.061037  0.062742  0.064577  0.066556  \n",
+      "3   0.080914  0.081827  0.069855  0.071105  0.072370  0.073634  0.074868  \n",
+      "4   0.081223  0.082527  0.069291  0.070700  0.072150  0.073630  0.075121  \n",
+      "5   0.081100  0.082877  0.068201  0.069782  0.071437  0.073163  0.074951  \n",
+      "6   0.074653  0.075047  0.066209  0.067233  0.068239  0.069203  0.070092  \n",
+      "7   0.075779  0.076403  0.066728  0.067873  0.069013  0.070127  0.071185  \n",
+      "8   0.076819  0.077713  0.067089  0.068365  0.069654  0.070936  0.072186  \n",
+      "9   0.077713  0.078926  0.067224  0.068643  0.070096  0.071567  0.073038  \n",
+      "10  0.067089  0.067224  0.060603  0.061468  0.062298  0.063070  0.063750  \n",
+      "11  0.068365  0.068643  0.061468  0.062424  0.063353  0.064232  0.065030  \n",
+      "12  0.069654  0.070096  0.062298  0.063353  0.064390  0.065389  0.066317  \n",
+      "13  0.070936  0.071567  0.063070  0.064232  0.065389  0.066519  0.067593  \n",
+      "14  0.072186  0.073038  0.063750  0.065030  0.066317  0.067593  0.068832  \n"
      ]
     }
    ],
@@ -916,10 +916,10 @@
      "output_type": "stream",
      "text": [
       "          0         1\n",
-      "0  3.956454  1.972286\n",
-      "1  1.972286  1.977089\n",
-      "[[3.95645365 1.97228638]\n",
-      " [1.97228638 1.97708897]]\n"
+      "0  4.059118  2.009163\n",
+      "1  2.009163  2.004788\n",
+      "[[4.05911793 2.00916336]\n",
+      " [2.00916336 2.00478786]]\n"
      ]
     }
    ],
@@ -949,13 +949,13 @@
      "output_type": "stream",
      "text": [
       "Centered covariance using own code\n",
-      "[[3.95645365 1.97228638]\n",
-      " [1.97228638 1.97708897]]\n"
+      "[[4.05911793 2.00916336]\n",
+      " [2.00916336 2.00478786]]\n"
      ]
     },
     {
      "data": {
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\n",
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\n",
       "text/plain": [
        "
    "
       ]
@@ -1044,12 +1044,12 @@
      "output_type": "stream",
      "text": [
       "Eigenvalues of Covariance matrix\n",
-      "5.173439546289586\n",
-      "0.7601030735620569\n",
+      "5.288455813429108\n",
+      "0.7754499790100834\n",
       "First eigenvector\n",
-      "[0.85102768 0.52512084]\n",
+      "[0.85299536 0.52191849]\n",
       "Second eigenvector\n",
-      "[-0.52512084  0.85102768]\n"
+      "[-0.52191849  0.85299536]\n"
      ]
     },
     {
@@ -1057,7 +1057,7 @@
      "output_type": "stream",
      "text": [
       "Eigenvector of largest eigenvalue\n",
-      "[-0.85102768 -0.52512084]\n"
+      "[0.85299536 0.52191849]\n"
      ]
     }
    ],
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter8_65_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter8_65_1.png
index e3a1b1fe7..ef2c77722 100644
Binary files a/doc/LectureNotes/_build/jupyter_execute/chapter8_65_1.png and b/doc/LectureNotes/_build/jupyter_execute/chapter8_65_1.png differ
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb
index a104689f4..65abce3f6 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization.ipynb
@@ -924,14 +924,14 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20669/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16261/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n",
       "  ax = fig.gca(projection=\"3d\")\n"
      ]
     },
     {
      "data": {
       "text/plain": [
-       ""
+       ""
       ]
      },
      "execution_count": 1,
@@ -1101,7 +1101,7 @@
     {
      "data": {
       "text/plain": [
-       "[]"
+       "[]"
       ]
      },
      "execution_count": 5,
@@ -1802,16 +1802,16 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[0.29972182 4.52744746]\n",
-      "[[4.01247056]\n",
-      " [2.97656972]]\n",
-      "[[4.01247056]\n",
-      " [2.97656972]]\n"
+      "[0.28836053 4.52113415]\n",
+      "[[4.19528375]\n",
+      " [2.90383424]]\n",
+      "[[4.19528375]\n",
+      " [2.90383424]]\n"
      ]
     },
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "
    "
       ]
@@ -1896,9 +1896,9 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[4.02158709]\n",
-      " [2.93023603]]\n",
-      "[4.00882596] [2.93522293]\n"
+      "[[4.16575256]\n",
+      " [2.8620652 ]]\n",
+      "[4.11520281] [2.85097049]\n"
      ]
     }
    ],
@@ -2002,15 +2002,15 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[4.09781386]\n",
-      " [2.97980332]]\n",
-      "[[4.0527437 ]\n",
-      " [3.01510929]]\n"
+      "[[4.20793824]\n",
+      " [2.75460639]]\n",
+      "[[4.106971  ]\n",
+      " [2.83724637]]\n"
      ]
     },
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "
    "
       ]
@@ -2389,20 +2389,20 @@
      "output_type": "stream",
      "text": [
       "Own inversion\n",
-      "[[3.87618586]\n",
-      " [3.13847924]]\n",
-      "Eigenvalues of Hessian Matrix:[0.3313155  4.62759057]\n",
+      "[[4.31347523]\n",
+      " [2.69915639]]\n",
+      "Eigenvalues of Hessian Matrix:[0.28457442 4.43693489]\n",
       "theta from own gd\n",
-      "[[3.87618586]\n",
-      " [3.13847924]]\n",
+      "[[4.31347523]\n",
+      " [2.69915639]]\n",
       "theta from own sdg\n",
-      "[[3.87379129]\n",
-      " [3.15508406]]\n"
+      "[[4.2891298 ]\n",
+      " [2.67783138]]\n"
      ]
     },
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "
    "
       ]
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_123_1.png b/doc/LectureNotes/_build/jupyter_execute/chapteroptimization_123_1.png
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diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb
new file mode 100644
index 000000000..c4fd6a7a0
--- /dev/null
+++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb
@@ -0,0 +1,393 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "50b2fbbe",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "\n",
+    ""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "da488c0c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Exercises week 36\n",
+    "**September 4-8, 2023**\n",
+    "\n",
+    "Date: **Deadline is Sunday September 10 at midnight**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b84d6b13",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Overarching aims of the exercises this week\n",
+    "\n",
+    "This set of exercises form an important part of the first project. The\n",
+    "analytical exercises deal with the material covered last week on the\n",
+    "mathematical interpretations of ordinary least squares and of Ridge\n",
+    "regression. The numerical exercises can be seen as a continuation of\n",
+    "exercise 3 from week 35, with the inclusion of Ridge regression. This\n",
+    "material enters also the discussions of the first project."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "96c9c28e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Exercise 1: Analytical exercises\n",
+    "\n",
+    "The aim here is to derive the expression for the optimal parameters\n",
+    "using Ridge regression. Furthermore, using the singular value\n",
+    "decomposition, we will analyze the difference between the ordinary\n",
+    "least squares approach and Ridge regression.\n",
+    "\n",
+    "The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, was given by the\n",
+    "optimization problem"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "439f1456",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b51e09f7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which we can also write as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "02c45981",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cc0e91ea",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have used the definition of  a norm-2 vector, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b5805f35",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6e3095bf",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "By minimizing the above equation with respect to the parameters\n",
+    "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n",
+    "parameters $\\boldsymbol{\\beta}$.\n",
+    "\n",
+    "We can add a regularization parameter $\\lambda$ by\n",
+    "defining a new cost function to be optimized, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "da90fe04",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1a106e07",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to the Ridge regression minimization problem. One can require as part of the optimization problem \n",
+    "that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n",
+    "a finite number larger than zero. We will not implement that here."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3917877b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "### a) Expression for Ridge regression\n",
+    "\n",
+    "Show that the optimal parameters"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "78226f28",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "951dfffa",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "21d2770e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ec212498",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with $t$ a finite positive number. \n",
+    "\n",
+    "The ordinary least squares result is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4ffabf6c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f97a6f45",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "### b) The singular value decomposition\n",
+    "\n",
+    "Use the singular value decomposition of an n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8761ed23",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "92f8479e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n",
+    "$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n",
+    "$n\\times p$ matrix which contains the ingular values only. This material was discussed during the lectures of week 35.\n",
+    "\n",
+    "Show that you can write the \n",
+    "OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix  $\\boldsymbol{U}$ as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9df91bda",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{y}}_{\\mathrm{OLS}}=\\boldsymbol{X}\\boldsymbol{\\beta}  = \\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e6de0312",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "For Ridge regression, show that the corresponding equation is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8e09d132",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{y}}_{\\mathrm{Ridge}}=\\boldsymbol{X}\\boldsymbol{\\beta}_{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{\\Sigma}^2\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a9c924ab",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$ from the SVD of the matrix $\\boldsymbol{X}$. \n",
+    "\n",
+    "Give an interpretation of the results.  [Section 3.4 of Hastie et al's textbook gives a good discussion of the above results](https://link.springer.com/book/10.1007/978-0-387-84858-7)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b9328a1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Exercise 2: Adding Ridge Regression\n",
+    "\n",
+    "This exercise is a continuation of exercise 3 from week 35, see . We will use the same function to\n",
+    "generate our data set, still staying with a simple function $y(x)$\n",
+    "which we want to fit using linear regression, but now extending the\n",
+    "analysis to include the Ridge regression method.\n",
+    "\n",
+    "In this exercise you need to include the same elements from last week, that is\n",
+    "1. scale your data by subtracting the mean value from each column in the design matrix.\n",
+    "\n",
+    "2. perform a split of the data in a training set and a test set.\n",
+    "\n",
+    "The addition to the analysis this time is the introduction of the hyperparameter $\\lambda$ when introducing Ridge regression.\n",
+    "\n",
+    "Extend the code from exercise 3 from [week 35](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek35.html) to include Ridge regression with the hyperparameter $\\lambda$. The optimal parameters $\\hat{\\beta}$ for Ridge regression can be obtained by matrix inversion in a similar way as done for ordinary least squares. You need to add to your code the following equations"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5b54b7b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a7ebe31b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The ordinary least squares result you encoded last week is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "874e0dd3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9adcc39f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Use these results to compute the mean squared error for ordinary least\n",
+    "squares and Ridge regression first for a polynomial of degree five\n",
+    "with $n=100$ data points and five selected values of\n",
+    "$\\lambda=[0.0001,0.001, 0.01,0.1,1.0]$. Compute thereafter the mean\n",
+    "squared error for the same values of $\\lambda$ for polynomials of degree ten\n",
+    "and $15$. Discuss your results for the training MSE and test MSE with\n",
+    "Ridge regression and ordinary least squares."
+   ]
+  }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.txt b/doc/LectureNotes/_build/jupyter_execute/exercisesweek36.txt
new file mode 100644
index 000000000..e69de29bb
diff --git a/doc/LectureNotes/_build/jupyter_execute/linalg.ipynb b/doc/LectureNotes/_build/jupyter_execute/linalg.ipynb
index 9bb1be44f..2f5f96d52 100644
--- a/doc/LectureNotes/_build/jupyter_execute/linalg.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/linalg.ipynb
@@ -225,8 +225,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[-0.38763091 -2.70501534 -0.3581571  -0.96251494 -1.26223899  0.35309734\n",
-      "  2.28186376 -1.85104809 -0.37114298 -1.20893188]\n"
+      "[-0.87136737  1.4300745  -0.3322326  -0.66758934 -1.00283636 -0.27625974\n",
+      "  1.89249454 -0.25006757  0.73565195  0.33697405]\n"
      ]
     }
    ],
@@ -662,26 +662,26 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[0.78459267 0.75453081 0.05363779 0.57350724 0.69764852 0.65795279\n",
-      "  0.51507839 0.20461136 0.38788697 0.96496641]\n",
-      " [0.25028968 0.96081861 0.18931988 0.51108791 0.30337713 0.43036842\n",
-      "  0.52839842 0.15321987 0.78561443 0.09030825]\n",
-      " [0.10097956 0.50584526 0.34989509 0.55626454 0.69154964 0.2895238\n",
-      "  0.13393141 0.15503141 0.26015755 0.42902155]\n",
-      " [0.25789255 0.9492866  0.90252116 0.904221   0.51933924 0.14432948\n",
-      "  0.54445121 0.02699523 0.18657863 0.971688  ]\n",
-      " [0.13392097 0.27801122 0.50931378 0.04234339 0.22442417 0.44065609\n",
-      "  0.74943449 0.42451192 0.33736485 0.97952271]\n",
-      " [0.95824108 0.59950055 0.91346044 0.58042237 0.13228567 0.31519573\n",
-      "  0.12427889 0.64736858 0.60236782 0.18036103]\n",
-      " [0.95911004 0.82027884 0.27547877 0.84317815 0.89842298 0.68322599\n",
-      "  0.02377668 0.39943328 0.00162091 0.0525221 ]\n",
-      " [0.94076632 0.88335933 0.75492292 0.7860324  0.41923956 0.86181269\n",
-      "  0.45979894 0.44190304 0.07829878 0.00458014]\n",
-      " [0.42915006 0.68127341 0.23875722 0.31988705 0.54956992 0.24801014\n",
-      "  0.65335632 0.97713364 0.05635863 0.12160173]\n",
-      " [0.93386901 0.74935095 0.96534137 0.98400474 0.98581925 0.30313128\n",
-      "  0.41386599 0.88450476 0.87099757 0.22566121]]\n"
+      "[[0.52428467 0.89421873 0.57286194 0.35473061 0.34626037 0.77640601\n",
+      "  0.60707801 0.60515083 0.41335011 0.16061448]\n",
+      " [0.51508374 0.7651899  0.07657669 0.14045566 0.92863147 0.32541271\n",
+      "  0.62761922 0.20965182 0.55051175 0.14072037]\n",
+      " [0.70311568 0.03617972 0.22335823 0.45797073 0.2223808  0.86208133\n",
+      "  0.64452882 0.0770789  0.7866024  0.45739087]\n",
+      " [0.38469441 0.16105443 0.09873132 0.21913656 0.34561469 0.521986\n",
+      "  0.89261458 0.75628425 0.25070907 0.96504592]\n",
+      " [0.40506137 0.48241947 0.32158804 0.45112054 0.5783575  0.3267061\n",
+      "  0.96261297 0.25216021 0.94271589 0.60651016]\n",
+      " [0.91410101 0.48217977 0.39116042 0.99397049 0.80768431 0.68650491\n",
+      "  0.04184636 0.5174581  0.86144422 0.46794929]\n",
+      " [0.62405149 0.27498198 0.63550707 0.85902603 0.38644889 0.86858926\n",
+      "  0.68282653 0.65554133 0.81051173 0.00281835]\n",
+      " [0.66001778 0.64858537 0.90034533 0.62389761 0.5333734  0.75390337\n",
+      "  0.97926642 0.9893405  0.61605739 0.51905011]\n",
+      " [0.59277468 0.52678301 0.68347072 0.76707201 0.08821204 0.55220861\n",
+      "  0.14648477 0.17606773 0.59609892 0.83723539]\n",
+      " [0.40961677 0.07325301 0.34652523 0.72201591 0.66250644 0.6367311\n",
+      "  0.79577619 0.85212606 0.86811137 0.43001767]]\n"
      ]
     }
    ],
@@ -800,13 +800,13 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.0626202708457115\n",
-      "4.327883798133277\n",
-      "0.3178411477108273\n",
-      "[[ 1.01422853  3.21909039  2.82750687]\n",
-      " [ 3.21909039 11.42694395  9.22887861]\n",
-      " [ 2.82750687  9.22887861 17.2086376 ]]\n",
-      "[24.72855434  0.09533574  4.82592   ]\n"
+      "0.03499693245669077\n",
+      "4.187268182169147\n",
+      "-0.19627430151896047\n",
+      "[[ 1.0622197   3.16232481  2.98018358]\n",
+      " [ 3.16232481 10.48442771  8.58039072]\n",
+      " [ 2.98018358  8.58039072 13.66351927]]\n",
+      "[21.70244042  0.07978179  3.42794448]\n"
      ]
     }
    ],
diff --git a/doc/LectureNotes/_build/jupyter_execute/statistics.ipynb b/doc/LectureNotes/_build/jupyter_execute/statistics.ipynb
index 81e6aed7a..99b1a0c6b 100644
--- a/doc/LectureNotes/_build/jupyter_execute/statistics.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/statistics.ipynb
@@ -1344,37 +1344,27 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "3.0818000034712947\n",
-      "[[1.82949718e-02 3.31658671e-01 3.68886089e-01 6.49666527e-01\n",
-      "  1.49052317e-01 5.97305528e-01 4.57810898e-01 2.51578522e-01\n",
-      "  2.89396164e-01 5.90342093e-01]\n",
-      " [3.31658671e-01 6.01244295e+00 6.68731669e+00 1.17774184e+01\n",
-      "  2.70208087e+00 1.08281969e+01 8.29938171e+00 4.56071752e+00\n",
-      "  5.24629107e+00 1.07019610e+01]\n",
-      " [3.68886089e-01 6.68731669e+00 7.43794243e+00 1.30993886e+01\n",
-      "  3.00537912e+00 1.20436207e+01 9.23095558e+00 5.07264063e+00\n",
-      "  5.83516719e+00 1.19032152e+01]\n",
-      " [6.49666527e-01 1.17774184e+01 1.30993886e+01 2.30700873e+01\n",
-      "  5.29294618e+00 2.12107137e+01 1.62571673e+01 8.93371943e+00\n",
-      "  1.02766489e+01 2.09634375e+01]\n",
-      " [1.49052317e-01 2.70208087e+00 3.00537912e+00 5.29294618e+00\n",
-      "  1.21435514e+00 4.86635201e+00 3.72986500e+00 2.04965397e+00\n",
-      "  2.35776087e+00 4.80961969e+00]\n",
-      " [5.97305528e-01 1.08281969e+01 1.20436207e+01 2.12107137e+01\n",
-      "  4.86635201e+00 1.95011994e+01 1.49468927e+01 8.21369083e+00\n",
-      "  9.44838453e+00 1.92738529e+01]\n",
-      " [4.57810898e-01 8.29938171e+00 9.23095558e+00 1.62571673e+01\n",
-      "  3.72986500e+00 1.49468927e+01 1.14561980e+01 6.29546690e+00\n",
-      "  7.24181044e+00 1.47726407e+01]\n",
-      " [2.51578522e-01 4.56071752e+00 5.07264063e+00 8.93371943e+00\n",
-      "  2.04965397e+00 8.21369083e+00 6.29546690e+00 3.45951629e+00\n",
-      "  3.97955570e+00 8.11793498e+00]\n",
-      " [2.89396164e-01 5.24629107e+00 5.83516719e+00 1.02766489e+01\n",
-      "  2.35776087e+00 9.44838453e+00 7.24181044e+00 3.97955570e+00\n",
-      "  4.57776818e+00 9.33823452e+00]\n",
-      " [5.90342093e-01 1.07019610e+01 1.19032152e+01 2.09634375e+01\n",
-      "  4.80961969e+00 1.92738529e+01 1.47726407e+01 8.11793498e+00\n",
-      "  9.33823452e+00 1.90491568e+01]]\n"
+      "1.9474044318317307\n",
+      "[[16.27404277  4.07430684  4.90189742  9.80462258  9.27412395  4.40264255\n",
+      "   5.15060311 17.7265799  -0.62769765  9.00658717]\n",
+      " [ 4.07430684  1.02002781  1.22722021  2.45464765  2.32183405  1.10222868\n",
+      "   1.28948521  4.43795846 -0.15714797  2.25485457]\n",
+      " [ 4.90189742  1.22722021  1.47649841  2.95324615  2.79345488  1.32611807\n",
+      "   1.55141095  5.33941551 -0.18906854  2.71287025]\n",
+      " [ 9.80462258  2.45464765  2.95324615  5.90699099  5.58738148  2.65246006\n",
+      "   3.10308387 10.67973263 -0.37816901  5.42619859]\n",
+      " [ 9.27412395  2.32183405  2.79345488  5.58738148  5.28506507  2.50894343\n",
+      "   2.93518534 10.10188442 -0.35770742  5.13260331]\n",
+      " [ 4.40264255  1.10222868  1.32611807  2.65246006  2.50894343  1.19105386\n",
+      "   1.39340081  4.79559972 -0.16981204  2.43656628]\n",
+      " [ 5.15060311  1.28948521  1.55141095  3.10308387  2.93518534  1.39340081\n",
+      "   1.63012429  5.61031939 -0.19866124  2.85051211]\n",
+      " [17.7265799   4.43795846  5.33941551 10.67973263 10.10188442  4.79559972\n",
+      "   5.61031939 19.308763   -0.68372271  9.8104687 ]\n",
+      " [-0.62769765 -0.15714797 -0.18906854 -0.37816901 -0.35770742 -0.16981204\n",
+      "  -0.19866124 -0.68372271  0.0242106  -0.3473884 ]\n",
+      " [ 9.00658717  2.25485457  2.71287025  5.42619859  5.13260331  2.43656628\n",
+      "   2.85051211  9.8104687  -0.3473884   4.98453971]]\n"
      ]
     }
    ],
@@ -2011,15 +2001,15 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "-0.054244842835462305\n",
-      "4.000854409696581\n",
-      "0.13083543199018746\n",
-      "0.8437169762110144 8.948675162607389 10.317825933186352\n",
-      "2.601583341274718 2.1245596497124075 6.443538568902246\n",
-      "[[ 0.84371698  2.60158334  2.12455965]\n",
-      " [ 2.60158334  8.94867516  6.44353857]\n",
-      " [ 2.12455965  6.44353857 10.31782593]]\n",
-      "[16.80422999  0.07128434  3.23470374]\n"
+      "0.08549632935144091\n",
+      "4.438111004052204\n",
+      "0.15485374225770068\n",
+      "0.8225325933960339 7.887392472315955 7.062734874360835\n",
+      "2.364244035362507 1.814860456269878 5.4421034379901165\n",
+      "[[0.82253259 2.36424404 1.81486046]\n",
+      " [2.36424404 7.88739247 5.44210344]\n",
+      " [1.81486046 5.44210344 7.06273487]]\n",
+      "[13.62123034  0.09662862  2.05480098]\n"
      ]
     }
    ],
@@ -2648,7 +2638,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "
    "
       ]
@@ -2774,12 +2764,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "-0.048547423739546604 0.9959293935368551\n"
+      "0.029574060388349064 0.9577775794806141\n"
      ]
     },
     {
      "data": {
-      "image/png": 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\n",
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diff --git a/doc/LectureNotes/_build/jupyter_execute/statistics_188_1.png b/doc/LectureNotes/_build/jupyter_execute/statistics_188_1.png
index ba13a577f..7e785ef0c 100644
Binary files a/doc/LectureNotes/_build/jupyter_execute/statistics_188_1.png and b/doc/LectureNotes/_build/jupyter_execute/statistics_188_1.png differ
diff --git a/doc/LectureNotes/_build/jupyter_execute/week34.ipynb b/doc/LectureNotes/_build/jupyter_execute/week34.ipynb
index 07c22d5a8..da02b85ed 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week34.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/week34.ipynb
@@ -985,8 +985,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[-0.03483677 -1.06136773  0.56481272 -0.35947075 -1.64972151 -1.58590795\n",
-      " -1.17428293  0.7225597  -1.14061742 -1.22893751]\n"
+      "[-0.75408649 -0.05177092 -0.07339251  1.49372893 -1.21467644  0.24712854\n",
+      "  0.67554302 -0.45143018  0.34212496  0.63262164]\n"
      ]
     }
    ],
@@ -1424,26 +1424,26 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[0.24279008 0.63112036 0.80947943 0.97509292 0.19425617 0.3957482\n",
-      "  0.1655226  0.83760781 0.07995375 0.76400155]\n",
-      " [0.75621895 0.20893514 0.93082503 0.79419162 0.02783644 0.21296315\n",
-      "  0.64298419 0.34578026 0.60975366 0.46369869]\n",
-      " [0.77859437 0.23477043 0.35438626 0.63115792 0.2460037  0.35568525\n",
-      "  0.0825971  0.94117118 0.14900336 0.30035718]\n",
-      " [0.36692356 0.78972773 0.67655635 0.67160204 0.80108096 0.31507591\n",
-      "  0.21328866 0.41340248 0.3005849  0.40672425]\n",
-      " [0.21922061 0.88274486 0.86572911 0.06486061 0.07565581 0.26678445\n",
-      "  0.03265139 0.22090974 0.33135331 0.66973261]\n",
-      " [0.7221662  0.96941962 0.39707147 0.24929083 0.31531613 0.33079801\n",
-      "  0.06538944 0.42352791 0.94227931 0.27809912]\n",
-      " [0.07195822 0.31719317 0.47248297 0.18264218 0.64033527 0.51146442\n",
-      "  0.49545491 0.91936525 0.81656508 0.78329097]\n",
-      " [0.58666142 0.01646892 0.11029323 0.67442363 0.6914791  0.87902877\n",
-      "  0.98950411 0.27090196 0.08732305 0.89543736]\n",
-      " [0.25253892 0.57505712 0.24848907 0.9631064  0.46312791 0.96431281\n",
-      "  0.28744729 0.09772449 0.17676228 0.51656406]\n",
-      " [0.68664664 0.63281351 0.62806444 0.36809474 0.98129668 0.62914124\n",
-      "  0.37885034 0.6577093  0.57947706 0.24688732]]\n"
+      "[[0.11167579 0.88778091 0.24908246 0.32566092 0.37585988 0.61893735\n",
+      "  0.58574143 0.60081665 0.81065392 0.00509527]\n",
+      " [0.02840884 0.39600607 0.96810393 0.28213741 0.19237495 0.54219245\n",
+      "  0.85581869 0.17928538 0.43185254 0.37624303]\n",
+      " [0.82045496 0.22828694 0.52773891 0.54526769 0.84708672 0.59118929\n",
+      "  0.18221254 0.94640767 0.25328924 0.03192893]\n",
+      " [0.98328759 0.68996179 0.81168086 0.08086066 0.10659682 0.66599246\n",
+      "  0.78035464 0.57403561 0.96995249 0.78990036]\n",
+      " [0.9909606  0.39920034 0.59813773 0.85982436 0.66665241 0.60522226\n",
+      "  0.08889302 0.74433499 0.85630009 0.2745333 ]\n",
+      " [0.60826013 0.94413158 0.32305813 0.27011455 0.38636467 0.02568505\n",
+      "  0.0610602  0.92275042 0.09189927 0.67177934]\n",
+      " [0.72097126 0.35819312 0.98060869 0.46573851 0.70482138 0.51749473\n",
+      "  0.17012048 0.53752634 0.66726905 0.08874264]\n",
+      " [0.79974359 0.05023564 0.84150451 0.05491301 0.90643349 0.75505597\n",
+      "  0.7300372  0.57445493 0.39626502 0.22016672]\n",
+      " [0.44463741 0.01448144 0.01167264 0.85595734 0.09989861 0.81007682\n",
+      "  0.30844205 0.68075712 0.9067362  0.96317459]\n",
+      " [0.87327422 0.12086283 0.50504743 0.60715594 0.9440887  0.54041662\n",
+      "  0.7316348  0.42976761 0.63933182 0.22772039]]\n"
      ]
     }
    ],
@@ -1557,13 +1557,13 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "-0.19542686939553625\n",
-      "3.3929303193461995\n",
-      "-0.3886211842840971\n",
-      "[[ 0.96552318  2.67354673  2.55640013]\n",
-      " [ 2.67354673  8.3318361   7.41646135]\n",
-      " [ 2.55640013  7.41646135 11.19890453]]\n",
-      "[18.10686259  0.09289204  2.29650918]\n"
+      "-0.05279147850537411\n",
+      "3.9698377974429513\n",
+      "0.22403397554517984\n",
+      "[[ 1.03173163  3.03859372  3.15084741]\n",
+      " [ 3.03859372  9.96514849  9.33589475]\n",
+      " [ 3.15084741  9.33589475 15.3141149 ]]\n",
+      "[23.20313883  0.08502083  3.02283536]\n"
      ]
     }
    ],
@@ -1916,7 +1916,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20702/1326197715.py:6: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_16293/1326197715.py:6: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.\n",
       "  data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))\n"
      ]
     },
diff --git a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
index f69ef471e..cc1020ef3 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
@@ -1519,7 +1519,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.9949347794612255\n"
+      "0.9959033816551833\n"
      ]
     }
    ],
@@ -1550,7 +1550,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0.01028995427580139\n"
+      "0.009290411029763584\n"
      ]
     }
    ],
@@ -1585,31 +1585,23 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[3.27961873e-03 1.75320666e-02 3.97840267e-02 1.41369795e-02\n",
-      " 1.12890007e-02 3.55765644e-02 1.34914442e-03 9.92463154e-03\n",
-      " 3.86416637e-02 8.40062254e-03 1.58912508e-02 7.41091395e-02\n",
-      " 2.28266792e-02 1.29662593e-03 1.47977394e-02 1.48877744e-02\n",
-      " 3.81874188e-02 7.92322465e-02 3.43012535e-02 1.47732343e-02\n",
-      " 2.59567145e-02 1.77816763e-02 1.07826171e-02 3.45241645e-02\n",
-      " 3.01752962e-02 4.99180856e-03 2.49838167e-04 4.38432672e-03\n",
-      " 5.93034176e-03 2.54644646e-02 1.76834212e-02 1.85785878e-02\n",
-      " 3.70827971e-02 9.47008799e-03 6.04008203e-02 2.45865699e-02\n",
-      " 2.43113284e-03 2.65643756e-02 1.40768645e-01 4.45840226e-03\n",
-      " 1.23365145e-02 3.49001938e-02 3.65693278e-02 2.08966600e-02\n",
-      " 1.45112450e-02 1.27524966e-02 7.26193455e-02 3.00129543e-02\n",
-      " 5.95594445e-03 1.06826023e-02 1.32609560e-02 1.42729886e-02\n",
-      " 2.87131961e-02 6.39892216e-02 4.19650622e-02 2.78272242e-02\n",
-      " 1.19023414e-02 2.32474290e-02 5.36904006e-02 7.48259891e-03\n",
-      " 2.74997869e-02 3.08549564e-02 4.12211324e-02 1.06229862e-02\n",
-      " 4.41816468e-02 5.87890188e-04 4.41486312e-02 1.41083418e-02\n",
-      " 5.35759569e-03 2.89869600e-02 2.20517254e-02 2.65534935e-02\n",
-      " 1.04481443e-02 2.47355725e-02 1.00634741e-02 2.73185363e-02\n",
-      " 1.13375152e-02 4.40236659e-02 1.20669787e-02 7.63258368e-02\n",
-      " 1.13208672e-04 1.15746031e-02 9.91094169e-03 4.09789363e-02\n",
-      " 5.98604688e-02 1.47082726e-02 1.54957609e-02 3.71921767e-02\n",
-      " 2.85688243e-03 7.15752533e-04 4.77553346e-02 2.33370315e-03\n",
-      " 3.30638512e-05 7.19284552e-02 5.16049171e-02 7.79000942e-03\n",
-      " 2.70465373e-02 5.84696315e-02 5.47099765e-02 6.36462687e-02]\n"
+      "[0.02126934 0.07602623 0.0014937  0.01294617 0.05421908 0.02237669\n",
+      " 0.0026081  0.00564956 0.00399247 0.05763394 0.00040953 0.06861085\n",
+      " 0.0098629  0.01199845 0.0097577  0.03260201 0.020964   0.01933058\n",
+      " 0.02255062 0.01801774 0.04039087 0.00472303 0.01003763 0.01487742\n",
+      " 0.05554042 0.00886682 0.05110883 0.02944194 0.00806407 0.01028231\n",
+      " 0.03613949 0.03352185 0.0512238  0.01525206 0.00660801 0.01073938\n",
+      " 0.06353697 0.00700232 0.0391902  0.08741274 0.01227458 0.01049472\n",
+      " 0.04691549 0.00963223 0.0143088  0.05177527 0.00850988 0.01121347\n",
+      " 0.02768957 0.02259051 0.02233576 0.01322543 0.02143332 0.01400329\n",
+      " 0.00102864 0.01322099 0.00611932 0.01011376 0.13281267 0.00684221\n",
+      " 0.05358851 0.02232779 0.00695738 0.03054765 0.00554475 0.05748797\n",
+      " 0.03507211 0.00563446 0.03123832 0.00033779 0.01122997 0.1098906\n",
+      " 0.07003926 0.03718926 0.0695405  0.00605451 0.0456042  0.00477722\n",
+      " 0.01224109 0.01072866 0.04273116 0.01873409 0.02903947 0.01927709\n",
+      " 0.00819724 0.00628788 0.00086553 0.02341603 0.0525063  0.03546779\n",
+      " 0.03012368 0.05069808 0.00327082 0.00517074 0.00071305 0.01194406\n",
