From 7d6ccbed0d02f1b34bac4d1c16704c6b9bebe06f Mon Sep 17 00:00:00 2001
From: Morten Hjorth-Jensen <morten.hjorth-jensen@fys.uio.no>
Date: Mon, 9 Sep 2024 06:12:22 +0200
Subject: [PATCH] update week 37

---
 .../_build/.doctrees/environment.pickle       |  Bin 307233 -> 307633 bytes
 .../_build/.doctrees/week37.doctree           |  Bin 716097 -> 589435 bytes
 .../_build/html/_images/week37_121_0.png      |  Bin 0 -> 22216 bytes
 .../_build/html/_images/week37_137_1.png      |  Bin 0 -> 43341 bytes
 .../_build/html/_images/week37_139_2.png      |  Bin 0 -> 29329 bytes
 .../_build/html/_images/week37_142_1.png      |  Bin 0 -> 62168 bytes
 .../_build/html/_images/week37_146_0.png      |  Bin 0 -> 26449 bytes
 .../_build/html/_images/week37_148_9.png      |  Bin 0 -> 29882 bytes
 .../_build/html/_images/week37_151_1.png      |  Bin 0 -> 22705 bytes
 .../_build/html/_sources/week37.ipynb         | 1282 ++++-----------
 doc/LectureNotes/_build/html/searchindex.js   |    2 +-
 doc/LectureNotes/_build/html/week37.html      |  873 +++-------
 .../_build/jupyter_execute/week37.ipynb       | 1460 +++++------------
 .../_build/jupyter_execute/week37.py          |  630 ++-----
 .../_build/jupyter_execute/week37_121_0.png   |  Bin 0 -> 22216 bytes
 .../_build/jupyter_execute/week37_137_1.png   |  Bin 0 -> 43341 bytes
 .../_build/jupyter_execute/week37_139_2.png   |  Bin 0 -> 29329 bytes
 .../_build/jupyter_execute/week37_142_1.png   |  Bin 0 -> 62168 bytes
 .../_build/jupyter_execute/week37_146_0.png   |  Bin 0 -> 26449 bytes
 .../_build/jupyter_execute/week37_148_9.png   |  Bin 0 -> 29882 bytes
 .../_build/jupyter_execute/week37_151_1.png   |  Bin 0 -> 22705 bytes
 doc/LectureNotes/week37.ipynb                 | 1282 ++++-----------
 22 files changed, 1472 insertions(+), 4057 deletions(-)
 create mode 100644 doc/LectureNotes/_build/html/_images/week37_121_0.png
 create mode 100644 doc/LectureNotes/_build/html/_images/week37_137_1.png
 create mode 100644 doc/LectureNotes/_build/html/_images/week37_139_2.png
 create mode 100644 doc/LectureNotes/_build/html/_images/week37_142_1.png
 create mode 100644 doc/LectureNotes/_build/html/_images/week37_146_0.png
 create mode 100644 doc/LectureNotes/_build/html/_images/week37_148_9.png
 create mode 100644 doc/LectureNotes/_build/html/_images/week37_151_1.png
 create mode 100644 doc/LectureNotes/_build/jupyter_execute/week37_121_0.png
 create mode 100644 doc/LectureNotes/_build/jupyter_execute/week37_137_1.png
 create mode 100644 doc/LectureNotes/_build/jupyter_execute/week37_139_2.png
 create mode 100644 doc/LectureNotes/_build/jupyter_execute/week37_142_1.png
 create mode 100644 doc/LectureNotes/_build/jupyter_execute/week37_146_0.png
 create mode 100644 doc/LectureNotes/_build/jupyter_execute/week37_148_9.png
 create mode 100644 doc/LectureNotes/_build/jupyter_execute/week37_151_1.png

diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle
index b851d229703e1d34d0b43313ced16500b37c0cb8..12f8a63ddc15efe89bb9d783127644e9c2dd0a09 100644
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diff --git a/doc/LectureNotes/_build/.doctrees/week37.doctree b/doc/LectureNotes/_build/.doctrees/week37.doctree
index c4c91427d6c63e063f2b32b71a6f7eba9bde3f5a..1b2e31d89220874ac4572730c63b9053e6d4a609 100644
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diff --git a/doc/LectureNotes/_build/html/_sources/week37.ipynb b/doc/LectureNotes/_build/html/_sources/week37.ipynb
index 94e3e70ef..c24beccbd 100644
--- a/doc/LectureNotes/_build/html/_sources/week37.ipynb
+++ b/doc/LectureNotes/_build/html/_sources/week37.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "672fea2a",
+   "id": "7c0ddab1",
    "metadata": {
     "editable": true
    },
@@ -14,13 +14,13 @@
   },
   {
    "cell_type": "markdown",
-   "id": "479a6913",
+   "id": "6c033bf4",
    "metadata": {
     "editable": true
    },
    "source": [
     "# Week 37: Statistical interpretations and Resampling Methods\n",
-    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University\n",
+    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway\n",
     "\n",
     "Date: **September 9, 2024**\n",
     "\n",
@@ -29,419 +29,69 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10b8f456",
+   "id": "e4cd5fa1",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Plans for week 37\n",
+    "## Plans for week 37, lecture Monday\n",
     "\n",
     "**Material for the lecture on Monday September 9.**\n",
     "\n",
     "<!-- * [Video of Lecture](https://youtu.be/YOBBr_toYxc) -->\n",
     "<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf) -->\n",
-    "  * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff\n",
+    "  * Statistical interpretation of Ridge and Lasso regression, see also slides from last week\n",
     "\n",
-    "  * Statistical interpretation of Ridge and Lasso regression\n",
+    "  * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.\n",
     "\n",
     "  * Readings and Videos:\n",
     "\n",
-    "    * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). \n",
+    "    * Raschka et al, pages 175-192\n",
+    "\n",
+    "    * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See <https://link.springer.com/book/10.1007/978-0-387-84858-7>.\n",
     "\n",
     "    * [Video on cross validation](https://www.youtube.com/watch?v=fSytzGwwBVw)\n",
     "\n",
     "    * [Video on Bootstrapping](https://www.youtube.com/watch?v=Xz0x-8-cgaQ)\n",
     "\n",
-    "    * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)\n",
+    "    * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bf80a276",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Plans for week 37, lab sessions\n",
     "\n",
-    "**Material for the active learning sessions on Tuesday and Wednesday.**\n",
+    "**Material for the lab  sessions on Tuesday and Wednesday.**\n",
     "\n",
     "  * Calculations of expectation values\n",
     "\n",
+    "  * Discussion of resampling techniques\n",
+    "\n",
     "  * Exercise set for week 37\n",
     "\n",
     "  * Work on project 1\n",
     "\n",
-    "  * See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.\n",
-    "\n",
     "  * For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f7e43b1b",
+   "id": "0e1f0a90",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Material from last week and relevant for the weekly exercises"
+    "## Material for lecture Monday September 9"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "af6a447a",
-   "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": "b6b7ea38",
-   "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": "e53d8d41",
-   "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": "401644e7",
-   "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": "0603387c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "eb2b6edf",
-   "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": "f1a08194",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ad40fa0b",
-   "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": "d3a4907d",
-   "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": "d7908c9c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "while\n",
-    "its variance is"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "2528f74e",
-   "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": "525f3a59",
-   "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": "ead1d419",
-   "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": "e471f7ab",
-   "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": "7886e97f",
-   "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": "49c32ca5",
-   "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": "9bd27980",
-   "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": "184f9c3b",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d0f46be1",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "We see clearly that \n",
-    "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n",
-    "\n",
-    "We can also compute the variance as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "03e3ac5f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "13ec2524",
-   "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": "58c3f236",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "bfcfc2a0",
-   "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. \n",
-    "\n",
-    "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f3a9f084",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Material for lecture Thursday September 14"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a4338936",
+   "id": "1b290d64",
    "metadata": {
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@@ -464,7 +114,7 @@
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   {
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-   "id": "d7566301",
+   "id": "ceed3a82",
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@@ -476,7 +126,7 @@
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   {
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+   "id": "1d7e8f98",
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@@ -489,7 +139,7 @@
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   {
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@@ -501,7 +151,7 @@
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   {
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+   "id": "45120edc",
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@@ -513,7 +163,7 @@
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   {
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-   "id": "75c188ba",
+   "id": "4f3b680a",
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@@ -525,7 +175,7 @@
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   {
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+   "id": "08489797",
    "metadata": {
     "editable": true
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@@ -536,7 +186,7 @@
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   {
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+   "id": "a1c81f06",
    "metadata": {
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@@ -548,7 +198,7 @@
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   {
    "cell_type": "markdown",
-   "id": "56f25582",
+   "id": "4e092776",
    "metadata": {
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@@ -559,7 +209,7 @@
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   {
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@@ -571,7 +221,7 @@
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   {
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-   "id": "4112e053",
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@@ -581,7 +231,7 @@
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   {
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@@ -612,7 +262,7 @@
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   {
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@@ -624,7 +274,7 @@
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   {
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+   "id": "0fd1502e",
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@@ -636,7 +286,7 @@
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@@ -646,7 +296,7 @@
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@@ -658,7 +308,7 @@
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   {
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@@ -668,7 +318,7 @@
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   {
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+   "id": "30315a10",
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@@ -680,7 +330,7 @@
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   {
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+   "id": "fd8cdf51",
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@@ -690,7 +340,7 @@
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   {
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    "metadata": {
     "editable": true
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@@ -702,7 +352,7 @@
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   {
    "cell_type": "markdown",
-   "id": "9431eca4",
+   "id": "e21334ab",
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@@ -712,7 +362,7 @@
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   {
    "cell_type": "markdown",
-   "id": "1db6c222",
+   "id": "82974124",
    "metadata": {
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@@ -732,7 +382,7 @@
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   {
    "cell_type": "markdown",
-   "id": "98c457ce",
+   "id": "e053e43c",
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@@ -744,7 +394,7 @@
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   {
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-   "id": "b6a3893f",
+   "id": "94098e13",
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@@ -754,7 +404,7 @@
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   {
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-   "id": "17f0e52e",
+   "id": "27625b1f",
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@@ -766,7 +416,7 @@
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   {
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-   "id": "c2bfaea6",
+   "id": "8f3cc8b4",
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     "editable": true
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@@ -778,7 +428,7 @@
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   {
    "cell_type": "markdown",
-   "id": "377eefdf",
+   "id": "d7d5464d",
    "metadata": {
     "editable": true
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@@ -790,7 +440,7 @@
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   {
    "cell_type": "markdown",
-   "id": "6da6253e",
+   "id": "c292afee",
    "metadata": {
     "editable": true
    },
@@ -802,7 +452,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e5906b3e",
+   "id": "cdbbd444",
    "metadata": {
     "editable": true
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@@ -814,7 +464,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "792d292c",
+   "id": "100a8014",
    "metadata": {
     "editable": true
    },
@@ -826,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3bc0674",
+   "id": "d2e40c4d",
    "metadata": {
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@@ -838,7 +488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0faadf3f",
+   "id": "1e48456e",
    "metadata": {
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@@ -850,7 +500,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a2d478a7",
+   "id": "38ced5b6",
    "metadata": {
     "editable": true
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@@ -860,7 +510,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d51ac6ce",
+   "id": "1163c07b",
    "metadata": {
     "editable": true
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@@ -872,7 +522,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "91d489d9",
+   "id": "becc087f",
    "metadata": {
     "editable": true
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@@ -882,7 +532,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ebe26b44",
+   "id": "5ca2b09d",
    "metadata": {
     "editable": true
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@@ -901,7 +551,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a67660e2",
+   "id": "6572c77d",
    "metadata": {
     "editable": true
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@@ -922,7 +572,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40b7eac0",
+   "id": "2a3f65e5",
    "metadata": {
     "editable": true
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@@ -934,7 +584,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16a3b6ea",
+   "id": "2d264f77",
    "metadata": {
     "editable": true
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@@ -945,7 +595,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f41ccc6d",
+   "id": "06b8a13c",
    "metadata": {
     "editable": true
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@@ -958,7 +608,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ae81305b",
+   "id": "6d958ed3",
    "metadata": {
     "editable": true
    },
@@ -970,7 +620,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4829b81",
+   "id": "6330f3f2",
    "metadata": {
     "editable": true
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@@ -980,7 +630,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "812273af",
+   "id": "ed6a9f67",
    "metadata": {
     "editable": true
    },
@@ -992,7 +642,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31ecba0e",
+   "id": "de42ab8a",
    "metadata": {
     "editable": true
    },
@@ -1002,7 +652,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca6682ec",
+   "id": "4f36b126",
    "metadata": {
     "editable": true
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@@ -1014,7 +664,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "662eaffb",
+   "id": "0a7eef00",
    "metadata": {
     "editable": true
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@@ -1024,7 +674,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "08d5551d",
+   "id": "29940f94",
    "metadata": {
     "editable": true
    },
@@ -1038,7 +688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03f3de9c",
+   "id": "f2204679",
    "metadata": {
     "editable": true
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@@ -1050,7 +700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c57ace49",
+   "id": "d3f4597f",
    "metadata": {
     "editable": true
    },
@@ -1060,7 +710,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb014b2b",
+   "id": "c9e9ca01",
    "metadata": {
     "editable": true
    },
@@ -1072,7 +722,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1054d47e",
+   "id": "1c3062b6",
    "metadata": {
     "editable": true
    },
@@ -1082,7 +732,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1485d44c",
+   "id": "b455f707",
    "metadata": {
     "editable": true
    },
@@ -1094,7 +744,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f457f909",
+   "id": "5ac8fdd3",
    "metadata": {
     "editable": true
    },
@@ -1104,7 +754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59447b2a",
+   "id": "827927db",
    "metadata": {
     "editable": true
    },
@@ -1116,17 +766,17 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fe26100",
+   "id": "0c740a13",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!"
+    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta})$!"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d73c2d9c",
+   "id": "a855fc3c",
    "metadata": {
     "editable": true
    },
@@ -1142,7 +792,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8cdf2c97",
+   "id": "9dda8b9e",
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@@ -1154,7 +804,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7272ef9f",
+   "id": "d160205c",
    "metadata": {
     "editable": true
    },
@@ -1164,7 +814,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0dc5b2b7",
+   "id": "1b7501a6",
    "metadata": {
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@@ -1176,7 +826,7 @@
   },
   {
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-   "id": "b6c8c684",
+   "id": "d75a8c65",
    "metadata": {
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@@ -1189,7 +839,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f681ed50",
+   "id": "c9f718df",
    "metadata": {
     "editable": true
    },
@@ -1201,7 +851,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c3bc898",
+   "id": "b07b6425",
    "metadata": {
     "editable": true
    },
@@ -1211,7 +861,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3cea046",
+   "id": "215a8730",
    "metadata": {
     "editable": true
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@@ -1223,7 +873,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68276410",
+   "id": "7013d0f0",
    "metadata": {
     "editable": true
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@@ -1233,7 +883,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dfd55611",
+   "id": "d56b4ce0",
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@@ -1245,7 +895,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "83656dd8",
+   "id": "f9c6b14d",
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@@ -1257,7 +907,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b62e29fd",
+   "id": "0bde16cb",
    "metadata": {
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@@ -1267,7 +917,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1de8855c",
+   "id": "0a2869fc",
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@@ -1279,7 +929,7 @@
   },
   {
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-   "id": "212152f1",
+   "id": "75b83b9b",
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@@ -1291,7 +941,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afb042c4",
+   "id": "1e76d916",
    "metadata": {
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@@ -1303,7 +953,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab1fd3c5",
+   "id": "09dfe26d",
    "metadata": {
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@@ -1313,7 +963,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "43979108",
+   "id": "4037962f",
    "metadata": {
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@@ -1325,7 +975,7 @@
   },
   {
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-   "id": "96622b66",
+   "id": "39a85ba0",
    "metadata": {
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@@ -1335,7 +985,7 @@
   },
   {
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-   "id": "e17a5b6d",
+   "id": "b263a1fe",
    "metadata": {
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@@ -1354,7 +1004,7 @@
   },
   {
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-   "id": "24e35401",
+   "id": "41730165",
    "metadata": {
     "editable": true
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@@ -1382,7 +1032,7 @@
   },
   {
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-   "id": "a37f1d06",
+   "id": "4076cee3",
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@@ -1408,7 +1058,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58f54a23",
+   "id": "11139cb3",
    "metadata": {
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@@ -1425,7 +1075,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ed7c809",
+   "id": "b2bead9b",
    "metadata": {
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@@ -1445,7 +1095,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bed9bb2f",
+   "id": "8df8364e",
    "metadata": {
     "editable": true
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@@ -1474,7 +1124,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2cafa0a3",
+   "id": "6b5273f1",
    "metadata": {
     "editable": true
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@@ -1499,7 +1149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fefbb980",
+   "id": "a260b371",
    "metadata": {
     "editable": true
    },
@@ -1519,7 +1169,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4560153d",
+   "id": "2ff1e63c",
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@@ -1531,7 +1181,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ffcbdd5a",
+   "id": "ab4b2e8c",
    "metadata": {
     "editable": true
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@@ -1541,7 +1191,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d055ea55",
+   "id": "85b6fb62",
    "metadata": {
     "editable": true
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@@ -1556,7 +1206,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "004f0161",
+   "id": "42449344",
    "metadata": {
     "editable": true
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@@ -1569,7 +1219,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "039f4ad5",
+   "id": "35e4f511",
    "metadata": {
     "editable": true
    },
@@ -1582,7 +1232,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "413459f1",
+   "id": "c787234f",
    "metadata": {
     "editable": true
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@@ -1594,7 +1244,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bef022c8",
+   "id": "183711dc",
    "metadata": {
     "editable": true
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@@ -1607,7 +1257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3d53693d",
+   "id": "2351078d",
    "metadata": {
     "editable": true
    },
@@ -1618,7 +1268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e682c32a",
+   "id": "408ba3b5",
    "metadata": {
     "editable": true
    },
@@ -1632,7 +1282,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f44053ea",
+   "id": "3ea79849",
    "metadata": {
     "editable": true
    },
@@ -1642,7 +1292,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f10de930",
+   "id": "69439aad",
    "metadata": {
     "editable": true
    },
@@ -1656,7 +1306,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f91402a",
+   "id": "e6c29fd8",
    "metadata": {
     "editable": true
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@@ -1669,7 +1319,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d7a8e36",
+   "id": "4bb52549",
    "metadata": {
     "editable": true
    },
@@ -1682,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4c756ca2",
+   "id": "3430d548",
    "metadata": {
     "editable": true
    },
@@ -1692,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "908f782e",
+   "id": "9ba636b3",
    "metadata": {
     "editable": true
    },
@@ -1705,7 +1355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f12152da",
+   "id": "cb19e98f",
    "metadata": {
     "editable": true
    },
@@ -1715,7 +1365,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "85dd290d",
+   "id": "3cc840d7",
    "metadata": {
     "editable": true
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@@ -1728,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "57631d10",
+   "id": "a5062213",
    "metadata": {
     "editable": true
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@@ -1740,7 +1390,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d4d5ccb",
+   "id": "8d555244",
    "metadata": {
     "editable": true
    },
@@ -1759,7 +1409,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a29c8551",
+   "id": "c4356604",
    "metadata": {
     "editable": true
    },
@@ -1772,7 +1422,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6509ba36",
+   "id": "09dc08a1",
    "metadata": {
     "editable": true
    },
@@ -1784,7 +1434,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "733307b1",
+   "id": "dab5056a",
    "metadata": {
     "editable": true
    },
@@ -1797,7 +1447,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a69b7331",
+   "id": "8a1f8583",
    "metadata": {
     "editable": true
    },
@@ -1817,7 +1467,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ced659da",
+   "id": "1be08ebb",
    "metadata": {
     "editable": true
    },
@@ -1831,16 +1481,16 @@
     "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we found \n",
     "$\\mathbb{E}(\\boldsymbol{\\beta}) = \\boldsymbol{\\beta}$, which means that the estimator of the regression parameters is unbiased.\n",
     "\n",
-    "We found also that the variance of the estimate of the $j$-th regression coefficient is\n",
+    "In the exercises this week we show that the variance of the estimate of the $j$-th regression coefficient is\n",
     "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $.\n",
     "\n",
-    "This quantity will be used to\n",
+    "This quantity can be used to\n",
     "construct a confidence interval for the estimates."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2c0d0a71",
+   "id": "2a368353",
    "metadata": {
     "editable": true
    },
@@ -1855,7 +1505,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d3a4d693",
+   "id": "6fee4a1a",
    "metadata": {
     "editable": true
    },
@@ -1867,7 +1517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ed3d852",
+   "id": "1a97c9ef",
    "metadata": {
     "editable": true
    },
@@ -1887,7 +1537,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e826e70b",
+   "id": "0a70afb2",
    "metadata": {
     "editable": true
    },
@@ -1907,7 +1557,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10b5d4d5",
+   "id": "dc32eaea",
    "metadata": {
     "editable": true
    },
@@ -1931,7 +1581,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1645ce3c",
+   "id": "016d0ab3",
    "metadata": {
     "editable": true
    },
@@ -1952,7 +1602,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4986483a",
+   "id": "3f371f0d",
    "metadata": {
     "editable": true
    },
@@ -1982,7 +1632,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fe853f3b",
+   "id": "abc56fe7",
    "metadata": {
     "editable": true
    },
@@ -2006,7 +1656,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "81ec18f8",
+   "id": "a3d2885b",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2045,7 +1695,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0b5b588",
+   "id": "5db67125",
    "metadata": {
     "editable": true
    },
@@ -2055,7 +1705,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "262a95e8",
+   "id": "6f1b5417",
    "metadata": {
     "editable": true
    },
@@ -2066,7 +1716,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "d235b516",
+   "id": "9d98136a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2086,7 +1736,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1620bb7",
+   "id": "c87eac4e",
    "metadata": {
     "editable": true
    },
@@ -2104,7 +1754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fd3bd4b5",
+   "id": "2f0f4313",
    "metadata": {
     "editable": true
    },
@@ -2116,7 +1766,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a129d86",
+   "id": "7895ee8b",
    "metadata": {
     "editable": true
    },
@@ -2133,7 +1783,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff57b405",
+   "id": "be454b28",
    "metadata": {
     "editable": true
    },
@@ -2145,7 +1795,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5a13919e",
+   "id": "7475f8f9",
    "metadata": {
     "editable": true
    },
@@ -2155,7 +1805,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "45d4a7b6",
+   "id": "fd472029",
    "metadata": {
     "editable": true
    },
@@ -2167,7 +1817,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c837442b",
+   "id": "b43c8f67",
    "metadata": {
     "editable": true
    },
@@ -2184,7 +1834,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b3c7b8e",
+   "id": "c47cfb6d",
    "metadata": {
     "editable": true
    },
@@ -2196,7 +1846,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e6f977d",
+   "id": "593188cd",
    "metadata": {
     "editable": true
    },
@@ -2206,7 +1856,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "adeab0fd",
+   "id": "76de4d81",
    "metadata": {
     "editable": true
    },
@@ -2218,7 +1868,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63b18db5",
+   "id": "afc38c97",
    "metadata": {
     "editable": true
    },
@@ -2228,7 +1878,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c7eb00d",
+   "id": "b0f2971c",
    "metadata": {
     "editable": true
    },
@@ -2240,7 +1890,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3f4530b",
+   "id": "5a65e495",
    "metadata": {
     "editable": true
    },
@@ -2250,7 +1900,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "90792236",
+   "id": "31419298",
    "metadata": {
     "editable": true
    },
@@ -2266,7 +1916,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d73c7cd4",
+   "id": "217ddd53",
    "metadata": {
     "editable": true
    },
@@ -2277,7 +1927,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "58124207",
+   "id": "ece05002",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2342,7 +1992,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b4e7bf5",
+   "id": "de6dc4e2",
    "metadata": {
     "editable": true
    },
@@ -2353,7 +2003,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "47b138a9",
+   "id": "da218205",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2410,7 +2060,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37b66761",
+   "id": "5ea5df06",
    "metadata": {
     "editable": true
    },
@@ -2448,7 +2098,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b5c601e",
+   "id": "40f41d74",
    "metadata": {
     "editable": true
    },
@@ -2459,7 +2109,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "9bc32416",
+   "id": "03bb50ae",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2541,7 +2191,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eb934dbe",
+   "id": "cd11e9fd",
    "metadata": {
     "editable": true
    },
@@ -2566,7 +2216,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c65f802b",
+   "id": "6cf68ff2",
    "metadata": {
     "editable": true
    },
@@ -2594,7 +2244,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9accabd4",
+   "id": "c445bfb2",
    "metadata": {
     "editable": true
    },
@@ -2607,7 +2257,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "d5ed3f5c",
+   "id": "8e26035a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2707,7 +2357,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db1976f4",
+   "id": "86a6b012",
    "metadata": {
     "editable": true
    },
@@ -2718,7 +2368,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "3c35540c",
+   "id": "4a0e1afd",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2807,7 +2457,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e60fabdf",
+   "id": "bb1e0f15",
    "metadata": {
     "editable": true
    },
@@ -2817,7 +2467,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9b6750d6",
+   "id": "83c87d33",
    "metadata": {
     "editable": true
    },
@@ -2830,7 +2480,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "5e6b91e1",
+   "id": "74eaac59",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2908,622 +2558,364 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e83d7b10",
+   "id": "aa703354",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Notes on scaling with examples\n",
-    "\n",
-    "The programs here use both ordinrary least squares (OLS) and Ridge\n",
-    "regression with one value only for the hyperparameter $\\lambda$. The\n",
-    "first example has no scaling and includes the intercept as well and we\n",
-    "are trying to fit a second-order polynomial. The second code takes out\n",
-    "the intercept and subtracts the mean values of each column of the\n",
-    "design matrix and the mean value of the outputs.\n",
-    "\n",
-    "The third and final code uses **Scikit-Learn** as library in order to\n",
-    "calculate the optimal parameters for OLS and Ridge regression. Note\n",
-    "that it is highly recommended to not include the intercept in Ridge\n",
-    "and Lasso regression, in order to avoid penalizing the optimization by\n",
-    "the intercept. The second and third codes do thus not include the\n",
-    "intercept. In the second code we do the scaling ourselves while the\n",
-    "last code uses the standard scaler option included in **Scikit-Learn**, known as centering (where\n",
-    "we subtract the mean values)."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "id": "73807393",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "from sklearn.linear_model import LinearRegression\n",
-    "from sklearn.preprocessing import PolynomialFeatures\n",
-    "from sklearn.model_selection import train_test_split\n",
-    "from sklearn.preprocessing import StandardScaler\n",
-    "\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",
-    "def OLS_fit_beta(X, y):\n",
-    "    return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
-    "\n",
-    "def Ridge_fit_beta(X, y,L,d):\n",
-    "    I = np.eye(d,d)\n",
-    "    return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y\n",
-    "\n",
-    "# Same random numbers for each test.\n",
-    "np.random.seed(2018)\n",
-    "n = 100\n",
-    "d = 3\n",
-    "# hyperparameter lambda\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set, simple second-order polynomial\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "# The design matrix X includes the intercept and no scaling is made\n",
-    "X = np.zeros((len(x), d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p) \n",
-    "\n",
-    "\n",
-    "#Split data, no scaling is used and we include the intercept\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "\n",
-    "\n",
-    "#Calculate beta, own code\n",
-    "beta_OLS = OLS_fit_beta(X_train, y_train)\n",
-    "beta_Ridge = Ridge_fit_beta(X_train, y_train,Lambda,d)\n",
-    "print(beta_OLS)\n",
-    "print(beta_Ridge)\n",
-    "#predict value\n",
-    "ytilde_test_OLS = X_test @ beta_OLS\n",
-    "ytilde_test_Ridge = X_test @ beta_Ridge\n",
-    "\n",
-    "#Calculate MSE\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS:\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ beta_OLS,'*', label=\"OLS_Fit\")\n",
-    "plt.plot(x, X @ beta_Ridge, label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
+    "## Material for the lab sessions"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b09d20a2",
+   "id": "f8bffca6",
    "metadata": {
     "editable": true
    },
    "source": [
-    "In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the [lecture material](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#more-on-rescaling-data).\n",
-    "see also the weekly slides [for week 36](https://compphysics.github.io/MachineLearning/doc/pub/week36/html/._week36-bs029.html).\n",
-    "It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.\n",
+    "## Linking the regression analysis with a statistical interpretation\n",
     "\n",
-    "Before we discuss the code, we repeat some of the basic math from the slides of week 36.\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",
-    "Let us try to understand what this may imply mathematically when we\n",
-    "subtract the mean values, also known as *zero centering* or simply *centering*. For\n",
-    "simplicity, we will focus on  ordinary regression, as done in the above example.\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",
-    "The cost/loss function  for regression is"
+    "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": "ef1cec97",
+   "id": "109c07d6",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+    "\\begin{align*} \n",
+    "\\mbox{Cov}(\\varepsilon_{i_1},\n",
+    "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n",
+    "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2.  \\end{array} \\right.\n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e43c422b",
+   "id": "1ea48f22",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Recall also that we use the squared value. This expression can lead to an\n",
-    "increased penalty for higher differences between predicted and\n",
-    "output/target values.\n",
+    "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",
-    "What we have done is to single out the $\\beta_0$ term in the\n",
-    "definition of the mean squared error (MSE).  The design matrix $X$\n",
-    "does in this case not contain any intercept column.  When we take the\n",
-    "derivative with respect to $\\beta_0$, we want the derivative to obey"
+    "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": "852577bc",
+   "id": "212f90ec",
+   "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": "a019e7cb",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4edd229a",
+   "id": "2ca46c43",
    "metadata": {
     "editable": true
    },
    "source": [
-    "for all $j$. For $\\beta_0$ we have"
+    "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": "bb93d608",
+   "id": "c61ed9c8",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "5ef162e3",
+   "id": "96d84043",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Multiplying away the constant $2/n$, we obtain"
+    "## Expectation value and variance\n",
+    "\n",
+    "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b7c5467d",
+   "id": "e53b66df",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+    "\\begin{align*} \n",
+    "\\mathbb{E}(y_i) & =\n",
+    "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n",
+    "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "493c75d6",
+   "id": "71d542a3",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
-    "Our result for $\\beta_0$ simplifies then to"
+    "while\n",
+    "its variance is"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "862f2222",
+   "id": "01b18637",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+    "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
+    "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n",
+    "[\\mathbb{E}(y_i)]^2  \\\\  & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n",
+    "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n",
+    "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n",
+    "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n",
+    "\\ast} \\, \\beta)^2 \\\\  & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n",
+    "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n",
+    "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n",
+    "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n",
+    "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2.  \n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e0d0a4e0",
+   "id": "e35bdcc1",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We obtain then"
+    "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": "73d327de",
+   "id": "15b7ea3c",
+   "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": "a4c5f625",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
+    "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "8be27cbd",
+   "id": "450007e4",
    "metadata": {
     "editable": true
    },
    "source": [
-    "If we define"
+    "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": "d49a9bac",
+   "id": "3bcf81c1",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\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": "dd964823",