+      " 0.05454172 0.02480935 0.00577016 0.02925853]\n"
      ]
     }
    ],
@@ -1663,15 +1655,15 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 1.96667129  1.00622755  0.25455904  7.36570403 -3.55965719]\n",
+      "[ 2.04860436 -0.39444293  5.97533203 -0.78980112  0.09575221]\n",
       "Training R2\n",
-      "0.9967912002709148\n",
+      "0.9960913291755783\n",
       "Training MSE\n",
-      "0.007098683295819775\n",
+      "0.008823125370709272\n",
       "Test R2\n",
-      "0.9957763445154583\n",
+      "0.9906601091977738\n",
       "Test MSE\n",
-      "0.008754614401844201\n"
+      "0.01969004537138309\n"
      ]
     }
    ],
diff --git a/doc/LectureNotes/_build/jupyter_execute/week36.ipynb b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
new file mode 100644
index 000000000..20ea50cf2
--- /dev/null
+++ b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
@@ -0,0 +1,3439 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b49fd9eb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "\n",
+    ""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "138fe268",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Week 36: Statistical interpretation of Linear Regression and Resampling techniques\n",
+    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
+    "\n",
+    "Date: **September 4-8, 2023**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c20f461b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Plans for week 36\n",
+    "\n",
+    "* Material for the active learning sessions on Tuesday and Wednesday\n",
+    "\n",
+    "  * Summary from last week on discussion of SVD, Ridge and Lasso linear regression.\n",
+    "\n",
+    "  * Recommended Reading: Hastie et al chapter 3, see \n",
+    "\n",
+    "  * Presentation and discussion of first project\n",
+    "\n",
+    "* Material for the lecture on Thursday September 7\n",
+    "\n",
+    "  * Linear Regression and links with Statistics, Resampling methods\n",
+    "\n",
+    "  * Recommended Reading: Goodfellow et al chapter 3 on probability theory, see URL:\"\"\n",
+    "\n",
+    "  * See also Murphy, sections 2.4 (Gaussian distributions) and 3.2 (Bayesian Statistics, basis)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "55f0d080",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Material for the active learning sessions Tuesday and Wednesday\n",
+    "\n",
+    "The material here contains a summary from last Week and discussion of SVD, Ridge and Lasso regression with examples"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "69ce62a6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Linear Regression and  the SVD\n",
+    "\n",
+    "We used the SVD to analyse the matrix to invert in ordinary lineat regression"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a9aaa130",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{V}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2b1c0902",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Since the matrices here have dimension $p\\times p$, with $p$ corresponding to the singular values, we defined last week the matrix"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "093e3a8c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma} = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c0102cd2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where the tilde-matrix $\\tilde{\\boldsymbol{\\Sigma}}$ is a matrix of dimension $p\\times p$ containing only the singular values $\\sigma_i$, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "084138a1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{\\Sigma}}=\\begin{bmatrix} \\sigma_0 & 0 & 0 & \\dots & 0 & 0 \\\\\n",
+    "                                    0 & \\sigma_1 & 0 & \\dots & 0 & 0 \\\\\n",
+    "\t\t\t\t    0 & 0 & \\sigma_2 & \\dots & 0 & 0 \\\\\n",
+    "\t\t\t\t    0 & 0 & 0 & \\dots & \\sigma_{p-2} & 0 \\\\\n",
+    "\t\t\t\t    0 & 0 & 0 & \\dots & 0 & \\sigma_{p-1} \\\\\n",
+    "\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cb8ed4f2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "meaning we can write"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d7c1c21b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2\\boldsymbol{V}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "791e9c6d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Multiplying from the right with $\\boldsymbol{V}$ (using the orthogonality of $\\boldsymbol{V}$) we get"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "924b8081",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{V}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "949f6658",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## What does it mean?\n",
+    "\n",
+    "This means the vectors $\\boldsymbol{v}_i$ of the orthogonal matrix $\\boldsymbol{V}$\n",
+    "are the eigenvectors of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ with eigenvalues\n",
+    "given by the singular values squared, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "46a4a83c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{v}_i=\\boldsymbol{v}_i\\sigma_i^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ce019aa6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In other words, each non-zero singular value of $\\boldsymbol{X}$ is a positive\n",
+    "square root of an eigenvalue of $\\boldsymbol{X}^T\\boldsymbol{X}$.  It means also that\n",
+    "the columns of $\\boldsymbol{V}$ are the eigenvectors of\n",
+    "$\\boldsymbol{X}^T\\boldsymbol{X}$. Since we have ordered the singular values of\n",
+    "$\\boldsymbol{X}$ in a descending order, it means that the column vectors\n",
+    "$\\boldsymbol{v}_i$ are hierarchically ordered by how much correlation they\n",
+    "encode from the columns of $\\boldsymbol{X}$. \n",
+    "\n",
+    "Note that these are also the eigenvectors and eigenvalues of the\n",
+    "Hessian matrix.\n",
+    "\n",
+    "If we now recall the definition of the covariance matrix (not using\n",
+    "Bessel's correction) we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c662d90f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{C}[\\boldsymbol{X}]=\\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7df45c97",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "meaning that every squared non-singular value of $\\boldsymbol{X}$ divided by $n$ (\n",
+    "the number of samples) are the eigenvalues of the covariance\n",
+    "matrix. Every singular value of $\\boldsymbol{X}$ is thus a positive square\n",
+    "root of an eigenvalue of $\\boldsymbol{X}^T\\boldsymbol{X}$. If the matrix $\\boldsymbol{X}$ is\n",
+    "self-adjoint, the singular values of $\\boldsymbol{X}$ are equal to the\n",
+    "absolute value of the eigenvalues of $\\boldsymbol{X}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f0741872",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## And finally  $\\boldsymbol{X}\\boldsymbol{X}^T$\n",
+    "\n",
+    "For $\\boldsymbol{X}\\boldsymbol{X}^T$ we found"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "05530ccb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{U}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3048f911",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Since the matrices here have dimension $n\\times n$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9e3c2de1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\Sigma}\\boldsymbol{\\Sigma}^T = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}}  \\boldsymbol{0}\\\\ \\end{bmatrix}=\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "491923c1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "leading to"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "77d1dbe5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\boldsymbol{U}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fb7e2323",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Multiplying with $\\boldsymbol{U}$ from the right gives us the eigenvalue problem"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c9cca57a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U}=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "04e4b486",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "It means that the eigenvalues of $\\boldsymbol{X}\\boldsymbol{X}^T$ are again given by\n",
+    "the non-zero singular values plus now a series of zeros.  The column\n",
+    "vectors of $\\boldsymbol{U}$ are the eigenvectors of $\\boldsymbol{X}\\boldsymbol{X}^T$ and\n",
+    "measure how much correlations are contained in the rows of $\\boldsymbol{X}$.\n",
+    "\n",
+    "Since we will mainly be interested in the correlations among the features\n",
+    "of our data (the columns of $\\boldsymbol{X}$, the quantity of interest for us are the non-zero singular\n",
+    "values and the column vectors of $\\boldsymbol{V}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2b3c22fb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Code for SVD and Inversion of Matrices\n",
+    "\n",
+    "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n",
+    "The simple answer is to use the linear algebra function for pseudoinvers, that is"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "12637a17",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [
+    {
+     "ename": "NameError",
+     "evalue": "name 'np' is not defined",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
+      "Input \u001b[0;32mIn [1]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m Ainv \u001b[38;5;241m=\u001b[39m \u001b[43mnp\u001b[49m\u001b[38;5;241m.\u001b[39mlinlag\u001b[38;5;241m.\u001b[39mpinv(A)\n",
+      "\u001b[0;31mNameError\u001b[0m: name 'np' is not defined"
+     ]
+    }
+   ],
+   "source": [
+    "Ainv = np.linlag.pinv(A)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "904adaf8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "6b5465dc",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "# SVD inversion\n",
+    "def SVDinv(A):\n",
+    "    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).\n",
+    "    SVD is numerically more stable than the inversion algorithms provided by\n",
+    "    numpy and scipy.linalg at the cost of being slower.\n",
+    "    '''\n",
+    "    U, s, VT = np.linalg.svd(A)\n",
+    "    print('test U')\n",
+    "    print( (np.transpose(U) @ U - U @np.transpose(U)))\n",
+    "    print('test VT')\n",
+    "    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n",
+    "\n",
+    "\n",
+    "    D = np.zeros((len(U),len(VT)))\n",
+    "    D = np.diag(s)\n",
+    "    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)\n",
+    "    return np.matmul(V,np.matmul(invD,UT))\n",
+    "\n",
+    "\n",
+    "#X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])\n",
+    "# Non-singular square matrix\n",
+    "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n",
+    "print(X)\n",
+    "A = np.transpose(X) @ X\n",
+    "# Brute force inversion\n",
+    "B = np.linalg.inv(A)  # here we could use np.linalg.pinv(A)\n",
+    "C = SVDinv(A)\n",
+    "print(np.abs(B-C))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a80dcee4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Inverse of Rectangular Matrix\n",
+    "\n",
+    "Although our matrix to invert $\\boldsymbol{X}^T\\boldsymbol{X}$ is a square matrix, our matrix may be singular. \n",
+    "\n",
+    "The pseudoinverse is the generalization of the matrix inverse for square matrices to\n",
+    "rectangular matrices where the number of rows and columns are not equal.\n",
+    "\n",
+    "It is also called the the Moore-Penrose Inverse after two independent discoverers of the method or the Generalized Inverse.\n",
+    "It is used for the calculation of the inverse for singular or near singular matrices and for rectangular matrices.\n",
+    "\n",
+    "Using the SVD we can obtain the pseudoinverse of a matrix $\\boldsymbol{A}$ (labeled here as $\\boldsymbol{A}_{\\mathrm{PI}}$)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0c4052d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{A}_{\\mathrm{PI}}= \\boldsymbol{V}\\boldsymbol{D}_{\\mathrm{PI}}\\boldsymbol{U}^T,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "185d433f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $\\boldsymbol{D}_{\\mathrm{PI}}$ can be calculated by creating a diagonal matrix from $\\boldsymbol{\\Sigma}$ where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "418f8797",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "# SVD inversion\n",
+    "def SVDinv(A):\n",
+    "    U, s, VT = np.linalg.svd(A)\n",
+    "    # reciprocals of singular values of s\n",
+    "    d = 1.0 / s\n",
+    "    # create m x n D matrix\n",
+    "    D = np.zeros(A.shape)\n",
+    "    # populate D with n x n diagonal matrix\n",
+    "    D[:A.shape[1], :A.shape[1]] = np.diag(d)\n",
+    "    UT = np.transpose(U)\n",
+    "    V = np.transpose(VT)\n",
+    "    return np.matmul(V,np.matmul(D.T,UT))\n",
+    "\n",
+    "\n",
+    "A = np.array([ [0.3, 0.4], [0.5, 0.6], [0.7, 0.8],[0.9, 1.0]])\n",
+    "print(A)\n",
+    "# Brute force inversion of super-collinear matrix\n",
+    "B = np.linalg.pinv(A)\n",
+    "print(B)\n",
+    "# Compare our own algorithm with pinv\n",
+    "C = SVDinv(A)\n",
+    "print(np.abs(C-B))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bf5051ea",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "As you can see from this example, our own decomposition based on the SVD agrees with  the pseudoinverse algorithm provided by **Numpy**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "202d1f90",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge and LASSO Regression\n",
+    "\n",
+    "Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is \n",
+    "our optimization problem is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "df0a8718",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4ebd1d20",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "or we can state it as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40df3859",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "980001f5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have used the definition of  a norm-2 vector, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "699b1198",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ff01ef41",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## From OLS to Ridge and Lasso\n",
+    "\n",
+    "By minimizing the above equation with respect to the parameters\n",
+    "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n",
+    "parameters $\\boldsymbol{\\beta}$.  We can add a regularization parameter $\\lambda$ by\n",
+    "defining a new cost function to be optimized, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d7b9188f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "778790dc",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to the Ridge regression minimization problem where we\n",
+    "require that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n",
+    "a finite number larger than zero. We do not include such a constraints in the discussions here.\n",
+    "\n",
+    "By defining"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38dd7428",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6ed3deb0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "we have a new optimization equation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e0860754",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "66751140",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. \n",
+    "\n",
+    "Here we have defined the norm-1 as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e1fa2bdb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\vert\\vert \\boldsymbol{x}\\vert\\vert_1 = \\sum_i \\vert x_i\\vert.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "32131982",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Deriving the  Ridge Regression Equations\n",
+    "\n",
+    "Using the matrix-vector expression for Ridge regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c1979796",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\boldsymbol{\\beta}^T\\boldsymbol{\\beta},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d7eb56d7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and \n",
+    "taking the derivatives with respect to $\\boldsymbol{\\beta}$ we obtain then\n",
+    "a slightly modified matrix inversion problem which for finite values\n",
+    "of $\\lambda$ does not suffer from singularity problems. We obtain\n",
+    "the optimal parameters"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7c42f326",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6496e5c0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "837f7c04",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "55e9395f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with $t$ a finite positive number."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a943f34a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Note on Scikit-Learn\n",
+    "\n",
+    "Note well that a library like **Scikit-Learn** does not include the $1/n$ factor in the expression for the mean-squared error. If you include it, the optimal parameter $\\beta$ becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "63a227a9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+n\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8be2bd7b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In our codes where we compare our own codes with **Scikit-Learn**, we do thus not include the $1/n$ factor in the cost function."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "35509652",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Comparison with OLS\n",
+    "When we compare this with the ordinary least squares result we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2fbe21d0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "71e986f8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$.\n",
+    "\n",
+    "We see that Ridge regression is nothing but the standard OLS with a\n",
+    "modified diagonal term added to $\\boldsymbol{X}^T\\boldsymbol{X}$. The consequences, in\n",
+    "particular for our discussion of the bias-variance tradeoff are rather\n",
+    "interesting. We will see that for specific values of $\\lambda$, we may\n",
+    "even reduce the variance of the optimal parameters $\\boldsymbol{\\beta}$. These topics and other related ones, will be discussed after the more linear algebra oriented analysis here."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a5e5f40",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## SVD analysis\n",
+    "\n",
+    "Using our insights about the SVD of the design matrix $\\boldsymbol{X}$ \n",
+    "We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\\boldsymbol{U}$ as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "752b6ba2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{y}}_{\\mathrm{OLS}}=\\boldsymbol{X}\\boldsymbol{\\beta}  =\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4e724327",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "For Ridge regression this becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b15c2c90",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{y}}_{\\mathrm{Ridge}}=\\boldsymbol{X}\\boldsymbol{\\beta}_{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{\\Sigma}^2\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "82e70fa5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$ from the SVD of the matrix $\\boldsymbol{X}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3394e2e7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Interpreting the Ridge results\n",
+    "\n",
+    "Since $\\lambda \\geq 0$, it means that compared to OLS, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "722ffa86",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda} \\leq 1.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "720ed773",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Ridge regression finds the coordinates of $\\boldsymbol{y}$ with respect to the\n",
+    "orthonormal basis $\\boldsymbol{U}$, it then shrinks the coordinates by\n",
+    "$\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}$. Recall that the SVD has\n",
+    "eigenvalues ordered in a descending way, that is $\\sigma_i \\geq\n",
+    "\\sigma_{i+1}$.\n",
+    "\n",
+    "For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b57864f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## More interpretations\n",
+    "\n",
+    "For the sake of simplicity, let us assume that the design matrix is orthonormal, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8f36ff27",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1} =\\boldsymbol{I}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "70adc9ce",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In this case the standard OLS results in"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eb4a995f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\beta}^{\\mathrm{OLS}} = \\boldsymbol{X}^T\\boldsymbol{y}=\\sum_{i=0}^{n-1}\\boldsymbol{u}_i\\boldsymbol{u}_i^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3cd4b92",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bbf9b1de",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{I}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\left(1+\\lambda\\right)^{-1}\\boldsymbol{\\beta}^{\\mathrm{OLS}},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e30eff48",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\\lambda$, and\n",
+    "the Ridge estimator converges to zero when the hyperparameter goes to\n",
+    "infinity.\n",
+    "\n",
+    "We will come back to more interpreations after we have gone through some of the statistical analysis part. \n",
+    "\n",
+    "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n",
+    "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9a8f8e2e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Deriving the  Lasso Regression Equations\n",
+    "\n",
+    "Using the matrix-vector expression for Lasso regression, we have the following **cost** function"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3cb7928c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b1c6c813",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Taking the derivative with respect to $\\boldsymbol{\\beta}$ and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicity)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "da336775",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{d \\vert \\beta\\vert}{d \\beta}=\\mathrm{sgn}(\\beta)=\\left\\{\\begin{array}{cc} 1 & \\beta > 0 \\\\-1 & \\beta < 0, \\end{array}\\right.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2e0de141",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "we have that the derivative of the cost function is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40985ec4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C(\\boldsymbol{X},\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}=-\\frac{2}{n}\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})+\\lambda sgn(\\boldsymbol{\\beta})=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "31091d7b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and reordering we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e81d965e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta}+\\lambda sgn(\\boldsymbol{\\beta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9984b3a4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor $2/n$ in a redefinition of the parameter $\\lambda$. We will solve this type of problems using libraries like **scikit-learn**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "21ec3768",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression\n",
+    "\n",
+    "Let us assume that our design matrix is given by unit (identity) matrix, that is a square diagonal matrix with ones only along the\n",
+    "diagonal. In this case we have an equal number of rows and columns $n=p$.\n",
+    "\n",
+    "Our model approximation is just $\\tilde{\\boldsymbol{y}}=\\boldsymbol{\\beta}$ and the mean squared error and thereby the cost function for ordinary least sqquares (OLS) is then (we drop the term $1/n$)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6be0ec63",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "44c7278c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and minimizing we have that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "918a8cc3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\beta}_i^{\\mathrm{OLS}} = y_i.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3949af51",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge Regression\n",
+    "\n",
+    "For Ridge regression our cost function is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aa856d18",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\beta_i^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bd30d1d4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and minimizing we have that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "142de535",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\beta}_i^{\\mathrm{Ridge}} = \\frac{y_i}{1+\\lambda}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1efee0c3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Lasso Regression\n",
+    "\n",
+    "For Lasso regression our cost function is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6db019a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\vert\\beta_i\\vert=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\sqrt{\\beta_i^2},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0f38deda",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and minimizing we have that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7f9738a5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "-2\\sum_{i=0}^{p-1}(y_i-\\beta_i)+\\lambda \\sum_{i=0}^{p-1}\\frac{(\\beta_i)}{\\vert\\beta_i\\vert}=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4e8460f8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ae49ddce",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_i^{\\mathrm{Lasso}} = \\left\\{\\begin{array}{ccc}y_i-\\frac{\\lambda}{2} &\\mathrm{if} & y_i> \\frac{\\lambda}{2}\\\\\n",
+    "                                                          y_i+\\frac{\\lambda}{2} &\\mathrm{if} & y_i< -\\frac{\\lambda}{2}\\\\\n",
+    "\t\t\t\t\t\t\t  0 &\\mathrm{if} & \\vert y_i\\vert\\le  \\frac{\\lambda}{2}\\end{array}\\right.\\\\.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1efb96d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Plotting these results shows clearly that Lasso regression suppresses (sets to zero) values of $\\beta_i$ for specific values of $\\lambda$. Ridge regression reduces on the other hand the values of $\\beta_i$ as function of $\\lambda$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1a030291",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Yet another Example\n",
+    "\n",
+    "Let us assume we have a data set with outputs/targets given by the vector"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af486213",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{y}=\\begin{bmatrix}4 \\\\ 2 \\\\3\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "87e5470d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and our inputs as a $3\\times 2$ design matrix"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "182f9f51",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}=\\begin{bmatrix}2 & 0\\\\ 0 & 1 \\\\ 0 & 0\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "edc0ee3d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "meaning that we have two features and two unknown parameters $\\beta_0$ and $\\beta_1$ to be determined either by ordinary least squares, Ridge or Lasso regression."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "673c64a0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The OLS case\n",
+    "\n",
+    "For ordinary least squares (OLS) we know that the optimal solution is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c0554476",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cc9c761b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Inserting the above values we obtain that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b5c84ed1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\begin{bmatrix}2 \\\\ 2\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a8d2ba86",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The code which implements this simpler case is presented after the discussion of Ridge and Lasso."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9e407326",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The Ridge case\n",
+    "\n",
+    "For Ridge regression we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "36ce4758",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8665e33b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Inserting the above values we obtain that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "16dbcd82",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\begin{bmatrix}\\frac{8}{4+\\lambda} \\\\ \\frac{2}{1+\\lambda}\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "642f7768",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "There is normally a constraint on the value of $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2$ via the parameter $\\lambda$.\n",
+    "Let us for simplicity assume that $\\beta_0^2+\\beta_1^2=1$ as constraint. This will allow us to find an expression for the optimal values of $\\beta$ and $\\lambda$.\n",
+    "\n",
+    "To see this, let us write the cost function for Ridge regression."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1784c79c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Writing the Cost Function\n",
+    "\n",
+    "We define the MSE without the $1/n$ factor and have then, using that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "72b7600f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}\\boldsymbol{\\beta}=\\begin{bmatrix} 2\\beta_0 \\\\ \\beta_1 \\\\0 \\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84ca3171",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\beta_0^2+\\beta_1^2),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d31aa45e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and taking the derivative with respect to $\\beta_0$ we get"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1298ca6d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0=\\frac{8}{4+\\lambda},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fa2455d6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and for $\\beta_1$ we obtain"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8cd50073",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_1=\\frac{2}{1+\\lambda},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1f7e12c3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Using the constraint for $\\beta_0^2+\\beta_1^2=1$ we can constrain $\\lambda$ by solving"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a79821fd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\left(\\frac{8}{4+\\lambda}\\right)^2+\\left(\\frac{2}{1+\\lambda}\\right)^2=1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2ac13531",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which gives $\\lambda=4.571$ and $\\beta_0=0.933$ and $\\beta_1=0.359$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eed25f68",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Lasso case\n",