+   "id": "f69a019e",
    "metadata": {
     "editable": true
    },
    "source": [
-    "and the mean value of the outputs as"
+    "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": "09a87e67",
+   "id": "95b37da0",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9b8797a0",
+   "id": "241c2335",
    "metadata": {
     "editable": true
    },
    "source": [
-    "we have"
+    "We see clearly that \n",
+    "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n",
+    "\n",
+    "We can also compute the variance as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "30aeafdb",
+   "id": "cf794384",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
+    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "dc867cbb",
+   "id": "4cd5260a",
    "metadata": {
     "editable": true
    },
    "source": [
-    "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
+    "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": "f154321e",
+   "id": "7361e846",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
+    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "6015f267",
+   "id": "6a07837b",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We can rewrite the latter equation as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f181e88f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a6e6c0e7",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where we have defined"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "708652ea",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "b360788c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
+    "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. \n",
     "\n",
-    "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "8f473602",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "70f518a7",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "674d02d9",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "6715261c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
-    "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
-    "\n",
-    "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "17600ee3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a8c6f068",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Now we try to implement this."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "id": "1d89f80d",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "\n",
-    "np.random.seed(2018)\n",
-    "n = 100\n",
-    "# we do not include the intercept\n",
-    "d = 2\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set.\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "#Design matrix X does not include the intercept. \n",
-    "X = np.zeros((len(x), d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p+1)\n",
-    "\n",
-    "\n",
-    "#Split data in train and test\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "\n",
-    "# Scale data by subtracting mean value,own implementation\n",
-    "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
-    "X_train_mean = np.mean(X_train,axis=0)\n",
-    "#Center by removing mean from each feature\n",
-    "X_train_scaled = X_train - X_train_mean\n",
-    "X_test_scaled = X_test - X_train_mean\n",
-    "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered, note)\n",
-    "y_scaler = np.mean(y_train)\n",
-    "y_train_scaled = y_train - y_scaler\n",
-    "\n",
-    "\n",
-    "#Calculate beta\n",
-    "beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)\n",
-    "beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,Lambda,d)\n",
-    "print(beta_OLS)\n",
-    "print(beta_Ridge)\n",
-    "# calculate intercepts and print them\n",
-    "interceptOLS = y_scaler - X_train_mean @ beta_OLS\n",
-    "interceptRidge = y_scaler - X_train_mean @ beta_Ridge\n",
-    "print(interceptOLS)\n",
-    "print(interceptRidge)\n",
-    "\n",
-    "#predict value with intercept\n",
-    "ytilde_test_OLS = X_test_scaled @ beta_OLS+y_scaler\n",
-    "ytilde_test_Ridge = X_test_scaled @ beta_Ridge+y_scaler\n",
-    "\n",
-    "\n",
-    "#Calculate MSE\n",
-    "\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS:\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ beta_OLS+interceptOLS,'*', label=\"OLS_Fit\")\n",
-    "plt.plot(x, X @ beta_Ridge+interceptRidge, label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "522668c3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Finally, instead of using our own function we repeat the same example\n",
-    "using the **standardscaler** functionality of the library\n",
-    "**Scikit-Learn**.  Here we limit ourselves to Ridge regression only."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "id": "9954bcc1",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "from sklearn import linear_model\n",
-    "np.random.seed(2018)\n",
-    "n = 10\n",
-    "d = 2\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set.\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "# Design matrix X does not include the intercept. \n",
-    "X = np.zeros((n, d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p+1)\n",
-    "\n",
-    "#Split data in train and test\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "# Scale data by subtracting mean value of the input using scikit-learn\n",
-    "scaler = StandardScaler(with_std=False)\n",
-    "scaler.fit(X_train)\n",
-    "X_train_mean = np.mean(X_train,axis=0)\n",
-    "X_train_scaled = scaler.transform(X_train)\n",
-    "X_test_scaled = scaler.transform(X_test)\n",
-    "# We scale also the output, here by our own code\n",
-    "y_scaler = np.mean(y_train)\n",
-    "y_train_scaled = y_train - y_scaler\n",
-    "y_test_scaled = y_test- y_scaler\n",
-    "\n",
-    "#Calculate beta\n",
-    "OLS = LinearRegression()\n",
-    "betaOLS=OLS.fit(X_train_scaled,y_train_scaled)\n",
-    "ypredictOLS = OLS.predict(X_test_scaled)\n",
-    "linear_model.Ridge(Lambda)\n",
-    "RegRidge.fit(X_train_scaled,y_train_scaled)\n",
-    "ypredictRidge = RegRidge.predict(X_test_scaled)\n",
-    "betaOLS = OLS.coef_\n",
-    "betaRidge = RegRidge.coef_\n",
-    "print(betaOLS)\n",
-    "print(betaRidge)\n",
-    "interceptOLS = np.mean(y_train) - X_train_mean @ betaOLS\n",
-    "interceptRidge = y_scaler - X_train_mean @ betaRidge\n",
-    "print(interceptOLS)\n",
-    "print(interceptRidge)\n",
-    "#predict value \n",
-    "ytilde_test_Ridge = X_test_scaled @ betaRidge+y_scaler\n",
-    "ytilde_test_OLS = X_test_scaled @ betaOLS+y_scaler\n",
-    "\n",
-    "#Calculate MSE\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ RegRidge.coef_ + RegRidge.intercept_ , label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
+    "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
    ]
   }
  ],
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 5f95f4674..63fe57975 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
+++ b/doc/LectureNotes/_build/html/searchindex.js
@@ -1 +1 @@
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0,26],z_c:1,z_h:1,z_hidden:2,z_i:[1,12],z_j:[1,12],z_k:[12,27],z_m:1,z_mod:9,z_o:1,z_output:2,zaman:23,zaxi:[6,21],zero:[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,17,18,20,21,23,26,27,28,29],zeros_lik:4,zeroth:27,zfill:4,zip:[2,4,6],zm_h:[0,26],zn:0,zone:0,zoom:26,zx:[20,26],zy:[20,26],zz:[20,26]},titles:["<span class=\"section-number\">3. </span>Linear Regression","<span class=\"section-number\">14. </span>Building a Feed Forward Neural Network","<span class=\"section-number\">15. </span>Solving Differential Equations  with Deep Learning","<span class=\"section-number\">16. </span>Convolutional Neural Networks","<span class=\"section-number\">17. </span>Recurrent neural networks: Overarching view","<span class=\"section-number\">4. </span>Ridge and Lasso Regression","<span class=\"section-number\">5. </span>Resampling Methods","<span class=\"section-number\">6. </span>Logistic Regression","<span class=\"section-number\">8. </span>Support Vector Machines, overarching aims","<span class=\"section-number\">9. </span>Decision trees, overarching aims","<span class=\"section-number\">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","<span class=\"section-number\">11. </span>Basic ideas of the Principal Component Analysis (PCA)","<span class=\"section-number\">13. </span>Neural networks","<span class=\"section-number\">7. </span>Optimization, the central part of any Machine Learning algortithm","<span class=\"section-number\">12. </span>Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Applied Data Analysis and Machine Learning","<span class=\"section-number\">2. </span>Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 7 (midnight), 2024","Teaching schedule with links to material","<span class=\"section-number\">1. </span>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: Linear Regression and Statistical interpretations","Week 37: Statistical interpretations and Resampling 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class=\"section-number\">3. </span>Linear Regression","<span class=\"section-number\">14. </span>Building a Feed Forward Neural Network","<span class=\"section-number\">15. </span>Solving Differential Equations  with Deep Learning","<span class=\"section-number\">16. </span>Convolutional Neural Networks","<span class=\"section-number\">17. </span>Recurrent neural networks: Overarching view","<span class=\"section-number\">4. </span>Ridge and Lasso Regression","<span class=\"section-number\">5. </span>Resampling Methods","<span class=\"section-number\">6. </span>Logistic Regression","<span class=\"section-number\">8. </span>Support Vector Machines, overarching aims","<span class=\"section-number\">9. </span>Decision trees, overarching aims","<span class=\"section-number\">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","<span class=\"section-number\">11. </span>Basic ideas of the Principal Component Analysis (PCA)","<span class=\"section-number\">13. </span>Neural networks","<span class=\"section-number\">7. </span>Optimization, the central part of any Machine Learning algortithm","<span class=\"section-number\">12. </span>Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Applied Data Analysis and Machine Learning","<span class=\"section-number\">2. </span>Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 7 (midnight), 2024","Teaching schedule with links to material","<span class=\"section-number\">1. </span>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: Linear Regression and Statistical interpretations","Week 37: Statistical interpretations and Resampling Methods"],titleterms:{"1":[0,15,16,17,18,21,26,27],"14":[],"17":[],"18":[],"2":[0,15,16,17,18,26,27,28],"2023":24,"2024":21,"26":27,"3":[0,15,16,26,27],"34":[15,26],"35":[16,27],"36":[17,28],"37":[18,29],"4":[0,27],"5":0,"7":21,"9":29,"case":[8,10,23,27,28],"do":[1,28,29],"final":[12,27,28],"function":[0,1,6,7,8,10,11,12,13,21,23,26,27,28,29],"import":[5,20,26,27,28],"new":[4,28,29],A:[0,1,4,8,9,26,28,29],And:[26,27,28],For:27,In:24,Ising:6,The:[0,1,2,3,5,6,7,8,9,11,12,17,19,26,27,28,29],To:[26,27],With:[4,28],about:[26,27],abov:28,activ:[1,12,28],ad:[0,6,17,21,26,27],adaboost:10,adagrad:13,adam:13,adapt:10,adjust:1,adversari:4,again:[3,9],ai:[21,26],aim:[8,9,17,18,26],aka:[26,27],algebra:[20,26],algorithm:[9,10,11,12,27],algortithm:13,all:8,altern:[],an:[0,4,10,26],analys:[5,27],analysi:[0,5,6,11,19,21,23,26,27,28,29],analyt:[0,16,17],analyz:[],ani:13,anoth:[9,28,29],appli:19,approach:[0,8,14,26,29],approxim:12,architectur:1,arrai:[20,26],assist:24,assumpt:[28,29],august:27,autocorrel:23,autograd:[2,13],automat:13,b:[17,21],back:[1,11,12],background:[19,21,29],bag:10,base:[13,29],basic:[0,5,7,9,10,11,20,27,28,29],batch:1,bay:[5,28,29],befor:11,beta:[28,29],better:8,bia:[6,21,29],binari:1,bind:26,bird:10,boldsymbol:[27,28,29],boost:10,bootstrap:[6,10,29],boston:0,breast:1,brief:[26,29],bring:12,build:[1,3,9],c:[21,26],calcul:27,can:[26,29],cancer:[1,7,9,11],cart:9,center:27,central:[13,19,23,29],chain:12,chang:10,channel:26,chi:[0,26],choos:1,cifar01:3,classic:11,classif:[1,9,10],classifi:8,clip:1,cluster:14,cnn:3,code:[0,1,2,5,9,11,12,13,14,26,27,28,29],collect:[1,3],commun:26,compar:[2,10],comparison:28,complet:27,complex:[0,6,21,27],complic:6,compon:11,comput:9,computation:29,computerlab:26,con:9,concept:23,condit:[28,29],confid:29,conjug:13,contn:26,convex:[8,13],convolut:[3,12],correctli:[28,29],correl:[11,27],cost:[1,10,27,28,29],cours:[19,25,26],covari:[5,11,23,27],cover:26,critic:[],cross:[6,21,29],cython:26,d:21,data:[0,1,3,6,7,9,11,15,16,19,21,23,26,27,28],dataset:[1,3],deadlin:[21,26],decai:2,decemb:[],decis:[9,10],decomposit:[5,11,17,20,27],deep:[1,2,26],defin:[1,26],degre:[0,27],deliveri:21,delta:29,dens:[0,26],deriv:[5,12,27,28,29],descent:[2,10,13],descript:21,design:27,detail:[3,26],develop:1,diagon:11,differ:8,differenti:[2,13],diffus:2,dimension:[2,3,8,21,27],disadvantag:9,discret:23,distribut:[5,23,28,29],distrubut:29,doe:[27,28],domain:23,down:1,dropout:1,e:21,economi:27,electron:21,element:[0,23,26],elimin:20,energi:26,ensembl:10,entropi:9,environ:[0,15,26],equat:[0,2,12,26,27,28],error:[0,10,26,27,29],essenti:26,estim:[28,29],etc:26,euler:2,evalu:1,exampl:[0,1,2,3,4,6,7,8,9,10,26,27,28,29],exercis:[0,6,15,16,17,18,26,27],expect:[18,23,28,29],expens:29,experi:23,explor:[0,15,16,26],exponenti:2,express:[17,18,27],extrapol:4,extrem:[10,26],ey:10,f:21,fall:24,famili:[1,26,27],famou:20,fantast:27,featur:[9,20,27],feed:[1,12],find:29,fine:1,first:[4,12,26,27,28],fit:[0,10,26,28],fix:27,fold:29,forc:3,forest:10,format:[21,26],forward:[1,2,12],fourier:3,frank:[6,21,27],freedom:[0,27],frequent:27,frequentist:[0,26],from:[5,10,12,27,28,29],full:2,further:[3,5,27],g:21,gan:4,gaussian:20,gd:13,gener:[4,9,26],geometr:11,gini:9,good:[0,26],grade:[24,26],gradient:[1,2,10,13],growth:2,ha:19,handl:[20,26,27],happen:[28,29],hessian:27,hidden:2,histogram:29,hous:0,hyperparamet:1,hyperplan:8,i:1,id3:9,idea:11,ident:[28,29],identifi:29,ii:26,iid:[28,29],illustr:28,implement:1,implic:[5,27],improv:1,includ:13,increment:11,independ:[28,29],index:9,inform:24,input:2,instal:[19,21,26],instructor:24,intercept:27,interpret:[5,11,26,27,28,29],interv:29,introduc:[11,13,27],introduct:[0,6,19,20,21,26],invers:[5,20,28],invert:27,iter:10,its:27,jacobian:27,jax:13,julia:26,jungl:10,k:29,kera:[1,3],kernel:[8,11],lab:29,lagrangian:8,lasso:[5,6,21,27,28,29],last:27,later:[5,27],layer:[1,2,3,12],learn:[0,1,2,11,13,14,15,16,19,21,26,27,28,29],least:[5,6,18,21,26,27,28],lectur:[26,27,28,29],level:10,librari:[19,26],likelihood:[7,28,29],limit:[1,13,23,29],linear:[0,8,13,20,26,27,28],link:[5,11,22,25,27,28,29],literatur:21,logist:[7,26],loss:27,lu:20,machin:[0,8,13,19,21,26],made:[28,29],main:[23,26],make:[0,9,10,15,16,26,27],mani:[10,12],manipul:27,margin:[28,29],mass:26,materi:[21,22,26,27,28,29],math:[5,27],mathemat:[3,5,8,27],matric:[5,20,26,28],matrix:[1,5,11,12,20,26,27,28],matter:[0,26],max:27,maximum:[28,29],mean:[0,27,28],meet:[5,10,23,26,27],mercer:8,method:[6,9,10,13,21,26,29],midnight:21,min:27,minim:26,ml:26,mle:[28,29],mlp:12,mnist:[3,4],model:[0,1,4,6,12,26],momentum:13,mondai:[27,28,29],moon:[8,9],more:[3,6,20,21,26,27,28,29],multilay:12,multipl:[1,3],multipli:8,need:[21,26],network:[1,2,3,4,7,12,26],neural:[1,2,3,4,7,12,26],non:8,normal:[0,1,29],notat:12,note:[21,27,28],novemb:[],now:[1,9,13,28,29],nuclear:[0,26],numba:26,number:[0,2,23,27],numer:[2,21,23],numpi:[20,26],object:3,obtain:11,octob:21,od:2,off:[6,21],ol:[5,6,21,28,29],one:[2,12],oper:20,optim:[1,8,13,19,26,27],order:13,ordinari:[5,6,18,21,26,27,28],organ:[0,26],oslo:25,other:[4,9,11,12,20,21,26,27],our:[0,4,5,11,13,26,27],outcom:[19,26],output:2,overarch:[0,4,8,9,17,18,26,27],overview:[10,26],own:[0,10,11,15,16,26,27],packag:[20,26],panda:[26,27],paper:21,paramet:[26,27],part:[13,19,21,27],partial:2,pass:1,path:[],pca:11,pdf:23,pencil:21,perceptron:12,perform:[1,9],period:3,perspect:1,plan:[27,28,29],plot:29,point:4,poisson:2,polynomi:[3,28],popul:2,popular:26,practic:[13,24,26],pre:[1,3],preambl:21,predict:4,preprocess:27,prerequisit:[3,19,26],princip:11,principl:3,pro:9,probabl:[5,23,28,29],problem:[1,2,13,26,27,28],procedur:[9,26],process:[1,3],program:[2,13,21],project:[6,21,26],prop:13,propag:[1,12],properti:[5,23,27],python:[0,9,15,19,20,26],quick:8,r:26,random:[10,11,23],read:[9,26,27,29],real:[6,21,26],recommend:[26,27],rectangular:28,recurr:[4,12],reduc:[0,27],reduct:3,refer:21,reformul:2,regress:[0,5,6,7,9,10,13,17,18,21,26,27,28,29],regular:1,relat:27,relev:[25,27],relu:1,remark:3,remind:[6,8,26],repeat:27,replac:13,report:21,repositori:29,requir:[2,19],resampl:[6,21,29],rescal:[6,28],residu:27,resourc:2,result:[27,28],revisit:13,rewrit:[26,27,29],rgeress:[],ridg:[0,5,6,17,18,21,27,28,29],rm:13,rule:12,s:[8,10,29],same:[13,29],sampl:11,scale:27,schedul:[22,26],schemat:9,scheme:2,scienc:26,scikit:[0,1,11,15,16,26,27,28,29],second:13,semest:24,septemb:[28,29],session:[27,28,29],set:[0,2,3,9,12,15,26,27],sgd:13,should:1,similar:13,simpl:[0,4,9,13,26,27,28],singl:10,singular:[5,11,17,27],size:27,soft:8,softmax:1,softwar:[21,26],solv:[2,28],solver:13,some:[13,20,27],specifi:2,split:[0,15,16,26,27],squar:[0,5,6,10,18,21,26,27,28],standard:[13,27,29],state:[0,26],statist:[5,6,19,23,26,28,29],steepest:[10,13],step:29,still:27,stochast:[13,23],strongli:26,structur:[],subtract:27,suggest:26,sum:29,summari:[24,26],superposit:3,supervis:1,support:8,svd:[5,27,28],systemat:3,t:[27,28],teach:[22,24],teacher:[24,26],technic:28,techniqu:[6,11,21],technolog:19,tensorflow:[1,3],tent:26,term:29,test:[0,1,15,16,26,27,28],text:26,textbook:[25,26],theorem:[5,8,11,12,23,28,29],theori:23,thi:[17,18,26],think:27,thursdai:[],tip:13,togeth:12,tool:[21,26],top:1,topic:26,toward:11,trade:[6,21],tradeoff:[6,29],train:[0,1,4,15,16,26,27],transform:3,tree:[9,10],tuesdai:28,tune:1,two:[3,8,19,21,27],type:[2,4,12,26],uio:26,understand:29,univers:[12,25],unsupervis:14,up:[0,2,9,12,15,26,27,29],us:[0,1,2,3,7,13,19,21,26,27],usag:[28,29],valid:[6,21,29],valu:[5,11,17,18,23,27,28,29],variabl:23,varianc:[6,21,28,29],variou:[0,15,26,29],vector:[8,12,20,26,27],view:[0,4,10,27],visual:[1,9],vs:3,wai:[9,29],wave:2,we:26,wednesdai:28,week:[15,16,17,18,26,27,28,29],weekli:22,what:[0,26,27,28,29],which:1,why:[26,27,29],wisconsin:7,wrap:[27,29],write:[4,11,21,28],x:[27,28],xgboost:10,yet:28,your:[0,10,15,16,26,27],yourself:[]}})
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/week37.html b/doc/LectureNotes/_build/html/week37.html
index 32a4cc488..cd478a242 100644
--- a/doc/LectureNotes/_build/html/week37.html
+++ b/doc/LectureNotes/_build/html/week37.html
@@ -372,41 +372,18 @@ const thebe_selector_output = ".output, .cell_output"
             <nav id="bd-toc-nav" aria-label="Page">
                 <ul class="visible nav section-nav flex-column">
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#plans-for-week-37">
-   Plans for week 37
+  <a class="reference internal nav-link" href="#plans-for-week-37-lecture-monday">
+   Plans for week 37, lecture Monday
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#material-from-last-week-and-relevant-for-the-weekly-exercises">
-   Material from last week and relevant for the weekly exercises
+  <a class="reference internal nav-link" href="#plans-for-week-37-lab-sessions">
+   Plans for week 37, lab sessions
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
-   Linking the regression analysis with a statistical interpretation
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#assumptions-made">
-   Assumptions made
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#expectation-value-and-variance">
-   Expectation value and variance
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#expectation-value-and-variance-for-boldsymbol-beta">
-   Expectation value and variance for
-   <span class="math notranslate nohighlight">
-    \(\boldsymbol{\beta}\)
-   </span>
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#material-for-lecture-thursday-september-14">
-   Material for lecture Thursday September 14
+  <a class="reference internal nav-link" href="#material-for-lecture-monday-september-9">
+   Material for lecture Monday September 9
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
@@ -643,8 +620,31 @@ const thebe_selector_output = ".output, .cell_output"
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#notes-on-scaling-with-examples">
-   Notes on scaling with examples
+  <a class="reference internal nav-link" href="#material-for-the-lab-sessions">
+   Material for the lab sessions
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
+   Linking the regression analysis with a statistical interpretation
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#assumptions-made">
+   Assumptions made
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#expectation-value-and-variance">
+   Expectation value and variance
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#expectation-value-and-variance-for-boldsymbol-beta">
+   Expectation value and variance for
+   <span class="math notranslate nohighlight">
+    \(\boldsymbol{\beta}\)
+   </span>
   </a>
  </li>
 </ul>
@@ -668,41 +668,18 @@ const thebe_selector_output = ".output, .cell_output"
                         <nav aria-label="Page">
                             <ul class="visible nav section-nav flex-column">
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#plans-for-week-37">
-   Plans for week 37
+  <a class="reference internal nav-link" href="#plans-for-week-37-lecture-monday">
+   Plans for week 37, lecture Monday
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#material-from-last-week-and-relevant-for-the-weekly-exercises">
-   Material from last week and relevant for the weekly exercises
+  <a class="reference internal nav-link" href="#plans-for-week-37-lab-sessions">
+   Plans for week 37, lab sessions
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
-   Linking the regression analysis with a statistical interpretation
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#assumptions-made">
-   Assumptions made
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#expectation-value-and-variance">
-   Expectation value and variance
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#expectation-value-and-variance-for-boldsymbol-beta">
-   Expectation value and variance for
-   <span class="math notranslate nohighlight">
-    \(\boldsymbol{\beta}\)
-   </span>
-  </a>
- </li>
- <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#material-for-lecture-thursday-september-14">
-   Material for lecture Thursday September 14
+  <a class="reference internal nav-link" href="#material-for-lecture-monday-september-9">
+   Material for lecture Monday September 9
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
@@ -939,8 +916,31 @@ const thebe_selector_output = ".output, .cell_output"
   </a>
  </li>
  <li class="toc-h2 nav-item toc-entry">
-  <a class="reference internal nav-link" href="#notes-on-scaling-with-examples">
-   Notes on scaling with examples
+  <a class="reference internal nav-link" href="#material-for-the-lab-sessions">
+   Material for the lab sessions
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
+   Linking the regression analysis with a statistical interpretation
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#assumptions-made">
+   Assumptions made
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#expectation-value-and-variance">
+   Expectation value and variance
+  </a>
+ </li>
+ <li class="toc-h2 nav-item toc-entry">
+  <a class="reference internal nav-link" href="#expectation-value-and-variance-for-boldsymbol-beta">
+   Expectation value and variance for
+   <span class="math notranslate nohighlight">
+    \(\boldsymbol{\beta}\)
+   </span>
   </a>
  </li>
 </ul>
@@ -956,188 +956,40 @@ const thebe_selector_output = ".output, .cell_output"
 doconce format html week37.do.txt --no_mako -->
 <!-- dom:TITLE: Week 37: Statistical interpretations and Resampling Methods --><div class="tex2jax_ignore mathjax_ignore section" id="week-37-statistical-interpretations-and-resampling-methods">
 <h1>Week 37: Statistical interpretations and Resampling Methods<a class="headerlink" href="#week-37-statistical-interpretations-and-resampling-methods" title="Permalink to this headline">¶</a></h1>
-<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University</p>
+<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo, Norway</p>
 <p>Date: <strong>September 9, 2024</strong></p>
-<!-- todo add link to videos and add link to Van Wieringens notes --><div class="section" id="plans-for-week-37">
-<h2>Plans for week 37<a class="headerlink" href="#plans-for-week-37" title="Permalink to this headline">¶</a></h2>
+<!-- todo add link to videos and add link to Van Wieringens notes --><div class="section" id="plans-for-week-37-lecture-monday">
+<h2>Plans for week 37, lecture Monday<a class="headerlink" href="#plans-for-week-37-lecture-monday" title="Permalink to this headline">¶</a></h2>
 <p><strong>Material for the lecture on Monday September 9.</strong></p>
 <!-- * [Video of Lecture](https://youtu.be/YOBBr_toYxc) -->
 <!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf) -->
 <ul class="simple">
-<li><p>Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff</p></li>
-<li><p>Statistical interpretation of Ridge and Lasso regression</p></li>
+<li><p>Statistical interpretation of Ridge and Lasso regression, see also slides from last week</p></li>
+<li><p>Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.</p></li>
 <li><p>Readings and Videos:</p>
 <ul>
-<li><p>Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap).</p></li>
+<li><p>Raschka et al, pages 175-192</p></li>
+<li><p>Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See <a class="reference external" href="https://link.springer.com/book/10.1007/978-0-387-84858-7">https://link.springer.com/book/10.1007/978-0-387-84858-7</a>.</p></li>
 <li><p><a class="reference external" href="https://www.youtube.com/watch?v=fSytzGwwBVw">Video on cross validation</a></p></li>
 <li><p><a class="reference external" href="https://www.youtube.com/watch?v=Xz0x-8-cgaQ">Video on Bootstrapping</a></p></li>
 <li><p><a class="reference external" href="https://www.youtube.com/watch?v=EuBBz3bI-aA">Video on bias-variance tradeoff</a></p></li>
 </ul>
 </li>
 </ul>
-<p><strong>Material for the active learning sessions on Tuesday and Wednesday.</strong></p>
+</div>
+<div class="section" id="plans-for-week-37-lab-sessions">
+<h2>Plans for week 37, lab sessions<a class="headerlink" href="#plans-for-week-37-lab-sessions" title="Permalink to this headline">¶</a></h2>
+<p><strong>Material for the lab  sessions on Tuesday and Wednesday.</strong></p>
 <ul class="simple">
 <li><p>Calculations of expectation values</p></li>
+<li><p>Discussion of resampling techniques</p></li>
 <li><p>Exercise set for week 37</p></li>
 <li><p>Work on project 1</p></li>
-<li><p>See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.</p></li>
 <li><p>For more discussions of Ridge regression and calculation of averages, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.</p></li>
 </ul>
 </div>
-<div class="section" id="material-from-last-week-and-relevant-for-the-weekly-exercises">
-<h2>Material from last week and relevant for the weekly exercises<a class="headerlink" href="#material-from-last-week-and-relevant-for-the-weekly-exercises" title="Permalink to this headline">¶</a></h2>
-</div>
-<div class="section" id="linking-the-regression-analysis-with-a-statistical-interpretation">
-<h2>Linking the regression analysis with a statistical interpretation<a class="headerlink" href="#linking-the-regression-analysis-with-a-statistical-interpretation" title="Permalink to this headline">¶</a></h2>
-<p>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
-<span class="math notranslate nohighlight">\(\beta\)</span>.</p>
-<p>The
-advantage of doing linear regression is that we actually end up with
-analytical expressions for several statistical quantities.<br />
-Standard least squares and Ridge regression  allow us to
-derive quantities like the variance and other expectation values in a
-rather straightforward way.</p>
-<p>It is assumed that <span class="math notranslate nohighlight">\(\varepsilon_i
-\sim \mathcal{N}(0, \sigma^2)\)</span> and the <span class="math notranslate nohighlight">\(\varepsilon_{i}\)</span> are
-independent, i.e.:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-\begin{align*} 
-\mbox{Cov}(\varepsilon_{i_1},
-\varepsilon_{i_2}) &amp; = \left\{ \begin{array}{lcc} \sigma^2 &amp; \mbox{if}
-&amp; i_1 = i_2, \\ 0 &amp; \mbox{if} &amp; i_1 \not= i_2.  \end{array} \right.
-\end{align*}
-\end{split}\]</div>
-<p>The randomness of <span class="math notranslate nohighlight">\(\varepsilon_i\)</span> implies that
-<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> is also a random variable. In particular,
-<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> is normally distributed, because <span class="math notranslate nohighlight">\(\varepsilon_i \sim
-\mathcal{N}(0, \sigma^2)\)</span> and <span class="math notranslate nohighlight">\(\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}\)</span> is a
-non-random scalar. To specify the parameters of the distribution of
-<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> we need to calculate its first two moments.</p>
-<p>Recall that <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is a matrix of dimensionality <span class="math notranslate nohighlight">\(n\times p\)</span>. The
-notation above <span class="math notranslate nohighlight">\(\mathbf{X}_{i,\ast}\)</span> means that we are looking at the
-row number <span class="math notranslate nohighlight">\(i\)</span> and perform a sum over all values <span class="math notranslate nohighlight">\(p\)</span>.</p>
-</div>
-<div class="section" id="assumptions-made">
-<h2>Assumptions made<a class="headerlink" href="#assumptions-made" title="Permalink to this headline">¶</a></h2>
-<p>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 <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and  a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)\)</span>
-which describe our data</p>
-<div class="math notranslate nohighlight">
-\[
-\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
-\]</div>
-<p>We approximate this function with our model from the solution of the linear regression equations, that is our
-function <span class="math notranslate nohighlight">\(f\)</span> is approximated by <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> where we want to minimize <span class="math notranslate nohighlight">\((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\)</span>, our MSE, with</p>
-<div class="math notranslate nohighlight">
-\[
-\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
-\]</div>
-</div>
-<div class="section" id="expectation-value-and-variance">
-<h2>Expectation value and variance<a class="headerlink" href="#expectation-value-and-variance" title="Permalink to this headline">¶</a></h2>
-<p>We can calculate the expectation value of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> for a given element <span class="math notranslate nohighlight">\(i\)</span></p>
-<div class="math notranslate nohighlight">
-\[
-\begin{align*} 
-\mathbb{E}(y_i) &amp; =
-\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
-\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, 
-\end{align*}
-\]</div>
-<p>while
-its variance is</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-\begin{align*} \mbox{Var}(y_i) &amp; = \mathbb{E} \{ [y_i
-- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
-[\mathbb{E}(y_i)]^2  \\  &amp; = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
-\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &amp;
-= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
-\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
-\ast} \, \beta)^2 \\  &amp; = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
-\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
-\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 
-\\ &amp; = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
-\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.  
-\end{align*}
-\end{split}\]</div>
-<p>Hence, <span class="math notranslate nohighlight">\(y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\)</span>, that is <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> follows a normal distribution with
-mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbol{\beta}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</span> (not be confused with the singular values of the SVD).</p>
-</div>
-<div class="section" id="expectation-value-and-variance-for-boldsymbol-beta">
-<h2>Expectation value and variance for <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span><a class="headerlink" href="#expectation-value-and-variance-for-boldsymbol-beta" title="Permalink to this headline">¶</a></h2>
-<p>With the OLS expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> we can evaluate the expectation value</p>
-<div class="math notranslate nohighlight">
-\[
-\mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
-\]</div>
-<p>This means that the estimator of the regression parameters is unbiased.</p>
-<p>We can also calculate the variance</p>
-<p>The variance of the optimal value <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> is</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-\begin{eqnarray*}
-\mbox{Var}(\boldsymbol{\hat{\beta}}) &amp; = &amp; \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
-\\
-&amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}]^{T} \}
-\\
-% &amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-% \\
-% &amp; = &amp; \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} \, \mathbf{y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1}  \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-% \\
-&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{y} \, \mathbf{y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-\\
-&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \,  \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-% \\
-% &amp; = &amp; (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \,  \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
-% \\
-% &amp; &amp; + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T  \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
-\\
-&amp; = &amp; \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}  + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
-\end{eqnarray*}
-\end{split}\]</div>
-<p>where we have used  that <span class="math notranslate nohighlight">\(\mathbb{E} (\mathbf{y} \mathbf{y}^{T}) =
-\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} +
-\sigma^2 \, \mathbf{I}_{nn}\)</span>. From <span class="math notranslate nohighlight">\(\mbox{Var}(\boldsymbol{\beta}) = \sigma^2
-\, (\mathbf{X}^{T} \mathbf{X})^{-1}\)</span>, one obtains an estimate of the
-variance of the estimate of the <span class="math notranslate nohighlight">\(j\)</span>-th regression coefficient:
-<span class="math notranslate nohighlight">\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \)</span>. This may be used to
-construct a confidence interval for the estimates.</p>
-<p>In a similar way, we can obtain analytical expressions for say the
-expectation values of the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and their variance
-when we employ Ridge regression, allowing us again to define a confidence interval.</p>
-<p>It is rather straightforward to show that</p>
-<div class="math notranslate nohighlight">
-\[
-\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
-\]</div>
-<p>We see clearly that
-<span class="math notranslate nohighlight">\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\)</span> for any <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span>.</p>
-<p>We can also compute the variance as</p>
-<div class="math notranslate nohighlight">
-\[
-\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
-\]</div>
-<p>and it is easy to see that if the parameter <span class="math notranslate nohighlight">\(\lambda\)</span> goes to infinity then the variance of Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
-<p>With this, we can compute the difference</p>
-<div class="math notranslate nohighlight">
-\[
-\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}]-\mbox{Var}(\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
-\]</div>
-<p>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 <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span> be larger than the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.</p>
-<p>For more discussions of Ridge regression and calculation of averages, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.</p>
-</div>
-<div class="section" id="material-for-lecture-thursday-september-14">
-<h2>Material for lecture Thursday September 14<a class="headerlink" href="#material-for-lecture-thursday-september-14" title="Permalink to this headline">¶</a></h2>
+<div class="section" id="material-for-lecture-monday-september-9">
+<h2>Material for lecture Monday September 9<a class="headerlink" href="#material-for-lecture-monday-september-9" title="Permalink to this headline">¶</a></h2>
 </div>
 <div class="section" id="deriving-ols-from-a-probability-distribution">
 <h2>Deriving OLS from a probability distribution<a class="headerlink" href="#deriving-ols-from-a-probability-distribution" title="Permalink to this headline">¶</a></h2>
@@ -1348,7 +1200,7 @@ p(\boldsymbol{\beta}\vert\boldsymbol{D}).
 \[
 p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
 \]</div>
-<p>We have a model for <span class="math notranslate nohighlight">\(p(\boldsymbol{D}\vert\boldsymbol{\beta})\)</span> but need one for the <strong>prior</strong> <span class="math notranslate nohighlight">\(p(\boldsymbol{\beta}\)</span>!</p>
+<p>We have a model for <span class="math notranslate nohighlight">\(p(\boldsymbol{D}\vert\boldsymbol{\beta})\)</span> but need one for the <strong>prior</strong> <span class="math notranslate nohighlight">\(p(\boldsymbol{\beta})\)</span>!</p>
 </div>
 <div class="section" id="ridge-and-bayes">
 <h2>Ridge and Bayes<a class="headerlink" href="#ridge-and-bayes" title="Permalink to this headline">¶</a></h2>
@@ -1632,9 +1484,9 @@ computed from the observed data. This gives a range of values for an
 unknown parameter such as the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> from linear regression.</p>
 <p>With the OLS expressions for the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> we found
 <span class="math notranslate nohighlight">\(\mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta}\)</span>, which means that the estimator of the regression parameters is unbiased.</p>
-<p>We found also that the variance of the estimate of the <span class="math notranslate nohighlight">\(j\)</span>-th regression coefficient is
+<p>In the exercises this week we show that the variance of the estimate of the <span class="math notranslate nohighlight">\(j\)</span>-th regression coefficient is
 <span class="math notranslate nohighlight">\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \)</span>.</p>
-<p>This quantity will be used to
+<p>This quantity can be used to