+    "\n",
+    "For Lasso we need now, keeping a  constraint on $\\vert\\beta_0\\vert+\\vert\\beta_1\\vert=1$,  to take the derivative of the absolute values of $\\beta_0$\n",
+    "and $\\beta_1$. This gives us the following derivatives of the cost function"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f74d71ec",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\vert\\beta_0\\vert+\\vert\\beta_1\\vert),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f0797067",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_0}=-4(4-2\\beta_0)+\\lambda\\mathrm{sgn}(\\beta_0)=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3cc5d5e0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "04250e5c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_1}=-2(2-\\beta_1)+\\lambda\\mathrm{sgn}(\\beta_1)=0.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2b24d2fa",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We have now four cases to solve besides the trivial cases $\\beta_0$ and/or $\\beta_1$ are zero, namely\n",
+    "1. $\\beta_0 > 0$ and $\\beta_1 > 0$,\n",
+    "\n",
+    "2. $\\beta_0 > 0$ and $\\beta_1 < 0$,\n",
+    "\n",
+    "3. $\\beta_0 < 0$ and $\\beta_1 > 0$,\n",
+    "\n",
+    "4. $\\beta_0 < 0$ and $\\beta_1 < 0$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aa950760",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The first Case\n",
+    "\n",
+    "If we consider the first case, we have then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "475389ed",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "-4(4-2\\beta_0)+\\lambda=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "086558d9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1a3b2ba5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "-2(2-\\beta_1)+\\lambda=0.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c928bc46",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which yields"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "809a58f7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0=\\frac{16+\\lambda}{8},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b937bf4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6e5b2527",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_1=\\frac{4+\\lambda}{2}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c5f88a45",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Using the constraint on $\\beta_0$ and $\\beta_1$ we can then find the optimal value of $\\lambda$ for the different cases. We leave this as an exercise to you."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c57753b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Simple code for solving the above problem\n",
+    "\n",
+    "Here we set up the OLS, Ridge and Lasso functionality in order to study the above example. Note that here we have opted for a set of values of $\\lambda$, meaning that we need to perform a search in order to find the optimal values.\n",
+    "\n",
+    "First we study and compare the OLS and Ridge results.  The next code compares all three methods."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "392f218d",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "\n",
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "\n",
+    "X = np.array( [ [ 2, 0], [0, 1], [0,0]])\n",
+    "y = np.array( [4, 2, 3])\n",
+    "\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y\n",
+    "print(OLSbeta)\n",
+    "# and then make the prediction\n",
+    "ytildeOLS = X @ OLSbeta\n",
+    "print(\"Training MSE for OLS\")\n",
+    "print(MSE(y,ytildeOLS))\n",
+    "ypredictOLS = X @ OLSbeta\n",
+    "\n",
+    "# Repeat now for Ridge regression and various values of the regularization parameter\n",
+    "I = np.eye(2,2)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 100\n",
+    "MSEPredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 4, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y\n",
+    "#    print(Ridgebeta)\n",
+    "    # and then make the prediction\n",
+    "    ypredictRidge = X @ Ridgebeta\n",
+    "    MSEPredict[i] = MSE(y,ypredictRidge)\n",
+    "#    print(MSEPredict[i])\n",
+    "    # Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Train')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dd134ab1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We see here that we reach a plateau. What is actually happening?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8d9a61db",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## With Lasso Regression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "afba7c4a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "\n",
+    "X = np.array( [ [ 2, 0], [0, 1], [0,0]])\n",
+    "y = np.array( [4, 2, 3])\n",
+    "\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y\n",
+    "print(OLSbeta)\n",
+    "# and then make the prediction\n",
+    "ytildeOLS = X @ OLSbeta\n",
+    "print(\"Training MSE for OLS\")\n",
+    "print(MSE(y,ytildeOLS))\n",
+    "ypredictOLS = X @ OLSbeta\n",
+    "\n",
+    "# Repeat now for Ridge regression and various values of the regularization parameter\n",
+    "I = np.eye(2,2)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 100\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "MSELassoPredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 4, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y\n",
+    "    print(Ridgebeta)\n",
+    "    # and then make the prediction\n",
+    "    ypredictRidge = X @ Ridgebeta\n",
+    "    MSERidgePredict[i] = MSE(y,ypredictRidge)\n",
+    "    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n",
+    "    RegLasso.fit(X,y)\n",
+    "    ypredictLasso = RegLasso.predict(X)\n",
+    "    print(RegLasso.coef_)\n",
+    "    MSELassoPredict[i] = MSE(y,ypredictLasso)\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'r--', label = 'MSE Ridge Train')\n",
+    "plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Train')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4d63db0b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Another Example, now with a polynomial fit"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "81115144",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "np.random.seed(3155)\n",
+    "\n",
+    "x = np.random.rand(100)\n",
+    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+    "\n",
+    "# number of features p (here degree of polynomial\n",
+    "p = 3\n",
+    "#  The design matrix now as function of a given polynomial\n",
+    "X = np.zeros((len(x),p))\n",
+    "X[:,0] = 1.0\n",
+    "X[:,1] = x\n",
+    "X[:,2] = x*x\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train\n",
+    "print(OLSbeta)\n",
+    "# and then make the prediction\n",
+    "ytildeOLS = X_train @ OLSbeta\n",
+    "print(\"Training MSE for OLS\")\n",
+    "print(MSE(y_train,ytildeOLS))\n",
+    "ypredictOLS = X_test @ OLSbeta\n",
+    "print(\"Test MSE OLS\")\n",
+    "print(MSE(y_test,ypredictOLS))\n",
+    "\n",
+    "# Repeat now for Lasso and Ridge regression and various values of the regularization parameter\n",
+    "I = np.eye(p,p)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 100\n",
+    "MSEPredict = np.zeros(nlambdas)\n",
+    "MSETrain = np.zeros(nlambdas)\n",
+    "MSELassoPredict = np.zeros(nlambdas)\n",
+    "MSELassoTrain = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 4, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    Ridgebeta = np.linalg.inv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n",
+    "    # include lasso using Scikit-Learn\n",
+    "    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n",
+    "    RegLasso.fit(X_train,y_train)\n",
+    "    # and then make the prediction\n",
+    "    ytildeRidge = X_train @ Ridgebeta\n",
+    "    ypredictRidge = X_test @ Ridgebeta\n",
+    "    ytildeLasso = RegLasso.predict(X_train)\n",
+    "    ypredictLasso = RegLasso.predict(X_test)\n",
+    "    MSEPredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    MSETrain[i] = MSE(y_train,ytildeRidge)\n",
+    "    MSELassoPredict[i] = MSE(y_test,ypredictLasso)\n",
+    "    MSELassoTrain[i] = MSE(y_train,ytildeLasso)\n",
+    "\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSETrain, label = 'MSE Ridge train')\n",
+    "plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSELassoTrain, label = 'MSE Lasso train')\n",
+    "plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Test')\n",
+    "\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e076a968",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Material for lecture Thursday September 7"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "db822c65",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Linking the regression analysis with a statistical interpretation\n",
+    "\n",
+    "We will now couple the discussions of ordinary least squares, Ridge\n",
+    "and Lasso regression with a statistical interpretation, that is we\n",
+    "move from a linear algebra analysis to a statistical analysis. In\n",
+    "particular, we will focus on what the regularization terms can result\n",
+    "in.  We will amongst other things show that the regularization\n",
+    "parameter can reduce considerably the variance of the parameters\n",
+    "$\\beta$.\n",
+    "\n",
+    "The\n",
+    "advantage of doing linear regression is that we actually end up with\n",
+    "analytical expressions for several statistical quantities.  \n",
+    "Standard least squares and Ridge regression  allow us to\n",
+    "derive quantities like the variance and other expectation values in a\n",
+    "rather straightforward way.\n",
+    "\n",
+    "It is assumed that $\\varepsilon_i\n",
+    "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n",
+    "independent, i.e.:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fc4a06b6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{align*} \n",
+    "\\mbox{Cov}(\\varepsilon_{i_1},\n",
+    "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n",
+    "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2.  \\end{array} \\right.\n",
+    "\\end{align*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "43299fb3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The randomness of $\\varepsilon_i$ implies that\n",
+    "$\\mathbf{y}_i$ is also a random variable. In particular,\n",
+    "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n",
+    "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n",
+    "non-random scalar. To specify the parameters of the distribution of\n",
+    "$\\mathbf{y}_i$ we need to calculate its first two moments. \n",
+    "\n",
+    "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n",
+    "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n",
+    "row number $i$ and perform a sum over all values $p$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6cba00c7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Assumptions made\n",
+    "\n",
+    "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n",
+    "that there exists a function $f(\\boldsymbol{x})$ and  a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n",
+    "which describe our data"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c85f61d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a1901c32",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We approximate this function with our model from the solution of the linear regression equations, that is our\n",
+    "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c59e8272",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8cba4bae",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Expectation value and variance\n",
+    "\n",
+    "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5e18ebb5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{align*} \n",
+    "\\mathbb{E}(y_i) & =\n",
+    "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n",
+    "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n",
+    "\\end{align*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "10d86c58",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "while\n",
+    "its variance is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "efba1ba9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
+    "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n",
+    "[\\mathbb{E}(y_i)]^2  \\\\  & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n",
+    "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n",
+    "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n",
+    "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n",
+    "\\ast} \\, \\beta)^2 \\\\  & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n",
+    "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n",
+    "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n",
+    "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n",
+    "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2.  \n",
+    "\\end{align*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a3eccdab",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n",
+    "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "70b901b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Expectation value and variance for $\\boldsymbol{\\beta}$\n",
+    "\n",
+    "With the OLS expressions for the optimal parameters $\\boldsymbol{\\hat{\\beta}}$ we can evaluate the expectation value"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "752b72d2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "108aba79",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This means that the estimator of the regression parameters is unbiased.\n",
+    "\n",
+    "We can also calculate the variance\n",
+    "\n",
+    "The variance of the optimal value $\\boldsymbol{\\hat{\\beta}}$ is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "65801f3b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{eqnarray*}\n",
+    "\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n",
+    "\\\\\n",
+    "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}]^{T} \\}\n",
+    "\\\\\n",
+    "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "% \\\\\n",
+    "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} \\, \\mathbf{Y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}  \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "% \\\\\n",
+    "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{Y} \\, \\mathbf{Y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "\\\\\n",
+    "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\,  \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "% \\\\\n",
+    "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\,  \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n",
+    "% \\\\\n",
+    "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T  \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n",
+    "\\\\\n",
+    "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}  + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n",
+    "\\end{eqnarray*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "faf8da78",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have used  that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n",
+    "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n",
+    "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n",
+    "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n",
+    "variance of the estimate of the $j$-th regression coefficient:\n",
+    "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $. This may be used to\n",
+    "construct a confidence interval for the estimates.\n",
+    "\n",
+    "In a similar way, we can obtain analytical expressions for say the\n",
+    "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n",
+    "when we employ Ridge regression, allowing us again to define a confidence interval. \n",
+    "\n",
+    "It is rather straightforward to show that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1c71921e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cfb57087",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We see clearly that \n",
+    "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n",
+    "\n",
+    "We can also compute the variance as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "be4ad159",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f5e11f36",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n",
+    "\n",
+    "With this, we can compute the difference"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dbf2700f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0f6d760d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The difference is non-negative definite since each component of the\n",
+    "matrix product is non-negative definite. \n",
+    "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ce614956",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Deriving OLS from a probability distribution\n",
+    "\n",
+    "Our basic assumption when we derived the OLS equations was to assume\n",
+    "that our output is determined by a given continuous function\n",
+    "$f(\\boldsymbol{x})$ and a random noise $\\boldsymbol{\\epsilon}$ given by the normal\n",
+    "distribution with zero mean value and an undetermined variance\n",
+    "$\\sigma^2$.\n",
+    "\n",
+    "We found above that the outputs $\\boldsymbol{y}$ have a mean value given by\n",
+    "$\\boldsymbol{X}\\hat{\\boldsymbol{\\beta}}$ and variance $\\sigma^2$. Since the entries to\n",
+    "the design matrix are not stochastic variables, we can assume that the\n",
+    "probability distribution of our targets is also a normal distribution\n",
+    "but now with mean value $\\boldsymbol{X}\\hat{\\boldsymbol{\\beta}}$. This means that a\n",
+    "single output $y_i$ is given by the Gaussian distribution"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "086419d8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "y_i\\sim \\mathcal{N}(\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta}, \\sigma^2)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a228dcf4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Independent and Identically Distrubuted (iid)\n",
+    "\n",
+    "We assume now that the various $y_i$ values are stochastically distributed according to the above Gaussian distribution. \n",
+    "We define this distribution as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f04f6b76",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(y_i, \\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "642936a8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which reads as finding the likelihood of an event $y_i$ with the input variables $\\boldsymbol{X}$ given the parameters (to be determined) $\\boldsymbol{\\beta}$.\n",
+    "\n",
+    "Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event $\\boldsymbol{y}$ as the product of the single events, that is we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f137023",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{y},\\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}=\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "48fdd9f7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We will write this in a more compact form reserving $\\boldsymbol{D}$ for the domain of events, including the ouputs (targets) and the inputs. That is\n",
+    "in case we have a simple one-dimensional input and output case"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ae36f857",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})].\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d312d461",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In the more general case the various inputs should be replaced by the possible features represented by the input data set $\\boldsymbol{X}$. \n",
+    "We can now rewrite the above probability as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "59ea8b57",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b9be32b5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "It is a conditional probability (see below) and reads as the likelihood of a domain of events $\\boldsymbol{D}$ given a set of parameters $\\boldsymbol{\\beta}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b378b2c9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Maximum Likelihood Estimation (MLE)\n",
+    "\n",
+    "In statistics, maximum likelihood estimation (MLE) is a method of\n",
+    "estimating the parameters of an assumed probability distribution,\n",
+    "given some observed data. This is achieved by maximizing a likelihood\n",
+    "function so that, under the assumed statistical model, the observed\n",
+    "data is the most probable. \n",
+    "\n",
+    "We will assume here that our events are given by the above Gaussian\n",
+    "distribution and we will determine the optimal parameters $\\beta$ by\n",
+    "maximizing the above PDF. However, computing the derivatives of a\n",
+    "product function is cumbersome and can easily lead to overflow and/or\n",
+    "underflowproblems, with potentials for loss of numerical precision.\n",
+    "\n",
+    "In practice, it is more convenient to maximize the logarithm of the\n",
+    "PDF because it is a monotonically increasing function of the argument.\n",
+    "Alternatively, and this will be our option, we will minimize the\n",
+    "negative of the logarithm since this is a monotonically decreasing\n",
+    "function.\n",
+    "\n",
+    "Note also that maximization/minimization of the logarithm of the PDF\n",
+    "is equivalent to the maximization/minimization of the function itself."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6ddc5457",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## A new Cost Function\n",
+    "\n",
+    "We could now define a new cost function to minimize, namely the negative logarithm of the above PDF"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bb0f5795",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=-\\log{\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})}=-\\sum_{i=0}^{n-1}\\log{p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c969ba54",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "630a5421",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=\\frac{n}{2}\\log{2\\pi\\sigma^2}+\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "49969ea1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Taking the derivative of the *new* cost function with respect to the parameters $\\beta$ we recognize our familiar OLS equation, namely"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "93b3d19b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right) =0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f61d06bf",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to the well-known OLS equation for the optimal paramters $\\beta$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c25aa29",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}!\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "add3ac78",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2cb62527",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## More basic Statistics and Bayes' theorem\n",
+    "\n",
+    "A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry.\n",
+    "Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.\n",
+    "\n",
+    "Assume we have two domains of events $X=[x_0,x_1,\\dots,x_{n-1}]$ and $Y=[y_0,y_1,\\dots,y_{n-1}]$.\n",
+    "\n",
+    "We define also the likelihood for $X$ and $Y$ as $p(X)$ and $p(Y)$ respectively.\n",
+    "The likelihood of a specific event $x_i$ (or $y_i$) is then written as $p(X=x_i)$ or just $p(x_i)=p_i$. \n",
+    "\n",
+    "**Union of events is given by.**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3704ebb3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X \\cup Y)= p(X)+p(Y)-p(X \\cap Y).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "76f1d92c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "**The product rule (aka joint probability) is given by.**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "31ef87a3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X \\cup Y)= p(X,Y)= p(X\\vert Y)p(Y)=p(Y\\vert X)p(X),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9f87e53d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we read $p(X\\vert Y)$ as the likelihood of obtaining $X$ given $Y$.\n",
+    "\n",
+    "If we have independent events then $p(X,Y)=p(X)p(Y)$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b465eba1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Marginal Probability\n",
+    "\n",
+    "The marginal probability is defined in terms of only one of the set of variables $X,Y$. For a discrete probability we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b522f15f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X)=\\sum_{i=0}^{n-1}p(X,Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert Y=y_i)p(Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert y_i)p(y_i).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a8bf634f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Conditional  Probability\n",
+    "\n",
+    "The conditional  probability, if $p(Y) > 0$, is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51e8bfb6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)}=\\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bfb3c814",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Bayes' Theorem\n",
+    "\n",
+    "If we combine the conditional probability with the marginal probability and the standard product rule, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dfff0c7c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cad0634f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which we can rewrite as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1de5e07a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X\\vert Y)= \\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}=\\frac{p(Y\\vert X)p(X)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "43f1714e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which is Bayes' theorem. It allows us to evaluate the uncertainty in in $X$ after we have observed $Y$. We can easily interchange $X$ with $Y$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "166406ab",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Interpretations of Bayes' Theorem\n",
+    "\n",
+    "The quantity $p(Y\\vert X)$ on the right-hand side of the theorem is\n",
+    "evaluated for the observed data $Y$ and can be viewed as a function of\n",
+    "the parameter space represented by $X$. This function is not\n",
+    "necesseraly normalized and is normally called the likelihood function.\n",
+    "\n",
+    "The function $p(X)$ on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.\n",
+    "\n",
+    "Let us try to illustrate Bayes' theorem through an example."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d547f8dd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Example of Usage of Bayes' theorem\n",
+    "\n",
+    "Let us suppose that you are undergoing a series of mammography scans in\n",
+    "order to rule out possible breast cancer cases.  We define the\n",
+    "sensitivity for a positive event by the variable $X$. It takes binary\n",
+    "values with $X=1$ representing a positive event and $X=0$ being a\n",
+    "negative event. We reserve $Y$ as a classification parameter for\n",
+    "either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a  positive thing).\n",
+    "\n",
+    "We let $Y=1$ represent the the case of having breast cancer and $Y=0$ as not.\n",
+    "\n",
+    "Let us assume that if you have breast cancer, the test will be positive with a probability of $0.8$, that is we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f848f875",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X=1\\vert Y=1) =0.8.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c506872",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This obviously sounds  scary since many would conclude that if the test is positive, there is a likelihood of $80\\%$ for having cancer.\n",
+    "It is however not correct, as the following Bayesian analysis shows."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "317558b2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Doing it correctly\n",
+    "\n",
+    "If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number.\n",
+    "Let us assume that the prior probability in the population as a whole is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8f8f8c5c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(Y=1) =0.004.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "acb7607d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c5866ceb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X=1\\vert Y=0) =0.1.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3648f6b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0b2cef16",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(Y=1\\vert X=1)=\\frac{p(X=1\\vert Y=1)p(Y=1)}{p(X=1\\vert Y=1)p(Y=1)+p(X=1\\vert Y=0)p(Y=0)}=\\frac{0.8\\times 0.004}{0.8\\times 0.004+0.1\\times 0.996}=0.031.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "94233096",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "That is, in case of a positive test, there is only a $3\\%$ chance of having breast cancer!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "91284f6d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Bayes' Theorem and Ridge and Lasso Regression\n",
+    "\n",
+    "Hitherto we have discussed Ridge and Lasso regression in terms of a\n",
+    "linear analysis. This may to many of you feel rather technical and\n",
+    "perhaps not that intuitive. The question is whether we can develop a\n",
+    "more intuitive way of understanding what Ridge and Lasso express.\n",
+    "\n",
+    "Before we proceed let us perform a Ridge, Lasso  and OLS analysis of a polynomial fit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3debb528",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Test Function for what happens with OLS, Ridge and Lasso\n",
+    "\n",
+    "We will play around with a study of the values for the optimal\n",
+    "parameters $\\boldsymbol{\\beta}$ using OLS, Ridge and Lasso regression.  For\n",
+    "OLS, you will notice as function of the noise and polynomial degree,\n",
+    "that the parameters $\\beta$ will fluctuate from order to order in the\n",
+    "polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS.\n",
+    "\n",
+    "For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "a4cf2a42",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "# Make data set.\n",
+    "n = 10000\n",
+    "x = np.random.rand(n)\n",
+    "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)\n",
+    "\n",
+    "Maxpolydegree = 5\n",
+    "X = np.zeros((len(x),Maxpolydegree))\n",
+    "X[:,0] = 1.0\n",
+    "\n",
+    "for polydegree in range(1, Maxpolydegree):\n",
+    "    for degree in range(polydegree):\n",
+    "        X[:,degree] = x**(degree)\n",
+    "\n",
+    "\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train\n",
+    "print(OLSbeta)\n",
+    "ypredictOLS = X_test @ OLSbeta\n",
+    "print(\"Test MSE OLS\")\n",
+    "print(MSE(y_test,ypredictOLS))\n",
+    "# Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 4\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "MSELassoPredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-3, 1, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    # Make the fit using Ridge and Lasso\n",
+    "    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n",
+    "    RegRidge.fit(X_train,y_train)\n",
+    "    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n",
+    "    RegLasso.fit(X_train,y_train)\n",
+    "    # and then make the prediction\n",
+    "    ypredictRidge = RegRidge.predict(X_test)\n",
+    "    ypredictLasso = RegLasso.predict(X_test)\n",
+    "    # Compute the MSE and print it\n",
+    "    MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    MSELassoPredict[i] = MSE(y_test,ypredictLasso)\n",
+    "    print(lmb,RegRidge.coef_)\n",
+    "    print(lmb,RegLasso.coef_)\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'b', label = 'MSE Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSELassoPredict, 'r', label = 'MSE Lasso Test')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "26270da6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "How can we understand this?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d8f0e20a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Invoking Bayes' theorem\n",
+    "\n",
+    "Using Bayes' theorem we can gain a better intuition about Ridge and Lasso regression. \n",
+    "\n",
+    "For ordinary least squares we postulated that the maximum likelihood for the doamin of events $\\boldsymbol{D}$ (one-dimensional case)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6c93bc4b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})],\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c8b6cc8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f76e3a40",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af323d99",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set $\\boldsymbol{\\beta}$ given a domain of events $\\boldsymbol{D}$?  That is, how can we define the posterior probability"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "49a0988c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38568afe",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Bayes' theorem comes to our rescue here since (omitting the normalization constant)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c3d0eeb9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})\\propto p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})p(\\boldsymbol{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1d0ba152",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f936c1ce",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge and Bayes\n",
+    "\n",
+    "With the posterior probability defined by a likelihood which we have\n",
+    "already modeled and an unknown prior, we are now ready to make\n",
+    "additional models for the prior.\n",
+    "\n",
+    "We can, based on our discussions of the variance of $\\boldsymbol{\\beta}$ and the mean value, assume that the prior for the values $\\boldsymbol{\\beta}$ is given by a Gaussian with mean value zero and variance $\\tau^2$, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dd10cea6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2534eb1d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Our posterior probability becomes then (omitting the normalization factor which is just a constant)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "104af119",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta\\vert\\boldsymbol{D})}=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5e8e89cf",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We can now optimize this quantity with respect to $\\boldsymbol{\\beta}$. As we\n",
+    "did for OLS, this is most conveniently done by taking the negative\n",
+    "logarithm of the posterior probability. Doing so and leaving out the\n",
+    "constants terms that do not depend on $\\beta$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "467910cd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{2\\tau^2}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "16f15265",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and replacing $1/2\\tau^2$ with $\\lambda$ we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1cd22464",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "90676d1e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which is our Ridge cost function!  Nice, isn't it?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2082d057",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Lasso and Bayes\n",
+    "\n",
+    "To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution ([Laplace in this case](https://en.wikipedia.org/wiki/Laplace_distribution)) with zero mean value,  that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "544b15d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "592804c7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Our posterior probability becomes then (omitting the normalization factor which is just a constant)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c11e6e06",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fdc6e9b5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Taking the negative\n",
+    "logarithm of the posterior probability and leaving out the\n",
+    "constants terms that do not depend on $\\beta$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a0e8571d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{\\tau}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84eca415",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and replacing $1/\\tau$ with $\\lambda$ we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0924d8d4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7270fef7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which is our Lasso cost function!"
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.10"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/week36.py b/doc/LectureNotes/_build/jupyter_execute/week36.py
new file mode 100644
index 000000000..912ea6dbe
--- /dev/null
+++ b/doc/LectureNotes/_build/jupyter_execute/week36.py
@@ -0,0 +1,1408 @@
+#!/usr/bin/env python
+# coding: utf-8
+
+# 
+# 
+
+# # Week 36: Statistical interpretation of Linear Regression and Resampling techniques
+# **Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+# 
+# Date: **September 4-8, 2023**
+
+# ## Plans for week 36
+# 
+# * Material for the active learning sessions on Tuesday and Wednesday
+# 
+#   * Summary from last week on discussion of SVD, Ridge and Lasso linear regression.
+# 
+#   * Recommended Reading: Hastie et al chapter 3, see 
+# 
+#   * Presentation and discussion of first project
+# 
+# * Material for the lecture on Thursday September 7
+# 
+#   * Linear Regression and links with Statistics, Resampling methods
+# 
+#   * Recommended Reading: Goodfellow et al chapter 3 on probability theory, see URL:""
+# 
+#   * See also Murphy, sections 2.4 (Gaussian distributions) and 3.2 (Bayesian Statistics, basis)
+
+# ## Material for the active learning sessions Tuesday and Wednesday
+# 
+# The material here contains a summary from last Week and discussion of SVD, Ridge and Lasso regression with examples
+
+# ## Linear Regression and  the SVD
+# 
+# We used the SVD to analyse the matrix to invert in ordinary lineat regression
+
+# $$
+# \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T.
+# $$
+
+# Since the matrices here have dimension $p\times p$, with $p$ corresponding to the singular values, we defined last week the matrix
+
+# $$
+# \boldsymbol{\Sigma}^T\boldsymbol{\Sigma} = \begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0}\\ \end{bmatrix}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \\ \boldsymbol{0}\end{bmatrix},
+# $$
+
+# where the tilde-matrix $\tilde{\boldsymbol{\Sigma}}$ is a matrix of dimension $p\times p$ containing only the singular values $\sigma_i$, that is
+
+# $$
+# \tilde{\boldsymbol{\Sigma}}=\begin{bmatrix} \sigma_0 & 0 & 0 & \dots & 0 & 0 \\
+#                                     0 & \sigma_1 & 0 & \dots & 0 & 0 \\
+# 				    0 & 0 & \sigma_2 & \dots & 0 & 0 \\
+# 				    0 & 0 & 0 & \dots & \sigma_{p-2} & 0 \\
+# 				    0 & 0 & 0 & \dots & 0 & \sigma_{p-1} \\
+# \end{bmatrix},
+# $$
+
+# meaning we can write
+
+# $$
+# \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^2\boldsymbol{V}^T.
+# $$
+
+# Multiplying from the right with $\boldsymbol{V}$ (using the orthogonality of $\boldsymbol{V}$) we get
+
+# $$
+# \left(\boldsymbol{X}^T\boldsymbol{X}\right)\boldsymbol{V}=\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^2.
+# $$
+
+# ## What does it mean?
+# 
+# This means the vectors $\boldsymbol{v}_i$ of the orthogonal matrix $\boldsymbol{V}$
+# are the eigenvectors of the matrix $\boldsymbol{X}^T\boldsymbol{X}$ with eigenvalues
+# given by the singular values squared, that is
+
+# $$
+# \left(\boldsymbol{X}^T\boldsymbol{X}\right)\boldsymbol{v}_i=\boldsymbol{v}_i\sigma_i^2.
+# $$
+
+# In other words, each non-zero singular value of $\boldsymbol{X}$ is a positive
+# square root of an eigenvalue of $\boldsymbol{X}^T\boldsymbol{X}$.  It means also that
+# the columns of $\boldsymbol{V}$ are the eigenvectors of
+# $\boldsymbol{X}^T\boldsymbol{X}$. Since we have ordered the singular values of
+# $\boldsymbol{X}$ in a descending order, it means that the column vectors
+# $\boldsymbol{v}_i$ are hierarchically ordered by how much correlation they
+# encode from the columns of $\boldsymbol{X}$. 
+# 
+# Note that these are also the eigenvectors and eigenvalues of the
+# Hessian matrix.
+# 
+# If we now recall the definition of the covariance matrix (not using
+# Bessel's correction) we have
+
+# $$
+# \boldsymbol{C}[\boldsymbol{X}]=\frac{1}{n}\boldsymbol{X}^T\boldsymbol{X},
+# $$
+
+# meaning that every squared non-singular value of $\boldsymbol{X}$ divided by $n$ (
+# the number of samples) are the eigenvalues of the covariance
+# matrix. Every singular value of $\boldsymbol{X}$ is thus a positive square
+# root of an eigenvalue of $\boldsymbol{X}^T\boldsymbol{X}$. If the matrix $\boldsymbol{X}$ is
+# self-adjoint, the singular values of $\boldsymbol{X}$ are equal to the
+# absolute value of the eigenvalues of $\boldsymbol{X}$.
+
+# ## And finally  $\boldsymbol{X}\boldsymbol{X}^T$
+# 
+# For $\boldsymbol{X}\boldsymbol{X}^T$ we found
+
+# $$
+# \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T=\boldsymbol{U}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{U}^T.
+# $$
+
+# Since the matrices here have dimension $n\times n$, we have
+
+# $$
+# \boldsymbol{\Sigma}\boldsymbol{\Sigma}^T = \begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \\ \boldsymbol{0}\\ \end{bmatrix}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}}  \boldsymbol{0}\\ \end{bmatrix}=\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix},
+# $$
+
+# leading to
+
+# $$
+# \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}\boldsymbol{U}^T.
+# $$
+
+# Multiplying with $\boldsymbol{U}$ from the right gives us the eigenvalue problem
+
+# $$
+# (\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U}=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}.
+# $$
+
+# It means that the eigenvalues of $\boldsymbol{X}\boldsymbol{X}^T$ are again given by
+# the non-zero singular values plus now a series of zeros.  The column
+# vectors of $\boldsymbol{U}$ are the eigenvectors of $\boldsymbol{X}\boldsymbol{X}^T$ and
+# measure how much correlations are contained in the rows of $\boldsymbol{X}$.
+# 
+# Since we will mainly be interested in the correlations among the features
+# of our data (the columns of $\boldsymbol{X}$, the quantity of interest for us are the non-zero singular
+# values and the column vectors of $\boldsymbol{V}$.
+
+# ## Code for SVD and Inversion of Matrices
+# 
+# How do we use the SVD to invert a matrix $\boldsymbol{X}^\boldsymbol{X}$ which is singular or near singular?
+# The simple answer is to use the linear algebra function for pseudoinvers, that is
+
+# In[1]:
+
+
+Ainv = np.linlag.pinv(A)
+
+
+# Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD.
+
+# In[2]:
+
+
+import numpy as np
+# SVD inversion
+def SVDinv(A):
+    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
+    SVD is numerically more stable than the inversion algorithms provided by
+    numpy and scipy.linalg at the cost of being slower.
+    '''
+    U, s, VT = np.linalg.svd(A)
+    print('test U')
+    print( (np.transpose(U) @ U - U @np.transpose(U)))
+    print('test VT')
+    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
+
+
+    D = np.zeros((len(U),len(VT)))
+    D = np.diag(s)
+    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
+    return np.matmul(V,np.matmul(invD,UT))
+
+
+#X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
+# Non-singular square matrix
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+print(X)
+A = np.transpose(X) @ X
+# Brute force inversion
+B = np.linalg.inv(A)  # here we could use np.linalg.pinv(A)
+C = SVDinv(A)
+print(np.abs(B-C))
+
+
+# ## Inverse of Rectangular Matrix
+# 
+# Although our matrix to invert $\boldsymbol{X}^T\boldsymbol{X}$ is a square matrix, our matrix may be singular. 
+# 
+# The pseudoinverse is the generalization of the matrix inverse for square matrices to
+# rectangular matrices where the number of rows and columns are not equal.
+# 
+# It is also called the the Moore-Penrose Inverse after two independent discoverers of the method or the Generalized Inverse.
+# It is used for the calculation of the inverse for singular or near singular matrices and for rectangular matrices.
+# 
+# Using the SVD we can obtain the pseudoinverse of a matrix $\boldsymbol{A}$ (labeled here as $\boldsymbol{A}_{\mathrm{PI}}$)
+
+# $$
+# \boldsymbol{A}_{\mathrm{PI}}= \boldsymbol{V}\boldsymbol{D}_{\mathrm{PI}}\boldsymbol{U}^T,
+# $$
+
+# where $\boldsymbol{D}_{\mathrm{PI}}$ can be calculated by creating a diagonal matrix from $\boldsymbol{\Sigma}$ where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD.
+
+# In[3]:
+
+
+import numpy as np
+# SVD inversion
+def SVDinv(A):
+    U, s, VT = np.linalg.svd(A)
+    # reciprocals of singular values of s
+    d = 1.0 / s
+    # create m x n D matrix
+    D = np.zeros(A.shape)
+    # populate D with n x n diagonal matrix
+    D[:A.shape[1], :A.shape[1]] = np.diag(d)
+    UT = np.transpose(U)
+    V = np.transpose(VT)
+    return np.matmul(V,np.matmul(D.T,UT))
+
+
+A = np.array([ [0.3, 0.4], [0.5, 0.6], [0.7, 0.8],[0.9, 1.0]])
+print(A)
+# Brute force inversion of super-collinear matrix
+B = np.linalg.pinv(A)
+print(B)
+# Compare our own algorithm with pinv
+C = SVDinv(A)
+print(np.abs(C-B))
+
+
+# As you can see from this example, our own decomposition based on the SVD agrees with  the pseudoinverse algorithm provided by **Numpy**.
+
+# ## Ridge and LASSO Regression
+# 
+# Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
+# our optimization problem is
+
+# $$
+# {\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
+# $$
+
+# or we can state it as
+
+# $$
+# {\displaystyle \min_{\boldsymbol{\beta}\in
+# {\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
+# $$
+
+# where we have used the definition of  a norm-2 vector, that is
+
+# $$
+# \vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
+# $$
+
+# ## From OLS to Ridge and Lasso
+# 
+# By minimizing the above equation with respect to the parameters
+# $\boldsymbol{\beta}$ we could then obtain an analytical expression for the
+# parameters $\boldsymbol{\beta}$.  We can add a regularization parameter $\lambda$ by
+# defining a new cost function to be optimized, that is
+
+# $$
+# {\displaystyle \min_{\boldsymbol{\beta}\in
+# {\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
+# $$
+
+# which leads to the Ridge regression minimization problem where we
+# require that $\vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t$, where $t$ is
+# a finite number larger than zero. We do not include such a constraints in the discussions here.
+# 
+# By defining
+
+# $$
+# C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,
+# $$
+
+# we have a new optimization equation
+
+# $$
+# {\displaystyle \min_{\boldsymbol{\beta}\in
+# {\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1
+# $$
+
+# which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. 
+# 
+# Here we have defined the norm-1 as
+
+# $$
+# \vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert.
+# $$
+
+# ## Deriving the  Ridge Regression Equations
+# 
+# Using the matrix-vector expression for Ridge regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have
+
+# $$
+# C(\boldsymbol{X},\boldsymbol{\beta})=\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta},
+# $$
+
+# and 
+# taking the derivatives with respect to $\boldsymbol{\beta}$ we obtain then
+# a slightly modified matrix inversion problem which for finite values
+# of $\lambda$ does not suffer from singularity problems. We obtain
+# the optimal parameters
+
+# $$
+# \hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+# $$
+
+# with $\boldsymbol{I}$ being a $p\times p$ identity matrix with the constraint that
+
+# $$
+# \sum_{i=0}^{p-1} \beta_i^2 \leq t,
+# $$
+
+# with $t$ a finite positive number.
+
+# ## Note on Scikit-Learn
+# 
+# Note well that a library like **Scikit-Learn** does not include the $1/n$ factor in the expression for the mean-squared error. If you include it, the optimal parameter $\beta$ becomes
+
+# $$
+# \hat{\boldsymbol{\beta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+n\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+# $$
+
+# In our codes where we compare our own codes with **Scikit-Learn**, we do thus not include the $1/n$ factor in the cost function.
+
+# ## Comparison with OLS
+# When we compare this with the ordinary least squares result we have
+
+# $$
+# \hat{\boldsymbol{\beta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+# $$
+
+# which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix $\boldsymbol{X}^T\boldsymbol{X}$.
+# 
+# We see that Ridge regression is nothing but the standard OLS with a
+# modified diagonal term added to $\boldsymbol{X}^T\boldsymbol{X}$. The consequences, in
+# particular for our discussion of the bias-variance tradeoff are rather
+# interesting. We will see that for specific values of $\lambda$, we may
+# even reduce the variance of the optimal parameters $\boldsymbol{\beta}$. These topics and other related ones, will be discussed after the more linear algebra oriented analysis here.
+
+# ## SVD analysis
+# 
+# Using our insights about the SVD of the design matrix $\boldsymbol{X}$ 
+# We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\boldsymbol{U}$ as
+
+# $$
+# \tilde{\boldsymbol{y}}_{\mathrm{OLS}}=\boldsymbol{X}\boldsymbol{\beta}  =\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}.
+# $$
+
+# For Ridge regression this becomes
+
+# $$
+# \tilde{\boldsymbol{y}}_{\mathrm{Ridge}}=\boldsymbol{X}\boldsymbol{\beta}_{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{\Sigma}^2\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
+# $$
+
+# with the vectors $\boldsymbol{u}_j$ being the columns of $\boldsymbol{U}$ from the SVD of the matrix $\boldsymbol{X}$.
+
+# ## Interpreting the Ridge results
+# 
+# Since $\lambda \geq 0$, it means that compared to OLS, we have
+
+# $$
+# \frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1.
+# $$
+
+# Ridge regression finds the coordinates of $\boldsymbol{y}$ with respect to the
+# orthonormal basis $\boldsymbol{U}$, it then shrinks the coordinates by
+# $\frac{\sigma_j^2}{\sigma_j^2+\lambda}$. Recall that the SVD has
+# eigenvalues ordered in a descending way, that is $\sigma_i \geq
+# \sigma_{i+1}$.
+# 
+# For small eigenvalues $\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.
+
+# ## More interpretations
+# 
+# For the sake of simplicity, let us assume that the design matrix is orthonormal, that is
+
+# $$
+# \boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}.
+# $$
+
+# In this case the standard OLS results in
+
+# $$
+# \boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
+# $$
+
+# and
+
+# $$
+# \boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}},
+# $$
+
+# that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\lambda$, and
+# the Ridge estimator converges to zero when the hyperparameter goes to
+# infinity.
+# 
+# We will come back to more interpreations after we have gone through some of the statistical analysis part. 
+# 
+# For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.
+# Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended.
+
+# ## Deriving the  Lasso Regression Equations
+# 
+# Using the matrix-vector expression for Lasso regression, we have the following **cost** function
+
+# $$
+# C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1,
+# $$
+
+# Taking the derivative with respect to $\boldsymbol{\beta}$ and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicity)
+
+# $$
+# \frac{d \vert \beta\vert}{d \beta}=\mathrm{sgn}(\beta)=\left\{\begin{array}{cc} 1 & \beta > 0 \\-1 & \beta < 0, \end{array}\right.
+# $$
+
+# we have that the derivative of the cost function is
+
+# $$
+# \frac{\partial C(\boldsymbol{X},\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}=-\frac{2}{n}\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})+\lambda sgn(\boldsymbol{\beta})=0,
+# $$
+
+# and reordering we have
+
+# $$
+# \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}+\lambda sgn(\boldsymbol{\beta})=\boldsymbol{X}^T\boldsymbol{y}.
+# $$
+
+# This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor $2/n$ in a redefinition of the parameter $\lambda$. We will solve this type of problems using libraries like **scikit-learn**.
+
+# ## Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression
+# 
+# Let us assume that our design matrix is given by unit (identity) matrix, that is a square diagonal matrix with ones only along the
+# diagonal. In this case we have an equal number of rows and columns $n=p$.
+# 
+# Our model approximation is just $\tilde{\boldsymbol{y}}=\boldsymbol{\beta}$ and the mean squared error and thereby the cost function for ordinary least sqquares (OLS) is then (we drop the term $1/n$)
+
+# $$
+# C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2,
+# $$
+
+# and minimizing we have that
+
+# $$
+# \hat{\beta}_i^{\mathrm{OLS}} = y_i.
+# $$
+
+# ## Ridge Regression
+# 
+# For Ridge regression our cost function is
+
+# $$
+# C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\beta_i^2,
+# $$
+
+# and minimizing we have that
+
+# $$
+# \hat{\beta}_i^{\mathrm{Ridge}} = \frac{y_i}{1+\lambda}.
+# $$
+
+# ## Lasso Regression
+# 
+# For Lasso regression our cost function is
+
+# $$
+# C(\boldsymbol{\beta})=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\vert\beta_i\vert=\sum_{i=0}^{p-1}(y_i-\beta_i)^2+\lambda\sum_{i=0}^{p-1}\sqrt{\beta_i^2},
+# $$
+
+# and minimizing we have that
+
+# $$
+# -2\sum_{i=0}^{p-1}(y_i-\beta_i)+\lambda \sum_{i=0}^{p-1}\frac{(\beta_i)}{\vert\beta_i\vert}=0,
+# $$
+
+# which leads to
+
+# $$
+# \hat{\boldsymbol{\beta}}_i^{\mathrm{Lasso}} = \left\{\begin{array}{ccc}y_i-\frac{\lambda}{2} &\mathrm{if} & y_i> \frac{\lambda}{2}\\
+#                                                           y_i+\frac{\lambda}{2} &\mathrm{if} & y_i< -\frac{\lambda}{2}\\
+# 							  0 &\mathrm{if} & \vert y_i\vert\le  \frac{\lambda}{2}\end{array}\right.\\.
+# $$
+
+# Plotting these results shows clearly that Lasso regression suppresses (sets to zero) values of $\beta_i$ for specific values of $\lambda$. Ridge regression reduces on the other hand the values of $\beta_i$ as function of $\lambda$.
+
+# ## Yet another Example
+# 
+# Let us assume we have a data set with outputs/targets given by the vector
+
+# $$
+# \boldsymbol{y}=\begin{bmatrix}4 \\ 2 \\3\end{bmatrix},
+# $$
+
+# and our inputs as a $3\times 2$ design matrix
+
+# $$
+# \boldsymbol{X}=\begin{bmatrix}2 & 0\\ 0 & 1 \\ 0 & 0\end{bmatrix},
+# $$
+
+# meaning that we have two features and two unknown parameters $\beta_0$ and $\beta_1$ to be determined either by ordinary least squares, Ridge or Lasso regression.
+
+# ## The OLS case
+# 
+# For ordinary least squares (OLS) we know that the optimal solution is
+
+# $$
+# \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left( \boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+# $$
+
+# Inserting the above values we obtain that
+
+# $$
+# \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\begin{bmatrix}2 \\ 2\end{bmatrix},
+# $$
+
+# The code which implements this simpler case is presented after the discussion of Ridge and Lasso.
+
+# ## The Ridge case
+# 
+# For Ridge regression we have
+
+# $$
+# \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}=\left( \boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+# $$
+
+# Inserting the above values we obtain that
+
+# $$
+# \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}=\begin{bmatrix}\frac{8}{4+\lambda} \\ \frac{2}{1+\lambda}\end{bmatrix},
+# $$
+
+# There is normally a constraint on the value of $\vert\vert \boldsymbol{\beta}\vert\vert_2$ via the parameter $\lambda$.
+# Let us for simplicity assume that $\beta_0^2+\beta_1^2=1$ as constraint. This will allow us to find an expression for the optimal values of $\beta$ and $\lambda$.
+# 
+# To see this, let us write the cost function for Ridge regression.
+
+# ## Writing the Cost Function
+# 
+# We define the MSE without the $1/n$ factor and have then, using that
+
+# $$
+# \boldsymbol{X}\boldsymbol{\beta}=\begin{bmatrix} 2\beta_0 \\ \beta_1 \\0 \end{bmatrix},
+# $$
+
+# $$
+# C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\beta_0^2+\beta_1^2),
+# $$
+
+# and taking the derivative with respect to $\beta_0$ we get
+
+# $$
+# \beta_0=\frac{8}{4+\lambda},
+# $$
+
+# and for $\beta_1$ we obtain
+
+# $$
+# \beta_1=\frac{2}{1+\lambda},
+# $$
+
+# Using the constraint for $\beta_0^2+\beta_1^2=1$ we can constrain $\lambda$ by solving
+
+# $$
+# \left(\frac{8}{4+\lambda}\right)^2+\left(\frac{2}{1+\lambda}\right)^2=1,
+# $$
+
+# which gives $\lambda=4.571$ and $\beta_0=0.933$ and $\beta_1=0.359$.
+
+# ## Lasso case
+# 
+# For Lasso we need now, keeping a  constraint on $\vert\beta_0\vert+\vert\beta_1\vert=1$,  to take the derivative of the absolute values of $\beta_0$
+# and $\beta_1$. This gives us the following derivatives of the cost function
+
+# $$
+# C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\vert\beta_1\vert),
+# $$
+
+# $$
+# \frac{\partial C(\boldsymbol{\beta})}{\partial \beta_0}=-4(4-2\beta_0)+\lambda\mathrm{sgn}(\beta_0)=0,
+# $$
+
+# and
+
+# $$
+# \frac{\partial C(\boldsymbol{\beta})}{\partial \beta_1}=-2(2-\beta_1)+\lambda\mathrm{sgn}(\beta_1)=0.
+# $$
+
+# We have now four cases to solve besides the trivial cases $\beta_0$ and/or $\beta_1$ are zero, namely
+# 1. $\beta_0 > 0$ and $\beta_1 > 0$,
+# 
+# 2. $\beta_0 > 0$ and $\beta_1 < 0$,
+# 
+# 3. $\beta_0 < 0$ and $\beta_1 > 0$,
+# 
+# 4. $\beta_0 < 0$ and $\beta_1 < 0$.
+
+# ## The first Case
+# 
+# If we consider the first case, we have then
+
+# $$
+# -4(4-2\beta_0)+\lambda=0,
+# $$
+
+# and
+
+# $$
+# -2(2-\beta_1)+\lambda=0.
+# $$
+
+# which yields
+
+# $$
+# \beta_0=\frac{16+\lambda}{8},
+# $$
+
+# and
+
+# $$
+# \beta_1=\frac{4+\lambda}{2}.
+# $$
+
+# Using the constraint on $\beta_0$ and $\beta_1$ we can then find the optimal value of $\lambda$ for the different cases. We leave this as an exercise to you.
+
+# ## Simple code for solving the above problem
+# 
+# Here we set up the OLS, Ridge and Lasso functionality in order to study the above example. Note that here we have opted for a set of values of $\lambda$, meaning that we need to perform a search in order to find the optimal values.
+# 
+# First we study and compare the OLS and Ridge results.  The next code compares all three methods.
+
+# In[4]:
+
+
+get_ipython().run_line_magic('matplotlib', 'inline')
+
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+
+X = np.array( [ [ 2, 0], [0, 1], [0,0]])
+y = np.array( [4, 2, 3])
+
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y
+print(OLSbeta)
+# and then make the prediction
+ytildeOLS = X @ OLSbeta
+print("Training MSE for OLS")
+print(MSE(y,ytildeOLS))
+ypredictOLS = X @ OLSbeta
+
+# Repeat now for Ridge regression and various values of the regularization parameter
+I = np.eye(2,2)
+# Decide which values of lambda to use
+nlambdas = 100
+MSEPredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 4, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y
+#    print(Ridgebeta)
+    # and then make the prediction
+    ypredictRidge = X @ Ridgebeta
+    MSEPredict[i] = MSE(y,ypredictRidge)
+#    print(MSEPredict[i])
+    # Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Train')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+# We see here that we reach a plateau. What is actually happening?
+
+# ## With Lasso Regression
+
+# In[5]:
+
+
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn import linear_model
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+
+X = np.array( [ [ 2, 0], [0, 1], [0,0]])
+y = np.array( [4, 2, 3])
+
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y
+print(OLSbeta)
+# and then make the prediction
+ytildeOLS = X @ OLSbeta
+print("Training MSE for OLS")
+print(MSE(y,ytildeOLS))
+ypredictOLS = X @ OLSbeta
+
+# Repeat now for Ridge regression and various values of the regularization parameter
+I = np.eye(2,2)
+# Decide which values of lambda to use
+nlambdas = 100
+MSERidgePredict = np.zeros(nlambdas)
+MSELassoPredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 4, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y
+    print(Ridgebeta)
+    # and then make the prediction
+    ypredictRidge = X @ Ridgebeta
+    MSERidgePredict[i] = MSE(y,ypredictRidge)
+    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
+    RegLasso.fit(X,y)
+    ypredictLasso = RegLasso.predict(X)
+    print(RegLasso.coef_)
+    MSELassoPredict[i] = MSE(y,ypredictLasso)
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSERidgePredict, 'r--', label = 'MSE Ridge Train')
+plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Train')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+# ## Another Example, now with a polynomial fit
+
+# In[6]:
+
+
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+
+# number of features p (here degree of polynomial
+p = 3
+#  The design matrix now as function of a given polynomial
+X = np.zeros((len(x),p))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x*x
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