 construct a confidence interval for the estimates.</p>
 </div>
 <div class="section" id="standard-approach-based-on-the-normal-distribution">
@@ -1766,7 +1618,7 @@ theorem.</p>
 <div class="cell_output docutils container">
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Bootstrap Statistics :
 original           bias      std. error
- 99.9841  15.0402        99.9828          0.1512
+ 99.7503  14.9999        99.7496        0.151068
 </pre></div>
 </div>
 </div>
@@ -1790,7 +1642,7 @@ original           bias      std. error
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/week37_144_0.png" src="_images/week37_144_0.png" />
+<img alt="_images/week37_121_0.png" src="_images/week37_121_0.png" />
 </div>
 </div>
 </div>
@@ -1918,7 +1770,7 @@ Var: 0.0010479245693919886
 0.013121574015585152 &gt;= 0.012073649446193166 + 0.0010479245693919886 = 0.013121574015585155
 </pre></div>
 </div>
-<img alt="_images/week37_160_1.png" src="_images/week37_160_1.png" />
+<img alt="_images/week37_137_1.png" src="_images/week37_137_1.png" />
 </div>
 </div>
 </div>
@@ -2006,9 +1858,7 @@ Error: 0.05227921801205686
 Bias^2: 0.0481872773043029
 Var: 0.004091940707753939
 0.05227921801205686 &gt;= 0.0481872773043029 + 0.004091940707753939 = 0.052279218012056844
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 6
+Polynomial degree: 6
 Error: 0.037813671417389005
 Bias^2: 0.033657685071527665
 Var: 0.00415598634586135
@@ -2028,14 +1878,14 @@ Error: 0.02660572763718093
 Bias^2: 0.010018312644137363
 Var: 0.016587414993043573
 0.02660572763718093 &gt;= 0.010018312644137363 + 0.016587414993043573 = 0.026605727637180936
-Polynomial degree: 10
+</pre></div>
+</div>
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 10
 Error: 0.021592704588025025
 Bias^2: 0.010516485576645508
 Var: 0.011076219011379514
 0.021592704588025025 &gt;= 0.010516485576645508 + 0.011076219011379514 = 0.021592704588025022
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 11
+Polynomial degree: 11
 Error: 0.07160048164233104
 Bias^2: 0.014436800088904942
 Var: 0.05716368155342608
@@ -2045,16 +1895,14 @@ Error: 0.11547777218872497
 Bias^2: 0.01628578269596628
 Var: 0.09919198949275869
 0.11547777218872497 &gt;= 0.01628578269596628 + 0.09919198949275869 = 0.11547777218872497
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 13
+Polynomial degree: 13
 Error: 0.22842468702219465
 Bias^2: 0.01975416527185249
 Var: 0.20867052175034223
 0.22842468702219465 &gt;= 0.01975416527185249 + 0.20867052175034223 = 0.2284246870221947
 </pre></div>
 </div>
-<img alt="_images/week37_162_4.png" src="_images/week37_162_4.png" />
+<img alt="_images/week37_139_2.png" src="_images/week37_139_2.png" />
 </div>
 </div>
 </div>
@@ -2185,7 +2033,7 @@ set, the higher, the less likely the model generalizes correctly from the
 training data.
 </pre></div>
 </div>
-<img alt="_images/week37_165_1.png" src="_images/week37_165_1.png" />
+<img alt="_images/week37_142_1.png" src="_images/week37_142_1.png" />
 </div>
 </div>
 </div>
@@ -2320,7 +2168,7 @@ cross-validation (LOOCV).</p>
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="_images/week37_169_0.png" src="_images/week37_169_0.png" />
+<img alt="_images/week37_146_0.png" src="_images/week37_146_0.png" />
 </div>
 </div>
 </div>
@@ -2436,12 +2284,12 @@ Mean squared error on test data: 1.53738247
 Degree of polynomial:   8
 Mean squared error on training data: 0.04926746
 Mean squared error on test data: 0.14629156
-Degree of polynomial:   9
-Mean squared error on training data: 0.02546675
-Mean squared error on test data: 0.11202337
 </pre></div>
 </div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  10
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:   9
+Mean squared error on training data: 0.02546675
+Mean squared error on test data: 0.11202337
+Degree of polynomial:  10
 Mean squared error on training data: 0.02424794
 Mean squared error on test data: 0.22467274
 Degree of polynomial:  11
@@ -2450,21 +2298,23 @@ Mean squared error on test data: 1.07641937
 Degree of polynomial:  12
 Mean squared error on training data: 0.00805074
 Mean squared error on test data: 0.04295757
-Degree of polynomial:  13
+</pre></div>
+</div>
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  13
 Mean squared error on training data: 0.00781918
 Mean squared error on test data: 0.56965674
 Degree of polynomial:  14
 Mean squared error on training data: 0.00465099
 Mean squared error on test data: 0.28443039
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  15
+Degree of polynomial:  15
 Mean squared error on training data: 0.00420072
 Mean squared error on test data: 568.47051432
 Degree of polynomial:  16
 Mean squared error on training data: 0.00325450
 Mean squared error on test data: 48.97630233
-Degree of polynomial:  17
+</pre></div>
+</div>
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  17
 Mean squared error on training data: 0.00242954
 Mean squared error on test data: 2.52780600
 Degree of polynomial:  18
@@ -2473,12 +2323,12 @@ Mean squared error on test data: 429.25695398
 Degree of polynomial:  19
 Mean squared error on training data: 0.00154853
 Mean squared error on test data: 239.97065359
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  20
+Degree of polynomial:  20
 Mean squared error on training data: 0.00140846
 Mean squared error on test data: 1350.24493666
-Degree of polynomial:  21
+</pre></div>
+</div>
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  21
 Mean squared error on training data: 0.00119688
 Mean squared error on test data: 1840.50530832
 Degree of polynomial:  22
@@ -2487,12 +2337,12 @@ Mean squared error on test data: 1184.60929685
 Degree of polynomial:  23
 Mean squared error on training data: 0.00089193
 Mean squared error on test data: 3892.17483760
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  24
+Degree of polynomial:  24
 Mean squared error on training data: 0.00083355
 Mean squared error on test data: 1332.46736215
-Degree of polynomial:  25
+</pre></div>
+</div>
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  25
 Mean squared error on training data: 0.00079904
 Mean squared error on test data: 7577.76690383
 Degree of polynomial:  26
@@ -2501,23 +2351,23 @@ Mean squared error on test data: 1079.36895644
 Degree of polynomial:  27
 Mean squared error on training data: 0.00068091
 Mean squared error on test data: 3207.25343155
-Degree of polynomial:  28
-Mean squared error on training data: 0.00063362
-Mean squared error on test data: 674.79633065
 </pre></div>
 </div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  29
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial:  28
+Mean squared error on training data: 0.00063362
+Mean squared error on test data: 674.79633065
+Degree of polynomial:  29
 Mean squared error on training data: 0.00063866
 Mean squared error on test data: 3099.60342978
 </pre></div>
 </div>
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_85796/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87621/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label=&#39;Training Error&#39;)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_85796/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87621/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label=&#39;Test Error&#39;)
 </pre></div>
 </div>
-<img alt="_images/week37_171_8.png" src="_images/week37_171_8.png" />
+<img alt="_images/week37_148_9.png" src="_images/week37_148_9.png" />
 </div>
 </div>
 <p>Note that we kept the intercept column in the fitting here. This means that we need to set the <strong>intercept</strong> in the call to the <strong>Scikit-Learn</strong> function as <strong>False</strong>. Alternatively, we could have set up the design matrix <span class="math notranslate nohighlight">\(X\)</span> without the first column of ones.</p>
@@ -2598,368 +2448,165 @@ Mean squared error on test data: 3099.60342978
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_85796/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87621/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label=&#39;Test Error&#39;)
 </pre></div>
 </div>
-<img alt="_images/week37_174_1.png" src="_images/week37_174_1.png" />
+<img alt="_images/week37_151_1.png" src="_images/week37_151_1.png" />
 </div>
 </div>
 </div>
-<div class="section" id="notes-on-scaling-with-examples">
-<h2>Notes on scaling with examples<a class="headerlink" href="#notes-on-scaling-with-examples" title="Permalink to this headline">¶</a></h2>
-<p>The programs here use both ordinrary least squares (OLS) and Ridge
-regression with one value only for the hyperparameter <span class="math notranslate nohighlight">\(\lambda\)</span>. The
-first example has no scaling and includes the intercept as well and we
-are trying to fit a second-order polynomial. The second code takes out
-the intercept and subtracts the mean values of each column of the
-design matrix and the mean value of the outputs.</p>
-<p>The third and final code uses <strong>Scikit-Learn</strong> as library in order to
-calculate the optimal parameters for OLS and Ridge regression. Note
-that it is highly recommended to not include the intercept in Ridge
-and Lasso regression, in order to avoid penalizing the optimization by
-the intercept. The second and third codes do thus not include the
-intercept. In the second code we do the scaling ourselves while the
-last code uses the standard scaler option included in <strong>Scikit-Learn</strong>, known as centering (where
-we subtract the mean values).</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
-<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
-<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
-<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
-
-<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
-
-<span class="k">def</span> <span class="nf">OLS_fit_beta</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span>
-
-<span class="k">def</span> <span class="nf">Ridge_fit_beta</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span><span class="n">L</span><span class="p">,</span><span class="n">d</span><span class="p">):</span>
-    <span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="n">d</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span> <span class="o">+</span> <span class="n">L</span><span class="o">*</span><span class="n">I</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span>
-
-<span class="c1"># Same random numbers for each test.</span>
-<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2018</span><span class="p">)</span>
-<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
-<span class="n">d</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="c1"># hyperparameter lambda</span>
-<span class="n">Lambda</span> <span class="o">=</span> <span class="mf">0.01</span>
-
-<span class="c1"># Make data set, simple second-order polynomial</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mf">5.0</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-<span class="c1"># The design matrix X includes the intercept and no scaling is made</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">d</span><span class="p">))</span>
-<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>     
-    <span class="n">X</span><span class="p">[:,</span> <span class="n">p</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">**</span> <span class="p">(</span><span class="n">p</span><span class="p">)</span> 
-
-
-<span class="c1">#Split data, no scaling is used and we include the intercept</span>
-<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
-
-
-<span class="c1">#Calculate beta, own code</span>
-<span class="n">beta_OLS</span> <span class="o">=</span> <span class="n">OLS_fit_beta</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
-<span class="n">beta_Ridge</span> <span class="o">=</span> <span class="n">Ridge_fit_beta</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span><span class="n">Lambda</span><span class="p">,</span><span class="n">d</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">beta_OLS</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">beta_Ridge</span><span class="p">)</span>
-<span class="c1">#predict value</span>
-<span class="n">ytilde_test_OLS</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">beta_OLS</span>
-<span class="n">ytilde_test_Ridge</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">beta_Ridge</span>
-
-<span class="c1">#Calculate MSE</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;  &quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;test MSE of OLS:&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ytilde_test_OLS</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;  &quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;test MSE of Ridge&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ytilde_test_Ridge</span><span class="p">))</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta_OLS</span><span class="p">,</span><span class="s1">&#39;*&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;OLS_Fit&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta_Ridge</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ridge_Fit&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-</pre></div>
+<div class="section" id="material-for-the-lab-sessions">
+<h2>Material for the lab sessions<a class="headerlink" href="#material-for-the-lab-sessions" title="Permalink to this headline">¶</a></h2>
 </div>
+<div class="section" id="linking-the-regression-analysis-with-a-statistical-interpretation">
+<h2>Linking the regression analysis with a statistical interpretation<a class="headerlink" href="#linking-the-regression-analysis-with-a-statistical-interpretation" title="Permalink to this headline">¶</a></h2>
+<p>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
+<span class="math notranslate nohighlight">\(\beta\)</span>.</p>
+<p>The
+advantage of doing linear regression is that we actually end up with
+analytical expressions for several statistical quantities.<br />
+Standard least squares and Ridge regression  allow us to
+derive quantities like the variance and other expectation values in a
+rather straightforward way.</p>
+<p>It is assumed that <span class="math notranslate nohighlight">\(\varepsilon_i
+\sim \mathcal{N}(0, \sigma^2)\)</span> and the <span class="math notranslate nohighlight">\(\varepsilon_{i}\)</span> are
+independent, i.e.:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align*} 
+\mbox{Cov}(\varepsilon_{i_1},
+\varepsilon_{i_2}) &amp; = \left\{ \begin{array}{lcc} \sigma^2 &amp; \mbox{if}
+&amp; i_1 = i_2, \\ 0 &amp; \mbox{if} &amp; i_1 \not= i_2.  \end{array} \right.
+\end{align*}
+\end{split}\]</div>
+<p>The randomness of <span class="math notranslate nohighlight">\(\varepsilon_i\)</span> implies that
+<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> is also a random variable. In particular,
+<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> is normally distributed, because <span class="math notranslate nohighlight">\(\varepsilon_i \sim
+\mathcal{N}(0, \sigma^2)\)</span> and <span class="math notranslate nohighlight">\(\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}\)</span> is a
+non-random scalar. To specify the parameters of the distribution of
+<span class="math notranslate nohighlight">\(\mathbf{y}_i\)</span> we need to calculate its first two moments.</p>
+<p>Recall that <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is a matrix of dimensionality <span class="math notranslate nohighlight">\(n\times p\)</span>. The
+notation above <span class="math notranslate nohighlight">\(\mathbf{X}_{i,\ast}\)</span> means that we are looking at the
+row number <span class="math notranslate nohighlight">\(i\)</span> and perform a sum over all values <span class="math notranslate nohighlight">\(p\)</span>.</p>
 </div>
-<div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[1.79934087 0.47179152 5.01549939]
-[1.79909592 0.47176716 5.01550546]
-  
-test MSE of OLS:
-1.139431112903922
-  
-test MSE of Ridge
-1.1395235273363669
-</pre></div>
-</div>
-<img alt="_images/week37_176_1.png" src="_images/week37_176_1.png" />
-</div>
-</div>
-<p>In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#more-on-rescaling-data">lecture material</a>.
-see also the weekly slides <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/week36/html/._week36-bs029.html">for week 36</a>.
-It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.</p>
-<p>Before we discuss the code, we repeat some of the basic math from the slides of week 36.</p>
-<p>Let us try to understand what this may imply mathematically when we
-subtract the mean values, also known as <em>zero centering</em> or simply <em>centering</em>. For
-simplicity, we will focus on  ordinary regression, as done in the above example.</p>
-<p>The cost/loss function  for regression is</p>
+<div class="section" id="assumptions-made">
+<h2>Assumptions made<a class="headerlink" href="#assumptions-made" title="Permalink to this headline">¶</a></h2>
+<p>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 <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and  a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)\)</span>
+which describe our data</p>
 <div class="math notranslate nohighlight">
 \[
-C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
+\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
 \]</div>
-<p>Recall also that we use the squared value. This expression can lead to an
-increased penalty for higher differences between predicted and
-output/target values.</p>
-<p>What we have done is to single out the <span class="math notranslate nohighlight">\(\beta_0\)</span> term in the
-definition of the mean squared error (MSE).  The design matrix <span class="math notranslate nohighlight">\(X\)</span>
-does in this case not contain any intercept column.  When we take the
-derivative with respect to <span class="math notranslate nohighlight">\(\beta_0\)</span>, we want the derivative to obey</p>
+<p>We approximate this function with our model from the solution of the linear regression equations, that is our
+function <span class="math notranslate nohighlight">\(f\)</span> is approximated by <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> where we want to minimize <span class="math notranslate nohighlight">\((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\)</span>, our MSE, with</p>
 <div class="math notranslate nohighlight">
 \[
-\frac{\partial C}{\partial \beta_j} = 0,
+\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
 \]</div>
-<p>for all <span class="math notranslate nohighlight">\(j\)</span>. For <span class="math notranslate nohighlight">\(\beta_0\)</span> we have</p>
-<div class="math notranslate nohighlight">
-\[
-\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
-\]</div>
-<p>Multiplying away the constant <span class="math notranslate nohighlight">\(2/n\)</span>, we obtain</p>
-<div class="math notranslate nohighlight">
-\[
-\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
-\]</div>
-<p>Let us specialize first to the case where we have only two parameters <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>.
-Our result for <span class="math notranslate nohighlight">\(\beta_0\)</span> simplifies then to</p>
-<div class="math notranslate nohighlight">
-\[
-n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
-\]</div>
-<p>We obtain then</p>
-<div class="math notranslate nohighlight">
-\[
-\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}.
-\]</div>
-<p>If we define</p>
-<div class="math notranslate nohighlight">
-\[
-\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1},
-\]</div>
-<p>and the mean value of the outputs as</p>
-<div class="math notranslate nohighlight">
-\[
-\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
-\]</div>
-<p>we have</p>
-<div class="math notranslate nohighlight">
-\[
-\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}.
-\]</div>
-<p>In the general case with more parameters than <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>, we have</p>
-<div class="math notranslate nohighlight">
-\[
-\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j.
-\]</div>
-<p>We can rewrite the latter equation as</p>
-<div class="math notranslate nohighlight">
-\[
-\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j,
-\]</div>
-<p>where we have defined</p>
-<div class="math notranslate nohighlight">
-\[
-\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij},
-\]</div>
-<p>the mean value for all elements of the column vector <span class="math notranslate nohighlight">\(\boldsymbol{x}_j\)</span>.</p>
-<p>Replacing <span class="math notranslate nohighlight">\(y_i\)</span> with <span class="math notranslate nohighlight">\(y_i - y_i - \overline{\boldsymbol{y}}\)</span> and centering also our design matrix results in a cost function (in vector-matrix disguise)</p>
-<div class="math notranslate nohighlight">
-\[
-C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
-\]</div>
-<p>If we minimize with respect to <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> we have then</p>
-<div class="math notranslate nohighlight">
-\[
-\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
-\]</div>
-<p>where <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}}\)</span>
-and <span class="math notranslate nohighlight">\(\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}\)</span>.</p>
-<p>For Ridge regression we need to add <span class="math notranslate nohighlight">\(\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}\)</span> to the cost function and get then</p>
-<div class="math notranslate nohighlight">
-\[
-\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
-\]</div>
-<p>Now we try to implement this.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2018</span><span class="p">)</span>
-<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
-<span class="c1"># we do not include the intercept</span>
-<span class="n">d</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="n">Lambda</span> <span class="o">=</span> <span class="mf">0.01</span>
-
-<span class="c1"># Make data set.</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mf">5.0</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-<span class="c1">#Design matrix X does not include the intercept. </span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">d</span><span class="p">))</span>
-<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>     
-    <span class="n">X</span><span class="p">[:,</span> <span class="n">p</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">**</span> <span class="p">(</span><span class="n">p</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-
-
-<span class="c1">#Split data in train and test</span>
-<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
-
-<span class="c1"># Scale data by subtracting mean value,own implementation</span>
-<span class="c1">#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable</span>
-<span class="n">X_train_mean</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-<span class="c1">#Center by removing mean from each feature</span>
-<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">X_train</span> <span class="o">-</span> <span class="n">X_train_mean</span>
-<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">-</span> <span class="n">X_train_mean</span>
-<span class="c1">#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered, note)</span>
-<span class="n">y_scaler</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span>
-<span class="n">y_train_scaled</span> <span class="o">=</span> <span class="n">y_train</span> <span class="o">-</span> <span class="n">y_scaler</span>
-
-
-<span class="c1">#Calculate beta</span>
-<span class="n">beta_OLS</span> <span class="o">=</span> <span class="n">OLS_fit_beta</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train_scaled</span><span class="p">)</span>
-<span class="n">beta_Ridge</span> <span class="o">=</span> <span class="n">Ridge_fit_beta</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train_scaled</span><span class="p">,</span><span class="n">Lambda</span><span class="p">,</span><span class="n">d</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">beta_OLS</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">beta_Ridge</span><span class="p">)</span>
-<span class="c1"># calculate intercepts and print them</span>
-<span class="n">interceptOLS</span> <span class="o">=</span> <span class="n">y_scaler</span> <span class="o">-</span> <span class="n">X_train_mean</span> <span class="o">@</span> <span class="n">beta_OLS</span>
-<span class="n">interceptRidge</span> <span class="o">=</span> <span class="n">y_scaler</span> <span class="o">-</span> <span class="n">X_train_mean</span> <span class="o">@</span> <span class="n">beta_Ridge</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">interceptOLS</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">interceptRidge</span><span class="p">)</span>
-
-<span class="c1">#predict value with intercept</span>
-<span class="n">ytilde_test_OLS</span> <span class="o">=</span> <span class="n">X_test_scaled</span> <span class="o">@</span> <span class="n">beta_OLS</span><span class="o">+</span><span class="n">y_scaler</span>
-<span class="n">ytilde_test_Ridge</span> <span class="o">=</span> <span class="n">X_test_scaled</span> <span class="o">@</span> <span class="n">beta_Ridge</span><span class="o">+</span><span class="n">y_scaler</span>
-
-
-<span class="c1">#Calculate MSE</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;  &quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;test MSE of OLS:&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ytilde_test_OLS</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;  &quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;test MSE of Ridge&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ytilde_test_Ridge</span><span class="p">))</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta_OLS</span><span class="o">+</span><span class="n">interceptOLS</span><span class="p">,</span><span class="s1">&#39;*&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;OLS_Fit&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta_Ridge</span><span class="o">+</span><span class="n">interceptRidge</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ridge_Fit&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.47179152 5.01549939]
-[0.47176783 5.01542292]
-1.7993408651198877
-1.7995707762668065
-  
-test MSE of OLS:
-1.1394311129039245
-  
-test MSE of Ridge
-1.1395084586525954
-</pre></div>
-</div>
-<img alt="_images/week37_208_1.png" src="_images/week37_208_1.png" />
-</div>
-</div>
-<p>Finally, instead of using our own function we repeat the same example
-using the <strong>standardscaler</strong> functionality of the library
-<strong>Scikit-Learn</strong>.  Here we limit ourselves to Ridge regression only.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
-<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2018</span><span class="p">)</span>
-<span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="n">d</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="n">Lambda</span> <span class="o">=</span> <span class="mf">0.01</span>
-
-<span class="c1"># Make data set.</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mf">5.0</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-<span class="c1"># Design matrix X does not include the intercept. </span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">))</span>
-<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>     
-    <span class="n">X</span><span class="p">[:,</span> <span class="n">p</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">**</span> <span class="p">(</span><span class="n">p</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-
-<span class="c1">#Split data in train and test</span>
-<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
-<span class="c1"># Scale data by subtracting mean value of the input using scikit-learn</span>
-<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">(</span><span class="n">with_std</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
-<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
-<span class="n">X_train_mean</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
-<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
-<span class="c1"># We scale also the output, here by our own code</span>
-<span class="n">y_scaler</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span>
-<span class="n">y_train_scaled</span> <span class="o">=</span> <span class="n">y_train</span> <span class="o">-</span> <span class="n">y_scaler</span>
-<span class="n">y_test_scaled</span> <span class="o">=</span> <span class="n">y_test</span><span class="o">-</span> <span class="n">y_scaler</span>
-
-<span class="c1">#Calculate beta</span>
-<span class="n">OLS</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
-<span class="n">betaOLS</span><span class="o">=</span><span class="n">OLS</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span><span class="n">y_train_scaled</span><span class="p">)</span>
-<span class="n">ypredictOLS</span> <span class="o">=</span> <span class="n">OLS</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
-<span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">Lambda</span><span class="p">)</span>
-<span class="n">RegRidge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span><span class="n">y_train_scaled</span><span class="p">)</span>
-<span class="n">ypredictRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
-<span class="n">betaOLS</span> <span class="o">=</span> <span class="n">OLS</span><span class="o">.</span><span class="n">coef_</span>
-<span class="n">betaRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">coef_</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">betaOLS</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">betaRidge</span><span class="p">)</span>
-<span class="n">interceptOLS</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span> <span class="o">-</span> <span class="n">X_train_mean</span> <span class="o">@</span> <span class="n">betaOLS</span>
-<span class="n">interceptRidge</span> <span class="o">=</span> <span class="n">y_scaler</span> <span class="o">-</span> <span class="n">X_train_mean</span> <span class="o">@</span> <span class="n">betaRidge</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">interceptOLS</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">interceptRidge</span><span class="p">)</span>
-<span class="c1">#predict value </span>
-<span class="n">ytilde_test_Ridge</span> <span class="o">=</span> <span class="n">X_test_scaled</span> <span class="o">@</span> <span class="n">betaRidge</span><span class="o">+</span><span class="n">y_scaler</span>
-<span class="n">ytilde_test_OLS</span> <span class="o">=</span> <span class="n">X_test_scaled</span> <span class="o">@</span> <span class="n">betaOLS</span><span class="o">+</span><span class="n">y_scaler</span>
-
-<span class="c1">#Calculate MSE</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;  &quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;test MSE of OLS&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ytilde_test_OLS</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;  &quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;test MSE of Ridge&quot;</span><span class="p">)</span>
-<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ytilde_test_Ridge</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">X</span> <span class="o">@</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">coef_</span> <span class="o">+</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">intercept_</span> <span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Ridge_Fit&quot;</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
-<span class="ne">NameError</span><span class="g g-Whitespace">                                 </span>Traceback (most recent call last)
-<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">11</span><span class="p">],</span> <span class="n">line</span> <span class="mi">34</span>
-<span class="g g-Whitespace">     </span><span class="mi">32</span> <span class="n">ypredictOLS</span> <span class="o">=</span> <span class="n">OLS</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
-<span class="g g-Whitespace">     </span><span class="mi">33</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">Lambda</span><span class="p">)</span>
-<span class="ne">---&gt; </span><span class="mi">34</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span><span class="n">y_train_scaled</span><span class="p">)</span>
-<span class="g g-Whitespace">     </span><span class="mi">35</span> <span class="n">ypredictRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
-<span class="g g-Whitespace">     </span><span class="mi">36</span> <span class="n">betaOLS</span> <span class="o">=</span> <span class="n">OLS</span><span class="o">.</span><span class="n">coef_</span>
-
-<span class="ne">NameError</span>: name &#39;RegRidge&#39; is not defined
-</pre></div>
-</div>
 </div>
+<div class="section" id="expectation-value-and-variance">
+<h2>Expectation value and variance<a class="headerlink" href="#expectation-value-and-variance" title="Permalink to this headline">¶</a></h2>
+<p>We can calculate the expectation value of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> for a given element <span class="math notranslate nohighlight">\(i\)</span></p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align*} 
+\mathbb{E}(y_i) &amp; =
+\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
+\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, 
+\end{align*}
+\]</div>
+<p>while
+its variance is</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align*} \mbox{Var}(y_i) &amp; = \mathbb{E} \{ [y_i
+- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
+[\mathbb{E}(y_i)]^2  \\  &amp; = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
+\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &amp;
+= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
+\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
+\ast} \, \beta)^2 \\  &amp; = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
+\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
+\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 
+\\ &amp; = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
+\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.  
+\end{align*}
+\end{split}\]</div>
+<p>Hence, <span class="math notranslate nohighlight">\(y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\)</span>, that is <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> follows a normal distribution with
+mean value <span class="math notranslate nohighlight">\(\boldsymbol{X}\boldsymbol{\beta}\)</span> and variance <span class="math notranslate nohighlight">\(\sigma^2\)</span> (not be confused with the singular values of the SVD).</p>
 </div>
+<div class="section" id="expectation-value-and-variance-for-boldsymbol-beta">
+<h2>Expectation value and variance for <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span><a class="headerlink" href="#expectation-value-and-variance-for-boldsymbol-beta" title="Permalink to this headline">¶</a></h2>
+<p>With the OLS expressions for the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> we can evaluate the expectation value</p>
+<div class="math notranslate nohighlight">
+\[
+\mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
+\]</div>
+<p>This means that the estimator of the regression parameters is unbiased.</p>
+<p>We can also calculate the variance</p>
+<p>The variance of the optimal value <span class="math notranslate nohighlight">\(\boldsymbol{\hat{\beta}}\)</span> is</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{eqnarray*}
+\mbox{Var}(\boldsymbol{\hat{\beta}}) &amp; = &amp; \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
+\\
+&amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} - \boldsymbol{\beta}]^{T} \}
+\\
+% &amp; = &amp; \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+% &amp; = &amp; \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{y} \, \mathbf{y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1}  \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{y} \, \mathbf{y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+\\
+&amp; = &amp; (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \,  \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+% &amp; = &amp; (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \,  \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
+% \\
+% &amp; &amp; + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T  \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
+\\
+&amp; = &amp; \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}  + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
+\end{eqnarray*}
+\end{split}\]</div>
+<p>where we have used  that <span class="math notranslate nohighlight">\(\mathbb{E} (\mathbf{y} \mathbf{y}^{T}) =
+\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} +
+\sigma^2 \, \mathbf{I}_{nn}\)</span>. From <span class="math notranslate nohighlight">\(\mbox{Var}(\boldsymbol{\beta}) = \sigma^2
+\, (\mathbf{X}^{T} \mathbf{X})^{-1}\)</span>, one obtains an estimate of the
+variance of the estimate of the <span class="math notranslate nohighlight">\(j\)</span>-th regression coefficient:
+<span class="math notranslate nohighlight">\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \)</span>. This may be used to
+construct a confidence interval for the estimates.</p>
+<p>In a similar way, we can obtain analytical expressions for say the
+expectation values of the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and their variance
+when we employ Ridge regression, allowing us again to define a confidence interval.</p>
+<p>It is rather straightforward to show that</p>
+<div class="math notranslate nohighlight">
+\[
+\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
+\]</div>
+<p>We see clearly that
+<span class="math notranslate nohighlight">\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\)</span> for any <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span>.</p>
+<p>We can also compute the variance as</p>
+<div class="math notranslate nohighlight">
+\[
+\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
+\]</div>
+<p>and it is easy to see that if the parameter <span class="math notranslate nohighlight">\(\lambda\)</span> goes to infinity then the variance of Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
+<p>With this, we can compute the difference</p>
+<div class="math notranslate nohighlight">
+\[
+\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}]-\mbox{Var}(\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
+\]</div>
+<p>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 <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span> be larger than the variance of <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.</p>
+<p>For more discussions of Ridge regression and calculation of averages, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.</p>
 </div>
 </div>
 