+print(OLSbeta)
+# and then make the prediction
+ytildeOLS = X_train @ OLSbeta
+print("Training MSE for OLS")
+print(MSE(y_train,ytildeOLS))
+ypredictOLS = X_test @ OLSbeta
+print("Test MSE OLS")
+print(MSE(y_test,ypredictOLS))
+
+# Repeat now for Lasso and Ridge regression and various values of the regularization parameter
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 100
+MSEPredict = np.zeros(nlambdas)
+MSETrain = np.zeros(nlambdas)
+MSELassoPredict = np.zeros(nlambdas)
+MSELassoTrain = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 4, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    Ridgebeta = np.linalg.inv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
+    # include lasso using Scikit-Learn
+    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
+    RegLasso.fit(X_train,y_train)
+    # and then make the prediction
+    ytildeRidge = X_train @ Ridgebeta
+    ypredictRidge = X_test @ Ridgebeta
+    ytildeLasso = RegLasso.predict(X_train)
+    ypredictLasso = RegLasso.predict(X_test)
+    MSEPredict[i] = MSE(y_test,ypredictRidge)
+    MSETrain[i] = MSE(y_train,ytildeRidge)
+    MSELassoPredict[i] = MSE(y_test,ypredictLasso)
+    MSELassoTrain[i] = MSE(y_train,ytildeLasso)
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSETrain, label = 'MSE Ridge train')
+plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Test')
+plt.plot(np.log10(lambdas), MSELassoTrain, label = 'MSE Lasso train')
+plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+# ## Material for lecture Thursday September 7
+
+# ## Linking the regression analysis with a statistical interpretation
+# 
+# We will now couple the discussions of ordinary least squares, Ridge
+# and Lasso regression with a statistical interpretation, that is we
+# move from a linear algebra analysis to a statistical analysis. In
+# particular, we will focus on what the regularization terms can result
+# in.  We will amongst other things show that the regularization
+# parameter can reduce considerably the variance of the parameters
+# $\beta$.
+# 
+# The
+# advantage of doing linear regression is that we actually end up with
+# analytical expressions for several statistical quantities.  
+# Standard least squares and Ridge regression  allow us to
+# derive quantities like the variance and other expectation values in a
+# rather straightforward way.
+# 
+# It is assumed that $\varepsilon_i
+# \sim \mathcal{N}(0, \sigma^2)$ and the $\varepsilon_{i}$ are
+# independent, i.e.:
+
+# $$
+# \begin{align*} 
+# \mbox{Cov}(\varepsilon_{i_1},
+# \varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
+# & i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2.  \end{array} \right.
+# \end{align*}
+# $$
+
+# The randomness of $\varepsilon_i$ implies that
+# $\mathbf{y}_i$ is also a random variable. In particular,
+# $\mathbf{y}_i$ is normally distributed, because $\varepsilon_i \sim
+# \mathcal{N}(0, \sigma^2)$ and $\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}$ is a
+# non-random scalar. To specify the parameters of the distribution of
+# $\mathbf{y}_i$ we need to calculate its first two moments. 
+# 
+# Recall that $\boldsymbol{X}$ is a matrix of dimensionality $n\times p$. The
+# notation above $\mathbf{X}_{i,\ast}$ means that we are looking at the
+# row number $i$ and perform a sum over all values $p$.
+
+# ## Assumptions made
+# 
+# The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)
+# that there exists a function $f(\boldsymbol{x})$ and  a normal distributed error $\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)$
+# which describe our data
+
+# $$
+# \boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
+# $$
+
+# We approximate this function with our model from the solution of the linear regression equations, that is our
+# function $f$ is approximated by $\boldsymbol{\tilde{y}}$ where we want to minimize $(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2$, our MSE, with
+
+# $$
+# \boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
+# $$
+
+# ## Expectation value and variance
+# 
+# We can calculate the expectation value of $\boldsymbol{y}$ for a given element $i$
+
+# $$
+# \begin{align*} 
+# \mathbb{E}(y_i) & =
+# \mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
+# \, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, 
+# \end{align*}
+# $$
+
+# while
+# its variance is
+
+# $$
+# \begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i
+# - \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
+# [\mathbb{E}(y_i)]^2  \\  & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
+# \beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &
+# = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
+# \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
+# \ast} \, \beta)^2 \\  & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
+# \mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
+# \mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 
+# \\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
+# \mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.  
+# \end{align*}
+# $$
+
+# Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)$, that is $\boldsymbol{y}$ follows a normal distribution with 
+# mean value $\boldsymbol{X}\boldsymbol{\beta}$ and variance $\sigma^2$ (not be confused with the singular values of the SVD).
+
+# ## Expectation value and variance for $\boldsymbol{\beta}$
+# 
+# With the OLS expressions for the optimal parameters $\boldsymbol{\hat{\beta}}$ we can evaluate the expectation value
+
+# $$
+# \mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
+# $$
+
+# This means that the estimator of the regression parameters is unbiased.
+# 
+# We can also calculate the variance
+# 
+# The variance of the optimal value $\boldsymbol{\hat{\beta}}$ is
+
+# $$
+# \begin{eqnarray*}
+# \mbox{Var}(\boldsymbol{\hat{\beta}}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
+# \\
+# & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \}
+# \\
+# % & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+# % \\
+# % & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1}  \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+# % \\
+# & = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+# \\
+# & = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \,  \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+# % \\
+# % & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \,  \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
+# % \\
+# % & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T  \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
+# \\
+# & = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}  + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+# \, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
+# \end{eqnarray*}
+# $$
+
+# where we have used  that $\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) =
+# \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} +
+# \sigma^2 \, \mathbf{I}_{nn}$. From $\mbox{Var}(\boldsymbol{\beta}) = \sigma^2
+# \, (\mathbf{X}^{T} \mathbf{X})^{-1}$, one obtains an estimate of the
+# variance of the estimate of the $j$-th regression coefficient:
+# $\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} $. This may be used to
+# construct a confidence interval for the estimates.
+# 
+# In a similar way, we can obtain analytical expressions for say the
+# expectation values of the parameters $\boldsymbol{\beta}$ and their variance
+# when we employ Ridge regression, allowing us again to define a confidence interval. 
+# 
+# It is rather straightforward to show that
+
+# $$
+# \mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}.
+# $$
+
+# We see clearly that 
+# $\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}$ for any $\lambda > 0$. We say then that the ridge estimator is biased.
+# 
+# We can also compute the variance as
+
+# $$
+# \mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
+# $$
+
+# and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of Ridge parameters $\boldsymbol{\beta}$ goes to zero. 
+# 
+# With this, we can compute the difference
+
+# $$
+# \mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
+# $$
+
+# The difference is non-negative definite since each component of the
+# matrix product is non-negative definite. 
+# This means the variance we obtain with the standard OLS will always for $\lambda > 0$ be larger than the variance of $\boldsymbol{\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.
+
+# ## Deriving OLS from a probability distribution
+# 
+# Our basic assumption when we derived the OLS equations was to assume
+# that our output is determined by a given continuous function
+# $f(\boldsymbol{x})$ and a random noise $\boldsymbol{\epsilon}$ given by the normal
+# distribution with zero mean value and an undetermined variance
+# $\sigma^2$.
+# 
+# We found above that the outputs $\boldsymbol{y}$ have a mean value given by
+# $\boldsymbol{X}\hat{\boldsymbol{\beta}}$ and variance $\sigma^2$. Since the entries to
+# the design matrix are not stochastic variables, we can assume that the
+# probability distribution of our targets is also a normal distribution
+# but now with mean value $\boldsymbol{X}\hat{\boldsymbol{\beta}}$. This means that a
+# single output $y_i$ is given by the Gaussian distribution
+
+# $$
+# y_i\sim \mathcal{N}(\boldsymbol{X}_{i,*}\boldsymbol{\beta}, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
+# $$
+
+# ## Independent and Identically Distrubuted (iid)
+# 
+# We assume now that the various $y_i$ values are stochastically distributed according to the above Gaussian distribution. 
+# We define this distribution as
+
+# $$
+# p(y_i, \boldsymbol{X}\vert\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]},
+# $$
+
+# which reads as finding the likelihood of an event $y_i$ with the input variables $\boldsymbol{X}$ given the parameters (to be determined) $\boldsymbol{\beta}$.
+# 
+# Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event $\boldsymbol{y}$ as the product of the single events, that is we have
+
+# $$
+# p(\boldsymbol{y},\boldsymbol{X}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}=\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta}).
+# $$
+
+# We will write this in a more compact form reserving $\boldsymbol{D}$ for the domain of events, including the ouputs (targets) and the inputs. That is
+# in case we have a simple one-dimensional input and output case
+
+# $$
+# \boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})].
+# $$
+
+# In the more general case the various inputs should be replaced by the possible features represented by the input data set $\boldsymbol{X}$. 
+# We can now rewrite the above probability as
+
+# $$
+# p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
+# $$
+
+# It is a conditional probability (see below) and reads as the likelihood of a domain of events $\boldsymbol{D}$ given a set of parameters $\boldsymbol{\beta}$.
+
+# ## Maximum Likelihood Estimation (MLE)
+# 
+# In statistics, maximum likelihood estimation (MLE) is a method of
+# estimating the parameters of an assumed probability distribution,
+# given some observed data. This is achieved by maximizing a likelihood
+# function so that, under the assumed statistical model, the observed
+# data is the most probable. 
+# 
+# We will assume here that our events are given by the above Gaussian
+# distribution and we will determine the optimal parameters $\beta$ by
+# maximizing the above PDF. However, computing the derivatives of a
+# product function is cumbersome and can easily lead to overflow and/or
+# underflowproblems, with potentials for loss of numerical precision.
+# 
+# In practice, it is more convenient to maximize the logarithm of the
+# PDF because it is a monotonically increasing function of the argument.
+# Alternatively, and this will be our option, we will minimize the
+# negative of the logarithm since this is a monotonically decreasing
+# function.
+# 
+# Note also that maximization/minimization of the logarithm of the PDF
+# is equivalent to the maximization/minimization of the function itself.
+
+# ## A new Cost Function
+# 
+# We could now define a new cost function to minimize, namely the negative logarithm of the above PDF
+
+# $$
+# C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})},
+# $$
+
+# which becomes
+
+# $$
+# C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}.
+# $$
+
+# Taking the derivative of the *new* cost function with respect to the parameters $\beta$ we recognize our familiar OLS equation, namely
+
+# $$
+# \boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0,
+# $$
+
+# which leads to the well-known OLS equation for the optimal paramters $\beta$
+
+# $$
+# \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}!
+# $$
+
+# Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics.
+
+# ## More basic Statistics and Bayes' theorem
+# 
+# A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry.
+# Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.
+# 
+# Assume we have two domains of events $X=[x_0,x_1,\dots,x_{n-1}]$ and $Y=[y_0,y_1,\dots,y_{n-1}]$.
+# 
+# We define also the likelihood for $X$ and $Y$ as $p(X)$ and $p(Y)$ respectively.
+# The likelihood of a specific event $x_i$ (or $y_i$) is then written as $p(X=x_i)$ or just $p(x_i)=p_i$. 
+# 
+# **Union of events is given by.**
+
+# $$
+# p(X \cup Y)= p(X)+p(Y)-p(X \cap Y).
+# $$
+
+# **The product rule (aka joint probability) is given by.**
+
+# $$
+# p(X \cup Y)= p(X,Y)= p(X\vert Y)p(Y)=p(Y\vert X)p(X),
+# $$
+
+# where we read $p(X\vert Y)$ as the likelihood of obtaining $X$ given $Y$.
+# 
+# If we have independent events then $p(X,Y)=p(X)p(Y)$.
+
+# ## Marginal Probability
+# 
+# The marginal probability is defined in terms of only one of the set of variables $X,Y$. For a discrete probability we have
+
+# $$
+# p(X)=\sum_{i=0}^{n-1}p(X,Y=y_i)=\sum_{i=0}^{n-1}p(X\vert Y=y_i)p(Y=y_i)=\sum_{i=0}^{n-1}p(X\vert y_i)p(y_i).
+# $$
+
+# ## Conditional  Probability
+# 
+# The conditional  probability, if $p(Y) > 0$, is
+
+# $$
+# p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}.
+# $$
+
+# ## Bayes' Theorem
+# 
+# If we combine the conditional probability with the marginal probability and the standard product rule, we have
+
+# $$
+# p(X\vert Y)= \frac{p(X,Y)}{p(Y)},
+# $$
+
+# which we can rewrite as
+
+# $$
+# p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)},
+# $$
+
+# which is Bayes' theorem. It allows us to evaluate the uncertainty in in $X$ after we have observed $Y$. We can easily interchange $X$ with $Y$.
+
+# ## Interpretations of Bayes' Theorem
+# 
+# The quantity $p(Y\vert X)$ on the right-hand side of the theorem is
+# evaluated for the observed data $Y$ and can be viewed as a function of
+# the parameter space represented by $X$. This function is not
+# necesseraly normalized and is normally called the likelihood function.
+# 
+# The function $p(X)$ on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.
+# 
+# Let us try to illustrate Bayes' theorem through an example.
+
+# ## Example of Usage of Bayes' theorem
+# 
+# Let us suppose that you are undergoing a series of mammography scans in
+# order to rule out possible breast cancer cases.  We define the
+# sensitivity for a positive event by the variable $X$. It takes binary
+# values with $X=1$ representing a positive event and $X=0$ being a
+# negative event. We reserve $Y$ as a classification parameter for
+# either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a  positive thing).
+# 
+# We let $Y=1$ represent the the case of having breast cancer and $Y=0$ as not.
+# 
+# Let us assume that if you have breast cancer, the test will be positive with a probability of $0.8$, that is we have
+
+# $$
+# p(X=1\vert Y=1) =0.8.
+# $$
+
+# This obviously sounds  scary since many would conclude that if the test is positive, there is a likelihood of $80\%$ for having cancer.
+# It is however not correct, as the following Bayesian analysis shows.
+
+# ## Doing it correctly
+# 
+# If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number.
+# Let us assume that the prior probability in the population as a whole is
+
+# $$
+# p(Y=1) =0.004.
+# $$
+
+# We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have
+
+# $$
+# p(X=1\vert Y=0) =0.1.
+# $$
+
+# Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute
+
+# $$
+# p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=0)p(Y=0)}=\frac{0.8\times 0.004}{0.8\times 0.004+0.1\times 0.996}=0.031.
+# $$
+
+# That is, in case of a positive test, there is only a $3\%$ chance of having breast cancer!
+
+# ## Bayes' Theorem and Ridge and Lasso Regression
+# 
+# Hitherto we have discussed Ridge and Lasso regression in terms of a
+# linear analysis. This may to many of you feel rather technical and
+# perhaps not that intuitive. The question is whether we can develop a
+# more intuitive way of understanding what Ridge and Lasso express.
+# 
+# Before we proceed let us perform a Ridge, Lasso  and OLS analysis of a polynomial fit.
+
+# ## Test Function for what happens with OLS, Ridge and Lasso
+# 
+# We will play around with a study of the values for the optimal
+# parameters $\boldsymbol{\beta}$ using OLS, Ridge and Lasso regression.  For
+# OLS, you will notice as function of the noise and polynomial degree,
+# that the parameters $\beta$ will fluctuate from order to order in the
+# polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS.
+# 
+# For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one.
+
+# In[7]:
+
+
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+# Make data set.
+n = 10000
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)
+
+Maxpolydegree = 5
+X = np.zeros((len(x),Maxpolydegree))
+X[:,0] = 1.0
+
+for polydegree in range(1, Maxpolydegree):
+    for degree in range(polydegree):
+        X[:,degree] = x**(degree)
+
+
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+# matrix inversion to find beta
+OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
+print(OLSbeta)
+ypredictOLS = X_test @ OLSbeta
+print("Test MSE OLS")
+print(MSE(y_test,ypredictOLS))
+# Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn
+# Decide which values of lambda to use
+nlambdas = 4
+MSERidgePredict = np.zeros(nlambdas)
+MSELassoPredict = np.zeros(nlambdas)
+lambdas = np.logspace(-3, 1, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    # Make the fit using Ridge and Lasso
+    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
+    RegRidge.fit(X_train,y_train)
+    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
+    RegLasso.fit(X_train,y_train)
+    # and then make the prediction
+    ypredictRidge = RegRidge.predict(X_test)
+    ypredictLasso = RegLasso.predict(X_test)
+    # Compute the MSE and print it
+    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+    MSELassoPredict[i] = MSE(y_test,ypredictLasso)
+    print(lmb,RegRidge.coef_)
+    print(lmb,RegLasso.coef_)
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSERidgePredict, 'b', label = 'MSE Ridge Test')
+plt.plot(np.log10(lambdas), MSELassoPredict, 'r', label = 'MSE Lasso Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+# How can we understand this?
+
+# ## Invoking Bayes' theorem
+# 
+# Using Bayes' theorem we can gain a better intuition about Ridge and Lasso regression. 
+# 
+# For ordinary least squares we postulated that the maximum likelihood for the doamin of events $\boldsymbol{D}$ (one-dimensional case)
+
+# $$
+# \boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})],
+# $$
+
+# is given by
+
+# $$
+# p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
+# $$
+
+# In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set $\boldsymbol{\beta}$ given a domain of events $\boldsymbol{D}$?  That is, how can we define the posterior probability
+
+# $$
+# p(\boldsymbol{\beta}\vert\boldsymbol{D}).
+# $$
+
+# Bayes' theorem comes to our rescue here since (omitting the normalization constant)
+
+# $$
+# p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
+# $$
+
+# We have a model for $p(\boldsymbol{D}\vert\boldsymbol{\beta})$ but need one for the **prior** $p(\boldsymbol{\beta}$!
+
+# ## Ridge and Bayes
+# 
+# With the posterior probability defined by a likelihood which we have
+# already modeled and an unknown prior, we are now ready to make
+# additional models for the prior.
+# 
+# We can, based on our discussions of the variance of $\boldsymbol{\beta}$ and the mean value, assume that the prior for the values $\boldsymbol{\beta}$ is given by a Gaussian with mean value zero and variance $\tau^2$, that is
+
+# $$
+# p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
+# $$
+
+# Our posterior probability becomes then (omitting the normalization factor which is just a constant)
+
+# $$
+# p(\boldsymbol{\beta\vert\boldsymbol{D})}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
+# $$
+
+# We can now optimize this quantity with respect to $\boldsymbol{\beta}$. As we
+# did for OLS, this is most conveniently done by taking the negative
+# logarithm of the posterior probability. Doing so and leaving out the
+# constants terms that do not depend on $\beta$, we have
+
+# $$
+# C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
+# $$
+
+# and replacing $1/2\tau^2$ with $\lambda$ we have
+
+# $$
+# C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
+# $$
+
+# which is our Ridge cost function!  Nice, isn't it?
+
+# ## Lasso and Bayes
+# 
+# To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution ([Laplace in this case](https://en.wikipedia.org/wiki/Laplace_distribution)) with zero mean value,  that is
+
+# $$
+# p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
+# $$
+
+# Our posterior probability becomes then (omitting the normalization factor which is just a constant)
+
+# $$
+# p(\boldsymbol{\beta}\vert\boldsymbol{D})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
+# $$
+
+# Taking the negative
+# logarithm of the posterior probability and leaving out the
+# constants terms that do not depend on $\beta$, we have
+
+# $$
+# C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1,
+# $$
+
+# and replacing $1/\tau$ with $\lambda$ we have
+
+# $$
+# C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1,
+# $$
+
+# which is our Lasso cost function!
diff --git a/doc/LectureNotes/_toc.yml b/doc/LectureNotes/_toc.yml
index 44f2bc06f..08d1a2377 100644
--- a/doc/LectureNotes/_toc.yml
+++ b/doc/LectureNotes/_toc.yml
@@ -45,3 +45,5 @@ parts:
     - file: week34.ipynb
     - file: exercisesweek35.ipynb
     - file: week35.ipynb
+    - file: exercisesweek36.ipynb
+    - file: week36.ipynb
diff --git a/doc/LectureNotes/gaussian.pdf b/doc/LectureNotes/gaussian.pdf
index 7e852ec73..ce18be03b 100644
Binary files a/doc/LectureNotes/gaussian.pdf and b/doc/LectureNotes/gaussian.pdf differ
diff --git a/doc/LectureNotes/week36.ipynb b/doc/LectureNotes/week36.ipynb
new file mode 100644
index 000000000..991b0f568
--- /dev/null
+++ b/doc/LectureNotes/week36.ipynb
@@ -0,0 +1,3414 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b49fd9eb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "\n",
+    ""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "138fe268",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Week 36: Statistical interpretation of Linear Regression and Resampling techniques\n",
+    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
+    "\n",
+    "Date: **September 4-8, 2023**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c20f461b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Plans for week 36\n",
+    "\n",
+    "* Material for the active learning sessions on Tuesday and Wednesday\n",
+    "\n",
+    "  * Summary from last week on discussion of SVD, Ridge and Lasso linear regression.\n",
+    "\n",
+    "  * Recommended Reading: Hastie et al chapter 3, see \n",
+    "\n",
+    "  * Presentation and discussion of first project\n",
+    "\n",
+    "* Material for the lecture on Thursday September 7\n",
+    "\n",
+    "  * Linear Regression and links with Statistics, Resampling methods\n",
+    "\n",
+    "  * Recommended Reading: Goodfellow et al chapter 3 on probability theory, see URL:\"\"\n",
+    "\n",
+    "  * See also Murphy, sections 2.4 (Gaussian distributions) and 3.2 (Bayesian Statistics, basis)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "55f0d080",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Material for the active learning sessions Tuesday and Wednesday\n",
+    "\n",
+    "The material here contains a summary from last Week and discussion of SVD, Ridge and Lasso regression with examples"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "69ce62a6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Linear Regression and  the SVD\n",
+    "\n",
+    "We used the SVD to analyse the matrix to invert in ordinary lineat regression"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a9aaa130",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T=\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{V}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2b1c0902",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Since the matrices here have dimension $p\\times p$, with $p$ corresponding to the singular values, we defined last week the matrix"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "093e3a8c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma} = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c0102cd2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where the tilde-matrix $\\tilde{\\boldsymbol{\\Sigma}}$ is a matrix of dimension $p\\times p$ containing only the singular values $\\sigma_i$, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "084138a1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{\\Sigma}}=\\begin{bmatrix} \\sigma_0 & 0 & 0 & \\dots & 0 & 0 \\\\\n",
+    "                                    0 & \\sigma_1 & 0 & \\dots & 0 & 0 \\\\\n",
+    "\t\t\t\t    0 & 0 & \\sigma_2 & \\dots & 0 & 0 \\\\\n",
+    "\t\t\t\t    0 & 0 & 0 & \\dots & \\sigma_{p-2} & 0 \\\\\n",
+    "\t\t\t\t    0 & 0 & 0 & \\dots & 0 & \\sigma_{p-1} \\\\\n",
+    "\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cb8ed4f2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "meaning we can write"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d7c1c21b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2\\boldsymbol{V}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "791e9c6d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Multiplying from the right with $\\boldsymbol{V}$ (using the orthogonality of $\\boldsymbol{V}$) we get"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "924b8081",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{V}=\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "949f6658",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## What does it mean?\n",
+    "\n",
+    "This means the vectors $\\boldsymbol{v}_i$ of the orthogonal matrix $\\boldsymbol{V}$\n",
+    "are the eigenvectors of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ with eigenvalues\n",
+    "given by the singular values squared, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "46a4a83c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)\\boldsymbol{v}_i=\\boldsymbol{v}_i\\sigma_i^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ce019aa6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In other words, each non-zero singular value of $\\boldsymbol{X}$ is a positive\n",
+    "square root of an eigenvalue of $\\boldsymbol{X}^T\\boldsymbol{X}$.  It means also that\n",
+    "the columns of $\\boldsymbol{V}$ are the eigenvectors of\n",
+    "$\\boldsymbol{X}^T\\boldsymbol{X}$. Since we have ordered the singular values of\n",
+    "$\\boldsymbol{X}$ in a descending order, it means that the column vectors\n",
+    "$\\boldsymbol{v}_i$ are hierarchically ordered by how much correlation they\n",
+    "encode from the columns of $\\boldsymbol{X}$. \n",
+    "\n",
+    "Note that these are also the eigenvectors and eigenvalues of the\n",
+    "Hessian matrix.\n",
+    "\n",
+    "If we now recall the definition of the covariance matrix (not using\n",
+    "Bessel's correction) we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c662d90f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{C}[\\boldsymbol{X}]=\\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7df45c97",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "meaning that every squared non-singular value of $\\boldsymbol{X}$ divided by $n$ (\n",
+    "the number of samples) are the eigenvalues of the covariance\n",
+    "matrix. Every singular value of $\\boldsymbol{X}$ is thus a positive square\n",
+    "root of an eigenvalue of $\\boldsymbol{X}^T\\boldsymbol{X}$. If the matrix $\\boldsymbol{X}$ is\n",
+    "self-adjoint, the singular values of $\\boldsymbol{X}$ are equal to the\n",
+    "absolute value of the eigenvalues of $\\boldsymbol{X}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f0741872",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## And finally  $\\boldsymbol{X}\\boldsymbol{X}^T$\n",
+    "\n",
+    "For $\\boldsymbol{X}\\boldsymbol{X}^T$ we found"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "05530ccb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T=\\boldsymbol{U}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{U}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3048f911",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Since the matrices here have dimension $n\\times n$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9e3c2de1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\Sigma}\\boldsymbol{\\Sigma}^T = \\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} \\\\ \\boldsymbol{0}\\\\ \\end{bmatrix}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}}  \\boldsymbol{0}\\\\ \\end{bmatrix}=\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "491923c1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "leading to"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "77d1dbe5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}\\boldsymbol{U}^T.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fb7e2323",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Multiplying with $\\boldsymbol{U}$ from the right gives us the eigenvalue problem"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c9cca57a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U}=\\boldsymbol{U}\\begin{bmatrix} \\tilde{\\boldsymbol{\\Sigma}} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\boldsymbol{0}\\\\ \\end{bmatrix}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "04e4b486",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "It means that the eigenvalues of $\\boldsymbol{X}\\boldsymbol{X}^T$ are again given by\n",
+    "the non-zero singular values plus now a series of zeros.  The column\n",
+    "vectors of $\\boldsymbol{U}$ are the eigenvectors of $\\boldsymbol{X}\\boldsymbol{X}^T$ and\n",
+    "measure how much correlations are contained in the rows of $\\boldsymbol{X}$.\n",
+    "\n",
+    "Since we will mainly be interested in the correlations among the features\n",
+    "of our data (the columns of $\\boldsymbol{X}$, the quantity of interest for us are the non-zero singular\n",
+    "values and the column vectors of $\\boldsymbol{V}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2b3c22fb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Code for SVD and Inversion of Matrices\n",
+    "\n",
+    "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n",
+    "The simple answer is to use the linear algebra function for pseudoinvers, that is"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "12637a17",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "Ainv = np.linlag.pinv(A)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "904adaf8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "6b5465dc",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "# SVD inversion\n",
+    "def SVDinv(A):\n",
+    "    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).\n",
+    "    SVD is numerically more stable than the inversion algorithms provided by\n",
+    "    numpy and scipy.linalg at the cost of being slower.\n",
+    "    '''\n",
+    "    U, s, VT = np.linalg.svd(A)\n",
+    "    print('test U')\n",
+    "    print( (np.transpose(U) @ U - U @np.transpose(U)))\n",
+    "    print('test VT')\n",
+    "    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n",
+    "\n",
+    "\n",
+    "    D = np.zeros((len(U),len(VT)))\n",
+    "    D = np.diag(s)\n",
+    "    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)\n",
+    "    return np.matmul(V,np.matmul(invD,UT))\n",
+    "\n",
+    "\n",
+    "#X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])\n",
+    "# Non-singular square matrix\n",
+    "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n",
+    "print(X)\n",
+    "A = np.transpose(X) @ X\n",
+    "# Brute force inversion\n",
+    "B = np.linalg.inv(A)  # here we could use np.linalg.pinv(A)\n",
+    "C = SVDinv(A)\n",
+    "print(np.abs(B-C))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a80dcee4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Inverse of Rectangular Matrix\n",
+    "\n",
+    "Although our matrix to invert $\\boldsymbol{X}^T\\boldsymbol{X}$ is a square matrix, our matrix may be singular. \n",
+    "\n",
+    "The pseudoinverse is the generalization of the matrix inverse for square matrices to\n",
+    "rectangular matrices where the number of rows and columns are not equal.\n",
+    "\n",
+    "It is also called the the Moore-Penrose Inverse after two independent discoverers of the method or the Generalized Inverse.\n",
+    "It is used for the calculation of the inverse for singular or near singular matrices and for rectangular matrices.\n",
+    "\n",
+    "Using the SVD we can obtain the pseudoinverse of a matrix $\\boldsymbol{A}$ (labeled here as $\\boldsymbol{A}_{\\mathrm{PI}}$)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0c4052d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{A}_{\\mathrm{PI}}= \\boldsymbol{V}\\boldsymbol{D}_{\\mathrm{PI}}\\boldsymbol{U}^T,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "185d433f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $\\boldsymbol{D}_{\\mathrm{PI}}$ can be calculated by creating a diagonal matrix from $\\boldsymbol{\\Sigma}$ where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "418f8797",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "# SVD inversion\n",
+    "def SVDinv(A):\n",
+    "    U, s, VT = np.linalg.svd(A)\n",
+    "    # reciprocals of singular values of s\n",
+    "    d = 1.0 / s\n",
+    "    # create m x n D matrix\n",
+    "    D = np.zeros(A.shape)\n",
+    "    # populate D with n x n diagonal matrix\n",
+    "    D[:A.shape[1], :A.shape[1]] = np.diag(d)\n",
+    "    UT = np.transpose(U)\n",
+    "    V = np.transpose(VT)\n",
+    "    return np.matmul(V,np.matmul(D.T,UT))\n",
+    "\n",
+    "\n",
+    "A = np.array([ [0.3, 0.4], [0.5, 0.6], [0.7, 0.8],[0.9, 1.0]])\n",
+    "print(A)\n",
+    "# Brute force inversion of super-collinear matrix\n",
+    "B = np.linalg.pinv(A)\n",
+    "print(B)\n",
+    "# Compare our own algorithm with pinv\n",
+    "C = SVDinv(A)\n",
+    "print(np.abs(C-B))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bf5051ea",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "As you can see from this example, our own decomposition based on the SVD agrees with  the pseudoinverse algorithm provided by **Numpy**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "202d1f90",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge and LASSO Regression\n",
+    "\n",
+    "Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is \n",
+    "our optimization problem is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "df0a8718",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4ebd1d20",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "or we can state it as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40df3859",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "980001f5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have used the definition of  a norm-2 vector, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "699b1198",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ff01ef41",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## From OLS to Ridge and Lasso\n",
+    "\n",
+    "By minimizing the above equation with respect to the parameters\n",
+    "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n",
+    "parameters $\\boldsymbol{\\beta}$.  We can add a regularization parameter $\\lambda$ by\n",
+    "defining a new cost function to be optimized, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d7b9188f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "778790dc",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to the Ridge regression minimization problem where we\n",
+    "require that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n",
+    "a finite number larger than zero. We do not include such a constraints in the discussions here.\n",
+    "\n",
+    "By defining"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38dd7428",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6ed3deb0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "we have a new optimization equation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e0860754",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n",
+    "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "66751140",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. \n",
+    "\n",
+    "Here we have defined the norm-1 as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e1fa2bdb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\vert\\vert \\boldsymbol{x}\\vert\\vert_1 = \\sum_i \\vert x_i\\vert.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "32131982",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Deriving the  Ridge Regression Equations\n",
+    "\n",
+    "Using the matrix-vector expression for Ridge regression and dropping the parameter $1/n$ in front of the standard means squared error equation, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c1979796",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\boldsymbol{\\beta}^T\\boldsymbol{\\beta},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d7eb56d7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and \n",
+    "taking the derivatives with respect to $\\boldsymbol{\\beta}$ we obtain then\n",
+    "a slightly modified matrix inversion problem which for finite values\n",
+    "of $\\lambda$ does not suffer from singularity problems. We obtain\n",
+    "the optimal parameters"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7c42f326",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6496e5c0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "837f7c04",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "55e9395f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with $t$ a finite positive number."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a943f34a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Note on Scikit-Learn\n",
+    "\n",
+    "Note well that a library like **Scikit-Learn** does not include the $1/n$ factor in the expression for the mean-squared error. If you include it, the optimal parameter $\\beta$ becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "63a227a9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+n\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8be2bd7b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In our codes where we compare our own codes with **Scikit-Learn**, we do thus not include the $1/n$ factor in the cost function."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "35509652",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Comparison with OLS\n",
+    "When we compare this with the ordinary least squares result we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2fbe21d0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_{\\mathrm{OLS}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "71e986f8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$.\n",
+    "\n",
+    "We see that Ridge regression is nothing but the standard OLS with a\n",
+    "modified diagonal term added to $\\boldsymbol{X}^T\\boldsymbol{X}$. The consequences, in\n",
+    "particular for our discussion of the bias-variance tradeoff are rather\n",
+    "interesting. We will see that for specific values of $\\lambda$, we may\n",
+    "even reduce the variance of the optimal parameters $\\boldsymbol{\\beta}$. These topics and other related ones, will be discussed after the more linear algebra oriented analysis here."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a5e5f40",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## SVD analysis\n",
+    "\n",
+    "Using our insights about the SVD of the design matrix $\\boldsymbol{X}$ \n",
+    "We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\\boldsymbol{U}$ as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "752b6ba2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{y}}_{\\mathrm{OLS}}=\\boldsymbol{X}\\boldsymbol{\\beta}  =\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4e724327",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "For Ridge regression this becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b15c2c90",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\tilde{\\boldsymbol{y}}_{\\mathrm{Ridge}}=\\boldsymbol{X}\\boldsymbol{\\beta}_{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{\\Sigma}^2\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "82e70fa5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$ from the SVD of the matrix $\\boldsymbol{X}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3394e2e7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Interpreting the Ridge results\n",
+    "\n",
+    "Since $\\lambda \\geq 0$, it means that compared to OLS, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "722ffa86",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda} \\leq 1.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "720ed773",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Ridge regression finds the coordinates of $\\boldsymbol{y}$ with respect to the\n",
+    "orthonormal basis $\\boldsymbol{U}$, it then shrinks the coordinates by\n",
+    "$\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}$. Recall that the SVD has\n",
+    "eigenvalues ordered in a descending way, that is $\\sigma_i \\geq\n",
+    "\\sigma_{i+1}$.\n",
+    "\n",
+    "For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b57864f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## More interpretations\n",
+    "\n",
+    "For the sake of simplicity, let us assume that the design matrix is orthonormal, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8f36ff27",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1} =\\boldsymbol{I}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "70adc9ce",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In this case the standard OLS results in"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eb4a995f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\beta}^{\\mathrm{OLS}} = \\boldsymbol{X}^T\\boldsymbol{y}=\\sum_{i=0}^{n-1}\\boldsymbol{u}_i\\boldsymbol{u}_i^T\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3cd4b92",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bbf9b1de",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{I}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\left(1+\\lambda\\right)^{-1}\\boldsymbol{\\beta}^{\\mathrm{OLS}},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e30eff48",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\\lambda$, and\n",
+    "the Ridge estimator converges to zero when the hyperparameter goes to\n",
+    "infinity.\n",
+    "\n",
+    "We will come back to more interpreations after we have gone through some of the statistical analysis part. \n",
+    "\n",
+    "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n",
+    "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9a8f8e2e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Deriving the  Lasso Regression Equations\n",
+    "\n",
+    "Using the matrix-vector expression for Lasso regression, we have the following **cost** function"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3cb7928c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b1c6c813",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Taking the derivative with respect to $\\boldsymbol{\\beta}$ and recalling that the derivative of the absolute value is (we drop the boldfaced vector symbol for simplicity)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "da336775",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{d \\vert \\beta\\vert}{d \\beta}=\\mathrm{sgn}(\\beta)=\\left\\{\\begin{array}{cc} 1 & \\beta > 0 \\\\-1 & \\beta < 0, \\end{array}\\right.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2e0de141",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "we have that the derivative of the cost function is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40985ec4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C(\\boldsymbol{X},\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}=-\\frac{2}{n}\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})+\\lambda sgn(\\boldsymbol{\\beta})=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "31091d7b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and reordering we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e81d965e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta}+\\lambda sgn(\\boldsymbol{\\beta})=\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9984b3a4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor $2/n$ in a redefinition of the parameter $\\lambda$. We will solve this type of problems using libraries like **scikit-learn**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "21ec3768",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression\n",
+    "\n",
+    "Let us assume that our design matrix is given by unit (identity) matrix, that is a square diagonal matrix with ones only along the\n",
+    "diagonal. In this case we have an equal number of rows and columns $n=p$.\n",
+    "\n",
+    "Our model approximation is just $\\tilde{\\boldsymbol{y}}=\\boldsymbol{\\beta}$ and the mean squared error and thereby the cost function for ordinary least sqquares (OLS) is then (we drop the term $1/n$)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6be0ec63",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "44c7278c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and minimizing we have that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "918a8cc3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\beta}_i^{\\mathrm{OLS}} = y_i.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3949af51",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge Regression\n",
+    "\n",
+    "For Ridge regression our cost function is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aa856d18",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\beta_i^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bd30d1d4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and minimizing we have that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "142de535",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\beta}_i^{\\mathrm{Ridge}} = \\frac{y_i}{1+\\lambda}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1efee0c3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Lasso Regression\n",
+    "\n",
+    "For Lasso regression our cost function is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6db019a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\vert\\beta_i\\vert=\\sum_{i=0}^{p-1}(y_i-\\beta_i)^2+\\lambda\\sum_{i=0}^{p-1}\\sqrt{\\beta_i^2},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0f38deda",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and minimizing we have that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7f9738a5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "-2\\sum_{i=0}^{p-1}(y_i-\\beta_i)+\\lambda \\sum_{i=0}^{p-1}\\frac{(\\beta_i)}{\\vert\\beta_i\\vert}=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4e8460f8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ae49ddce",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}_i^{\\mathrm{Lasso}} = \\left\\{\\begin{array}{ccc}y_i-\\frac{\\lambda}{2} &\\mathrm{if} & y_i> \\frac{\\lambda}{2}\\\\\n",
+    "                                                          y_i+\\frac{\\lambda}{2} &\\mathrm{if} & y_i< -\\frac{\\lambda}{2}\\\\\n",
+    "\t\t\t\t\t\t\t  0 &\\mathrm{if} & \\vert y_i\\vert\\le  \\frac{\\lambda}{2}\\end{array}\\right.\\\\.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1efb96d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Plotting these results shows clearly that Lasso regression suppresses (sets to zero) values of $\\beta_i$ for specific values of $\\lambda$. Ridge regression reduces on the other hand the values of $\\beta_i$ as function of $\\lambda$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1a030291",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Yet another Example\n",
+    "\n",
+    "Let us assume we have a data set with outputs/targets given by the vector"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af486213",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{y}=\\begin{bmatrix}4 \\\\ 2 \\\\3\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "87e5470d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and our inputs as a $3\\times 2$ design matrix"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "182f9f51",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}=\\begin{bmatrix}2 & 0\\\\ 0 & 1 \\\\ 0 & 0\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "edc0ee3d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "meaning that we have two features and two unknown parameters $\\beta_0$ and $\\beta_1$ to be determined either by ordinary least squares, Ridge or Lasso regression."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "673c64a0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The OLS case\n",
+    "\n",
+    "For ordinary least squares (OLS) we know that the optimal solution is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c0554476",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cc9c761b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Inserting the above values we obtain that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b5c84ed1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\begin{bmatrix}2 \\\\ 2\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a8d2ba86",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The code which implements this simpler case is presented after the discussion of Ridge and Lasso."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9e407326",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The Ridge case\n",
+    "\n",
+    "For Ridge regression we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "36ce4758",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\left( \\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8665e33b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Inserting the above values we obtain that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "16dbcd82",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}=\\begin{bmatrix}\\frac{8}{4+\\lambda} \\\\ \\frac{2}{1+\\lambda}\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "642f7768",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "There is normally a constraint on the value of $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2$ via the parameter $\\lambda$.\n",
+    "Let us for simplicity assume that $\\beta_0^2+\\beta_1^2=1$ as constraint. This will allow us to find an expression for the optimal values of $\\beta$ and $\\lambda$.\n",
+    "\n",
+    "To see this, let us write the cost function for Ridge regression."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1784c79c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Writing the Cost Function\n",
+    "\n",
+    "We define the MSE without the $1/n$ factor and have then, using that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "72b7600f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}\\boldsymbol{\\beta}=\\begin{bmatrix} 2\\beta_0 \\\\ \\beta_1 \\\\0 \\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84ca3171",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\beta_0^2+\\beta_1^2),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d31aa45e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and taking the derivative with respect to $\\beta_0$ we get"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1298ca6d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0=\\frac{8}{4+\\lambda},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fa2455d6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and for $\\beta_1$ we obtain"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8cd50073",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_1=\\frac{2}{1+\\lambda},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1f7e12c3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Using the constraint for $\\beta_0^2+\\beta_1^2=1$ we can constrain $\\lambda$ by solving"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a79821fd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\left(\\frac{8}{4+\\lambda}\\right)^2+\\left(\\frac{2}{1+\\lambda}\\right)^2=1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2ac13531",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which gives $\\lambda=4.571$ and $\\beta_0=0.933$ and $\\beta_1=0.359$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eed25f68",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Lasso case\n",
+    "\n",
+    "For Lasso we need now, keeping a  constraint on $\\vert\\beta_0\\vert+\\vert\\beta_1\\vert=1$,  to take the derivative of the absolute values of $\\beta_0$\n",
+    "and $\\beta_1$. This gives us the following derivatives of the cost function"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f74d71ec",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=(4-2\\beta_0)^2+(2-\\beta_1)^2+\\lambda(\\vert\\beta_0\\vert+\\vert\\beta_1\\vert),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f0797067",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_0}=-4(4-2\\beta_0)+\\lambda\\mathrm{sgn}(\\beta_0)=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3cc5d5e0",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "04250e5c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_1}=-2(2-\\beta_1)+\\lambda\\mathrm{sgn}(\\beta_1)=0.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2b24d2fa",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We have now four cases to solve besides the trivial cases $\\beta_0$ and/or $\\beta_1$ are zero, namely\n",
+    "1. $\\beta_0 > 0$ and $\\beta_1 > 0$,\n",
+    "\n",
+    "2. $\\beta_0 > 0$ and $\\beta_1 < 0$,\n",
+    "\n",
+    "3. $\\beta_0 < 0$ and $\\beta_1 > 0$,\n",
+    "\n",
+    "4. $\\beta_0 < 0$ and $\\beta_1 < 0$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aa950760",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The first Case\n",
+    "\n",
+    "If we consider the first case, we have then"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "475389ed",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "-4(4-2\\beta_0)+\\lambda=0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "086558d9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1a3b2ba5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "-2(2-\\beta_1)+\\lambda=0.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c928bc46",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which yields"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "809a58f7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_0=\\frac{16+\\lambda}{8},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b937bf4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6e5b2527",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_1=\\frac{4+\\lambda}{2}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c5f88a45",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Using the constraint on $\\beta_0$ and $\\beta_1$ we can then find the optimal value of $\\lambda$ for the different cases. We leave this as an exercise to you."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c57753b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Simple code for solving the above problem\n",
+    "\n",
+    "Here we set up the OLS, Ridge and Lasso functionality in order to study the above example. Note that here we have opted for a set of values of $\\lambda$, meaning that we need to perform a search in order to find the optimal values.\n",
+    "\n",
+    "First we study and compare the OLS and Ridge results.  The next code compares all three methods."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "392f218d",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "\n",
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "\n",
+    "X = np.array( [ [ 2, 0], [0, 1], [0,0]])\n",
+    "y = np.array( [4, 2, 3])\n",
+    "\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y\n",
+    "print(OLSbeta)\n",
+    "# and then make the prediction\n",
+    "ytildeOLS = X @ OLSbeta\n",
+    "print(\"Training MSE for OLS\")\n",
+    "print(MSE(y,ytildeOLS))\n",
+    "ypredictOLS = X @ OLSbeta\n",
+    "\n",
+    "# Repeat now for Ridge regression and various values of the regularization parameter\n",
+    "I = np.eye(2,2)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 100\n",
+    "MSEPredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 4, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y\n",
+    "#    print(Ridgebeta)\n",
+    "    # and then make the prediction\n",
+    "    ypredictRidge = X @ Ridgebeta\n",
+    "    MSEPredict[i] = MSE(y,ypredictRidge)\n",
+    "#    print(MSEPredict[i])\n",
+    "    # Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Train')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dd134ab1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We see here that we reach a plateau. What is actually happening?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8d9a61db",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## With Lasso Regression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "afba7c4a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "\n",
+    "X = np.array( [ [ 2, 0], [0, 1], [0,0]])\n",
+    "y = np.array( [4, 2, 3])\n",
+    "\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.inv(X.T @ X) @ X.T @ y\n",
+    "print(OLSbeta)\n",
+    "# and then make the prediction\n",
+    "ytildeOLS = X @ OLSbeta\n",
+    "print(\"Training MSE for OLS\")\n",
+    "print(MSE(y,ytildeOLS))\n",
+    "ypredictOLS = X @ OLSbeta\n",
+    "\n",
+    "# Repeat now for Ridge regression and various values of the regularization parameter\n",
+    "I = np.eye(2,2)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 100\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "MSELassoPredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 4, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    Ridgebeta = np.linalg.inv(X.T @ X+lmb*I) @ X.T @ y\n",
+    "    print(Ridgebeta)\n",
+    "    # and then make the prediction\n",
+    "    ypredictRidge = X @ Ridgebeta\n",
+    "    MSERidgePredict[i] = MSE(y,ypredictRidge)\n",
+    "    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n",
+    "    RegLasso.fit(X,y)\n",
+    "    ypredictLasso = RegLasso.predict(X)\n",
+    "    print(RegLasso.coef_)\n",
+    "    MSELassoPredict[i] = MSE(y,ypredictLasso)\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'r--', label = 'MSE Ridge Train')\n",
+    "plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Train')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4d63db0b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Another Example, now with a polynomial fit"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "81115144",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "# Useful for eventual debugging.\n",
+    "np.random.seed(3155)\n",
+    "\n",
+    "x = np.random.rand(100)\n",
+    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+    "\n",
+    "# number of features p (here degree of polynomial\n",
+    "p = 3\n",
+    "#  The design matrix now as function of a given polynomial\n",
+    "X = np.zeros((len(x),p))\n",
+    "X[:,0] = 1.0\n",
+    "X[:,1] = x\n",
+    "X[:,2] = x*x\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train\n",
+    "print(OLSbeta)\n",
+    "# and then make the prediction\n",
+    "ytildeOLS = X_train @ OLSbeta\n",
+    "print(\"Training MSE for OLS\")\n",
+    "print(MSE(y_train,ytildeOLS))\n",
+    "ypredictOLS = X_test @ OLSbeta\n",
+    "print(\"Test MSE OLS\")\n",
+    "print(MSE(y_test,ypredictOLS))\n",
+    "\n",
+    "# Repeat now for Lasso and Ridge regression and various values of the regularization parameter\n",
+    "I = np.eye(p,p)\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 100\n",
+    "MSEPredict = np.zeros(nlambdas)\n",
+    "MSETrain = np.zeros(nlambdas)\n",
+    "MSELassoPredict = np.zeros(nlambdas)\n",
+    "MSELassoTrain = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-4, 4, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    Ridgebeta = np.linalg.inv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n",
+    "    # include lasso using Scikit-Learn\n",
+    "    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n",
+    "    RegLasso.fit(X_train,y_train)\n",
+    "    # and then make the prediction\n",
+    "    ytildeRidge = X_train @ Ridgebeta\n",
+    "    ypredictRidge = X_test @ Ridgebeta\n",
+    "    ytildeLasso = RegLasso.predict(X_train)\n",
+    "    ypredictLasso = RegLasso.predict(X_test)\n",
+    "    MSEPredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    MSETrain[i] = MSE(y_train,ytildeRidge)\n",
+    "    MSELassoPredict[i] = MSE(y_test,ypredictLasso)\n",
+    "    MSELassoTrain[i] = MSE(y_train,ytildeLasso)\n",