diff --git a/doc/LectureNotes/_build/jupyter_execute/week37.ipynb b/doc/LectureNotes/_build/jupyter_execute/week37.ipynb
index 02de1a2cf..e421a77b9 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week37.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/week37.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "672fea2a",
+   "id": "7c0ddab1",
    "metadata": {
     "editable": true
    },
@@ -14,13 +14,13 @@
   },
   {
    "cell_type": "markdown",
-   "id": "479a6913",
+   "id": "6c033bf4",
    "metadata": {
     "editable": true
    },
    "source": [
     "# Week 37: Statistical interpretations and Resampling Methods\n",
-    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University\n",
+    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway\n",
     "\n",
     "Date: **September 9, 2024**\n",
     "\n",
@@ -29,419 +29,69 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10b8f456",
+   "id": "e4cd5fa1",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Plans for week 37\n",
+    "## Plans for week 37, lecture Monday\n",
     "\n",
     "**Material for the lecture on Monday September 9.**\n",
     "\n",
     "<!-- * [Video of Lecture](https://youtu.be/YOBBr_toYxc) -->\n",
     "<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf) -->\n",
-    "  * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff\n",
+    "  * Statistical interpretation of Ridge and Lasso regression, see also slides from last week\n",
     "\n",
-    "  * Statistical interpretation of Ridge and Lasso regression\n",
+    "  * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.\n",
     "\n",
     "  * Readings and Videos:\n",
     "\n",
-    "    * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). \n",
+    "    * Raschka et al, pages 175-192\n",
+    "\n",
+    "    * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See <https://link.springer.com/book/10.1007/978-0-387-84858-7>.\n",
     "\n",
     "    * [Video on cross validation](https://www.youtube.com/watch?v=fSytzGwwBVw)\n",
     "\n",
     "    * [Video on Bootstrapping](https://www.youtube.com/watch?v=Xz0x-8-cgaQ)\n",
     "\n",
-    "    * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)\n",
+    "    * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bf80a276",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Plans for week 37, lab sessions\n",
     "\n",
-    "**Material for the active learning sessions on Tuesday and Wednesday.**\n",
+    "**Material for the lab  sessions on Tuesday and Wednesday.**\n",
     "\n",
     "  * Calculations of expectation values\n",
     "\n",
+    "  * Discussion of resampling techniques\n",
+    "\n",
     "  * Exercise set for week 37\n",
     "\n",
     "  * Work on project 1\n",
     "\n",
-    "  * See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.\n",
-    "\n",
     "  * For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f7e43b1b",
+   "id": "0e1f0a90",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Material from last week and relevant for the weekly exercises"
+    "## Material for lecture Monday September 9"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "af6a447a",
-   "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": "b6b7ea38",
-   "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": "e53d8d41",
-   "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": "401644e7",
-   "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": "0603387c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "eb2b6edf",
-   "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": "f1a08194",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ad40fa0b",
-   "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": "d3a4907d",
-   "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": "d7908c9c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "while\n",
-    "its variance is"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "2528f74e",
-   "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": "525f3a59",
-   "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": "ead1d419",
-   "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": "e471f7ab",
-   "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": "7886e97f",
-   "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": "49c32ca5",
-   "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": "9bd27980",
-   "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": "184f9c3b",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d0f46be1",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "We see clearly that \n",
-    "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n",
-    "\n",
-    "We can also compute the variance as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "03e3ac5f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "13ec2524",
-   "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": "58c3f236",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "bfcfc2a0",
-   "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. \n",
-    "\n",
-    "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f3a9f084",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Material for lecture Thursday September 14"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a4338936",
+   "id": "1b290d64",
    "metadata": {
     "editable": true
    },
@@ -464,7 +114,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7566301",
+   "id": "ceed3a82",
    "metadata": {
     "editable": true
    },
@@ -476,7 +126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8af7fae",
+   "id": "1d7e8f98",
    "metadata": {
     "editable": true
    },
@@ -489,7 +139,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "959e33e8",
+   "id": "58722d7d",
    "metadata": {
     "editable": true
    },
@@ -501,7 +151,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c847b78b",
+   "id": "45120edc",
    "metadata": {
     "editable": true
    },
@@ -513,7 +163,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "75c188ba",
+   "id": "4f3b680a",
    "metadata": {
     "editable": true
    },
@@ -525,7 +175,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60b89ba3",
+   "id": "08489797",
    "metadata": {
     "editable": true
    },
@@ -536,7 +186,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fd5d5a54",
+   "id": "a1c81f06",
    "metadata": {
     "editable": true
    },
@@ -548,7 +198,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "56f25582",
+   "id": "4e092776",
    "metadata": {
     "editable": true
    },
@@ -559,7 +209,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9974b303",
+   "id": "79e5358a",
    "metadata": {
     "editable": true
    },
@@ -571,7 +221,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4112e053",
+   "id": "e5a4f865",
    "metadata": {
     "editable": true
    },
@@ -581,7 +231,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7be4a14c",
+   "id": "10b81ba7",
    "metadata": {
     "editable": true
    },
@@ -612,7 +262,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35078f48",
+   "id": "3349cb27",
    "metadata": {
     "editable": true
    },
@@ -624,7 +274,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb25ead1",
+   "id": "0fd1502e",
    "metadata": {
     "editable": true
    },
@@ -636,7 +286,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5cc80036",
+   "id": "43cba0e8",
    "metadata": {
     "editable": true
    },
@@ -646,7 +296,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ebc6099",
+   "id": "d6f4e7dc",
    "metadata": {
     "editable": true
    },
@@ -658,7 +308,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3160dff4",
+   "id": "50c2d673",
    "metadata": {
     "editable": true
    },
@@ -668,7 +318,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8a3aaa69",
+   "id": "30315a10",
    "metadata": {
     "editable": true
    },
@@ -680,7 +330,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7af6708c",
+   "id": "fd8cdf51",
    "metadata": {
     "editable": true
    },
@@ -690,7 +340,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c82e095e",
+   "id": "7323296b",
    "metadata": {
     "editable": true
    },
@@ -702,7 +352,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9431eca4",
+   "id": "e21334ab",
    "metadata": {
     "editable": true
    },
@@ -712,7 +362,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1db6c222",
+   "id": "82974124",
    "metadata": {
     "editable": true
    },
@@ -732,7 +382,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "98c457ce",
+   "id": "e053e43c",
    "metadata": {
     "editable": true
    },
@@ -744,7 +394,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6a3893f",
+   "id": "94098e13",
    "metadata": {
     "editable": true
    },
@@ -754,7 +404,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "17f0e52e",
+   "id": "27625b1f",
    "metadata": {
     "editable": true
    },
@@ -766,7 +416,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c2bfaea6",
+   "id": "8f3cc8b4",
    "metadata": {
     "editable": true
    },
@@ -778,7 +428,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "377eefdf",
+   "id": "d7d5464d",
    "metadata": {
     "editable": true
    },
@@ -790,7 +440,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6da6253e",
+   "id": "c292afee",
    "metadata": {
     "editable": true
    },
@@ -802,7 +452,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e5906b3e",
+   "id": "cdbbd444",
    "metadata": {
     "editable": true
    },
@@ -814,7 +464,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "792d292c",
+   "id": "100a8014",
    "metadata": {
     "editable": true
    },
@@ -826,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3bc0674",
+   "id": "d2e40c4d",
    "metadata": {
     "editable": true
    },
@@ -838,7 +488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0faadf3f",
+   "id": "1e48456e",
    "metadata": {
     "editable": true
    },
@@ -850,7 +500,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a2d478a7",
+   "id": "38ced5b6",
    "metadata": {
     "editable": true
    },
@@ -860,7 +510,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d51ac6ce",
+   "id": "1163c07b",
    "metadata": {
     "editable": true
    },
@@ -872,7 +522,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "91d489d9",
+   "id": "becc087f",
    "metadata": {
     "editable": true
    },
@@ -882,7 +532,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ebe26b44",
+   "id": "5ca2b09d",
    "metadata": {
     "editable": true
    },
@@ -901,7 +551,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a67660e2",
+   "id": "6572c77d",
    "metadata": {
     "editable": true
    },
@@ -922,7 +572,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40b7eac0",
+   "id": "2a3f65e5",
    "metadata": {
     "editable": true
    },
@@ -934,7 +584,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16a3b6ea",
+   "id": "2d264f77",
    "metadata": {
     "editable": true
    },
@@ -945,7 +595,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f41ccc6d",
+   "id": "06b8a13c",
    "metadata": {
     "editable": true
    },
@@ -958,7 +608,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ae81305b",
+   "id": "6d958ed3",
    "metadata": {
     "editable": true
    },
@@ -970,7 +620,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4829b81",
+   "id": "6330f3f2",
    "metadata": {
     "editable": true
    },
@@ -980,7 +630,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "812273af",
+   "id": "ed6a9f67",
    "metadata": {
     "editable": true
    },
@@ -992,7 +642,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31ecba0e",
+   "id": "de42ab8a",
    "metadata": {
     "editable": true
    },
@@ -1002,7 +652,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca6682ec",
+   "id": "4f36b126",
    "metadata": {
     "editable": true
    },
@@ -1014,7 +664,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "662eaffb",
+   "id": "0a7eef00",
    "metadata": {
     "editable": true
    },
@@ -1024,7 +674,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "08d5551d",
+   "id": "29940f94",
    "metadata": {
     "editable": true
    },
@@ -1038,7 +688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03f3de9c",
+   "id": "f2204679",
    "metadata": {
     "editable": true
    },
@@ -1050,7 +700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c57ace49",
+   "id": "d3f4597f",
    "metadata": {
     "editable": true
    },
@@ -1060,7 +710,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb014b2b",
+   "id": "c9e9ca01",
    "metadata": {
     "editable": true
    },
@@ -1072,7 +722,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1054d47e",
+   "id": "1c3062b6",
    "metadata": {
     "editable": true
    },
@@ -1082,7 +732,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1485d44c",
+   "id": "b455f707",
    "metadata": {
     "editable": true
    },
@@ -1094,7 +744,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f457f909",
+   "id": "5ac8fdd3",
    "metadata": {
     "editable": true
    },
@@ -1104,7 +754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59447b2a",
+   "id": "827927db",
    "metadata": {
     "editable": true
    },
@@ -1116,17 +766,17 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fe26100",
+   "id": "0c740a13",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!"
+    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta})$!"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d73c2d9c",
+   "id": "a855fc3c",
    "metadata": {
     "editable": true
    },
@@ -1142,7 +792,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8cdf2c97",
+   "id": "9dda8b9e",
    "metadata": {
     "editable": true
    },
@@ -1154,7 +804,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7272ef9f",
+   "id": "d160205c",
    "metadata": {
     "editable": true
    },
@@ -1164,7 +814,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0dc5b2b7",
+   "id": "1b7501a6",
    "metadata": {
     "editable": true
    },
@@ -1176,7 +826,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6c8c684",
+   "id": "d75a8c65",
    "metadata": {
     "editable": true
    },
@@ -1189,7 +839,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f681ed50",
+   "id": "c9f718df",
    "metadata": {
     "editable": true
    },
@@ -1201,7 +851,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c3bc898",
+   "id": "b07b6425",
    "metadata": {
     "editable": true
    },
@@ -1211,7 +861,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3cea046",
+   "id": "215a8730",
    "metadata": {
     "editable": true
    },
@@ -1223,7 +873,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68276410",
+   "id": "7013d0f0",
    "metadata": {
     "editable": true
    },
@@ -1233,7 +883,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dfd55611",
+   "id": "d56b4ce0",
    "metadata": {
     "editable": true
    },
@@ -1245,7 +895,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "83656dd8",
+   "id": "f9c6b14d",
    "metadata": {
     "editable": true
    },
@@ -1257,7 +907,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b62e29fd",
+   "id": "0bde16cb",
    "metadata": {
     "editable": true
    },
@@ -1267,7 +917,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1de8855c",
+   "id": "0a2869fc",
    "metadata": {
     "editable": true
    },
@@ -1279,7 +929,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "212152f1",
+   "id": "75b83b9b",
    "metadata": {
     "editable": true
    },
@@ -1291,7 +941,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afb042c4",
+   "id": "1e76d916",
    "metadata": {
     "editable": true
    },
@@ -1303,7 +953,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab1fd3c5",
+   "id": "09dfe26d",
    "metadata": {
     "editable": true
    },
@@ -1313,7 +963,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "43979108",
+   "id": "4037962f",
    "metadata": {
     "editable": true
    },
@@ -1325,7 +975,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "96622b66",
+   "id": "39a85ba0",
    "metadata": {
     "editable": true
    },
@@ -1335,7 +985,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e17a5b6d",
+   "id": "b263a1fe",
    "metadata": {
     "editable": true
    },
@@ -1354,7 +1004,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24e35401",
+   "id": "41730165",
    "metadata": {
     "editable": true
    },
@@ -1382,7 +1032,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a37f1d06",
+   "id": "4076cee3",
    "metadata": {
     "editable": true
    },
@@ -1408,7 +1058,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58f54a23",
+   "id": "11139cb3",
    "metadata": {
     "editable": true
    },
@@ -1425,7 +1075,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ed7c809",
+   "id": "b2bead9b",
    "metadata": {
     "editable": true
    },
@@ -1445,7 +1095,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bed9bb2f",
+   "id": "8df8364e",
    "metadata": {
     "editable": true
    },
@@ -1474,7 +1124,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2cafa0a3",
+   "id": "6b5273f1",
    "metadata": {
     "editable": true
    },
@@ -1499,7 +1149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fefbb980",
+   "id": "a260b371",
    "metadata": {
     "editable": true
    },
@@ -1519,7 +1169,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4560153d",
+   "id": "2ff1e63c",
    "metadata": {
     "editable": true
    },
@@ -1531,7 +1181,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ffcbdd5a",
+   "id": "ab4b2e8c",
    "metadata": {
     "editable": true
    },
@@ -1541,7 +1191,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d055ea55",
+   "id": "85b6fb62",
    "metadata": {
     "editable": true
    },
@@ -1556,7 +1206,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "004f0161",
+   "id": "42449344",
    "metadata": {
     "editable": true
    },
@@ -1569,7 +1219,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "039f4ad5",
+   "id": "35e4f511",
    "metadata": {
     "editable": true
    },
@@ -1582,7 +1232,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "413459f1",
+   "id": "c787234f",
    "metadata": {
     "editable": true
    },
@@ -1594,7 +1244,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bef022c8",
+   "id": "183711dc",
    "metadata": {
     "editable": true
    },
@@ -1607,7 +1257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3d53693d",
+   "id": "2351078d",
    "metadata": {
     "editable": true
    },
@@ -1618,7 +1268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e682c32a",
+   "id": "408ba3b5",
    "metadata": {
     "editable": true
    },
@@ -1632,7 +1282,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f44053ea",
+   "id": "3ea79849",
    "metadata": {
     "editable": true
    },
@@ -1642,7 +1292,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f10de930",
+   "id": "69439aad",
    "metadata": {
     "editable": true
    },
@@ -1656,7 +1306,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f91402a",
+   "id": "e6c29fd8",
    "metadata": {
     "editable": true
    },
@@ -1669,7 +1319,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d7a8e36",
+   "id": "4bb52549",
    "metadata": {
     "editable": true
    },
@@ -1682,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4c756ca2",
+   "id": "3430d548",
    "metadata": {
     "editable": true
    },
@@ -1692,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "908f782e",
+   "id": "9ba636b3",
    "metadata": {
     "editable": true
    },
@@ -1705,7 +1355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f12152da",
+   "id": "cb19e98f",
    "metadata": {
     "editable": true
    },
@@ -1715,7 +1365,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "85dd290d",
+   "id": "3cc840d7",
    "metadata": {
     "editable": true
    },
@@ -1728,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "57631d10",
+   "id": "a5062213",
    "metadata": {
     "editable": true
    },
@@ -1740,7 +1390,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d4d5ccb",
+   "id": "8d555244",
    "metadata": {
     "editable": true
    },
@@ -1759,7 +1409,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a29c8551",
+   "id": "c4356604",
    "metadata": {
     "editable": true
    },
@@ -1772,7 +1422,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6509ba36",
+   "id": "09dc08a1",
    "metadata": {
     "editable": true
    },
@@ -1784,7 +1434,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "733307b1",
+   "id": "dab5056a",
    "metadata": {
     "editable": true
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@@ -1797,7 +1447,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a69b7331",
+   "id": "8a1f8583",
    "metadata": {
     "editable": true
    },
@@ -1817,7 +1467,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ced659da",
+   "id": "1be08ebb",
    "metadata": {
     "editable": true
    },
@@ -1831,16 +1481,16 @@
     "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we found \n",
     "$\\mathbb{E}(\\boldsymbol{\\beta}) = \\boldsymbol{\\beta}$, which means that the estimator of the regression parameters is unbiased.\n",
     "\n",
-    "We found also that the variance of the estimate of the $j$-th regression coefficient is\n",
+    "In the exercises this week we show that the variance of the estimate of the $j$-th regression coefficient is\n",
     "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $.\n",
     "\n",
-    "This quantity will be used to\n",
+    "This quantity can be used to\n",
     "construct a confidence interval for the estimates."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2c0d0a71",
+   "id": "2a368353",
    "metadata": {
     "editable": true
    },
@@ -1855,7 +1505,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d3a4d693",
+   "id": "6fee4a1a",
    "metadata": {
     "editable": true
    },
@@ -1867,7 +1517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ed3d852",
+   "id": "1a97c9ef",
    "metadata": {
     "editable": true
    },
@@ -1887,7 +1537,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e826e70b",
+   "id": "0a70afb2",
    "metadata": {
     "editable": true
    },
@@ -1907,7 +1557,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10b5d4d5",
+   "id": "dc32eaea",
    "metadata": {
     "editable": true
    },
@@ -1931,7 +1581,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1645ce3c",
+   "id": "016d0ab3",
    "metadata": {
     "editable": true
    },
@@ -1952,7 +1602,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4986483a",
+   "id": "3f371f0d",
    "metadata": {
     "editable": true
    },
@@ -1982,7 +1632,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fe853f3b",
+   "id": "abc56fe7",
    "metadata": {
     "editable": true
    },
@@ -2006,7 +1656,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "81ec18f8",
+   "id": "a3d2885b",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2018,7 +1668,7 @@
      "text": [
       "Bootstrap Statistics :\n",
       "original           bias      std. error\n",
-      " 99.9841  15.0402        99.9828          0.1512\n"
+      " 99.7503  14.9999        99.7496        0.151068\n"
      ]
     }
    ],
@@ -2055,7 +1705,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0b5b588",
+   "id": "5db67125",
    "metadata": {
     "editable": true
    },
@@ -2065,7 +1715,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "262a95e8",
+   "id": "6f1b5417",
    "metadata": {
     "editable": true
    },
@@ -2076,7 +1726,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "d235b516",
+   "id": "9d98136a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2084,14 +1734,14 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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",
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",
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_144_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_121_0.png"
       }
      },
      "output_type": "display_data"
@@ -2111,7 +1761,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1620bb7",
+   "id": "c87eac4e",
    "metadata": {
     "editable": true
    },
@@ -2129,7 +1779,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fd3bd4b5",
+   "id": "2f0f4313",
    "metadata": {
     "editable": true
    },
@@ -2141,7 +1791,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a129d86",
+   "id": "7895ee8b",
    "metadata": {
     "editable": true
    },
@@ -2158,7 +1808,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff57b405",
+   "id": "be454b28",
    "metadata": {
     "editable": true
    },
@@ -2170,7 +1820,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5a13919e",
+   "id": "7475f8f9",
    "metadata": {
     "editable": true
    },
@@ -2180,7 +1830,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "45d4a7b6",
+   "id": "fd472029",
    "metadata": {
     "editable": true
    },
@@ -2192,7 +1842,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c837442b",
+   "id": "b43c8f67",
    "metadata": {
     "editable": true
    },
@@ -2209,7 +1859,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b3c7b8e",
+   "id": "c47cfb6d",
    "metadata": {
     "editable": true
    },
@@ -2221,7 +1871,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e6f977d",
+   "id": "593188cd",
    "metadata": {
     "editable": true
    },
@@ -2231,7 +1881,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "adeab0fd",
+   "id": "76de4d81",
    "metadata": {
     "editable": true
    },
@@ -2243,7 +1893,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63b18db5",
+   "id": "afc38c97",
    "metadata": {
     "editable": true
    },
@@ -2253,7 +1903,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c7eb00d",
+   "id": "b0f2971c",
    "metadata": {
     "editable": true
    },
@@ -2265,7 +1915,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3f4530b",
+   "id": "5a65e495",
    "metadata": {
     "editable": true
    },
@@ -2275,7 +1925,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "90792236",
+   "id": "31419298",
    "metadata": {
     "editable": true
    },
@@ -2291,7 +1941,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d73c7cd4",
+   "id": "217ddd53",
    "metadata": {
     "editable": true
    },
@@ -2302,7 +1952,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "58124207",
+   "id": "ece05002",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2327,7 +1977,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_160_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_137_1.png"
       }
      },
      "output_type": "display_data"
@@ -2392,7 +2042,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b4e7bf5",
+   "id": "de6dc4e2",
    "metadata": {
     "editable": true
    },
@@ -2403,7 +2053,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "47b138a9",
+   "id": "da218205",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2442,13 +2092,7 @@
       "Error: 0.05227921801205686\n",
       "Bias^2: 0.0481872773043029\n",
       "Var: 0.004091940707753939\n",
-      "0.05227921801205686 >= 0.0481872773043029 + 0.004091940707753939 = 0.052279218012056844\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "0.05227921801205686 >= 0.0481872773043029 + 0.004091940707753939 = 0.052279218012056844\n",
       "Polynomial degree: 6\n",
       "Error: 0.037813671417389005\n",
       "Bias^2: 0.033657685071527665\n",
@@ -2468,18 +2112,18 @@
       "Error: 0.02660572763718093\n",
       "Bias^2: 0.010018312644137363\n",
       "Var: 0.016587414993043573\n",
-      "0.02660572763718093 >= 0.010018312644137363 + 0.016587414993043573 = 0.026605727637180936\n",
-      "Polynomial degree: 10\n",
-      "Error: 0.021592704588025025\n",
-      "Bias^2: 0.010516485576645508\n",
-      "Var: 0.011076219011379514\n",
-      "0.021592704588025025 >= 0.010516485576645508 + 0.011076219011379514 = 0.021592704588025022\n"
+      "0.02660572763718093 >= 0.010018312644137363 + 0.016587414993043573 = 0.026605727637180936\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Polynomial degree: 10\n",
+      "Error: 0.021592704588025025\n",
+      "Bias^2: 0.010516485576645508\n",
+      "Var: 0.011076219011379514\n",
+      "0.021592704588025025 >= 0.010516485576645508 + 0.011076219011379514 = 0.021592704588025022\n",
       "Polynomial degree: 11\n",
       "Error: 0.07160048164233104\n",
       "Bias^2: 0.014436800088904942\n",
@@ -2489,13 +2133,7 @@
       "Error: 0.11547777218872497\n",
       "Bias^2: 0.01628578269596628\n",
       "Var: 0.09919198949275869\n",
-      "0.11547777218872497 >= 0.01628578269596628 + 0.09919198949275869 = 0.11547777218872497\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "0.11547777218872497 >= 0.01628578269596628 + 0.09919198949275869 = 0.11547777218872497\n",
       "Polynomial degree: 13\n",
       "Error: 0.22842468702219465\n",
       "Bias^2: 0.01975416527185249\n",
@@ -2512,7 +2150,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_162_4.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_139_2.png"
       }
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      "output_type": "display_data"
@@ -2569,7 +2207,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37b66761",
+   "id": "5ea5df06",
    "metadata": {
     "editable": true
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@@ -2607,7 +2245,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b5c601e",
+   "id": "40f41d74",
    "metadata": {
     "editable": true
    },
@@ -2618,7 +2256,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "9bc32416",
+   "id": "03bb50ae",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2660,7 +2298,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_165_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_142_1.png"
       }
      },
      "output_type": "display_data"
@@ -2742,7 +2380,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eb934dbe",
+   "id": "cd11e9fd",
    "metadata": {
     "editable": true
    },
@@ -2767,7 +2405,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c65f802b",
+   "id": "6cf68ff2",
    "metadata": {
     "editable": true
    },
@@ -2795,7 +2433,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9accabd4",
+   "id": "c445bfb2",
    "metadata": {
     "editable": true
    },
@@ -2808,7 +2446,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "d5ed3f5c",
+   "id": "8e26035a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2823,7 +2461,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_169_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_146_0.png"
       }
      },
      "output_type": "display_data"
@@ -2923,7 +2561,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db1976f4",
+   "id": "86a6b012",
    "metadata": {
     "editable": true
    },
@@ -2934,7 +2572,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "3c35540c",
+   "id": "4a0e1afd",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2973,16 +2611,16 @@
       "Mean squared error on test data: 1.53738247\n",
       "Degree of polynomial:   8\n",
       "Mean squared error on training data: 0.04926746\n",
-      "Mean squared error on test data: 0.14629156\n",
-      "Degree of polynomial:   9\n",
-      "Mean squared error on training data: 0.02546675\n",
-      "Mean squared error on test data: 0.11202337\n"
+      "Mean squared error on test data: 0.14629156\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Degree of polynomial:   9\n",
+      "Mean squared error on training data: 0.02546675\n",
+      "Mean squared error on test data: 0.11202337\n",
       "Degree of polynomial:  10\n",
       "Mean squared error on training data: 0.02424794\n",
       "Mean squared error on test data: 0.22467274\n",
@@ -2991,25 +2629,31 @@
       "Mean squared error on test data: 1.07641937\n",
       "Degree of polynomial:  12\n",
       "Mean squared error on training data: 0.00805074\n",
-      "Mean squared error on test data: 0.04295757\n",
-      "Degree of polynomial:  13\n",
-      "Mean squared error on training data: 0.00781918\n",
-      "Mean squared error on test data: 0.56965674\n",
-      "Degree of polynomial:  14\n",
-      "Mean squared error on training data: 0.00465099\n",
-      "Mean squared error on test data: 0.28443039\n"
+      "Mean squared error on test data: 0.04295757\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Degree of polynomial:  13\n",
+      "Mean squared error on training data: 0.00781918\n",
+      "Mean squared error on test data: 0.56965674\n",
+      "Degree of polynomial:  14\n",
+      "Mean squared error on training data: 0.00465099\n",
+      "Mean squared error on test data: 0.28443039\n",
       "Degree of polynomial:  15\n",
       "Mean squared error on training data: 0.00420072\n",
       "Mean squared error on test data: 568.47051432\n",
       "Degree of polynomial:  16\n",
       "Mean squared error on training data: 0.00325450\n",
-      "Mean squared error on test data: 48.97630233\n",
+      "Mean squared error on test data: 48.97630233\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
       "Degree of polynomial:  17\n",
       "Mean squared error on training data: 0.00242954\n",
       "Mean squared error on test data: 2.52780600\n",
@@ -3018,16 +2662,16 @@
       "Mean squared error on test data: 429.25695398\n",
       "Degree of polynomial:  19\n",
       "Mean squared error on training data: 0.00154853\n",
-      "Mean squared error on test data: 239.97065359\n"
+      "Mean squared error on test data: 239.97065359\n",
+      "Degree of polynomial:  20\n",
+      "Mean squared error on training data: 0.00140846\n",
+      "Mean squared error on test data: 1350.24493666\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Degree of polynomial:  20\n",
-      "Mean squared error on training data: 0.00140846\n",
-      "Mean squared error on test data: 1350.24493666\n",
       "Degree of polynomial:  21\n",
       "Mean squared error on training data: 0.00119688\n",
       "Mean squared error on test data: 1840.50530832\n",
@@ -3036,16 +2680,16 @@
       "Mean squared error on test data: 1184.60929685\n",
       "Degree of polynomial:  23\n",
       "Mean squared error on training data: 0.00089193\n",
-      "Mean squared error on test data: 3892.17483760\n"
+      "Mean squared error on test data: 3892.17483760\n",
+      "Degree of polynomial:  24\n",
+      "Mean squared error on training data: 0.00083355\n",
+      "Mean squared error on test data: 1332.46736215\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Degree of polynomial:  24\n",
-      "Mean squared error on training data: 0.00083355\n",
-      "Mean squared error on test data: 1332.46736215\n",
       "Degree of polynomial:  25\n",
       "Mean squared error on training data: 0.00079904\n",
       "Mean squared error on test data: 7577.76690383\n",
@@ -3054,16 +2698,16 @@
       "Mean squared error on test data: 1079.36895644\n",
       "Degree of polynomial:  27\n",
       "Mean squared error on training data: 0.00068091\n",
-      "Mean squared error on test data: 3207.25343155\n",
-      "Degree of polynomial:  28\n",
-      "Mean squared error on training data: 0.00063362\n",
-      "Mean squared error on test data: 674.79633065\n"
+      "Mean squared error on test data: 3207.25343155\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Degree of polynomial:  28\n",
+      "Mean squared error on training data: 0.00063362\n",
+      "Mean squared error on test data: 674.79633065\n",
       "Degree of polynomial:  29\n",
       "Mean squared error on training data: 0.00063866\n",
       "Mean squared error on test data: 3099.60342978\n"
@@ -3073,9 +2717,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_85796/626635268.py:73: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87621/626635268.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_85796/626635268.py:74: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87621/626635268.py:74: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(testerror), label='Test Error')\n"
      ]
     },
@@ -3088,7 +2732,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_171_8.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_148_9.png"
       }
      },
      "output_type": "display_data"
@@ -3177,7 +2821,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e60fabdf",
+   "id": "bb1e0f15",
    "metadata": {
     "editable": true
    },
@@ -3187,7 +2831,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9b6750d6",
+   "id": "83c87d33",
    "metadata": {
     "editable": true
    },
@@ -3200,7 +2844,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "5e6b91e1",
+   "id": "74eaac59",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3210,7 +2854,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_85796/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87621/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n"
      ]
     },
@@ -3223,7 +2867,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_174_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_151_1.png"
       }
      },
      "output_type": "display_data"
@@ -3301,694 +2945,364 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e83d7b10",
+   "id": "aa703354",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Notes on scaling with examples\n",
-    "\n",
-    "The programs here use both ordinrary least squares (OLS) and Ridge\n",
-    "regression with one value only for the hyperparameter $\\lambda$. The\n",
-    "first example has no scaling and includes the intercept as well and we\n",
-    "are trying to fit a second-order polynomial. The second code takes out\n",
-    "the intercept and subtracts the mean values of each column of the\n",
-    "design matrix and the mean value of the outputs.\n",
-    "\n",
-    "The third and final code uses **Scikit-Learn** as library in order to\n",
-    "calculate the optimal parameters for OLS and Ridge regression. Note\n",
-    "that it is highly recommended to not include the intercept in Ridge\n",
-    "and Lasso regression, in order to avoid penalizing the optimization by\n",
-    "the intercept. The second and third codes do thus not include the\n",
-    "intercept. In the second code we do the scaling ourselves while the\n",
-    "last code uses the standard scaler option included in **Scikit-Learn**, known as centering (where\n",
-    "we subtract the mean values)."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "id": "73807393",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[1.79934087 0.47179152 5.01549939]\n",
-      "[1.79909592 0.47176716 5.01550546]\n",
-      "  \n",
-      "test MSE of OLS:\n",
-      "1.139431112903922\n",
-      "  \n",
-      "test MSE of Ridge\n",
-      "1.1395235273363669\n"
-     ]
-    },
-    {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 640x480 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_176_1.png"
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "from sklearn.linear_model import LinearRegression\n",
-    "from sklearn.preprocessing import PolynomialFeatures\n",
-    "from sklearn.model_selection import train_test_split\n",
-    "from sklearn.preprocessing import StandardScaler\n",
-    "\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",
-    "def OLS_fit_beta(X, y):\n",
-    "    return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
-    "\n",
-    "def Ridge_fit_beta(X, y,L,d):\n",
-    "    I = np.eye(d,d)\n",
-    "    return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y\n",
-    "\n",
-    "# Same random numbers for each test.\n",
-    "np.random.seed(2018)\n",
-    "n = 100\n",
-    "d = 3\n",
-    "# hyperparameter lambda\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set, simple second-order polynomial\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "# The design matrix X includes the intercept and no scaling is made\n",
-    "X = np.zeros((len(x), d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p) \n",
-    "\n",
-    "\n",
-    "#Split data, no scaling is used and we include the intercept\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "\n",
-    "\n",
-    "#Calculate beta, own code\n",
-    "beta_OLS = OLS_fit_beta(X_train, y_train)\n",
-    "beta_Ridge = Ridge_fit_beta(X_train, y_train,Lambda,d)\n",
-    "print(beta_OLS)\n",
-    "print(beta_Ridge)\n",
-    "#predict value\n",
-    "ytilde_test_OLS = X_test @ beta_OLS\n",
-    "ytilde_test_Ridge = X_test @ beta_Ridge\n",
-    "\n",
-    "#Calculate MSE\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS:\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ beta_OLS,'*', label=\"OLS_Fit\")\n",
-    "plt.plot(x, X @ beta_Ridge, label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
+    "## Material for the lab sessions"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b09d20a2",
+   "id": "f8bffca6",
    "metadata": {
     "editable": true
    },
    "source": [
-    "In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the [lecture material](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#more-on-rescaling-data).\n",
-    "see also the weekly slides [for week 36](https://compphysics.github.io/MachineLearning/doc/pub/week36/html/._week36-bs029.html).\n",
-    "It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.\n",
+    "## Linking the regression analysis with a statistical interpretation\n",
     "\n",
-    "Before we discuss the code, we repeat some of the basic math from the slides of week 36.\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",
-    "Let us try to understand what this may imply mathematically when we\n",
-    "subtract the mean values, also known as *zero centering* or simply *centering*. For\n",
-    "simplicity, we will focus on  ordinary regression, as done in the above example.\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",
-    "The cost/loss function  for regression is"
+    "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": "ef1cec97",
+   "id": "109c07d6",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+    "\\begin{align*} \n",
+    "\\mbox{Cov}(\\varepsilon_{i_1},\n",
+    "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n",
+    "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2.  \\end{array} \\right.\n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e43c422b",
+   "id": "1ea48f22",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Recall also that we use the squared value. This expression can lead to an\n",
-    "increased penalty for higher differences between predicted and\n",
-    "output/target values.\n",
+    "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",
-    "What we have done is to single out the $\\beta_0$ term in the\n",
-    "definition of the mean squared error (MSE).  The design matrix $X$\n",
-    "does in this case not contain any intercept column.  When we take the\n",
-    "derivative with respect to $\\beta_0$, we want the derivative to obey"
+    "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": "852577bc",
+   "id": "212f90ec",
+   "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": "a019e7cb",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4edd229a",
+   "id": "2ca46c43",
    "metadata": {
     "editable": true
    },
    "source": [
-    "for all $j$. For $\\beta_0$ we have"
+    "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": "bb93d608",
+   "id": "c61ed9c8",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "5ef162e3",
+   "id": "96d84043",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Multiplying away the constant $2/n$, we obtain"
+    "## Expectation value and variance\n",
+    "\n",
+    "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b7c5467d",
+   "id": "e53b66df",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+    "\\begin{align*} \n",
+    "\\mathbb{E}(y_i) & =\n",
+    "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n",
+    "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "493c75d6",
+   "id": "71d542a3",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
-    "Our result for $\\beta_0$ simplifies then to"
+    "while\n",
+    "its variance is"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "862f2222",
+   "id": "01b18637",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+    "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
+    "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n",
+    "[\\mathbb{E}(y_i)]^2  \\\\  & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n",
+    "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n",
+    "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n",
+    "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n",
+    "\\ast} \\, \\beta)^2 \\\\  & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n",
+    "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n",
+    "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n",
+    "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n",
+    "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2.  \n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e0d0a4e0",
+   "id": "e35bdcc1",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We obtain then"
+    "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": "73d327de",
+   "id": "15b7ea3c",
+   "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": "a4c5f625",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
+    "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "8be27cbd",
+   "id": "450007e4",
    "metadata": {
     "editable": true
    },
    "source": [
-    "If we define"
+    "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": "d49a9bac",
+   "id": "3bcf81c1",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\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": "dd964823",
+   "id": "f69a019e",
    "metadata": {
     "editable": true
    },
    "source": [
-    "and the mean value of the outputs as"
+    "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": "09a87e67",
+   "id": "95b37da0",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9b8797a0",
+   "id": "241c2335",
    "metadata": {
     "editable": true
    },
    "source": [
-    "we have"
+    "We see clearly that \n",
+    "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n",
+    "\n",
+    "We can also compute the variance as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "30aeafdb",
+   "id": "cf794384",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
+    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "dc867cbb",
+   "id": "4cd5260a",
    "metadata": {
     "editable": true
    },
    "source": [
-    "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
+    "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": "f154321e",
+   "id": "7361e846",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
+    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "6015f267",
+   "id": "6a07837b",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We can rewrite the latter equation as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f181e88f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a6e6c0e7",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where we have defined"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "708652ea",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "b360788c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
+    "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. \n",
     "\n",
-    "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "8f473602",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "70f518a7",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "674d02d9",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "6715261c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
-    "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
-    "\n",
-    "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "17600ee3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a8c6f068",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Now we try to implement this."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "id": "1d89f80d",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[0.47179152 5.01549939]\n",
-      "[0.47176783 5.01542292]\n",
-      "1.7993408651198877\n",
-      "1.7995707762668065\n",
-      "  \n",
-      "test MSE of OLS:\n",
-      "1.1394311129039245\n",
-      "  \n",
-      "test MSE of Ridge\n",
-      "1.1395084586525954\n"
-     ]
-    },
-    {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 640x480 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week37_208_1.png"
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "\n",
-    "np.random.seed(2018)\n",
-    "n = 100\n",
-    "# we do not include the intercept\n",
-    "d = 2\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set.\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "#Design matrix X does not include the intercept. \n",
-    "X = np.zeros((len(x), d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p+1)\n",
-    "\n",
-    "\n",
-    "#Split data in train and test\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "\n",
-    "# Scale data by subtracting mean value,own implementation\n",
-    "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
-    "X_train_mean = np.mean(X_train,axis=0)\n",
-    "#Center by removing mean from each feature\n",
-    "X_train_scaled = X_train - X_train_mean\n",
-    "X_test_scaled = X_test - X_train_mean\n",
-    "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered, note)\n",
-    "y_scaler = np.mean(y_train)\n",
-    "y_train_scaled = y_train - y_scaler\n",
-    "\n",
-    "\n",
-    "#Calculate beta\n",
-    "beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)\n",
-    "beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,Lambda,d)\n",
-    "print(beta_OLS)\n",
-    "print(beta_Ridge)\n",
-    "# calculate intercepts and print them\n",
-    "interceptOLS = y_scaler - X_train_mean @ beta_OLS\n",
-    "interceptRidge = y_scaler - X_train_mean @ beta_Ridge\n",
-    "print(interceptOLS)\n",
-    "print(interceptRidge)\n",
-    "\n",
-    "#predict value with intercept\n",
-    "ytilde_test_OLS = X_test_scaled @ beta_OLS+y_scaler\n",
-    "ytilde_test_Ridge = X_test_scaled @ beta_Ridge+y_scaler\n",
-    "\n",
-    "\n",
-    "#Calculate MSE\n",
-    "\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS:\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ beta_OLS+interceptOLS,'*', label=\"OLS_Fit\")\n",
-    "plt.plot(x, X @ beta_Ridge+interceptRidge, label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "522668c3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Finally, instead of using our own function we repeat the same example\n",
-    "using the **standardscaler** functionality of the library\n",
-    "**Scikit-Learn**.  Here we limit ourselves to Ridge regression only."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "id": "9954bcc1",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [
-    {
-     "ename": "NameError",
-     "evalue": "name 'RegRidge' is not defined",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
-      "Cell \u001b[0;32mIn[11], line 34\u001b[0m\n\u001b[1;32m     32\u001b[0m ypredictOLS \u001b[38;5;241m=\u001b[39m OLS\u001b[38;5;241m.\u001b[39mpredict(X_test_scaled)\n\u001b[1;32m     33\u001b[0m linear_model\u001b[38;5;241m.\u001b[39mRidge(Lambda)\n\u001b[0;32m---> 34\u001b[0m \u001b[43mRegRidge\u001b[49m\u001b[38;5;241m.\u001b[39mfit(X_train_scaled,y_train_scaled)\n\u001b[1;32m     35\u001b[0m ypredictRidge \u001b[38;5;241m=\u001b[39m RegRidge\u001b[38;5;241m.\u001b[39mpredict(X_test_scaled)\n\u001b[1;32m     36\u001b[0m betaOLS \u001b[38;5;241m=\u001b[39m OLS\u001b[38;5;241m.\u001b[39mcoef_\n",
-      "\u001b[0;31mNameError\u001b[0m: name 'RegRidge' is not defined"
-     ]
-    }
-   ],
-   "source": [
-    "from sklearn import linear_model\n",
-    "np.random.seed(2018)\n",
-    "n = 10\n",
-    "d = 2\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set.\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "# Design matrix X does not include the intercept. \n",
-    "X = np.zeros((n, d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p+1)\n",
-    "\n",
-    "#Split data in train and test\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "# Scale data by subtracting mean value of the input using scikit-learn\n",
-    "scaler = StandardScaler(with_std=False)\n",
-    "scaler.fit(X_train)\n",
-    "X_train_mean = np.mean(X_train,axis=0)\n",
-    "X_train_scaled = scaler.transform(X_train)\n",
-    "X_test_scaled = scaler.transform(X_test)\n",
-    "# We scale also the output, here by our own code\n",
-    "y_scaler = np.mean(y_train)\n",
-    "y_train_scaled = y_train - y_scaler\n",
-    "y_test_scaled = y_test- y_scaler\n",
-    "\n",
-    "#Calculate beta\n",
-    "OLS = LinearRegression()\n",
-    "betaOLS=OLS.fit(X_train_scaled,y_train_scaled)\n",
-    "ypredictOLS = OLS.predict(X_test_scaled)\n",
-    "linear_model.Ridge(Lambda)\n",
-    "RegRidge.fit(X_train_scaled,y_train_scaled)\n",
-    "ypredictRidge = RegRidge.predict(X_test_scaled)\n",
-    "betaOLS = OLS.coef_\n",
-    "betaRidge = RegRidge.coef_\n",
-    "print(betaOLS)\n",
-    "print(betaRidge)\n",
-    "interceptOLS = np.mean(y_train) - X_train_mean @ betaOLS\n",
-    "interceptRidge = y_scaler - X_train_mean @ betaRidge\n",
-    "print(interceptOLS)\n",
-    "print(interceptRidge)\n",
-    "#predict value \n",
-    "ytilde_test_Ridge = X_test_scaled @ betaRidge+y_scaler\n",
-    "ytilde_test_OLS = X_test_scaled @ betaOLS+y_scaler\n",
-    "\n",
-    "#Calculate MSE\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ RegRidge.coef_ + RegRidge.intercept_ , label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
+    "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
    ]
   }
  ],
diff --git a/doc/LectureNotes/_build/jupyter_execute/week37.py b/doc/LectureNotes/_build/jupyter_execute/week37.py
index 8be12b1d5..b1ad40141 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week37.py
+++ b/doc/LectureNotes/_build/jupyter_execute/week37.py
@@ -6,215 +6,49 @@
 # <!-- dom:TITLE: Week 37: Statistical interpretations and Resampling Methods -->
 
 # # Week 37: Statistical interpretations and Resampling Methods
-# **Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University
+# **Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway
 # 
 # Date: **September 9, 2024**
 # 
 # <!-- todo add link to videos and add link to Van Wieringens notes -->
 
-# ## Plans for week 37
+# ## Plans for week 37, lecture Monday
 # 
 # **Material for the lecture on Monday September 9.**
 # 
 # <!-- * [Video of Lecture](https://youtu.be/YOBBr_toYxc) -->
 # <!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf) -->
-#   * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff
+#   * Statistical interpretation of Ridge and Lasso regression, see also slides from last week
 # 
-#   * Statistical interpretation of Ridge and Lasso regression
+#   * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.
 # 
 #   * Readings and Videos:
 # 
-#     * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). 
+#     * Raschka et al, pages 175-192
+# 
+#     * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See <https://link.springer.com/book/10.1007/978-0-387-84858-7>.
 # 
 #     * [Video on cross validation](https://www.youtube.com/watch?v=fSytzGwwBVw)
 # 
 #     * [Video on Bootstrapping](https://www.youtube.com/watch?v=Xz0x-8-cgaQ)
 # 
 #     * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)
+
+# ## Plans for week 37, lab sessions
 # 
-# **Material for the active learning sessions on Tuesday and Wednesday.**
+# **Material for the lab  sessions on Tuesday and Wednesday.**
 # 
 #   * Calculations of expectation values
 # 
+#   * Discussion of resampling techniques
+# 
 #   * Exercise set for week 37
 # 
 #   * Work on project 1
 # 
-#   * See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.
-# 
 #   * For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.
 
-# ## Material from last week and relevant for the weekly exercises
-
-# ## 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
-
-# $$
-# \mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
-# $$
-
-# We see clearly that 
-# $\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}$ for any $\lambda > 0$.
-# 
-# We can also compute the variance as
-
-# $$
-# \mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
-# $$
-
-# 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
-
-# $$
-# \mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}]-\mbox{Var}(\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
-# $$
-
-# 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. 
-# 
-# For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.
-
-# ## Material for lecture Thursday September 14
+# ## Material for lecture Monday September 9
 
 # ## Deriving OLS from a probability distribution
 # 
@@ -459,7 +293,7 @@
 # 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** $p(\boldsymbol{\beta}$!
+# We have a model for $p(\boldsymbol{D}\vert\boldsymbol{\beta})$ but need one for the **prior** $p(\boldsymbol{\beta})$!
 
 # ## Ridge and Bayes
 # 
@@ -769,10 +603,10 @@
 # With the OLS expressions for the parameters $\boldsymbol{\beta}$ we found 
 # $\mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta}$, which means that the estimator of the regression parameters is unbiased.
 # 
-# We found also that the variance of the estimate of the $j$-th regression coefficient is
+# In the exercises this week we show that the variance of the estimate of the $j$-th regression coefficient is
 # $\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} $.
 # 
-# This quantity will be used to
+# This quantity can be used to
 # construct a confidence interval for the estimates.
 
 # ## Standard Approach based on the Normal Distribution
@@ -1515,336 +1349,172 @@ plt.legend()
 plt.show()
 
 
-# ## Notes on scaling with examples
+# ## Material for the lab sessions
+
+# ## Linking the regression analysis with a statistical interpretation
 # 
-# The programs here use both ordinrary least squares (OLS) and Ridge
-# regression with one value only for the hyperparameter $\lambda$. The
-# first example has no scaling and includes the intercept as well and we
-# are trying to fit a second-order polynomial. The second code takes out
-# the intercept and subtracts the mean values of each column of the
-# design matrix and the mean value of the outputs.
+# 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 third and final code uses **Scikit-Learn** as library in order to
-# calculate the optimal parameters for OLS and Ridge regression. Note
-# that it is highly recommended to not include the intercept in Ridge
-# and Lasso regression, in order to avoid penalizing the optimization by
-# the intercept. The second and third codes do thus not include the
-# intercept. In the second code we do the scaling ourselves while the
-# last code uses the standard scaler option included in **Scikit-Learn**, known as centering (where
-# we subtract the mean values).
-
-# In[9]:
-
-
-import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.linear_model import LinearRegression
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.model_selection import train_test_split
-from sklearn.preprocessing import StandardScaler
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-def OLS_fit_beta(X, y):
-    return np.linalg.pinv(X.T @ X) @ X.T @ y
-
-def Ridge_fit_beta(X, y,L,d):
-    I = np.eye(d,d)
-    return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y
-
-# Same random numbers for each test.
-np.random.seed(2018)
-n = 100
-d = 3
-# hyperparameter lambda
-Lambda = 0.01
-
-# Make data set, simple second-order polynomial
-x = np.linspace(-3, 3, n)
-y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)
-
-# The design matrix X includes the intercept and no scaling is made
-X = np.zeros((len(x), d))
-for p in range(d):     
-    X[:, p] = x ** (p) 
-
-
-#Split data, no scaling is used and we include the intercept
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-
-#Calculate beta, own code
-beta_OLS = OLS_fit_beta(X_train, y_train)
-beta_Ridge = Ridge_fit_beta(X_train, y_train,Lambda,d)
-print(beta_OLS)
-print(beta_Ridge)
-#predict value
-ytilde_test_OLS = X_test @ beta_OLS
-ytilde_test_Ridge = X_test @ beta_Ridge
-
-#Calculate MSE
-print("  ")
-print("test MSE of OLS:")
-print(MSE(y_test,ytilde_test_OLS))
-print("  ")
-print("test MSE of Ridge")
-print(MSE(y_test,ytilde_test_Ridge))
-
-plt.scatter(x,y,label='Data')
-plt.plot(x, X @ beta_OLS,'*', label="OLS_Fit")
-plt.plot(x, X @ beta_Ridge, label="Ridge_Fit")
-plt.grid()
-plt.legend()
-plt.show()
-
-
-# In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the [lecture material](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#more-on-rescaling-data).
-# see also the weekly slides [for week 36](https://compphysics.github.io/MachineLearning/doc/pub/week36/html/._week36-bs029.html).
-# It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.
+# 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.
 # 
-# Before we discuss the code, we repeat some of the basic math from the slides of week 36.
+# 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. 
 # 
-# Let us try to understand what this may imply mathematically when we
-# subtract the mean values, also known as *zero centering* or simply *centering*. For
-# simplicity, we will focus on  ordinary regression, as done in the above example.
+# 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 cost/loss function  for regression is
+# 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
 
 # $$
-# C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
+# \boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
 # $$
 
-# Recall also that we use the squared value. This expression can lead to an
-# increased penalty for higher differences between predicted and
-# output/target values.
+# 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
 # 
-# What we have done is to single out the $\beta_0$ term in the
-# definition of the mean squared error (MSE).  The design matrix $X$
-# does in this case not contain any intercept column.  When we take the
-# derivative with respect to $\beta_0$, we want the derivative to obey
+# We can calculate the expectation value of $\boldsymbol{y}$ for a given element $i$
 
 # $$
-# \frac{\partial C}{\partial \beta_j} = 0,
+# \begin{align*} 
+# \mathbb{E}(y_i) & =
+# \mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
+# \, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, 
+# \end{align*}
 # $$
 
-# for all $j$. For $\beta_0$ we have
+# while
+# its variance is
 
 # $$
-# \frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
+# \begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i
+# - \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
+# [\mathbb{E}(y_i)]^2  \\  & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
+# \beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &
+# = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
+# \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
+# \ast} \, \beta)^2 \\  & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
+# \mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
+# \mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 
+# \\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
+# \mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.  
+# \end{align*}
 # $$
 
-# Multiplying away the constant $2/n$, we obtain
+# 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).
 
-# $$
-# \sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
-# $$
-
-# Let us specialize first to the case where we have only two parameters $\beta_0$ and $\beta_1$.
-# Our result for $\beta_0$ simplifies then to
-
-# $$
-# n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
-# $$
-
-# We obtain then
-
-# $$
-# \beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}.
-# $$
-
-# If we define
-
-# $$
-# \mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1},
-# $$
-
-# and the mean value of the outputs as
-
-# $$
-# \mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
-# $$
-
-# we have
-
-# $$
-# \beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}.
-# $$
-
-# In the general case with more parameters than $\beta_0$ and $\beta_1$, we have
-
-# $$
-# \beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j.
-# $$
-
-# We can rewrite the latter equation as
-
-# $$
-# \beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j,
-# $$
-
-# where we have defined
-
-# $$
-# \mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij},
-# $$
-
-# the mean value for all elements of the column vector $\boldsymbol{x}_j$.
+# ## Expectation value and variance for $\boldsymbol{\beta}$
 # 
-# Replacing $y_i$ with $y_i - y_i - \overline{\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)
+# With the OLS expressions for the optimal parameters $\boldsymbol{\hat{\beta}}$ we can evaluate the expectation value
 
 # $$
-# C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
+# \mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
 # $$
 
-# If we minimize with respect to $\boldsymbol{\beta}$ we have then
-
-# $$
-# \hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
-# $$
-
-# where $\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}}$
-# and $\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}$.
+# This means that the estimator of the regression parameters is unbiased.
 # 
-# For Ridge regression we need to add $\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}$ to the cost function and get then
+# We can also calculate the variance
+# 
+# The variance of the optimal value $\boldsymbol{\hat{\beta}}$ is
 
 # $$
-# \hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
+# \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*}
 # $$
 
-# Now we try to implement this.
+# 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
 
-# In[10]:
+# $$
+# \mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}.
+# $$
 
+# We see clearly that 
+# $\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \hat{\boldsymbol{\beta}}^{\mathrm{OLS}}$ for any $\lambda > 0$.
+# 
+# We can also compute the variance as
 
-np.random.seed(2018)
-n = 100
-# we do not include the intercept
-d = 2
-Lambda = 0.01
+# $$
+# \mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
+# $$
 
-# Make data set.
-x = np.linspace(-3, 3, n)
-y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)
+# 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
 
-#Design matrix X does not include the intercept. 
-X = np.zeros((len(x), d))
-for p in range(d):     
-    X[:, p] = x ** (p+1)
-
-
-#Split data in train and test
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# Scale data by subtracting mean value,own implementation
-#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
-X_train_mean = np.mean(X_train,axis=0)
-#Center by removing mean from each feature
-X_train_scaled = X_train - X_train_mean
-X_test_scaled = X_test - X_train_mean
-#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered, note)
-y_scaler = np.mean(y_train)
-y_train_scaled = y_train - y_scaler
-
-
-#Calculate beta
-beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)
-beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,Lambda,d)
-print(beta_OLS)
-print(beta_Ridge)
-# calculate intercepts and print them
-interceptOLS = y_scaler - X_train_mean @ beta_OLS
-interceptRidge = y_scaler - X_train_mean @ beta_Ridge
-print(interceptOLS)
-print(interceptRidge)
-
-#predict value with intercept
-ytilde_test_OLS = X_test_scaled @ beta_OLS+y_scaler
-ytilde_test_Ridge = X_test_scaled @ beta_Ridge+y_scaler
-
-
-#Calculate MSE
-
-print("  ")
-print("test MSE of OLS:")
-print(MSE(y_test,ytilde_test_OLS))
-print("  ")
-print("test MSE of Ridge")
-print(MSE(y_test,ytilde_test_Ridge))
-
-plt.scatter(x,y,label='Data')
-plt.plot(x, X @ beta_OLS+interceptOLS,'*', label="OLS_Fit")
-plt.plot(x, X @ beta_Ridge+interceptRidge, label="Ridge_Fit")
-plt.grid()
-plt.legend()
-plt.show()
-
-
-# Finally, instead of using our own function we repeat the same example
-# using the **standardscaler** functionality of the library
-# **Scikit-Learn**.  Here we limit ourselves to Ridge regression only.
-
-# In[11]:
-
-
-from sklearn import linear_model
-np.random.seed(2018)
-n = 10
-d = 2
-Lambda = 0.01
-
-# Make data set.
-x = np.linspace(-3, 3, n)
-y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)
-
-# Design matrix X does not include the intercept. 
-X = np.zeros((n, d))
-for p in range(d):     
-    X[:, p] = x ** (p+1)
-
-#Split data in train and test
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-# Scale data by subtracting mean value of the input using scikit-learn
-scaler = StandardScaler(with_std=False)
-scaler.fit(X_train)
-X_train_mean = np.mean(X_train,axis=0)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# We scale also the output, here by our own code
-y_scaler = np.mean(y_train)
-y_train_scaled = y_train - y_scaler
-y_test_scaled = y_test- y_scaler
-
-#Calculate beta
-OLS = LinearRegression()
-betaOLS=OLS.fit(X_train_scaled,y_train_scaled)
-ypredictOLS = OLS.predict(X_test_scaled)
-linear_model.Ridge(Lambda)
-RegRidge.fit(X_train_scaled,y_train_scaled)
-ypredictRidge = RegRidge.predict(X_test_scaled)
-betaOLS = OLS.coef_
-betaRidge = RegRidge.coef_
-print(betaOLS)
-print(betaRidge)
-interceptOLS = np.mean(y_train) - X_train_mean @ betaOLS
-interceptRidge = y_scaler - X_train_mean @ betaRidge
-print(interceptOLS)
-print(interceptRidge)
-#predict value 
-ytilde_test_Ridge = X_test_scaled @ betaRidge+y_scaler
-ytilde_test_OLS = X_test_scaled @ betaOLS+y_scaler
-
-#Calculate MSE
-print("  ")
-print("test MSE of OLS")
-print(MSE(y_test,ytilde_test_OLS))
-print("  ")
-print("test MSE of Ridge")
-print(MSE(y_test,ytilde_test_Ridge))
-plt.scatter(x,y,label='Data')
-plt.plot(x, X @ RegRidge.coef_ + RegRidge.intercept_ , label="Ridge_Fit")
-plt.grid()
-plt.legend()
-plt.show()
+# $$
+# \mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}]-\mbox{Var}(\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
+# $$
 
+# 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. 
+# 
+# For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.
diff --git a/doc/LectureNotes/_build/jupyter_execute/week37_121_0.png b/doc/LectureNotes/_build/jupyter_execute/week37_121_0.png
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diff --git a/doc/LectureNotes/week37.ipynb b/doc/LectureNotes/week37.ipynb
index 94e3e70ef..c24beccbd 100644
--- a/doc/LectureNotes/week37.ipynb
+++ b/doc/LectureNotes/week37.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "672fea2a",
+   "id": "7c0ddab1",
    "metadata": {
     "editable": true
    },
@@ -14,13 +14,13 @@
   },
   {
    "cell_type": "markdown",
-   "id": "479a6913",
+   "id": "6c033bf4",
    "metadata": {
     "editable": true
    },
    "source": [
     "# Week 37: Statistical interpretations and Resampling Methods\n",
-    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University\n",
+    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway\n",
     "\n",
     "Date: **September 9, 2024**\n",
     "\n",
@@ -29,419 +29,69 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10b8f456",
+   "id": "e4cd5fa1",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Plans for week 37\n",
+    "## Plans for week 37, lecture Monday\n",
     "\n",
     "**Material for the lecture on Monday September 9.**\n",
     "\n",
     "<!-- * [Video of Lecture](https://youtu.be/YOBBr_toYxc) -->\n",
     "<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf) -->\n",
-    "  * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff\n",
+    "  * Statistical interpretation of Ridge and Lasso regression, see also slides from last week\n",
     "\n",
-    "  * Statistical interpretation of Ridge and Lasso regression\n",
+    "  * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.\n",
     "\n",
     "  * Readings and Videos:\n",
     "\n",
-    "    * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). \n",
+    "    * Raschka et al, pages 175-192\n",
+    "\n",
+    "    * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See <https://link.springer.com/book/10.1007/978-0-387-84858-7>.\n",
     "\n",
     "    * [Video on cross validation](https://www.youtube.com/watch?v=fSytzGwwBVw)\n",
     "\n",
     "    * [Video on Bootstrapping](https://www.youtube.com/watch?v=Xz0x-8-cgaQ)\n",
     "\n",
-    "    * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)\n",
+    "    * [Video on bias-variance tradeoff](https://www.youtube.com/watch?v=EuBBz3bI-aA)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bf80a276",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Plans for week 37, lab sessions\n",
     "\n",
-    "**Material for the active learning sessions on Tuesday and Wednesday.**\n",
+    "**Material for the lab  sessions on Tuesday and Wednesday.**\n",
     "\n",
     "  * Calculations of expectation values\n",
     "\n",
+    "  * Discussion of resampling techniques\n",
+    "\n",
     "  * Exercise set for week 37\n",
     "\n",
     "  * Work on project 1\n",
     "\n",
-    "  * See also additional note on scaling (jupyter-notebook) sent separately. This will be discussed during the first hour of each session. This note is added at the end of these slides.\n",
-    "\n",
     "  * For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f7e43b1b",
+   "id": "0e1f0a90",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Material from last week and relevant for the weekly exercises"
+    "## Material for lecture Monday September 9"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "af6a447a",
-   "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": "b6b7ea38",
-   "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": "e53d8d41",
-   "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": "401644e7",
-   "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": "0603387c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "eb2b6edf",
-   "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": "f1a08194",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ad40fa0b",
-   "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": "d3a4907d",
-   "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": "d7908c9c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "while\n",
-    "its variance is"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "2528f74e",
-   "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": "525f3a59",
-   "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": "ead1d419",
-   "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": "e471f7ab",
-   "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": "7886e97f",
-   "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": "49c32ca5",
-   "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": "9bd27980",
-   "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": "184f9c3b",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d0f46be1",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "We see clearly that \n",
-    "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n",
-    "\n",
-    "We can also compute the variance as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "03e3ac5f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "13ec2524",
-   "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": "58c3f236",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "bfcfc2a0",
-   "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. \n",
-    "\n",
-    "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f3a9f084",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Material for lecture Thursday September 14"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a4338936",
+   "id": "1b290d64",
    "metadata": {
     "editable": true
    },
@@ -464,7 +114,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7566301",
+   "id": "ceed3a82",
    "metadata": {
     "editable": true
    },
@@ -476,7 +126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8af7fae",
+   "id": "1d7e8f98",
    "metadata": {
     "editable": true
    },
@@ -489,7 +139,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "959e33e8",
+   "id": "58722d7d",
    "metadata": {
     "editable": true
    },
@@ -501,7 +151,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c847b78b",
+   "id": "45120edc",
    "metadata": {
     "editable": true
    },
@@ -513,7 +163,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "75c188ba",
+   "id": "4f3b680a",
    "metadata": {
     "editable": true
    },
@@ -525,7 +175,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60b89ba3",
+   "id": "08489797",
    "metadata": {
     "editable": true
    },
@@ -536,7 +186,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fd5d5a54",
+   "id": "a1c81f06",
    "metadata": {
     "editable": true
    },
@@ -548,7 +198,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "56f25582",
+   "id": "4e092776",
    "metadata": {
     "editable": true
    },
@@ -559,7 +209,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9974b303",
+   "id": "79e5358a",
    "metadata": {
     "editable": true
    },
@@ -571,7 +221,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4112e053",
+   "id": "e5a4f865",
    "metadata": {
     "editable": true
    },
@@ -581,7 +231,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7be4a14c",
+   "id": "10b81ba7",
    "metadata": {
     "editable": true
    },
@@ -612,7 +262,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35078f48",
+   "id": "3349cb27",
    "metadata": {
     "editable": true
    },
@@ -624,7 +274,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb25ead1",
+   "id": "0fd1502e",
    "metadata": {
     "editable": true
    },
@@ -636,7 +286,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5cc80036",
+   "id": "43cba0e8",
    "metadata": {
     "editable": true
    },
@@ -646,7 +296,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ebc6099",
+   "id": "d6f4e7dc",
    "metadata": {
     "editable": true
    },
@@ -658,7 +308,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3160dff4",
+   "id": "50c2d673",
    "metadata": {
     "editable": true
    },
@@ -668,7 +318,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8a3aaa69",
+   "id": "30315a10",
    "metadata": {
     "editable": true
    },
@@ -680,7 +330,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7af6708c",
+   "id": "fd8cdf51",
    "metadata": {
     "editable": true
    },
@@ -690,7 +340,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c82e095e",
+   "id": "7323296b",
    "metadata": {
     "editable": true
    },
@@ -702,7 +352,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9431eca4",
+   "id": "e21334ab",
    "metadata": {
     "editable": true
    },
@@ -712,7 +362,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1db6c222",
+   "id": "82974124",
    "metadata": {
     "editable": true
    },
@@ -732,7 +382,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "98c457ce",
+   "id": "e053e43c",
    "metadata": {
     "editable": true
    },
@@ -744,7 +394,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6a3893f",
+   "id": "94098e13",
    "metadata": {
     "editable": true
    },
@@ -754,7 +404,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "17f0e52e",
+   "id": "27625b1f",
    "metadata": {
     "editable": true
    },
@@ -766,7 +416,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c2bfaea6",
+   "id": "8f3cc8b4",
    "metadata": {
     "editable": true
    },
@@ -778,7 +428,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "377eefdf",
+   "id": "d7d5464d",
    "metadata": {
     "editable": true
    },
@@ -790,7 +440,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6da6253e",
+   "id": "c292afee",
    "metadata": {
     "editable": true
    },
@@ -802,7 +452,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e5906b3e",
+   "id": "cdbbd444",
    "metadata": {
     "editable": true
    },
@@ -814,7 +464,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "792d292c",
+   "id": "100a8014",
    "metadata": {
     "editable": true
    },
@@ -826,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3bc0674",
+   "id": "d2e40c4d",
    "metadata": {
     "editable": true
    },
@@ -838,7 +488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0faadf3f",
+   "id": "1e48456e",
    "metadata": {
     "editable": true
    },
@@ -850,7 +500,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a2d478a7",
+   "id": "38ced5b6",
    "metadata": {
     "editable": true
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@@ -860,7 +510,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d51ac6ce",
+   "id": "1163c07b",
    "metadata": {
     "editable": true
    },
@@ -872,7 +522,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "91d489d9",
+   "id": "becc087f",
    "metadata": {
     "editable": true
    },
@@ -882,7 +532,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ebe26b44",
+   "id": "5ca2b09d",
    "metadata": {
     "editable": true
    },
@@ -901,7 +551,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a67660e2",
+   "id": "6572c77d",
    "metadata": {
     "editable": true
    },
@@ -922,7 +572,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40b7eac0",
+   "id": "2a3f65e5",
    "metadata": {
     "editable": true
    },
@@ -934,7 +584,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16a3b6ea",
+   "id": "2d264f77",
    "metadata": {
     "editable": true
    },
@@ -945,7 +595,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f41ccc6d",
+   "id": "06b8a13c",
    "metadata": {
     "editable": true
    },
@@ -958,7 +608,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ae81305b",
+   "id": "6d958ed3",
    "metadata": {
     "editable": true
    },
@@ -970,7 +620,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4829b81",
+   "id": "6330f3f2",
    "metadata": {
     "editable": true
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@@ -980,7 +630,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "812273af",
+   "id": "ed6a9f67",
    "metadata": {
     "editable": true
    },
@@ -992,7 +642,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31ecba0e",
+   "id": "de42ab8a",
    "metadata": {
     "editable": true
    },
@@ -1002,7 +652,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca6682ec",
+   "id": "4f36b126",
    "metadata": {
     "editable": true
    },
@@ -1014,7 +664,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "662eaffb",
+   "id": "0a7eef00",
    "metadata": {
     "editable": true
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@@ -1024,7 +674,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "08d5551d",
+   "id": "29940f94",
    "metadata": {
     "editable": true
    },
@@ -1038,7 +688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03f3de9c",
+   "id": "f2204679",
    "metadata": {
     "editable": true
    },
@@ -1050,7 +700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c57ace49",
+   "id": "d3f4597f",
    "metadata": {
     "editable": true
    },
@@ -1060,7 +710,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb014b2b",
+   "id": "c9e9ca01",
    "metadata": {
     "editable": true
    },
@@ -1072,7 +722,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1054d47e",
+   "id": "1c3062b6",
    "metadata": {
     "editable": true
    },
@@ -1082,7 +732,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1485d44c",
+   "id": "b455f707",
    "metadata": {
     "editable": true
    },
@@ -1094,7 +744,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f457f909",
+   "id": "5ac8fdd3",
    "metadata": {
     "editable": true
    },
@@ -1104,7 +754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59447b2a",
+   "id": "827927db",
    "metadata": {
     "editable": true
    },
@@ -1116,17 +766,17 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fe26100",
+   "id": "0c740a13",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta}$!"
+    "We have a model for $p(\\boldsymbol{D}\\vert\\boldsymbol{\\beta})$ but need one for the **prior** $p(\\boldsymbol{\\beta})$!"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d73c2d9c",
+   "id": "a855fc3c",
    "metadata": {
     "editable": true
    },
@@ -1142,7 +792,7 @@
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   {
    "cell_type": "markdown",
-   "id": "8cdf2c97",
+   "id": "9dda8b9e",
    "metadata": {
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@@ -1154,7 +804,7 @@
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   {
    "cell_type": "markdown",
-   "id": "7272ef9f",
+   "id": "d160205c",
    "metadata": {
     "editable": true
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@@ -1164,7 +814,7 @@
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   {
    "cell_type": "markdown",
-   "id": "0dc5b2b7",
+   "id": "1b7501a6",
    "metadata": {
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@@ -1176,7 +826,7 @@
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   {
    "cell_type": "markdown",
-   "id": "b6c8c684",
+   "id": "d75a8c65",
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@@ -1189,7 +839,7 @@
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   {
    "cell_type": "markdown",
-   "id": "f681ed50",
+   "id": "c9f718df",
    "metadata": {
     "editable": true
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@@ -1201,7 +851,7 @@
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   {
    "cell_type": "markdown",
-   "id": "5c3bc898",
+   "id": "b07b6425",
    "metadata": {
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@@ -1211,7 +861,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a3cea046",
+   "id": "215a8730",
    "metadata": {
     "editable": true
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@@ -1223,7 +873,7 @@
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   {
    "cell_type": "markdown",
-   "id": "68276410",
+   "id": "7013d0f0",
    "metadata": {
     "editable": true
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@@ -1233,7 +883,7 @@
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   {
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-   "id": "dfd55611",
+   "id": "d56b4ce0",
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@@ -1245,7 +895,7 @@
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   {
    "cell_type": "markdown",
-   "id": "83656dd8",
+   "id": "f9c6b14d",
    "metadata": {
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@@ -1257,7 +907,7 @@
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   {
    "cell_type": "markdown",
-   "id": "b62e29fd",
+   "id": "0bde16cb",
    "metadata": {
     "editable": true
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@@ -1267,7 +917,7 @@
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   {
    "cell_type": "markdown",
-   "id": "1de8855c",
+   "id": "0a2869fc",
    "metadata": {
     "editable": true
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@@ -1279,7 +929,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "212152f1",
+   "id": "75b83b9b",
    "metadata": {
     "editable": true
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@@ -1291,7 +941,7 @@
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   {
    "cell_type": "markdown",
-   "id": "afb042c4",
+   "id": "1e76d916",
    "metadata": {
     "editable": true
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@@ -1303,7 +953,7 @@
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   {
    "cell_type": "markdown",
-   "id": "ab1fd3c5",
+   "id": "09dfe26d",
    "metadata": {
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@@ -1313,7 +963,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "43979108",
+   "id": "4037962f",
    "metadata": {
     "editable": true
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@@ -1325,7 +975,7 @@
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   {
    "cell_type": "markdown",
-   "id": "96622b66",
+   "id": "39a85ba0",
    "metadata": {
     "editable": true
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@@ -1335,7 +985,7 @@
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   {
    "cell_type": "markdown",
-   "id": "e17a5b6d",
+   "id": "b263a1fe",
    "metadata": {
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@@ -1354,7 +1004,7 @@
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   {
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-   "id": "24e35401",
+   "id": "41730165",
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     "editable": true
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@@ -1382,7 +1032,7 @@
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   {
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-   "id": "a37f1d06",
+   "id": "4076cee3",
    "metadata": {
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@@ -1408,7 +1058,7 @@
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   {
    "cell_type": "markdown",
-   "id": "58f54a23",
+   "id": "11139cb3",
    "metadata": {
     "editable": true
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@@ -1425,7 +1075,7 @@
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   {
    "cell_type": "markdown",
-   "id": "6ed7c809",
+   "id": "b2bead9b",
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@@ -1445,7 +1095,7 @@
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   {
    "cell_type": "markdown",
-   "id": "bed9bb2f",
+   "id": "8df8364e",
    "metadata": {
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@@ -1474,7 +1124,7 @@
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   {
    "cell_type": "markdown",
-   "id": "2cafa0a3",
+   "id": "6b5273f1",
    "metadata": {
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@@ -1499,7 +1149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fefbb980",
+   "id": "a260b371",
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@@ -1519,7 +1169,7 @@
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   {
    "cell_type": "markdown",
-   "id": "4560153d",
+   "id": "2ff1e63c",
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@@ -1531,7 +1181,7 @@
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   {
    "cell_type": "markdown",
-   "id": "ffcbdd5a",
+   "id": "ab4b2e8c",
    "metadata": {
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@@ -1541,7 +1191,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d055ea55",
+   "id": "85b6fb62",
    "metadata": {
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@@ -1556,7 +1206,7 @@
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   {
    "cell_type": "markdown",
-   "id": "004f0161",
+   "id": "42449344",
    "metadata": {
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@@ -1569,7 +1219,7 @@
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   {
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-   "id": "039f4ad5",
+   "id": "35e4f511",
    "metadata": {
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@@ -1582,7 +1232,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "413459f1",
+   "id": "c787234f",
    "metadata": {
     "editable": true
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@@ -1594,7 +1244,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bef022c8",
+   "id": "183711dc",
    "metadata": {
     "editable": true
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@@ -1607,7 +1257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3d53693d",
+   "id": "2351078d",
    "metadata": {
     "editable": true
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@@ -1618,7 +1268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e682c32a",
+   "id": "408ba3b5",
    "metadata": {
     "editable": true
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@@ -1632,7 +1282,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f44053ea",
+   "id": "3ea79849",
    "metadata": {
     "editable": true
    },
@@ -1642,7 +1292,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f10de930",
+   "id": "69439aad",
    "metadata": {
     "editable": true
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@@ -1656,7 +1306,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f91402a",
+   "id": "e6c29fd8",
    "metadata": {
     "editable": true
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@@ -1669,7 +1319,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d7a8e36",
+   "id": "4bb52549",
    "metadata": {
     "editable": true
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@@ -1682,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4c756ca2",
+   "id": "3430d548",
    "metadata": {
     "editable": true
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@@ -1692,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "908f782e",
+   "id": "9ba636b3",
    "metadata": {
     "editable": true
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@@ -1705,7 +1355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f12152da",
+   "id": "cb19e98f",
    "metadata": {
     "editable": true
    },
@@ -1715,7 +1365,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "85dd290d",
+   "id": "3cc840d7",
    "metadata": {
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@@ -1728,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "57631d10",
+   "id": "a5062213",
    "metadata": {
     "editable": true
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@@ -1740,7 +1390,7 @@
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   {
    "cell_type": "markdown",
-   "id": "2d4d5ccb",
+   "id": "8d555244",
    "metadata": {
     "editable": true
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@@ -1759,7 +1409,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a29c8551",
+   "id": "c4356604",
    "metadata": {
     "editable": true
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@@ -1772,7 +1422,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6509ba36",
+   "id": "09dc08a1",
    "metadata": {
     "editable": true
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@@ -1784,7 +1434,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "733307b1",
+   "id": "dab5056a",
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@@ -1797,7 +1447,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a69b7331",
+   "id": "8a1f8583",
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@@ -1817,7 +1467,7 @@
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   {
    "cell_type": "markdown",
-   "id": "ced659da",
+   "id": "1be08ebb",
    "metadata": {
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@@ -1831,16 +1481,16 @@
     "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we found \n",
     "$\\mathbb{E}(\\boldsymbol{\\beta}) = \\boldsymbol{\\beta}$, which means that the estimator of the regression parameters is unbiased.\n",
     "\n",
-    "We found also that the variance of the estimate of the $j$-th regression coefficient is\n",
+    "In the exercises this week we show that the variance of the estimate of the $j$-th regression coefficient is\n",
     "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 [(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} $.\n",
     "\n",
-    "This quantity will be used to\n",
+    "This quantity can be used to\n",
     "construct a confidence interval for the estimates."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2c0d0a71",
+   "id": "2a368353",
    "metadata": {
     "editable": true
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@@ -1855,7 +1505,7 @@
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   {
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-   "id": "d3a4d693",
+   "id": "6fee4a1a",
    "metadata": {
     "editable": true
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@@ -1867,7 +1517,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ed3d852",
+   "id": "1a97c9ef",
    "metadata": {
     "editable": true
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@@ -1887,7 +1537,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e826e70b",
+   "id": "0a70afb2",
    "metadata": {
     "editable": true
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@@ -1907,7 +1557,7 @@
   },
   {
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-   "id": "10b5d4d5",
+   "id": "dc32eaea",
    "metadata": {
     "editable": true
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@@ -1931,7 +1581,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1645ce3c",
+   "id": "016d0ab3",
    "metadata": {
     "editable": true
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@@ -1952,7 +1602,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4986483a",
+   "id": "3f371f0d",
    "metadata": {
     "editable": true
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@@ -1982,7 +1632,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fe853f3b",
+   "id": "abc56fe7",
    "metadata": {
     "editable": true
    },
@@ -2006,7 +1656,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "81ec18f8",
+   "id": "a3d2885b",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2045,7 +1695,7 @@
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   {
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-   "id": "f0b5b588",
+   "id": "5db67125",
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     "editable": true
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@@ -2055,7 +1705,7 @@
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   {
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-   "id": "262a95e8",
+   "id": "6f1b5417",
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@@ -2066,7 +1716,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "d235b516",
+   "id": "9d98136a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2086,7 +1736,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1620bb7",
+   "id": "c87eac4e",
    "metadata": {
     "editable": true
    },
@@ -2104,7 +1754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fd3bd4b5",
+   "id": "2f0f4313",
    "metadata": {
     "editable": true
    },
@@ -2116,7 +1766,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a129d86",
+   "id": "7895ee8b",
    "metadata": {
     "editable": true
    },
@@ -2133,7 +1783,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff57b405",
+   "id": "be454b28",
    "metadata": {
     "editable": true
    },
@@ -2145,7 +1795,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5a13919e",
+   "id": "7475f8f9",
    "metadata": {
     "editable": true
    },
@@ -2155,7 +1805,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "45d4a7b6",
+   "id": "fd472029",
    "metadata": {
     "editable": true
    },
@@ -2167,7 +1817,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c837442b",
+   "id": "b43c8f67",
    "metadata": {
     "editable": true
    },
@@ -2184,7 +1834,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b3c7b8e",
+   "id": "c47cfb6d",
    "metadata": {
     "editable": true
    },
@@ -2196,7 +1846,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7e6f977d",
+   "id": "593188cd",
    "metadata": {
     "editable": true
    },
@@ -2206,7 +1856,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "adeab0fd",
+   "id": "76de4d81",
    "metadata": {
     "editable": true
    },
@@ -2218,7 +1868,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63b18db5",
+   "id": "afc38c97",
    "metadata": {
     "editable": true
    },
@@ -2228,7 +1878,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c7eb00d",
+   "id": "b0f2971c",
    "metadata": {
     "editable": true
    },
@@ -2240,7 +1890,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3f4530b",
+   "id": "5a65e495",
    "metadata": {
     "editable": true
    },
@@ -2250,7 +1900,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "90792236",
+   "id": "31419298",
    "metadata": {
     "editable": true
    },
@@ -2266,7 +1916,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d73c7cd4",
+   "id": "217ddd53",
    "metadata": {
     "editable": true
    },
@@ -2277,7 +1927,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "58124207",
+   "id": "ece05002",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2342,7 +1992,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b4e7bf5",
+   "id": "de6dc4e2",
    "metadata": {
     "editable": true
    },
@@ -2353,7 +2003,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "47b138a9",
+   "id": "da218205",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2410,7 +2060,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "37b66761",
+   "id": "5ea5df06",
    "metadata": {
     "editable": true
    },
@@ -2448,7 +2098,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b5c601e",
+   "id": "40f41d74",
    "metadata": {
     "editable": true
    },
@@ -2459,7 +2109,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "9bc32416",
+   "id": "03bb50ae",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2541,7 +2191,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eb934dbe",
+   "id": "cd11e9fd",
    "metadata": {
     "editable": true
    },
@@ -2566,7 +2216,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c65f802b",
+   "id": "6cf68ff2",
    "metadata": {
     "editable": true
    },
@@ -2594,7 +2244,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9accabd4",
+   "id": "c445bfb2",
    "metadata": {
     "editable": true
    },
@@ -2607,7 +2257,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "d5ed3f5c",
+   "id": "8e26035a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2707,7 +2357,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db1976f4",
+   "id": "86a6b012",
    "metadata": {
     "editable": true
    },
@@ -2718,7 +2368,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "3c35540c",
+   "id": "4a0e1afd",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2807,7 +2457,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e60fabdf",
+   "id": "bb1e0f15",
    "metadata": {
     "editable": true
    },
@@ -2817,7 +2467,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9b6750d6",
+   "id": "83c87d33",
    "metadata": {
     "editable": true
    },
@@ -2830,7 +2480,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "5e6b91e1",
+   "id": "74eaac59",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2908,622 +2558,364 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e83d7b10",
+   "id": "aa703354",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Notes on scaling with examples\n",
-    "\n",
-    "The programs here use both ordinrary least squares (OLS) and Ridge\n",
-    "regression with one value only for the hyperparameter $\\lambda$. The\n",
-    "first example has no scaling and includes the intercept as well and we\n",
-    "are trying to fit a second-order polynomial. The second code takes out\n",
-    "the intercept and subtracts the mean values of each column of the\n",
-    "design matrix and the mean value of the outputs.\n",
-    "\n",
-    "The third and final code uses **Scikit-Learn** as library in order to\n",
-    "calculate the optimal parameters for OLS and Ridge regression. Note\n",
-    "that it is highly recommended to not include the intercept in Ridge\n",
-    "and Lasso regression, in order to avoid penalizing the optimization by\n",
-    "the intercept. The second and third codes do thus not include the\n",
-    "intercept. In the second code we do the scaling ourselves while the\n",
-    "last code uses the standard scaler option included in **Scikit-Learn**, known as centering (where\n",
-    "we subtract the mean values)."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "id": "73807393",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "from sklearn.linear_model import LinearRegression\n",
-    "from sklearn.preprocessing import PolynomialFeatures\n",
-    "from sklearn.model_selection import train_test_split\n",
-    "from sklearn.preprocessing import StandardScaler\n",
-    "\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",
-    "def OLS_fit_beta(X, y):\n",
-    "    return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
-    "\n",
-    "def Ridge_fit_beta(X, y,L,d):\n",
-    "    I = np.eye(d,d)\n",
-    "    return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y\n",
-    "\n",
-    "# Same random numbers for each test.\n",
-    "np.random.seed(2018)\n",
-    "n = 100\n",
-    "d = 3\n",
-    "# hyperparameter lambda\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set, simple second-order polynomial\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "# The design matrix X includes the intercept and no scaling is made\n",
-    "X = np.zeros((len(x), d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p) \n",
-    "\n",
-    "\n",
-    "#Split data, no scaling is used and we include the intercept\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "\n",
-    "\n",
-    "#Calculate beta, own code\n",
-    "beta_OLS = OLS_fit_beta(X_train, y_train)\n",
-    "beta_Ridge = Ridge_fit_beta(X_train, y_train,Lambda,d)\n",
-    "print(beta_OLS)\n",
-    "print(beta_Ridge)\n",
-    "#predict value\n",
-    "ytilde_test_OLS = X_test @ beta_OLS\n",
-    "ytilde_test_Ridge = X_test @ beta_Ridge\n",
-    "\n",
-    "#Calculate MSE\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS:\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ beta_OLS,'*', label=\"OLS_Fit\")\n",
-    "plt.plot(x, X @ beta_Ridge, label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
+    "## Material for the lab sessions"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b09d20a2",
+   "id": "f8bffca6",
    "metadata": {
     "editable": true
    },
    "source": [
-    "In this example we do not include the intercept and we scale the data by subtracting the mean values. This follows the discussion in the [lecture material](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#more-on-rescaling-data).\n",
-    "see also the weekly slides [for week 36](https://compphysics.github.io/MachineLearning/doc/pub/week36/html/._week36-bs029.html).\n",
-    "It is recommended whrn we use Ridge and Lasso regression to not include the intercept in the optimization process.\n",
+    "## Linking the regression analysis with a statistical interpretation\n",
     "\n",
-    "Before we discuss the code, we repeat some of the basic math from the slides of week 36.\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",
-    "Let us try to understand what this may imply mathematically when we\n",
-    "subtract the mean values, also known as *zero centering* or simply *centering*. For\n",
-    "simplicity, we will focus on  ordinary regression, as done in the above example.\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",
-    "The cost/loss function  for regression is"
+    "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": "ef1cec97",
+   "id": "109c07d6",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+    "\\begin{align*} \n",
+    "\\mbox{Cov}(\\varepsilon_{i_1},\n",
+    "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n",
+    "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2.  \\end{array} \\right.\n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e43c422b",
+   "id": "1ea48f22",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Recall also that we use the squared value. This expression can lead to an\n",
-    "increased penalty for higher differences between predicted and\n",
-    "output/target values.\n",
+    "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",
-    "What we have done is to single out the $\\beta_0$ term in the\n",
-    "definition of the mean squared error (MSE).  The design matrix $X$\n",
-    "does in this case not contain any intercept column.  When we take the\n",
-    "derivative with respect to $\\beta_0$, we want the derivative to obey"
+    "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": "852577bc",
+   "id": "212f90ec",
+   "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": "a019e7cb",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+    "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4edd229a",
+   "id": "2ca46c43",
    "metadata": {
     "editable": true
    },
    "source": [
-    "for all $j$. For $\\beta_0$ we have"
+    "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": "bb93d608",
+   "id": "c61ed9c8",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+    "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "5ef162e3",
+   "id": "96d84043",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Multiplying away the constant $2/n$, we obtain"
+    "## Expectation value and variance\n",
+    "\n",
+    "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b7c5467d",
+   "id": "e53b66df",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+    "\\begin{align*} \n",
+    "\\mathbb{E}(y_i) & =\n",
+    "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n",
+    "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "493c75d6",
+   "id": "71d542a3",
    "metadata": {
     "editable": true
    },
    "source": [
-    "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
-    "Our result for $\\beta_0$ simplifies then to"
+    "while\n",
+    "its variance is"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "862f2222",
+   "id": "01b18637",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+    "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
+    "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n",
+    "[\\mathbb{E}(y_i)]^2  \\\\  & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n",
+    "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n",
+    "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n",
+    "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n",
+    "\\ast} \\, \\beta)^2 \\\\  & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n",
+    "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n",
+    "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n",
+    "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n",
+    "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2.  \n",
+    "\\end{align*}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e0d0a4e0",
+   "id": "e35bdcc1",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We obtain then"
+    "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": "73d327de",
+   "id": "15b7ea3c",
+   "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": "a4c5f625",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
+    "\\mathbb{E}(\\boldsymbol{\\hat{\\beta}}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "8be27cbd",
+   "id": "450007e4",
    "metadata": {
     "editable": true
    },
    "source": [
-    "If we define"
+    "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": "d49a9bac",
+   "id": "3bcf81c1",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\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": "dd964823",
+   "id": "f69a019e",
    "metadata": {
     "editable": true
    },
    "source": [
-    "and the mean value of the outputs as"
+    "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": "09a87e67",
+   "id": "95b37da0",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+    "\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9b8797a0",
+   "id": "241c2335",
    "metadata": {
     "editable": true
    },
    "source": [
-    "we have"
+    "We see clearly that \n",
+    "$\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}$ for any $\\lambda > 0$.\n",
+    "\n",
+    "We can also compute the variance as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "30aeafdb",
+   "id": "cf794384",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
+    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "dc867cbb",
+   "id": "4cd5260a",
    "metadata": {
     "editable": true
    },
    "source": [
-    "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
+    "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": "f154321e",
+   "id": "7361e846",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
+    "\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}]-\\mbox{Var}(\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "6015f267",
+   "id": "6a07837b",
    "metadata": {
     "editable": true
    },
    "source": [
-    "We can rewrite the latter equation as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f181e88f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a6e6c0e7",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where we have defined"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "708652ea",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "b360788c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
+    "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. \n",
     "\n",
-    "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "8f473602",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "70f518a7",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "674d02d9",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "6715261c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
-    "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
-    "\n",
-    "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "17600ee3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a8c6f068",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Now we try to implement this."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "id": "1d89f80d",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "\n",
-    "np.random.seed(2018)\n",
-    "n = 100\n",
-    "# we do not include the intercept\n",
-    "d = 2\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set.\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "#Design matrix X does not include the intercept. \n",
-    "X = np.zeros((len(x), d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p+1)\n",
-    "\n",
-    "\n",
-    "#Split data in train and test\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "\n",
-    "# Scale data by subtracting mean value,own implementation\n",
-    "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
-    "X_train_mean = np.mean(X_train,axis=0)\n",
-    "#Center by removing mean from each feature\n",
-    "X_train_scaled = X_train - X_train_mean\n",
-    "X_test_scaled = X_test - X_train_mean\n",
-    "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered, note)\n",
-    "y_scaler = np.mean(y_train)\n",
-    "y_train_scaled = y_train - y_scaler\n",
-    "\n",
-    "\n",
-    "#Calculate beta\n",
-    "beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)\n",
-    "beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,Lambda,d)\n",
-    "print(beta_OLS)\n",
-    "print(beta_Ridge)\n",
-    "# calculate intercepts and print them\n",
-    "interceptOLS = y_scaler - X_train_mean @ beta_OLS\n",
-    "interceptRidge = y_scaler - X_train_mean @ beta_Ridge\n",
-    "print(interceptOLS)\n",
-    "print(interceptRidge)\n",
-    "\n",
-    "#predict value with intercept\n",
-    "ytilde_test_OLS = X_test_scaled @ beta_OLS+y_scaler\n",
-    "ytilde_test_Ridge = X_test_scaled @ beta_Ridge+y_scaler\n",
-    "\n",
-    "\n",
-    "#Calculate MSE\n",
-    "\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS:\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ beta_OLS+interceptOLS,'*', label=\"OLS_Fit\")\n",
-    "plt.plot(x, X @ beta_Ridge+interceptRidge, label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "522668c3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Finally, instead of using our own function we repeat the same example\n",
-    "using the **standardscaler** functionality of the library\n",
-    "**Scikit-Learn**.  Here we limit ourselves to Ridge regression only."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "id": "9954bcc1",
-   "metadata": {
-    "collapsed": false,
-    "editable": true
-   },
-   "outputs": [],
-   "source": [
-    "from sklearn import linear_model\n",
-    "np.random.seed(2018)\n",
-    "n = 10\n",
-    "d = 2\n",
-    "Lambda = 0.01\n",
-    "\n",
-    "# Make data set.\n",
-    "x = np.linspace(-3, 3, n)\n",
-    "y = 2.0 + 0.5*x + 5.0*(x**2)+ np.random.randn(n)\n",
-    "\n",
-    "# Design matrix X does not include the intercept. \n",
-    "X = np.zeros((n, d))\n",
-    "for p in range(d):     \n",
-    "    X[:, p] = x ** (p+1)\n",
-    "\n",
-    "#Split data in train and test\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "# Scale data by subtracting mean value of the input using scikit-learn\n",
-    "scaler = StandardScaler(with_std=False)\n",
-    "scaler.fit(X_train)\n",
-    "X_train_mean = np.mean(X_train,axis=0)\n",
-    "X_train_scaled = scaler.transform(X_train)\n",
-    "X_test_scaled = scaler.transform(X_test)\n",
-    "# We scale also the output, here by our own code\n",
-    "y_scaler = np.mean(y_train)\n",
-    "y_train_scaled = y_train - y_scaler\n",
-    "y_test_scaled = y_test- y_scaler\n",
-    "\n",
-    "#Calculate beta\n",
-    "OLS = LinearRegression()\n",
-    "betaOLS=OLS.fit(X_train_scaled,y_train_scaled)\n",
-    "ypredictOLS = OLS.predict(X_test_scaled)\n",
-    "linear_model.Ridge(Lambda)\n",
-    "RegRidge.fit(X_train_scaled,y_train_scaled)\n",
-    "ypredictRidge = RegRidge.predict(X_test_scaled)\n",
-    "betaOLS = OLS.coef_\n",
-    "betaRidge = RegRidge.coef_\n",
-    "print(betaOLS)\n",
-    "print(betaRidge)\n",
-    "interceptOLS = np.mean(y_train) - X_train_mean @ betaOLS\n",
-    "interceptRidge = y_scaler - X_train_mean @ betaRidge\n",
-    "print(interceptOLS)\n",
-    "print(interceptRidge)\n",
-    "#predict value \n",
-    "ytilde_test_Ridge = X_test_scaled @ betaRidge+y_scaler\n",
-    "ytilde_test_OLS = X_test_scaled @ betaOLS+y_scaler\n",
-    "\n",
-    "#Calculate MSE\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of OLS\")\n",
-    "print(MSE(y_test,ytilde_test_OLS))\n",
-    "print(\"  \")\n",
-    "print(\"test MSE of Ridge\")\n",
-    "print(MSE(y_test,ytilde_test_Ridge))\n",
-    "plt.scatter(x,y,label='Data')\n",
-    "plt.plot(x, X @ RegRidge.coef_ + RegRidge.intercept_ , label=\"Ridge_Fit\")\n",
-    "plt.grid()\n",
-    "plt.legend()\n",
-    "plt.show()"
+    "For more discussions of Ridge regression and calculation of averages, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended."
    ]
   }
  ],