+    "\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSETrain, label = 'MSE Ridge train')\n",
+    "plt.plot(np.log10(lambdas), MSEPredict, 'r--', label = 'MSE Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSELassoTrain, label = 'MSE Lasso train')\n",
+    "plt.plot(np.log10(lambdas), MSELassoPredict, 'r--', label = 'MSE Lasso Test')\n",
+    "\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e076a968",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Material for lecture Thursday September 7"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "db822c65",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Linking the regression analysis with a statistical interpretation\n",
+    "\n",
+    "We will now couple the discussions of ordinary least squares, Ridge\n",
+    "and Lasso regression with a statistical interpretation, that is we\n",
+    "move from a linear algebra analysis to a statistical analysis. In\n",
+    "particular, we will focus on what the regularization terms can result\n",
+    "in.  We will amongst other things show that the regularization\n",
+    "parameter can reduce considerably the variance of the parameters\n",
+    "$\\beta$.\n",
+    "\n",
+    "The\n",
+    "advantage of doing linear regression is that we actually end up with\n",
+    "analytical expressions for several statistical quantities.  \n",
+    "Standard least squares and Ridge regression  allow us to\n",
+    "derive quantities like the variance and other expectation values in a\n",
+    "rather straightforward way.\n",
+    "\n",
+    "It is assumed that $\\varepsilon_i\n",
+    "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n",
+    "independent, i.e.:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fc4a06b6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{align*} \n",
+    "\\mbox{Cov}(\\varepsilon_{i_1},\n",
+    "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n",
+    "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2.  \\end{array} \\right.\n",
+    "\\end{align*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "43299fb3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The randomness of $\\varepsilon_i$ implies that\n",
+    "$\\mathbf{y}_i$ is also a random variable. In particular,\n",
+    "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n",
+    "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n",
+    "non-random scalar. To specify the parameters of the distribution of\n",
+    "$\\mathbf{y}_i$ we need to calculate its first two moments. \n",
+    "\n",
+    "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n",
+    "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n",
+    "row number $i$ and perform a sum over all values $p$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6cba00c7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Assumptions made\n",
+    "\n",
+    "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n",
+    "that there exists a function $f(\\boldsymbol{x})$ and  a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n",
+    "which describe our data"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c85f61d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a1901c32",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We approximate this function with our model from the solution of the linear regression equations, that is our\n",
+    "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c59e8272",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8cba4bae",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Expectation value and variance\n",
+    "\n",
+    "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5e18ebb5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{align*} \n",
+    "\\mathbb{E}(y_i) & =\n",
+    "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n",
+    "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n",
+    "\\end{align*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "10d86c58",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "while\n",
+    "its variance is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "efba1ba9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
+    "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n",
+    "[\\mathbb{E}(y_i)]^2  \\\\  & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n",
+    "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n",
+    "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n",
+    "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n",
+    "\\ast} \\, \\beta)^2 \\\\  & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n",
+    "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n",
+    "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n",
+    "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n",
+    "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2.  \n",
+    "\\end{align*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a3eccdab",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n",
+    "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "70b901b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Expectation value and variance for $\\boldsymbol{\\beta}$\n",
+    "\n",
+    "With the OLS expressions for the optimal parameters $\\boldsymbol{\\hat{\\beta}}$ we can evaluate the expectation value"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "752b72d2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "108aba79",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This means that the estimator of the regression parameters is unbiased.\n",
+    "\n",
+    "We can also calculate the variance\n",
+    "\n",
+    "The variance of the optimal value $\\boldsymbol{\\hat{\\beta}}$ is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "65801f3b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\begin{eqnarray*}\n",
+    "\\mbox{Var}(\\boldsymbol{\\hat{\\beta}}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n",
+    "\\\\\n",
+    "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}]^{T} \\}\n",
+    "\\\\\n",
+    "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "% \\\\\n",
+    "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} \\, \\mathbf{Y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}  \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "% \\\\\n",
+    "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{Y} \\, \\mathbf{Y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "\\\\\n",
+    "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\,  \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "% \\\\\n",
+    "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\,  \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n",
+    "% \\\\\n",
+    "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T  \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n",
+    "\\\\\n",
+    "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}  + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n",
+    "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n",
+    "\\end{eqnarray*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "faf8da78",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have used  that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n",
+    "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n",
+    "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n",
+    "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n",
+    "variance of the estimate of the $j$-th regression coefficient:\n",
+    "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $. This may be used to\n",
+    "construct a confidence interval for the estimates.\n",
+    "\n",
+    "In a similar way, we can obtain analytical expressions for say the\n",
+    "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n",
+    "when we employ Ridge regression, allowing us again to define a confidence interval. \n",
+    "\n",
+    "It is rather straightforward to show that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1c71921e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cfb57087",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We see clearly that \n",
+    "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n",
+    "\n",
+    "We can also compute the variance as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "be4ad159",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f5e11f36",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n",
+    "\n",
+    "With this, we can compute the difference"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dbf2700f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0f6d760d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "The difference is non-negative definite since each component of the\n",
+    "matrix product is non-negative definite. \n",
+    "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ce614956",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Deriving OLS from a probability distribution\n",
+    "\n",
+    "Our basic assumption when we derived the OLS equations was to assume\n",
+    "that our output is determined by a given continuous function\n",
+    "$f(\\boldsymbol{x})$ and a random noise $\\boldsymbol{\\epsilon}$ given by the normal\n",
+    "distribution with zero mean value and an undetermined variance\n",
+    "$\\sigma^2$.\n",
+    "\n",
+    "We found above that the outputs $\\boldsymbol{y}$ have a mean value given by\n",
+    "$\\boldsymbol{X}\\hat{\\boldsymbol{\\beta}}$ and variance $\\sigma^2$. Since the entries to\n",
+    "the design matrix are not stochastic variables, we can assume that the\n",
+    "probability distribution of our targets is also a normal distribution\n",
+    "but now with mean value $\\boldsymbol{X}\\hat{\\boldsymbol{\\beta}}$. This means that a\n",
+    "single output $y_i$ is given by the Gaussian distribution"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "086419d8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "y_i\\sim \\mathcal{N}(\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta}, \\sigma^2)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a228dcf4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Independent and Identically Distrubuted (iid)\n",
+    "\n",
+    "We assume now that the various $y_i$ values are stochastically distributed according to the above Gaussian distribution. \n",
+    "We define this distribution as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f04f6b76",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(y_i, \\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "642936a8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which reads as finding the likelihood of an event $y_i$ with the input variables $\\boldsymbol{X}$ given the parameters (to be determined) $\\boldsymbol{\\beta}$.\n",
+    "\n",
+    "Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event $\\boldsymbol{y}$ as the product of the single events, that is we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f137023",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{y},\\boldsymbol{X}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}=\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "48fdd9f7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We will write this in a more compact form reserving $\\boldsymbol{D}$ for the domain of events, including the ouputs (targets) and the inputs. That is\n",
+    "in case we have a simple one-dimensional input and output case"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ae36f857",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})].\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d312d461",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In the more general case the various inputs should be replaced by the possible features represented by the input data set $\\boldsymbol{X}$. \n",
+    "We can now rewrite the above probability as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "59ea8b57",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b9be32b5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "It is a conditional probability (see below) and reads as the likelihood of a domain of events $\\boldsymbol{D}$ given a set of parameters $\\boldsymbol{\\beta}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b378b2c9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Maximum Likelihood Estimation (MLE)\n",
+    "\n",
+    "In statistics, maximum likelihood estimation (MLE) is a method of\n",
+    "estimating the parameters of an assumed probability distribution,\n",
+    "given some observed data. This is achieved by maximizing a likelihood\n",
+    "function so that, under the assumed statistical model, the observed\n",
+    "data is the most probable. \n",
+    "\n",
+    "We will assume here that our events are given by the above Gaussian\n",
+    "distribution and we will determine the optimal parameters $\\beta$ by\n",
+    "maximizing the above PDF. However, computing the derivatives of a\n",
+    "product function is cumbersome and can easily lead to overflow and/or\n",
+    "underflowproblems, with potentials for loss of numerical precision.\n",
+    "\n",
+    "In practice, it is more convenient to maximize the logarithm of the\n",
+    "PDF because it is a monotonically increasing function of the argument.\n",
+    "Alternatively, and this will be our option, we will minimize the\n",
+    "negative of the logarithm since this is a monotonically decreasing\n",
+    "function.\n",
+    "\n",
+    "Note also that maximization/minimization of the logarithm of the PDF\n",
+    "is equivalent to the maximization/minimization of the function itself."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6ddc5457",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## A new Cost Function\n",
+    "\n",
+    "We could now define a new cost function to minimize, namely the negative logarithm of the above PDF"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bb0f5795",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=-\\log{\\prod_{i=0}^{n-1}p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})}=-\\sum_{i=0}^{n-1}\\log{p(y_i,\\boldsymbol{X}\\vert\\boldsymbol{\\beta})},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c969ba54",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which becomes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "630a5421",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=\\frac{n}{2}\\log{2\\pi\\sigma^2}+\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "49969ea1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Taking the derivative of the *new* cost function with respect to the parameters $\\beta$ we recognize our familiar OLS equation, namely"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "93b3d19b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right) =0,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f61d06bf",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which leads to the well-known OLS equation for the optimal paramters $\\beta$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c25aa29",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}!\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "add3ac78",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2cb62527",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## More basic Statistics and Bayes' theorem\n",
+    "\n",
+    "A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry.\n",
+    "Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.\n",
+    "\n",
+    "Assume we have two domains of events $X=[x_0,x_1,\\dots,x_{n-1}]$ and $Y=[y_0,y_1,\\dots,y_{n-1}]$.\n",
+    "\n",
+    "We define also the likelihood for $X$ and $Y$ as $p(X)$ and $p(Y)$ respectively.\n",
+    "The likelihood of a specific event $x_i$ (or $y_i$) is then written as $p(X=x_i)$ or just $p(x_i)=p_i$. \n",
+    "\n",
+    "**Union of events is given by.**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3704ebb3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X \\cup Y)= p(X)+p(Y)-p(X \\cap Y).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "76f1d92c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "**The product rule (aka joint probability) is given by.**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "31ef87a3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X \\cup Y)= p(X,Y)= p(X\\vert Y)p(Y)=p(Y\\vert X)p(X),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9f87e53d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we read $p(X\\vert Y)$ as the likelihood of obtaining $X$ given $Y$.\n",
+    "\n",
+    "If we have independent events then $p(X,Y)=p(X)p(Y)$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b465eba1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Marginal Probability\n",
+    "\n",
+    "The marginal probability is defined in terms of only one of the set of variables $X,Y$. For a discrete probability we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b522f15f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X)=\\sum_{i=0}^{n-1}p(X,Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert Y=y_i)p(Y=y_i)=\\sum_{i=0}^{n-1}p(X\\vert y_i)p(y_i).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a8bf634f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Conditional  Probability\n",
+    "\n",
+    "The conditional  probability, if $p(Y) > 0$, is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51e8bfb6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)}=\\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bfb3c814",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Bayes' Theorem\n",
+    "\n",
+    "If we combine the conditional probability with the marginal probability and the standard product rule, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dfff0c7c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X\\vert Y)= \\frac{p(X,Y)}{p(Y)},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cad0634f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which we can rewrite as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1de5e07a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X\\vert Y)= \\frac{p(X,Y)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)}=\\frac{p(Y\\vert X)p(X)}{\\sum_{i=0}^{n-1}p(Y\\vert X=x_i)p(x_i)},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "43f1714e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which is Bayes' theorem. It allows us to evaluate the uncertainty in in $X$ after we have observed $Y$. We can easily interchange $X$ with $Y$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "166406ab",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Interpretations of Bayes' Theorem\n",
+    "\n",
+    "The quantity $p(Y\\vert X)$ on the right-hand side of the theorem is\n",
+    "evaluated for the observed data $Y$ and can be viewed as a function of\n",
+    "the parameter space represented by $X$. This function is not\n",
+    "necesseraly normalized and is normally called the likelihood function.\n",
+    "\n",
+    "The function $p(X)$ on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.\n",
+    "\n",
+    "Let us try to illustrate Bayes' theorem through an example."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d547f8dd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Example of Usage of Bayes' theorem\n",
+    "\n",
+    "Let us suppose that you are undergoing a series of mammography scans in\n",
+    "order to rule out possible breast cancer cases.  We define the\n",
+    "sensitivity for a positive event by the variable $X$. It takes binary\n",
+    "values with $X=1$ representing a positive event and $X=0$ being a\n",
+    "negative event. We reserve $Y$ as a classification parameter for\n",
+    "either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a  positive thing).\n",
+    "\n",
+    "We let $Y=1$ represent the the case of having breast cancer and $Y=0$ as not.\n",
+    "\n",
+    "Let us assume that if you have breast cancer, the test will be positive with a probability of $0.8$, that is we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f848f875",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X=1\\vert Y=1) =0.8.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c506872",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "This obviously sounds  scary since many would conclude that if the test is positive, there is a likelihood of $80\\%$ for having cancer.\n",
+    "It is however not correct, as the following Bayesian analysis shows."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "317558b2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Doing it correctly\n",
+    "\n",
+    "If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number.\n",
+    "Let us assume that the prior probability in the population as a whole is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8f8f8c5c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(Y=1) =0.004.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "acb7607d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c5866ceb",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(X=1\\vert Y=0) =0.1.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3648f6b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0b2cef16",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(Y=1\\vert X=1)=\\frac{p(X=1\\vert Y=1)p(Y=1)}{p(X=1\\vert Y=1)p(Y=1)+p(X=1\\vert Y=0)p(Y=0)}=\\frac{0.8\\times 0.004}{0.8\\times 0.004+0.1\\times 0.996}=0.031.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "94233096",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "That is, in case of a positive test, there is only a $3\\%$ chance of having breast cancer!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "91284f6d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Bayes' Theorem and Ridge and Lasso Regression\n",
+    "\n",
+    "Hitherto we have discussed Ridge and Lasso regression in terms of a\n",
+    "linear analysis. This may to many of you feel rather technical and\n",
+    "perhaps not that intuitive. The question is whether we can develop a\n",
+    "more intuitive way of understanding what Ridge and Lasso express.\n",
+    "\n",
+    "Before we proceed let us perform a Ridge, Lasso  and OLS analysis of a polynomial fit."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3debb528",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Test Function for what happens with OLS, Ridge and Lasso\n",
+    "\n",
+    "We will play around with a study of the values for the optimal\n",
+    "parameters $\\boldsymbol{\\beta}$ using OLS, Ridge and Lasso regression.  For\n",
+    "OLS, you will notice as function of the noise and polynomial degree,\n",
+    "that the parameters $\\beta$ will fluctuate from order to order in the\n",
+    "polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS.\n",
+    "\n",
+    "For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "a4cf2a42",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn import linear_model\n",
+    "\n",
+    "def R2(y_data, y_model):\n",
+    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
+    "def MSE(y_data,y_model):\n",
+    "    n = np.size(y_model)\n",
+    "    return np.sum((y_data-y_model)**2)/n\n",
+    "\n",
+    "# Make data set.\n",
+    "n = 10000\n",
+    "x = np.random.rand(n)\n",
+    "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)\n",
+    "\n",
+    "Maxpolydegree = 5\n",
+    "X = np.zeros((len(x),Maxpolydegree))\n",
+    "X[:,0] = 1.0\n",
+    "\n",
+    "for polydegree in range(1, Maxpolydegree):\n",
+    "    for degree in range(polydegree):\n",
+    "        X[:,degree] = x**(degree)\n",
+    "\n",
+    "\n",
+    "# We split the data in test and training data\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+    "\n",
+    "# matrix inversion to find beta\n",
+    "OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train\n",
+    "print(OLSbeta)\n",
+    "ypredictOLS = X_test @ OLSbeta\n",
+    "print(\"Test MSE OLS\")\n",
+    "print(MSE(y_test,ypredictOLS))\n",
+    "# Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 4\n",
+    "MSERidgePredict = np.zeros(nlambdas)\n",
+    "MSELassoPredict = np.zeros(nlambdas)\n",
+    "lambdas = np.logspace(-3, 1, nlambdas)\n",
+    "for i in range(nlambdas):\n",
+    "    lmb = lambdas[i]\n",
+    "    # Make the fit using Ridge and Lasso\n",
+    "    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n",
+    "    RegRidge.fit(X_train,y_train)\n",
+    "    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)\n",
+    "    RegLasso.fit(X_train,y_train)\n",
+    "    # and then make the prediction\n",
+    "    ypredictRidge = RegRidge.predict(X_test)\n",
+    "    ypredictLasso = RegLasso.predict(X_test)\n",
+    "    # Compute the MSE and print it\n",
+    "    MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+    "    MSELassoPredict[i] = MSE(y_test,ypredictLasso)\n",
+    "    print(lmb,RegRidge.coef_)\n",
+    "    print(lmb,RegLasso.coef_)\n",
+    "# Now plot the results\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), MSERidgePredict, 'b', label = 'MSE Ridge Test')\n",
+    "plt.plot(np.log10(lambdas), MSELassoPredict, 'r', label = 'MSE Lasso Test')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "26270da6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "How can we understand this?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d8f0e20a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Invoking Bayes' theorem\n",
+    "\n",
+    "Using Bayes' theorem we can gain a better intuition about Ridge and Lasso regression. \n",
+    "\n",
+    "For ordinary least squares we postulated that the maximum likelihood for the doamin of events $\\boldsymbol{D}$ (one-dimensional case)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6c93bc4b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\\dots, (x_{n-1},y_{n-1})],\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8c8b6cc8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f76e3a40",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af323d99",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set $\\boldsymbol{\\beta}$ given a domain of events $\\boldsymbol{D}$?  That is, how can we define the posterior probability"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "49a0988c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "38568afe",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Bayes' theorem comes to our rescue here since (omitting the normalization constant)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c3d0eeb9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})\\propto p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})p(\\boldsymbol{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1d0ba152",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f936c1ce",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Ridge and Bayes\n",
+    "\n",
+    "With the posterior probability defined by a likelihood which we have\n",
+    "already modeled and an unknown prior, we are now ready to make\n",
+    "additional models for the prior.\n",
+    "\n",
+    "We can, based on our discussions of the variance of $\\boldsymbol{\\beta}$ and the mean value, assume that the prior for the values $\\boldsymbol{\\beta}$ is given by a Gaussian with mean value zero and variance $\\tau^2$, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dd10cea6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2534eb1d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Our posterior probability becomes then (omitting the normalization factor which is just a constant)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "104af119",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta\\vert\\boldsymbol{D})}=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\beta_j^2}{2\\tau^2}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5e8e89cf",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We can now optimize this quantity with respect to $\\boldsymbol{\\beta}$. As we\n",
+    "did for OLS, this is most conveniently done by taking the negative\n",
+    "logarithm of the posterior probability. Doing so and leaving out the\n",
+    "constants terms that do not depend on $\\beta$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "467910cd",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{2\\tau^2}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "16f15265",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and replacing $1/2\\tau^2$ with $\\lambda$ we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1cd22464",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta})=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_2^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "90676d1e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which is our Ridge cost function!  Nice, isn't it?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2082d057",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Lasso and Bayes\n",
+    "\n",
+    "To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution ([Laplace in this case](https://en.wikipedia.org/wiki/Laplace_distribution)) with zero mean value,  that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "544b15d3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta})=\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "592804c7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Our posterior probability becomes then (omitting the normalization factor which is just a constant)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c11e6e06",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "p(\\boldsymbol{\\beta}\\vert\\boldsymbol{D})=\\prod_{i=0}^{n-1}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp{\\left[-\\frac{(y_i-\\boldsymbol{X}_{i,*}\\boldsymbol{\\beta})^2}{2\\sigma^2}\\right]}\\prod_{j=0}^{p-1}\\exp{\\left(-\\frac{\\vert\\beta_j\\vert}{\\tau}\\right)}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fdc6e9b5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Taking the negative\n",
+    "logarithm of the posterior probability and leaving out the\n",
+    "constants terms that do not depend on $\\beta$, we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a0e8571d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\frac{1}{\\tau}\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84eca415",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "and replacing $1/\\tau$ with $\\lambda$ we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0924d8d4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\boldsymbol{\\beta}=\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}+\\lambda\\vert\\vert\\boldsymbol{\\beta}\\vert\\vert_1,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7270fef7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "which is our Lasso cost function!"
+   ]
+  }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/doc/pub/week36/html/._week36-bs006.html b/doc/pub/week36/html/._week36-bs006.html
index d9cbf4d39..79d2fbf9a 100644
--- a/doc/pub/week36/html/._week36-bs006.html
+++ b/doc/pub/week36/html/._week36-bs006.html
@@ -263,7 +263,7 @@ MathJax.Hub.Config({
 
    Code for SVD and Inversion of Matrices 
    
 
 
    How do we use the SVD to invert a matrix \( \boldsymbol{X}^\boldsymbol{X} \) which is singular or near singular?
-The simple answer is to use the linear algebra function for pseudoinvers, that is
+The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix \( \boldsymbol{X} \), that is
 
    
 
 
@@ -272,7 +272,9 @@ The simple answer is to use the linear algebra function for pseudoinvers, that i
     
    
       
    
         
    
-  
    Ainv = np.linlag.pinv(A)
+  
    import numpy as np
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+Xinv = np.linlag.pinv(X)
 
    
 
    
       
    
diff --git a/doc/pub/week36/html/week36-reveal.html b/doc/pub/week36/html/week36-reveal.html
index f056e4b72..7e512154d 100644
--- a/doc/pub/week36/html/week36-reveal.html
+++ b/doc/pub/week36/html/week36-reveal.html
@@ -366,7 +366,7 @@ values and the column vectors of \( \boldsymbol{V} \).
 
    Code for SVD and Inversion of Matrices 
    
 
 
    How do we use the SVD to invert a matrix \( \boldsymbol{X}^\boldsymbol{X} \) which is singular or near singular?
-The simple answer is to use the linear algebra function for pseudoinvers, that is
+The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix \( \boldsymbol{X} \), that is
 
    
 
 
@@ -375,7 +375,9 @@ The simple answer is to use the linear algebra function for pseudoinvers, that i
     
    
       
    
         
    
-  
    Ainv = np.linlag.pinv(A)
+  
    import numpy as np
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+Xinv = np.linlag.pinv(X)
 
    
 
    
       
    
diff --git a/doc/pub/week36/html/week36-solarized.html b/doc/pub/week36/html/week36-solarized.html
index 0cb35d965..30f312fc6 100644
--- a/doc/pub/week36/html/week36-solarized.html
+++ b/doc/pub/week36/html/week36-solarized.html
@@ -371,7 +371,7 @@ values and the column vectors of \( \boldsymbol{V} \).
 
    Code for SVD and Inversion of Matrices 
    
 
 
    How do we use the SVD to invert a matrix \( \boldsymbol{X}^\boldsymbol{X} \) which is singular or near singular?
-The simple answer is to use the linear algebra function for pseudoinvers, that is
+The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix \( \boldsymbol{X} \), that is
 
    
 
 
@@ -380,7 +380,9 @@ The simple answer is to use the linear algebra function for pseudoinvers, that i
     
    
       
    
         
    
-  
    Ainv = np.linlag.pinv(A)
+  
    import numpy as np
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+Xinv = np.linlag.pinv(X)
 
    
 
    
       
    
diff --git a/doc/pub/week36/html/week36.html b/doc/pub/week36/html/week36.html
index 60c4ef6ab..92a3f2f78 100644
--- a/doc/pub/week36/html/week36.html
+++ b/doc/pub/week36/html/week36.html
@@ -448,7 +448,7 @@ values and the column vectors of \( \boldsymbol{V} \).
 
    Code for SVD and Inversion of Matrices 
    
 
 
    How do we use the SVD to invert a matrix \( \boldsymbol{X}^\boldsymbol{X} \) which is singular or near singular?
-The simple answer is to use the linear algebra function for pseudoinvers, that is
+The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix \( \boldsymbol{X} \), that is
 
    
 
 
@@ -457,7 +457,9 @@ The simple answer is to use the linear algebra function for pseudoinvers, that i
     
    
       
    
         
    
-  
    Ainv = np.linlag.pinv(A)
+  
    import numpy as np
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+Xinv = np.linlag.pinv(X)
 
    
 
    
       
    
diff --git a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz
index cfd38406f..a8594ef89 100644
Binary files a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz and b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz differ
diff --git a/doc/pub/week36/ipynb/week36.ipynb b/doc/pub/week36/ipynb/week36.ipynb
index 991b0f568..bdaee795a 100644
--- a/doc/pub/week36/ipynb/week36.ipynb
+++ b/doc/pub/week36/ipynb/week36.ipynb
@@ -2,7 +2,7 @@
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     "## Code for SVD and Inversion of Matrices\n",
     "\n",
     "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n",
-    "The simple answer is to use the linear algebra function for pseudoinvers, that is"
+    "The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\\boldsymbol{X}$, that is"
    ]
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    "source": [
-    "Ainv = np.linlag.pinv(A)"
+    "import numpy as np\n",
+    "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n",
+    "Xinv = np.linlag.pinv(X)"
    ]
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@@ -1265,7 +1267,7 @@
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@@ -1277,7 +1279,7 @@
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@@ -1287,7 +1289,7 @@
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@@ -1301,7 +1303,7 @@
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@@ -1311,7 +1313,7 @@
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@@ -1323,7 +1325,7 @@
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@@ -1335,7 +1337,7 @@
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@@ -1345,7 +1347,7 @@
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@@ -1357,7 +1359,7 @@
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@@ -1367,7 +1369,7 @@
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@@ -1379,7 +1381,7 @@
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@@ -1391,7 +1393,7 @@
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@@ -1401,7 +1403,7 @@
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@@ -1413,7 +1415,7 @@
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@@ -1423,7 +1425,7 @@
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@@ -1435,7 +1437,7 @@
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@@ -1447,7 +1449,7 @@
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@@ -1457,7 +1459,7 @@
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@@ -1469,7 +1471,7 @@
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@@ -1482,7 +1484,7 @@
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@@ -1494,7 +1496,7 @@
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@@ -1506,7 +1508,7 @@
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@@ -1518,7 +1520,7 @@
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@@ -1528,7 +1530,7 @@
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@@ -1540,7 +1542,7 @@
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@@ -1550,7 +1552,7 @@
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@@ -1562,7 +1564,7 @@
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@@ -1572,7 +1574,7 @@
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@@ -1584,7 +1586,7 @@
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@@ -1594,7 +1596,7 @@
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@@ -1607,7 +1609,7 @@
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@@ -1619,7 +1621,7 @@
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@@ -1631,7 +1633,7 @@
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@@ -1641,7 +1643,7 @@
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@@ -1653,7 +1655,7 @@
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@@ -1670,7 +1672,7 @@
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@@ -1682,7 +1684,7 @@
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@@ -1694,7 +1696,7 @@
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@@ -1704,7 +1706,7 @@
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@@ -1716,7 +1718,7 @@
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@@ -1726,7 +1728,7 @@
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@@ -1738,7 +1740,7 @@
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@@ -1748,7 +1750,7 @@
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@@ -1760,7 +1762,7 @@
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@@ -1770,7 +1772,7 @@
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@@ -1785,7 +1787,7 @@
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@@ -1847,7 +1849,7 @@
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@@ -1857,7 +1859,7 @@
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@@ -1868,7 +1870,7 @@
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@@ -1935,7 +1937,7 @@
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@@ -1946,7 +1948,7 @@
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@@ -2035,7 +2037,7 @@
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@@ -2045,7 +2047,7 @@
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@@ -2074,7 +2076,7 @@
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@@ -2090,7 +2092,7 @@
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@@ -2109,7 +2111,7 @@
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@@ -2123,7 +2125,7 @@
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@@ -2135,7 +2137,7 @@
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@@ -2146,7 +2148,7 @@
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@@ -2158,7 +2160,7 @@
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@@ -2170,7 +2172,7 @@
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@@ -2186,7 +2188,7 @@
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@@ -2197,7 +2199,7 @@
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@@ -2220,7 +2222,7 @@
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@@ -2231,7 +2233,7 @@
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@@ -2243,7 +2245,7 @@
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@@ -2255,7 +2257,7 @@
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@@ -2269,7 +2271,7 @@
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@@ -2300,7 +2302,7 @@
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@@ -2322,7 +2324,7 @@
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@@ -2334,7 +2336,7 @@
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@@ -2347,7 +2349,7 @@
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@@ -2359,7 +2361,7 @@
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@@ -2371,7 +2373,7 @@
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@@ -2383,7 +2385,7 @@
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@@ -2395,7 +2397,7 @@
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@@ -2418,7 +2420,7 @@
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@@ -2430,7 +2432,7 @@
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@@ -2443,7 +2445,7 @@
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@@ -2455,7 +2457,7 @@
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@@ -2467,7 +2469,7 @@
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@@ -2479,7 +2481,7 @@
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@@ -2490,7 +2492,7 @@
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@@ -2502,7 +2504,7 @@
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@@ -2513,7 +2515,7 @@
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@@ -2525,7 +2527,7 @@
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@@ -2535,7 +2537,7 @@
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@@ -2566,7 +2568,7 @@
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@@ -2578,7 +2580,7 @@
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@@ -2590,7 +2592,7 @@
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@@ -2600,7 +2602,7 @@
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@@ -2612,7 +2614,7 @@
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@@ -2622,7 +2624,7 @@
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@@ -2634,7 +2636,7 @@
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@@ -2644,7 +2646,7 @@
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@@ -2656,7 +2658,7 @@
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@@ -2666,7 +2668,7 @@
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@@ -2686,7 +2688,7 @@
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@@ -2698,7 +2700,7 @@
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@@ -2708,7 +2710,7 @@
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@@ -2720,7 +2722,7 @@
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@@ -2732,7 +2734,7 @@
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    "cell_type": "markdown",
-   "id": "b465eba1",
+   "id": "333869a5",
    "metadata": {
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@@ -2744,7 +2746,7 @@
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   {
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-   "id": "b522f15f",
+   "id": "bf8de993",
    "metadata": {
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@@ -2756,7 +2758,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a8bf634f",
+   "id": "1c5349d1",
    "metadata": {
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@@ -2768,7 +2770,7 @@
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   {
    "cell_type": "markdown",
-   "id": "51e8bfb6",
+   "id": "dbede038",
    "metadata": {
     "editable": true
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@@ -2780,7 +2782,7 @@
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   {
    "cell_type": "markdown",
-   "id": "bfb3c814",
+   "id": "c1334a9c",
    "metadata": {
     "editable": true
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@@ -2792,7 +2794,7 @@
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   {
    "cell_type": "markdown",
-   "id": "dfff0c7c",
+   "id": "ac8dcf04",
    "metadata": {
     "editable": true
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@@ -2804,7 +2806,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cad0634f",
+   "id": "1992e354",
    "metadata": {
     "editable": true
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@@ -2814,7 +2816,7 @@
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   {
    "cell_type": "markdown",
-   "id": "1de5e07a",
+   "id": "24e5179f",
    "metadata": {
     "editable": true
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@@ -2826,7 +2828,7 @@
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   {
    "cell_type": "markdown",
-   "id": "43f1714e",
+   "id": "42979e5a",
    "metadata": {
     "editable": true
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@@ -2836,7 +2838,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "166406ab",
+   "id": "86a8cd07",
    "metadata": {
     "editable": true
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@@ -2855,7 +2857,7 @@
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   {
    "cell_type": "markdown",
-   "id": "d547f8dd",
+   "id": "b9a4f9a9",
    "metadata": {
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@@ -2876,7 +2878,7 @@
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   {
    "cell_type": "markdown",
-   "id": "f848f875",
+   "id": "888c64ec",
    "metadata": {
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@@ -2888,7 +2890,7 @@
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   {
    "cell_type": "markdown",
-   "id": "8c506872",
+   "id": "5086b046",
    "metadata": {
     "editable": true
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@@ -2899,7 +2901,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "317558b2",
+   "id": "609ea2b0",
    "metadata": {
     "editable": true
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@@ -2912,7 +2914,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8f8f8c5c",
+   "id": "687bfdf3",
    "metadata": {
     "editable": true
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@@ -2924,7 +2926,7 @@
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   {
    "cell_type": "markdown",
-   "id": "acb7607d",
+   "id": "c2af1d2d",
    "metadata": {
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@@ -2934,7 +2936,7 @@
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   {
    "cell_type": "markdown",
-   "id": "c5866ceb",
+   "id": "b8ec8d93",
    "metadata": {
     "editable": true
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@@ -2946,7 +2948,7 @@
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   {
    "cell_type": "markdown",
-   "id": "e3648f6b",
+   "id": "48a6fb5b",
    "metadata": {
     "editable": true
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@@ -2956,7 +2958,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0b2cef16",
+   "id": "ba37d4fd",
    "metadata": {
     "editable": true
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@@ -2968,7 +2970,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94233096",
+   "id": "824f1789",
    "metadata": {
     "editable": true
    },
@@ -2978,7 +2980,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "91284f6d",
+   "id": "07f89fd5",
    "metadata": {
     "editable": true
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@@ -2995,7 +2997,7 @@
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   {
    "cell_type": "markdown",
-   "id": "3debb528",
+   "id": "8fcd73d9",
    "metadata": {
     "editable": true
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@@ -3014,7 +3016,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "a4cf2a42",
+   "id": "4402e113",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3088,7 +3090,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "26270da6",
+   "id": "b700e45e",
    "metadata": {
     "editable": true
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@@ -3098,7 +3100,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d8f0e20a",
+   "id": "69659d9d",
    "metadata": {
     "editable": true
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@@ -3112,7 +3114,7 @@
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   {
    "cell_type": "markdown",
-   "id": "6c93bc4b",
+   "id": "80b423eb",
    "metadata": {
     "editable": true
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@@ -3124,7 +3126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c8b6cc8",
+   "id": "de547949",
    "metadata": {
     "editable": true
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@@ -3134,7 +3136,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f76e3a40",
+   "id": "99d9476a",
    "metadata": {
     "editable": true
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@@ -3146,7 +3148,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "af323d99",
+   "id": "aa5613ea",
    "metadata": {
     "editable": true
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@@ -3156,7 +3158,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "49a0988c",
+   "id": "7dea4e11",
    "metadata": {
     "editable": true
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@@ -3168,7 +3170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "38568afe",
+   "id": "99ab1d57",
    "metadata": {
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@@ -3178,7 +3180,7 @@
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   {
    "cell_type": "markdown",
-   "id": "c3d0eeb9",
+   "id": "170ee68b",
    "metadata": {
     "editable": true
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@@ -3190,7 +3192,7 @@
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   {
    "cell_type": "markdown",
-   "id": "1d0ba152",
+   "id": "3388d6a4",
    "metadata": {
     "editable": true
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@@ -3200,7 +3202,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f936c1ce",
+   "id": "a5bdaa16",
    "metadata": {
     "editable": true
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@@ -3216,7 +3218,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dd10cea6",
+   "id": "cbf8e7e6",
    "metadata": {
     "editable": true
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@@ -3228,7 +3230,7 @@
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   {
    "cell_type": "markdown",
-   "id": "2534eb1d",
+   "id": "52c9e48f",
    "metadata": {
     "editable": true
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@@ -3238,7 +3240,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "104af119",
+   "id": "1ae37a89",
    "metadata": {
     "editable": true
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@@ -3250,7 +3252,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e8e89cf",
+   "id": "dd8fee27",
    "metadata": {
     "editable": true
    },
@@ -3263,7 +3265,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "467910cd",
+   "id": "de603b44",
    "metadata": {
     "editable": true
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@@ -3275,7 +3277,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16f15265",
+   "id": "3f08e4a0",
    "metadata": {
     "editable": true
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@@ -3285,7 +3287,7 @@
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   {
    "cell_type": "markdown",
-   "id": "1cd22464",
+   "id": "b65ee2f0",
    "metadata": {
     "editable": true
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@@ -3297,7 +3299,7 @@
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   {
    "cell_type": "markdown",
-   "id": "90676d1e",
+   "id": "6526b426",
    "metadata": {
     "editable": true
    },
@@ -3307,7 +3309,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2082d057",
+   "id": "29190fe5",
    "metadata": {
     "editable": true
    },
@@ -3319,7 +3321,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "544b15d3",
+   "id": "71b1585c",
    "metadata": {
     "editable": true
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@@ -3331,7 +3333,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "592804c7",
+   "id": "e18baaa9",
    "metadata": {
     "editable": true
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@@ -3341,7 +3343,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c11e6e06",
+   "id": "1c58b8db",
    "metadata": {
     "editable": true
    },
@@ -3353,7 +3355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fdc6e9b5",
+   "id": "238ee8c5",
    "metadata": {
     "editable": true
    },
@@ -3365,7 +3367,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0e8571d",
+   "id": "46389b82",
    "metadata": {
     "editable": true
    },
@@ -3377,7 +3379,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "84eca415",
+   "id": "0ecbb2d2",
    "metadata": {
     "editable": true
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@@ -3387,7 +3389,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0924d8d4",
+   "id": "27368dfb",
    "metadata": {
     "editable": true
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@@ -3399,7 +3401,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7270fef7",
+   "id": "27082f48",
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diff --git a/doc/src/week36/week36.do.txt b/doc/src/week36/week36.do.txt
index cfa3050eb..adf259ff5 100644
--- a/doc/src/week36/week36.do.txt
+++ b/doc/src/week36/week36.do.txt
@@ -147,9 +147,11 @@ values and the column vectors of $\bm{V}$.
 =====  Code for SVD and Inversion of Matrices =====
 
 How do we use the SVD to invert a matrix $\bm{X}^\bm{X}$ which is singular or near singular?
-The simple answer is to use the linear algebra function for pseudoinvers, that is
+The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\bm{X}$, that is
 !bc pycod
-Ainv = np.linlag.pinv(A)
+import numpy as np
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+Xinv = np.linlag.pinv(X)
 !ec
 
 Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD.