diff --git a/doc/BookChapters/chapter3.do.txt b/doc/BookChapters/chapter3.do.txt index 68153e5e4..9813418c7 100644 --- a/doc/BookChapters/chapter3.do.txt +++ b/doc/BookChapters/chapter3.do.txt @@ -134,7 +134,7 @@ With the OLS expressions for the parameters $\bm{\beta}$ we can evaluate the exp \] !et This means that the estimator of the regression parameters is unbiased. -v + We can also calculate the variance The variance of $\bm{\beta}$ is @@ -222,7 +222,7 @@ we discuss the o prediction error or simply the _test error_ $\mathrm{Err_{Test}}$, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the o training error $\mathrm{Err_{Train}}$, which is the average loss over the training data. -As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. +As our model becomes more and more complex, more of the training data tends to be used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. For a certain level of complexity the test error will reach minimum, before starting to increase again. The training error reaches a saturation. @@ -362,7 +362,7 @@ estimate any statistics thereof. In principle you never draw the histogram of the relative frequency of $\widehat{\beta}^*$. Instead you use the estimators corresponding to the statistic of interest. For example, if you are interested in estimating the variance of $\widehat -\beta$, apply the etsimator $\widehat \sigma^2$ to the values +\beta$, apply the estimator $\widehat \sigma^2$ to the values $\widehat \beta^*$. Before we proceed however, we need to remind ourselves about a central @@ -604,13 +604,13 @@ We can rewrite this as \] !et -The three terms represent the square of the bias of the learning +The first term represents the square of the bias of the learning method, which can be thought of as the error caused by the simplifying assumptions built into the method. The second term represents the variance of the chosen model and finally the last terms is variance of the error $\bm{\epsilon}$. -To derive this equation, we need to recall that the variance of $\bm{y}$ and $\bm{\epsilon}$ are both equal to $\sigma^2$. The mean value of $\bm{\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\bm{\tilde{y}}$. +To derive this equation, we need to recall that the variance of $\bm{y}$ and $\bm{\epsilon}$ are both equal to $\sigma^2$. The mean value of $\bm{\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastic variable, idem for $\bm{\tilde{y}}$. We use a more compact notation in terms of the expectation value !bt \[ @@ -1170,7 +1170,7 @@ plt.show() !ec -Note that we have kept the intercept in the first column of design matrix $\bm{X}$. When we call the corresponding _Scikit-Learn_ function we need thus to set the intercept to _False_. Libraries like _Scikit-Learn_ normally scale the design matrix and does not fit intercept. See the discussions below. +Note that we have kept the intercept in the first column of design matrix $\bm{X}$. When we call the corresponding _Scikit-Learn_ function we need thus to set the intercept to _False_. Libraries like _Scikit-Learn_ normally scale the design matrix and do not fit intercept. See the discussions below. ===== More on Rescaling data ===== @@ -1472,7 +1472,7 @@ the way we treat the intercept may give a larger or smaller MSE, meaning that the MSE can be penalized by the value of the intercept. Not including the intercept in the fit, means that the regularization term does not include $\beta_0$. For different values -of $\lambda$, this may lead to differeing MSE values. +of $\lambda$, this may lead to different MSE values. To remind the reader, the regularization term, with the intercept in Ridge regression, is given by !bt diff --git a/doc/BookChapters/chapter4.do.txt b/doc/BookChapters/chapter4.do.txt index 4bf3929dc..0fa9fc831 100644 --- a/doc/BookChapters/chapter4.do.txt +++ b/doc/BookChapters/chapter4.do.txt @@ -222,7 +222,7 @@ p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. !et Note that $1-p(t)= p(-t)$. -===== Examples of likelihood functions used in logistic regression and nueral networks ===== +===== Examples of likelihood functions used in logistic regression and neural networks ===== The following code plots the logistic function, the step function and other functions we will encounter from here and on. diff --git a/doc/LectureNotes/_build/.doctrees/chapter3.doctree b/doc/LectureNotes/_build/.doctrees/chapter3.doctree index 39c66b93c..c637b2aff 100644 Binary files a/doc/LectureNotes/_build/.doctrees/chapter3.doctree and 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100644 --- a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapter3.ipynb @@ -2,10 +2,32 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "51186d57", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "79a63fb4", + "metadata": { + "editable": true + }, + "source": [ + "# Resampling Methods" + ] + }, + { + "cell_type": "markdown", + "id": "f3d916b4", + "metadata": { + "editable": true + }, "source": [ - "# Resampling Methods\n", - "\n", "## Introduction\n", "\n", "Resampling methods are an indispensable tool in modern\n", @@ -27,7 +49,6 @@ "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n", "cross-validation and the bootstrap method. \n", "\n", - "\n", "Resampling approaches can be computationally expensive, because they\n", "involve fitting the same statistical method multiple times using\n", "different subsets of the training data. However, due to recent\n", @@ -44,23 +65,28 @@ "level of flexibility for a model is known as model selection. The\n", "bootstrap is widely used.\n", "\n", - "\n", "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n", "\n", "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n", "\n", - "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n", - "\n", + "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors." + ] + }, + { + "cell_type": "markdown", + "id": "46cb3279", + "metadata": { + "editable": true + }, + "source": [ "## Reminder on Statistics\n", "\n", - "\n", "* As in other experiments, many numerical experiments have two classes of errors:\n", "\n", " * Statistical errors\n", "\n", " * Systematical errors\n", "\n", - "\n", "* Statistical errors can be estimated using standard tools from statistics\n", "\n", "* Systematical errors are method specific and must be treated differently from case to case. \n", @@ -72,7 +98,6 @@ "derive quantities like the variance and other expectation values in a\n", "rather straightforward way.\n", "\n", - "\n", "It is assumed that $\\varepsilon_i\n", "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n", "independent, i.e.:" @@ -80,7 +105,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0fe38e07", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*} \n", @@ -93,7 +121,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d9d6955b", + "metadata": { + "editable": true + }, "source": [ "The randomness of $\\varepsilon_i$ implies that\n", "$\\mathbf{y}_i$ is also a random variable. In particular,\n", @@ -106,7 +137,6 @@ "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n", "row number $i$ and perform a sum over all values $p$.\n", "\n", - "\n", "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n", "that there exists a function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n", "which describe our data" @@ -114,7 +144,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f41f7049", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n", @@ -123,7 +156,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2ee172f4", + "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" @@ -131,7 +167,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1aecc768", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n", @@ -140,14 +179,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "12e9bdea", + "metadata": { + "editable": true + }, "source": [ "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "8f9db9db", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*} \n", @@ -160,7 +205,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b10abe89", + "metadata": { + "editable": true + }, "source": [ "while\n", "its variance is" @@ -168,7 +216,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bec51521", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n", @@ -188,18 +239,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4859640c", + "metadata": { + "editable": true + }, "source": [ "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n", "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD). \n", "\n", - "\n", "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we can evaluate the expectation value" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "69978823", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbb{E}(\\boldsymbol{\\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", @@ -208,10 +264,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5caee9d9", + "metadata": { + "editable": true + }, "source": [ "This means that the estimator of the regression parameters is unbiased.\n", - "v\n", + "\n", "We can also calculate the variance\n", "\n", "The variance of $\\boldsymbol{\\beta}$ is" @@ -219,7 +278,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6791e5b4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{eqnarray*}\n", @@ -247,7 +309,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4f747992", + "metadata": { + "editable": true + }, "source": [ "where we have used that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n", "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", @@ -258,7 +323,6 @@ "[(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} }$. This may be used to\n", "construct a confidence interval for the estimates.\n", "\n", - "\n", "In a similar way, we can obtain analytical expressions for say the\n", "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n", "when we employ Ridge regression, allowing us again to define a confidence interval. \n", @@ -268,7 +332,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d4679496", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n", @@ -277,7 +344,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bd3534e1", + "metadata": { + "editable": true + }, "source": [ "We see clearly that \n", "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n", @@ -287,7 +357,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "125b139e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", @@ -296,7 +369,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "98b59ea0", + "metadata": { + "editable": true + }, "source": [ "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n", "\n", @@ -305,7 +381,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "dcfd7a6c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", @@ -314,14 +393,23 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "30f43be3", + "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", - "\n", - "\n", + "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below." + ] + }, + { + "cell_type": "markdown", + "id": "e3c1bf3c", + "metadata": { + "editable": true + }, + "source": [ "## Resampling methods\n", "\n", "With all these analytical equations for both the OLS and Ridge\n", @@ -339,12 +427,10 @@ "\n", "2. training error $\\mathrm{Err_{Train}}$, which is the average loss over the training data.\n", "\n", - "As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n", + "As our model becomes more and more complex, more of the training data tends to be used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n", "For a certain level of complexity the test error will reach minimum, before starting to increase again. The\n", "training error reaches a saturation.\n", "\n", - "\n", - "\n", "Two famous\n", "resampling methods are the **independent bootstrap** and **the jackknife**. \n", "\n", @@ -360,7 +446,6 @@ "variance of $\\overline{X}$ (which often is the case), then there is no\n", "need for bootstrapping. \n", "\n", - "\n", "The Jackknife works by making many replicas of the estimator $\\widehat{\\beta}$. \n", "The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\\boldsymbol{x} = (x_1,x_2,\\cdots,X_n)$. \n", "Let $\\boldsymbol{x}_i$ denote the vector" @@ -368,7 +453,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3925f435", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{x}_i = (x_1,x_2,\\cdots,x_{i-1},x_{i+1},\\cdots,x_n),\n", @@ -377,7 +465,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "cc6328ee", + "metadata": { + "editable": true + }, "source": [ "which equals the vector $\\boldsymbol{x}$ with the exception that observation\n", "number $i$ is left out. Using this notation, define\n", @@ -387,7 +478,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 1, + "id": "b751a941", "metadata": { "collapsed": false, "editable": true @@ -426,7 +518,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "25ff562a", + "metadata": { + "editable": true + }, "source": [ "### Bootstrap\n", "\n", @@ -452,8 +547,6 @@ "estimate all the interesting parameters of $p(\\boldsymbol{t})$ using point\n", "estimators. \n", "\n", - "\n", - "\n", "In the case that $\\widehat{\\beta}$ has\n", "more than one component, and the components are independent, we use the\n", "same estimator on each component separately. If the probability\n", @@ -468,7 +561,6 @@ "idea is to use the relative frequency of $\\widehat{\\beta}^*$\n", "(think of a histogram) as an estimate of $p(\\boldsymbol{t})$.\n", "\n", - "\n", "But\n", "unless there is enough information available about the process that\n", "generated $X_1,X_2,\\cdots,X_n$, $p(x)$ is in general\n", @@ -478,13 +570,11 @@ "the relative frequency of the observations, will we obtain the same\n", "result in some asymptotic sense? The answer is yes.\n", "\n", - "\n", "Instead of generating the histogram for the relative\n", "frequency of the observation $X_i$, just draw the values\n", "$(X_1^*,X_2^*,\\cdots,X_n^*)$ with replacement from the vector\n", "$\\boldsymbol{X}$. \n", "\n", - "\n", "The independent bootstrap works like this: \n", "\n", "1. Draw with replacement $n$ numbers for the observed variables $\\boldsymbol{x} = (x_1,x_2,\\cdots,x_n)$. \n", @@ -502,7 +592,7 @@ "histogram of the relative frequency of $\\widehat{\\beta}^*$. Instead\n", "you use the estimators corresponding to the statistic of interest. For\n", "example, if you are interested in estimating the variance of $\\widehat\n", - "\\beta$, apply the etsimator $\\widehat \\sigma^2$ to the values\n", + "\\beta$, apply the estimator $\\widehat \\sigma^2$ to the values\n", "$\\widehat \\beta^*$.\n", "\n", "Before we proceed however, we need to remind ourselves about a central\n", @@ -511,7 +601,6 @@ "(and other resampling methods) work so well on independent and\n", "identically distributed variables.\n", "\n", - "\n", "Suppose we have a PDF $p(x)$ from which we generate a series $N$\n", "of averages $\\langle x_i \\rangle$. Each mean value $\\langle x_i \\rangle$\n", "is viewed as the average of a specific measurement, e.g., throwing \n", @@ -525,7 +614,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fa55ab5a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z=\\frac{x_1+x_2+\\dots+x_m}{m},\n", @@ -534,11 +626,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "697c0c94", + "metadata": { + "editable": true + }, "source": [ "the question we pose is which is the PDF of the new variable $z$.\n", "\n", - "\n", "The probability of obtaining an average value $z$ is the product of the \n", "probabilities of obtaining arbitrary individual mean values $x_i$,\n", "but with the constraint that the average is $z$. We can express this through\n", @@ -547,7 +641,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bd26bbd1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\tilde{p}(z)=\\int dx_1p(x_1)\\int dx_2p(x_2)\\dots\\int dx_mp(x_m)\n", @@ -557,21 +654,25 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "68664e4f", + "metadata": { + "editable": true + }, "source": [ "where the $\\delta$-function enbodies the constraint that the mean is $z$.\n", "All measurements that lead to each individual $x_i$ are expected to\n", "be independent, which in turn means that we can express $\\tilde{p}$ as the \n", "product of individual $p(x_i)$. The independence assumption is important in the derivation of the central limit theorem.\n", "\n", - "\n", - "\n", "If we use the integral expression for the $\\delta$-function" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "742a107a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta(z-\\frac{x_1+x_2+\\dots+x_m}{m})=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n", @@ -581,7 +682,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b6d624c1", + "metadata": { + "editable": true + }, "source": [ "and inserting $e^{i\\mu q-i\\mu q}$ where $\\mu$ is the mean value\n", "we arrive at" @@ -589,7 +693,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "46458586", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\tilde{p}(z)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n", @@ -600,14 +707,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ca119461", + "metadata": { + "editable": true + }, "source": [ "with the integral over $x$ resulting in" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f75b40fc", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}=\n", @@ -618,7 +731,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a4b64e20", + "metadata": { + "editable": true + }, "source": [ "The second term on the rhs disappears since this is just the mean and \n", "employing the definition of $\\sigma^2$ we have" @@ -626,7 +742,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fc76951f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\int_{-\\infty}^{\\infty}dxp(x)e^{\\left(iq(\\mu-x)/m\\right)}=\n", @@ -636,14 +755,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "76979572", + "metadata": { + "editable": true + }, "source": [ "resulting in" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3edff7d2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\left[\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}\\right]^m\\approx\n", @@ -653,14 +778,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "73a9341f", + "metadata": { + "editable": true + }, "source": [ "and in the limit $m\\rightarrow \\infty$ we obtain" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "88f1cc30", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\tilde{p}(z)=\\frac{1}{\\sqrt{2\\pi}(\\sigma/\\sqrt{m})}\n", @@ -670,13 +801,15 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1aec913e", + "metadata": { + "editable": true + }, "source": [ "which is the normal distribution with variance\n", "$\\sigma^2_m=\\sigma^2/m$, where $\\sigma$ is the variance of the PDF $p(x)$\n", "and $\\mu$ is also the mean of the PDF $p(x)$. \n", "\n", - "\n", "Thus, the central limit theorem states that the PDF $\\tilde{p}(z)$ of\n", "the average of $m$ random values corresponding to a PDF $p(x)$ \n", "is a normal distribution whose mean is the \n", @@ -689,7 +822,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "42b317e7", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\sigma_m=\n", @@ -699,7 +835,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e5baf71d", + "metadata": { + "editable": true + }, "source": [ "The latter is true only if the average value is known exactly. This is obtained in the limit\n", "$m\\rightarrow \\infty$ only. Because the mean and the variance are measured quantities we obtain \n", @@ -708,7 +847,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8e2b39e5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\sigma_m\\approx \n", @@ -718,7 +860,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "472e7c2c", + "metadata": { + "editable": true + }, "source": [ "In many cases however the above estimate for the standard deviation,\n", "in particular if correlations are strong, may be too simplistic. Keep\n", @@ -728,13 +873,10 @@ "we generate in various calculations do always exhibit some\n", "correlations.\n", "\n", - "\n", - "\n", "The theorem is satisfied by a large class of PDFs. Note however that for a\n", "finite $m$, it is not always possible to find a closed form /analytic expression for\n", "$\\tilde{p}(x)$.\n", "\n", - "\n", "The following code starts with a Gaussian distribution with mean value\n", "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n", "used in the bootstrap analysis. The bootstrap analysis returns a data\n", @@ -751,7 +893,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 2, + "id": "0ff7b796", "metadata": { "collapsed": false, "editable": true @@ -790,7 +933,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "22680159", + "metadata": { + "editable": true + }, "source": [ "We see that our new variance and from that the standard deviation, agrees with the central limit theorem.\n", "\n", @@ -799,7 +945,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, + "id": "fe018bb3", "metadata": { "collapsed": false, "editable": true @@ -819,11 +966,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5bbd613c", + "metadata": { + "editable": true + }, "source": [ "## The bias-variance tradeoff\n", "\n", - "\n", "We will discuss the bias-variance tradeoff in the context of\n", "continuous predictions such as regression. However, many of the\n", "intuitions and ideas discussed here also carry over to classification\n", @@ -835,7 +984,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ca3fde4a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}\n", @@ -844,7 +996,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8e11f84b", + "metadata": { + "editable": true + }, "source": [ "where $\\epsilon$ is normally distributed with mean zero and standard deviation $\\sigma^2$.\n", "\n", @@ -858,7 +1013,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "026a65c8", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n", @@ -867,14 +1025,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e59918c7", + "metadata": { + "editable": true + }, "source": [ "We can rewrite this as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "2fd3f73c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n", @@ -883,21 +1047,27 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7daf46c9", + "metadata": { + "editable": true + }, "source": [ - "The three terms represent the square of the bias of the learning\n", + "The first term represents the square of the bias of the learning\n", "method, which can be thought of as the error caused by the simplifying\n", "assumptions built into the method. The second term represents the\n", "variance of the chosen model and finally the last terms is variance of\n", "the error $\\boldsymbol{\\epsilon}$.\n", "\n", - "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\\boldsymbol{\\tilde{y}}$.\n", + "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastic variable, idem for $\\boldsymbol{\\tilde{y}}$.\n", "We use a more compact notation in terms of the expectation value" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6094266b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}})^2\\right],\n", @@ -906,14 +1076,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "643e0047", + "metadata": { + "editable": true + }, "source": [ "and adding and subtracting $\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]$ we get" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "1319bde5", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}}+\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right],\n", @@ -922,14 +1098,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9c6d6da1", + "metadata": { + "editable": true + }, "source": [ "which, using the abovementioned expectation values can be rewritten as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "855756ef", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right]+\\mathrm{Var}\\left[\\boldsymbol{\\tilde{y}}\\right]+\\sigma^2,\n", @@ -938,14 +1120,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "34d24717", + "metadata": { + "editable": true + }, "source": [ "that is the rewriting in terms of the so-called bias, the variance of the model $\\boldsymbol{\\tilde{y}}$ and the variance of $\\boldsymbol{\\epsilon}$." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, + "id": "d51b6100", "metadata": { "collapsed": false, "editable": true @@ -1010,7 +1196,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, + "id": "bd636def", "metadata": { "collapsed": false, "editable": true @@ -1067,7 +1254,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8cf88b3a", + "metadata": { + "editable": true + }, "source": [ "The bias-variance tradeoff summarizes the fundamental tension in\n", "machine learning, particularly supervised learning, between the\n", @@ -1078,8 +1268,6 @@ "train and less sensitive to sampling noise arising from having a\n", "finite-sized training dataset (smaller variance). \n", "\n", - "\n", - "\n", "The above equations tell us that in\n", "order to minimize the expected test error, we need to select a\n", "statistical learning method that simultaneously achieves low variance\n", @@ -1087,7 +1275,6 @@ "and squared bias is also nonnegative. Hence, we see that the expected\n", "test MSE can never lie below $Var(\\epsilon)$, the irreducible error.\n", "\n", - "\n", "What do we mean by the variance and bias of a statistical learning\n", "method? The variance refers to the amount by which our model would change if we\n", "estimated it using a different training data set. Since the training\n", @@ -1098,13 +1285,13 @@ "the training data can result in large changes in the model. In general, more\n", "flexible statistical methods have higher variance.\n", "\n", - "\n", "You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, + "id": "86bfc49a", "metadata": { "collapsed": false, "editable": true @@ -1186,7 +1373,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, + "id": "0c2a183a", "metadata": { "collapsed": false, "editable": true @@ -1275,7 +1463,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2c6c9e89", + "metadata": { + "editable": true + }, "source": [ "## Cross-validation\n", "\n", @@ -1294,7 +1485,6 @@ "choosing $k=n$. This particular case is referred to as leave-one-out\n", "cross-validation (LOOCV). \n", "\n", - "\n", "* Define a range of interest for the penalty parameter.\n", "\n", "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n", @@ -1304,7 +1494,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "71738b2a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -1317,7 +1510,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "14db46b6", + "metadata": { + "editable": true + }, "source": [ "* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", "\n", @@ -1328,7 +1524,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "95e5c8e4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -1339,7 +1538,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7e60f51d", + "metadata": { + "editable": true + }, "source": [ "For the various values of $k$\n", "\n", @@ -1357,7 +1559,6 @@ "\n", "d. Retain the evaluation score and discard the model\n", "\n", - "\n", "5. Summarize the model using the sample of model evaluation scores\n", "\n", "The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial." @@ -1365,7 +1566,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, + "id": "2cef0eb7", "metadata": { "collapsed": false, "editable": true @@ -1465,14 +1667,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f501c9cf", + "metadata": { + "editable": true + }, "source": [ "More examples of the application of cross-validation follow here." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 9, + "id": "30e1e320", "metadata": { "collapsed": false, "editable": true @@ -1550,10 +1756,21 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "383e5c2a", + "metadata": { + "editable": true + }, + "source": [ + "Note that we have kept the intercept in the first column of design matrix $\\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and do not fit intercept. See the discussions below." + ] + }, + { + "cell_type": "markdown", + "id": "1f9cd409", + "metadata": { + "editable": true + }, "source": [ - "Note that we have kept the intercept in the first column of design matrix $\\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and does not fit intercept. See the discussions below.\n", - "\n", "## More on Rescaling data\n", "\n", "We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.\n", @@ -1576,7 +1793,6 @@ "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n", "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n", "\n", - "\n", "If our predictors represent different scales, then it is important to\n", "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n", "column from the corresponding column and dividing the column with its\n", @@ -1602,15 +1818,14 @@ "coefficient term, than if measured in millimeters.\n", "This can clearly lead to problems in evaluating the cost/loss functions.\n", "\n", - "\n", - "\n", "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n", "on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 10, + "id": "eab81633", "metadata": { "collapsed": false, "editable": true @@ -1637,7 +1852,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0ca51a54", + "metadata": { + "editable": true + }, "source": [ "Let us try to understand what this may imply mathematically when we\n", "subtract the mean values, also known as *zero centering*. For\n", @@ -1648,7 +1866,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fa3a4110", + "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", @@ -1657,7 +1878,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9f499302", + "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", @@ -1671,7 +1895,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b1f7ba52", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n", @@ -1680,14 +1907,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7ff75b7f", + "metadata": { + "editable": true + }, "source": [ "for all $j$. For $\\beta_0$ we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "eac37d3c", + "metadata": { + "editable": true + }, "source": [ "$$\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", @@ -1696,14 +1929,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6e653464", + "metadata": { + "editable": true + }, "source": [ "Multiplying away the constant $2/n$, we obtain" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "8afa60a1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n", @@ -1712,7 +1951,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "055b7975", + "metadata": { + "editable": true + }, "source": [ "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n", "Our result for $\\beta_0$ simplifies then to" @@ -1720,7 +1962,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2bf6e5d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n", @@ -1729,14 +1974,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e15afbd5", + "metadata": { + "editable": true + }, "source": [ "We obtain then" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "d2f9a064", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n", @@ -1745,14 +1996,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8e8035d8", + "metadata": { + "editable": true + }, "source": [ "If we define" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "0bbcef44", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n", @@ -1761,14 +2018,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "919462c2", + "metadata": { + "editable": true + }, "source": [ "and the mean value of the outputs as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3a0899ec", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n", @@ -1777,14 +2040,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9b255ec5", + "metadata": { + "editable": true + }, "source": [ "we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7716aca1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n", @@ -1793,14 +2062,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "24863f48", + "metadata": { + "editable": true + }, "source": [ "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6c10f940", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n", @@ -1809,14 +2084,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7afefe95", + "metadata": { + "editable": true + }, "source": [ "We can rewrite the latter equation as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a2e7bc1a", + "metadata": { + "editable": true + }, "source": [ "$$\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", @@ -1825,14 +2106,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5ded7d78", + "metadata": { + "editable": true + }, "source": [ "where we have defined" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "bcac7366", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n", @@ -1841,18 +2128,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bafe8156", + "metadata": { + "editable": true + }, "source": [ "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n", "\n", - "\n", - "\n", "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7c91d280", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n", @@ -1861,14 +2152,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1b81d6de", + "metadata": { + "editable": true + }, "source": [ "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7d6d497e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n", @@ -1877,7 +2174,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "772593b2", + "metadata": { + "editable": true + }, "source": [ "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n", "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n", @@ -1887,7 +2187,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "3edbdb56", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n", @@ -1896,18 +2199,21 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7de9723e", + "metadata": { + "editable": true + }, "source": [ "What does this mean? And why do we insist on all this? Let us look at some examples.\n", "\n", - "\n", "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n", "Note also that we do not split the data into training and test." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 11, + "id": "2e4a0363", "metadata": { "collapsed": false, "editable": true @@ -2004,7 +2310,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c6d3a071", + "metadata": { + "editable": true + }, "source": [ "The intercept is the value of our output/target variable\n", "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n", @@ -2015,14 +2324,17 @@ "meaning that the MSE can be penalized by the value of the\n", "intercept. Not including the intercept in the fit, means that the\n", "regularization term does not include $\\beta_0$. For different values\n", - "of $\\lambda$, this may lead to differeing MSE values. \n", + "of $\\lambda$, this may lead to different MSE values. \n", "\n", "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "db803b77", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n", @@ -2031,14 +2343,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d7a9c9d8", + "metadata": { + "editable": true + }, "source": [ "but when we take out the intercept, this equation becomes" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "e08af763", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n", @@ -2047,14 +2365,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "10fe24e4", + "metadata": { + "editable": true + }, "source": [ "For Lasso regression we have" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6feb3bdd", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n", @@ -2063,7 +2387,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "657a0777", + "metadata": { + "editable": true + }, "source": [ "It means that, when scaling the design matrix and the outputs/targets,\n", "by subtracting the mean values, we have an optimization problem which\n", @@ -2072,13 +2399,13 @@ "back the intercept, we will get a MSE which then contains the\n", "intercept.\n", "\n", - "\n", "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known vanilla data set." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 12, + "id": "9f21317d", "metadata": { "collapsed": false, "editable": true @@ -2154,7 +2481,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "20de82fd", + "metadata": { + "editable": true + }, "source": [ "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n", "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n", @@ -2164,7 +2494,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 13, + "id": "4138ed50", "metadata": { "collapsed": false, "editable": true @@ -2253,7 +2584,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7e9dfda5", + "metadata": { + "editable": true + }, "source": [ "We see here, when compared to the code which includes explicitely the\n", "intercept column, that our MSE value is actually smaller. This is\n", @@ -2264,8 +2598,16 @@ "also that the problem with the intercept occurs mainly in these type\n", "of polynomial fitting problem.\n", "\n", - "The next example is indeed an example where all these discussions about the role of intercept are not present.\n", - "\n", + "The next example is indeed an example where all these discussions about the role of intercept are not present." + ] + }, + { + "cell_type": "markdown", + "id": "a7eb252c", + "metadata": { + "editable": true + }, + "source": [ "## More complicated Example: The Ising model\n", "\n", "The one-dimensional Ising model with nearest neighbor interaction, no\n", @@ -2274,7 +2616,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "990ff280", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2289,7 +2634,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5ea46b31", + "metadata": { + "editable": true + }, "source": [ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n", "in the system is determined by $L$. For the one-dimensional system\n", @@ -2302,7 +2650,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, + "id": "fa60aaab", "metadata": { "collapsed": false, "editable": true @@ -2333,7 +2682,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b9e8214e", + "metadata": { + "editable": true + }, "source": [ "Here we use ordinary least squares\n", "regression to predict the energy for the nearest neighbor\n", @@ -2346,7 +2698,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a578cd63", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2361,7 +2716,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d90dcfee", + "metadata": { + "editable": true + }, "source": [ "Here we allow for interactions beyond the nearest neighbors and a state dependent\n", "coupling constant. This latter expression can be formulated as\n", @@ -2370,7 +2728,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c4eec69a", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2385,7 +2746,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ae46a11b", + "metadata": { + "editable": true + }, "source": [ "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n", "elements $-J_{jk}$. This form of writing the energy fits perfectly\n", @@ -2394,7 +2758,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "87ebb9b8", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2409,14 +2776,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "856efd3f", + "metadata": { + "editable": true + }, "source": [ "We split the data in training and test data as discussed in the previous example" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 15, + "id": "5ae2828f", "metadata": { "collapsed": false, "editable": true @@ -2432,14 +2803,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0d6201d9", + "metadata": { + "editable": true + }, "source": [ "In the ordinary least squares method we choose the cost function" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "4af380a7", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2454,7 +2831,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8e8eefa3", + "metadata": { + "editable": true + }, "source": [ "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n", "This yields the expression for $\\boldsymbol{\\beta}$ to be" @@ -2462,7 +2842,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a1c0765c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n", @@ -2471,7 +2854,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c754e28b", + "metadata": { + "editable": true + }, "source": [ "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n", "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n", @@ -2481,7 +2867,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 16, + "id": "ff056cf9", "metadata": { "collapsed": false, "editable": true @@ -2500,7 +2887,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "524f259e", + "metadata": { + "editable": true + }, "source": [ "Doing the inversion directly turns out to be a bad idea since the matrix\n", "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n", @@ -2510,7 +2900,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "52be4553", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n", @@ -2519,14 +2912,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c37ff09a", + "metadata": { + "editable": true + }, "source": [ "where the pseudoinverse of $\\boldsymbol{X}$ is given by" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "ea49a105", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n", @@ -2535,7 +2934,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bf31067a", + "metadata": { + "editable": true + }, "source": [ "Using singular value decomposition we can decompose the matrix $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n", "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n", @@ -2545,7 +2947,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5ef2c839", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2560,7 +2965,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "96bbf134", + "metadata": { + "editable": true + }, "source": [ "Note that solving this equation by actually doing the pseudoinverse\n", "(which is what we will do) is not a good idea as this operation scales\n", @@ -2572,7 +2980,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 17, + "id": "5e729b1e", "metadata": { "collapsed": false, "editable": true @@ -2586,7 +2995,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 18, + "id": "80b923d1", "metadata": { "collapsed": false, "editable": true @@ -2598,14 +3008,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b4cf2f1a", + "metadata": { + "editable": true + }, "source": [ "When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 19, + "id": "93be2c0d", "metadata": { "collapsed": false, "editable": true @@ -2617,14 +3031,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ae869089", + "metadata": { + "editable": true + }, "source": [ "A way of looking at the coefficients in $J$ is to plot the matrices as images." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 20, + "id": "8d6d8152", "metadata": { "collapsed": false, "editable": true @@ -2643,7 +3061,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8092ac84", + "metadata": { + "editable": true + }, "source": [ "It is interesting to note that OLS\n", "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n", @@ -2653,10 +3074,6 @@ "\n", "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n", "\n", - "\n", - "\n", - "\n", - "\n", "Let us now \n", "focus on Ridge and Lasso regression as well. We repeat some of the\n", "basic parts of the Ising model and the setup of the training and test\n", @@ -2667,7 +3084,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d9251cde", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2682,7 +3102,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "abe454c9", + "metadata": { + "editable": true + }, "source": [ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n", "\n", @@ -2691,7 +3114,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 21, + "id": "89f8fbd0", "metadata": { "collapsed": false, "editable": true @@ -2723,14 +3147,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e06639d4", + "metadata": { + "editable": true + }, "source": [ "A more general form for the one-dimensional Ising model is" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "43ef0f40", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2745,7 +3175,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "523dc03b", + "metadata": { + "editable": true + }, "source": [ "Here we allow for interactions beyond the nearest neighbors and a more\n", "adaptive coupling matrix. This latter expression can be formulated as\n", @@ -2754,7 +3187,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0f389eef", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2769,7 +3205,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b6e00f24", + "metadata": { + "editable": true + }, "source": [ "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n", "elements $-J_{jk}$. This form of writing the energy fits perfectly\n", @@ -2778,7 +3217,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5ea1084d", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2793,14 +3235,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5551bb05", + "metadata": { + "editable": true + }, "source": [ "We organize the data as we did above" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 22, + "id": "f5dd7795", "metadata": { "collapsed": false, "editable": true @@ -2826,14 +3272,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f36c807b", + "metadata": { + "editable": true + }, "source": [ "We will do all fitting with **Scikit-Learn**," ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 23, + "id": "eb701dd4", "metadata": { "collapsed": false, "editable": true @@ -2845,14 +3295,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bd356ada", + "metadata": { + "editable": true + }, "source": [ "When extracting the $J$-matrix we make sure to remove the intercept" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 24, + "id": "ae737db8", "metadata": { "collapsed": false, "editable": true @@ -2864,14 +3318,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d152a32a", + "metadata": { + "editable": true + }, "source": [ "And then we plot the results" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 25, + "id": "8d713977", "metadata": { "collapsed": false, "editable": true @@ -2890,11 +3348,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ede30c18", + "metadata": { + "editable": true + }, "source": [ "The results agree perfectly with our previous discussion where we used our own code.\n", "\n", - "\n", "Having explored the ordinary least squares we move on to ridge\n", "regression. In ridge regression we include a **regularizer**. This\n", "involves a new cost function which leads to a new estimate for the\n", @@ -2904,31 +3364,26 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eac772e9", + "metadata": { + "editable": true + }, "source": [ - "6\n", - "0\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}.\n", + "\\label{_auto11} \\tag{11}\n", + "\\end{equation}\n", + "$$" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 26, + "id": "b66675f0", "metadata": { "collapsed": false, "editable": true @@ -2951,14 +3406,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "71aa37b5", + "metadata": { + "editable": true + }, "source": [ "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function." ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "8b70ef66", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -2973,14 +3434,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "079981ff", + "metadata": { + "editable": true + }, "source": [ "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 27, + "id": "c8d4a180", "metadata": { "collapsed": false, "editable": true @@ -3002,21 +3467,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "57839941", + "metadata": { + "editable": true + }, "source": [ "It is quite striking how LASSO breaks the symmetry of the coupling\n", "constant as opposed to ridge and OLS. We get a sparse solution with\n", "$J_{j, j + 1} = -1$.\n", "\n", - "\n", - "\n", - "\n", "We see how the different models perform for a different set of values for $\\lambda$." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 28, + "id": "0e3bbb4e", "metadata": { "collapsed": false, "editable": true @@ -3063,14 +3529,16 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "14711abd", + "metadata": { + "editable": true + }, "source": [ "We see that LASSO reaches a good solution for low\n", "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n", "much. Ridge is more stable over a larger range of values for\n", "$\\lambda$, but eventually also fades away.\n", "\n", - "\n", "To determine which value of $\\lambda$ is best we plot the accuracy of\n", "the models when predicting the training and the testing set. We expect\n", "the accuracy of the training set to be quite good, but if the accuracy\n", @@ -3081,7 +3549,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 29, + "id": "9ff29a72", "metadata": { "collapsed": false, "editable": true @@ -3122,26 +3591,29 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9c10be56", + "metadata": { + "editable": true + }, "source": [ "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n", "achieves a very good accuracy on the test set. This by far surpasses the\n", - "other models for all values of $\\lambda$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", + "other models for all values of $\\lambda$." + ] + }, + { + "cell_type": "markdown", + "id": "efce3b63", + "metadata": { + "editable": true + }, + "source": [ "## Exercises and Projects\n", "\n", - "\n", - "\n", "The main aim of this project is to study in more detail various\n", "regression methods, including the Ordinary Least Squares (OLS) method,\n", "The total score is **100** points. Each subtask has its own final score.\n", "\n", - "\n", "We will first study how to fit polynomials to a specific\n", "two-dimensional function called [Franke's\n", "function](http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf). This\n", @@ -3152,13 +3624,15 @@ "proper assessment of our models. We will also study in detail the\n", "so-called Bias-Variance trade off.\n", "\n", - "\n", "The Franke function, which is a weighted sum of four exponentials reads as follows" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "ef5a4df0", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -3170,7 +3644,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b085eff4", + "metadata": { + "editable": true + }, "source": [ "The function will be defined for $x,y\\in [0,1]$. Our first step will\n", "be to perform an OLS regression analysis of this function, trying out\n", @@ -3191,13 +3668,13 @@ "beyond the second-order polynomials metioned above and explore \n", "which polynomial fits the data best.\n", "\n", - "\n", "The Python code for the Franke function is included here (it performs also a three-dimensional plot of it)" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 30, + "id": "5056dccb", "metadata": { "collapsed": false, "editable": true @@ -3247,7 +3724,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f5e72aef", + "metadata": { + "editable": true + }, "source": [ "### Exercise: Ordinary Least Square (OLS) on the Franke function\n", "\n", @@ -3267,7 +3747,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0ba33237", + "metadata": { + "editable": true + }, "source": [ "$$\n", "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n", @@ -3277,7 +3760,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1f9af2ef", + "metadata": { + "editable": true + }, "source": [ "and the $R^2$ score function. If $\\tilde{\\hat{y}}_i$ is the predicted\n", "value of the $i-th$ sample and $y_i$ is the corresponding true value,\n", @@ -3286,7 +3772,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "77d5b0f2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n", @@ -3295,14 +3784,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6c1f9df9", + "metadata": { + "editable": true + }, "source": [ "where we have defined the mean value of $\\hat{y}$ as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "1d06102c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n", @@ -3311,7 +3806,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8fb40b08", + "metadata": { + "editable": true + }, "source": [ "Your code has to include a scaling of the data (for example by\n", "subtracting the mean value), and\n", @@ -3328,11 +3826,16 @@ "data and say test data. An accepted rule of thumb is to use\n", "approximately $2/3$ to $4/5$ of the data as training data.\n", "\n", - "\n", - "You can easily reuse the solutions to your exercises from week 35 and week 36.\n", - "\n", - "\n", - "\n", + "You can easily reuse the solutions to your exercises from week 35 and week 36." + ] + }, + { + "cell_type": "markdown", + "id": "1701de47", + "metadata": { + "editable": true + }, + "source": [ "### Exercise: Bias-variance trade-off and resampling techniques\n", "\n", "Our aim here is to study the bias-variance trade-off by implementing the **bootstrap** resampling technique.\n", @@ -3360,7 +3863,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "15110cdf", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}.\n", @@ -3369,7 +3875,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "33046595", + "metadata": { + "editable": true + }, "source": [ "Here $\\epsilon$ is normally distributed with mean zero and standard\n", "deviation $\\sigma^2$.\n", @@ -3385,7 +3894,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "84527747", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n", @@ -3394,7 +3906,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "da2f876d", + "metadata": { + "editable": true + }, "source": [ "Here the expected value $\\mathbb{E}$ is the sample value. \n", "\n", @@ -3403,7 +3918,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1845a7dc", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n", @@ -3412,7 +3930,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d2f1d7d7", + "metadata": { + "editable": true + }, "source": [ "Explain what the terms mean, which one is the bias and which one is\n", "the variance and discuss their interpretations.\n", @@ -3424,9 +3945,16 @@ "of your model complexity (the degree of the polynomial) and the number\n", "of data points, and possibly also your training and test data using the **bootstrap** resampling method.\n", "\n", - "Note also that when you calculate the bias, in all applications you don't know the function values $f_i$. You would hence replace them with the actual data points $y_i$.\n", - "\n", - "\n", + "Note also that when you calculate the bias, in all applications you don't know the function values $f_i$. You would hence replace them with the actual data points $y_i$." + ] + }, + { + "cell_type": "markdown", + "id": "e898c902", + "metadata": { + "editable": true + }, + "source": [ "### Exercise: Cross-validation as resampling techniques, adding more complexity\n", "\n", "The aim here is to write your own code for another widely popular\n", @@ -3442,9 +3970,16 @@ "Compare the MSE you get from your cross-validation code with the one\n", "you got from your **bootstrap** code. Comment your results. Try $5-10$\n", "folds. You can also compare your own cross-validation code with the\n", - "one provided by **Scikit-Learn**.\n", - "\n", - "\n", + "one provided by **Scikit-Learn**." + ] + }, + { + "cell_type": "markdown", + "id": "d23dc734", + "metadata": { + "editable": true + }, + "source": [ "### Exercise: Ridge Regression on the Franke function with resampling\n", "\n", "Write your own code for the Ridge method, either using matrix\n", @@ -3455,8 +3990,16 @@ "dependence on $\\lambda$.\n", "\n", "Study also the bias-variance trade-off as function of various values of\n", - "the parameter $\\lambda$. For the bias-variance trade-off, use the **bootstrap** resampling method. Comment your results. \n", - "\n", + "the parameter $\\lambda$. For the bias-variance trade-off, use the **bootstrap** resampling method. Comment your results." + ] + }, + { + "cell_type": "markdown", + "id": "39a35330", + "metadata": { + "editable": true + }, + "source": [ "### Exercise: Lasso Regression on the Franke function with resampling\n", "\n", "This exercise is essentially a repeat of the previous two ones, but now\n", @@ -3464,8 +4007,16 @@ "you can also use the functionalities of **Scikit-Learn** (recommended). \n", "Give a\n", "critical discussion of the three methods and a judgement of which\n", - "model fits the data best. Perform here as well an analysis of the bias-variance trade-off using the **bootstrap** resampling technique and an analysis of the mean squared error using cross-validation. \n", - "\n", + "model fits the data best. Perform here as well an analysis of the bias-variance trade-off using the **bootstrap** resampling technique and an analysis of the mean squared error using cross-validation." + ] + }, + { + "cell_type": "markdown", + "id": "44d9e821", + "metadata": { + "editable": true + }, + "source": [ "### Exercise: Analysis of real data\n", "\n", "With our codes functioning and having been tested properly on a\n", @@ -3488,7 +4039,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 31, + "id": "0a4e6d7e", "metadata": { "collapsed": false, "editable": true @@ -3500,7 +4052,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "16a73292", + "metadata": { + "editable": true + }, "source": [ "Here is a simple part of a Python code which reads and plots the data\n", "from such files" @@ -3508,7 +4063,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 32, + "id": "168356a4", "metadata": { "collapsed": false, "editable": true @@ -3536,7 +4092,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e9ec3e68", + "metadata": { + "editable": true + }, "source": [ "If you should have problems in downloading the digital terrain data,\n", "we provide two examples under the data folder of project 1. One is\n", @@ -3544,10 +4103,8 @@ "Austfjell, again in Norway.\n", "Feel free to produce your own terrain data.\n", "\n", - "\n", "Alternatively, if you would like to use another data set, feel free to do so. This could be data close to your reseach area or simply a data set you found interesting. See for example [kaggle.com](https://www.kaggle.com/datasets) for examples.\n", "\n", - "\n", "Our final part deals with the parameterization of your digital terrain\n", "data (or your own data). We will apply all three methods for linear regression, the same type (or higher order) of polynomial\n", "approximation and cross-validation as resampling technique to evaluate which\n", @@ -3561,5 +4118,5 @@ ], "metadata": {}, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } diff --git a/doc/LectureNotes/_build/html/_sources/chapter4.ipynb b/doc/LectureNotes/_build/html/_sources/chapter4.ipynb index faee62767..bdfb97b2b 100644 --- a/doc/LectureNotes/_build/html/_sources/chapter4.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapter4.ipynb @@ -2,12 +2,32 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "f63b8c1a", + "metadata": { + "editable": true + }, + "source": [ + "" + ] + }, + { + "cell_type": "markdown", + "id": "9a284cf5", + "metadata": { + "editable": true + }, + "source": [ + "# Logistic Regression" + ] + }, + { + "cell_type": "markdown", + "id": "d78b73e2", + "metadata": { + "editable": true + }, "source": [ - "# Logistic Regression\n", - "\n", - "\n", - "\n", "## Logistic Regression\n", "\n", "In linear regression our main interest was centered on learning the\n", @@ -22,7 +42,6 @@ "the product of the design matrices, linear regression gives then a\n", "simple recipe for fitting our data.\n", "\n", - "\n", "Classification problems, however, are concerned with outcomes taking\n", "the form of discrete variables (i.e. categories). We may for example,\n", "on the basis of DNA sequencing for a number of patients, like to find\n", @@ -39,7 +58,6 @@ "binary outcome, true or false, positive or negative, success or\n", "failure etc.\n", "\n", - "\n", "Logistic regression will also serve as our stepping stone towards\n", "neural network algorithms and supervised deep learning. For logistic\n", "learning, the minimization of the cost function leads to a non-linear\n", @@ -52,10 +70,16 @@ "\n", "We note also that many of the topics discussed here on logistic \n", "regression are also commonly used in modern supervised Deep Learning\n", - "models, as we will see later.\n", - "\n", - "\n", - "\n", + "models, as we will see later." + ] + }, + { + "cell_type": "markdown", + "id": "d0c829b4", + "metadata": { + "editable": true + }, + "source": [ "## Basics\n", "\n", "We consider the case where the dependent variables, also called the\n", @@ -76,7 +100,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f1f4d5f2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\ 1 & \\mathrm{yes} \\end{bmatrix}.\n", @@ -85,7 +112,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "8e2f836b", + "metadata": { + "editable": true + }, "source": [ "Before moving to the logistic model, let us try to use our linear\n", "regression model to classify these two outcomes. We could for example\n", @@ -98,7 +128,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6b58f1ba", + "metadata": { + "editable": true + }, "source": [ "\n", "
\n", @@ -113,12 +146,14 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "54e3604b", + "metadata": { + "editable": true + }, "source": [ "where $\\boldsymbol{y}$ is a vector representing the possible outcomes, $\\boldsymbol{X}$ is our\n", "$n\\times p$ design matrix and $\\boldsymbol{\\beta}$ represents our estimators/predictors.\n", "\n", - "\n", "The main problem with our function is that it takes values on the\n", "entire real axis. In the case of logistic regression, however, the\n", "labels $y_i$ are discrete variables. A typical example is the credit\n", @@ -134,13 +169,13 @@ "favorable to use a ``soft\" classifier that outputs\n", "the probability of a given category. This leads us to the logistic function.\n", "\n", - "\n", "The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 1, + "id": "cd1dd1c5", "metadata": { "collapsed": false, "editable": true @@ -207,14 +242,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2e06ee82", + "metadata": { + "editable": true + }, "source": [ "What we could attempt however is to plot the mean value for each group." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 2, + "id": "a774f001", "metadata": { "collapsed": false, "editable": true @@ -233,7 +272,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e102fa56", + "metadata": { + "editable": true + }, "source": [ "We are now trying to find a function $f(y\\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.\n", "In standard linear regression with a linear dependence on $x$, we would write this in terms of our model" @@ -241,7 +283,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "75d995ba", + "metadata": { + "editable": true + }, "source": [ "$$\n", "f(y_i\\vert x_i)=\\beta_0+\\beta_1 x_i.\n", @@ -250,7 +295,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "e2cacca5", + "metadata": { + "editable": true + }, "source": [ "This expression implies however that $f(y_i\\vert x_i)$ could take any\n", "value from minus infinity to plus infinity. If we however let\n", @@ -261,9 +309,16 @@ "us to a very popular model for the function $f$, namely the so-called\n", "Sigmoid function or logistic model. We will consider this function as\n", "representing the probability for finding a value of $y_i$ with a given\n", - "$x_i$.\n", - "\n", - "\n", + "$x_i$." + ] + }, + { + "cell_type": "markdown", + "id": "af9ac586", + "metadata": { + "editable": true + }, + "source": [ "## The logistic function\n", "\n", "Another widely studied model, is the so-called \n", @@ -281,7 +336,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "80aa3cbe", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n", @@ -290,19 +348,30 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "28d49915", + "metadata": { + "editable": true + }, "source": [ - "Note that $1-p(t)= p(-t)$.\n", - "\n", - "## Examples of likelihood functions used in logistic regression and nueral networks\n", - "\n", + "Note that $1-p(t)= p(-t)$." + ] + }, + { + "cell_type": "markdown", + "id": "eeff6cd9", + "metadata": { + "editable": true + }, + "source": [ + "## Examples of likelihood functions used in logistic regression and neural networks\n", "\n", "The following code plots the logistic function, the step function and other functions we will encounter from here and on." ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, + "id": "5ecec05e", "metadata": { "collapsed": false, "editable": true @@ -367,14 +436,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5802efe3", + "metadata": { + "editable": true + }, "source": [ "We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f85d14ea", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -386,7 +461,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6c4e0334", + "metadata": { + "editable": true + }, "source": [ "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", "\n", @@ -395,7 +473,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "df7facc9", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(y_i=0\\vert x_i, \\boldsymbol{\\beta}) = 1-p(y_i=1\\vert x_i, \\boldsymbol{\\beta}).\n", @@ -404,7 +485,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a920e6d8", + "metadata": { + "editable": true + }, "source": [ "In order to define the total likelihood for all possible outcomes from a \n", "dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n", @@ -416,7 +500,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a780bfe1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{align*}\n", @@ -427,14 +514,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9f810a44", + "metadata": { + "editable": true + }, "source": [ "from which we obtain the log-likelihood and our **cost/loss** function" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3fac4ef4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\boldsymbol{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\boldsymbol{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\boldsymbol{\\beta}))\\right]\\right).\n", @@ -443,14 +536,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "75aa2c14", + "metadata": { + "editable": true + }, "source": [ "Reordering the logarithms, we can rewrite the **cost/loss** function as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "780f2038", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\boldsymbol{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", @@ -459,7 +558,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c8c940aa", + "metadata": { + "editable": true + }, "source": [ "The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\beta$.\n", "Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that" @@ -467,7 +569,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9d4a527b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathcal{C}(\\boldsymbol{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", @@ -476,23 +581,27 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c1706606", + "metadata": { + "editable": true + }, "source": [ "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n", "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n", "\n", - "\n", "The cross entropy is a convex function of the weights $\\boldsymbol{\\beta}$ and,\n", "therefore, any local minimizer is a global minimizer. \n", "\n", - "\n", "Minimizing this\n", "cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "fe8fb387", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n", @@ -501,14 +610,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bb564e81", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "700c5443", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n", @@ -517,7 +632,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f442c956", + "metadata": { + "editable": true + }, "source": [ "Let us now define a vector $\\boldsymbol{y}$ with $n$ elements $y_i$, an\n", "$n\\times p$ matrix $\\boldsymbol{X}$ which contains the $x_i$ values and a\n", @@ -527,7 +645,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "83937f6f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = -\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{p}\\right).\n", @@ -536,7 +657,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "71819c1d", + "metadata": { + "editable": true + }, "source": [ "If we in addition define a diagonal matrix $\\boldsymbol{W}$ with elements \n", "$p(y_i\\vert x_i,\\boldsymbol{\\beta})(1-p(y_i\\vert x_i,\\boldsymbol{\\beta})$, we can obtain a compact expression of the second derivative as" @@ -544,7 +668,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "23dfd975", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T} = \\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X}.\n", @@ -553,14 +680,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6ca9f74b", + "metadata": { + "editable": true + }, "source": [ "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "ddb3d93a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\log{ \\frac{p(\\boldsymbol{\\beta}\\boldsymbol{x})}{1-p(\\boldsymbol{\\beta}\\boldsymbol{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n", @@ -569,14 +702,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0cfae560", + "metadata": { + "editable": true + }, "source": [ "Here we defined $\\boldsymbol{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\boldsymbol{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7c427fc3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(\\boldsymbol{\\beta}\\boldsymbol{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n", @@ -585,7 +724,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "6a6814bf", + "metadata": { + "editable": true + }, "source": [ "Till now we have mainly focused on two classes, the so-called binary\n", "system. Suppose we wish to extend to $K$ classes. Let us for the sake\n", @@ -594,7 +736,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "85282137", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\log{\\frac{p(C=1\\vert x)}{p(K\\vert x)}} = \\beta_{10}+\\beta_{11}x_1,\n", @@ -603,14 +748,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4864f76d", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "390e9a55", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n", @@ -619,14 +770,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c3f39c44", + "metadata": { + "editable": true + }, "source": [ "and so on till the class $C=K-1$ class" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "92f32a03", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n", @@ -635,13 +792,14 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "677a5c4d", + "metadata": { + "editable": true + }, "source": [ "and the model is specified in term of $K-1$ so-called log-odds or\n", "**logit** transformations.\n", "\n", - "\n", - "\n", "In our discussion of neural networks we will encounter the above again\n", "in terms of a slightly modified function, the so-called **Softmax** function.\n", "\n", @@ -658,7 +816,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fa70e3bf", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n", @@ -667,14 +828,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4c59400c", + "metadata": { + "editable": true + }, "source": [ "It is easy to extend to more predictors. The final class is" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f4073652", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n", @@ -683,7 +850,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "810e872a", + "metadata": { + "editable": true + }, "source": [ "and they sum to one. Our earlier discussions were all specialized to\n", "the case with two classes only. It is easy to see from the above that\n", @@ -692,8 +862,16 @@ "To find the optimal parameters we would typically use a gradient\n", "descent method. Newton's method and gradient descent methods are\n", "discussed in the material on [optimization\n", - "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", + "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html)." + ] + }, + { + "cell_type": "markdown", + "id": "f4ebc1da", + "metadata": { + "editable": true + }, + "source": [ "## Wisconsin Cancer Data\n", "\n", "We show here how we can use a simple regression case on the breast\n", @@ -703,7 +881,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, + "id": "b7a8cbac", "metadata": { "collapsed": false, "editable": true @@ -739,7 +918,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "10f74b93", + "metadata": { + "editable": true + }, "source": [ "In addition to the above scores, we could also study the covariance (and the correlation matrix).\n", "We use **Pandas** to compute the correlation matrix." @@ -747,7 +929,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, + "id": "d0b8025d", "metadata": { "collapsed": false, "editable": true @@ -792,7 +975,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9d84a2aa", + "metadata": { + "editable": true + }, "source": [ "In the above example we note two things. In the first plot we display\n", "the overlap of benign and malignant tumors as functions of the various\n", @@ -811,7 +997,8 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, + "id": "6144ea0a", "metadata": { "collapsed": false, "editable": true @@ -823,14 +1010,18 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5627f5ae", + "metadata": { + "editable": true + }, "source": [ "and then" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, + "id": "a7c8662a", "metadata": { "collapsed": false, "editable": true @@ -842,17 +1033,46 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d03ee2a7", + "metadata": { + "editable": true + }, "source": [ "Diagonalizing this matrix we can in turn say something about which\n", "features are of relevance and which are not. This leads us to\n", "the classical Principal Component Analysis (PCA) theorem with\n", - "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html))." + "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)).\n", + "\n", + "Here we present a further way to present our results in terms of a so-called **confusion matrix**, the cumulative gain and the **ROC** curve.\n", + "This way of displaying our data are based upon different ways to classify our possible outcomes. Before we proceed we need some definitions.\n", + "1. **TP**: true positive or in other words, something equivalent with a proper classification\n", + "\n", + "2. **TN**: true negative, which is equivalent with a correct rejection\n", + "\n", + "3. **FP**: false positive, or in simpler words something that is equivalent with a false alarm\n", + "\n", + "4. **FN**: false negative, which is mean to be equivalent with a miss.\n", + "\n", + "The total data set is then the sum of the true positive and true negative targets or outputs, labeled by $n$.\n", + "Based on this we can then define the accuracy score as the sum of correctly predicted **TP** and **TN** cases divided by the sum of true positive and treue negative events in our data set, or as" + ] + }, + { + "cell_type": "markdown", + "id": "44ce5095", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathrm{Accuracy} = \\frac{\\sum_{i=0}^{n-1}I(y_i=\\tilde{y}_i)}{n}.\n", + "$$" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, + "id": "24ff3dd3", "metadata": { "collapsed": false, "editable": true @@ -908,5 +1128,5 @@ ], "metadata": {}, "nbformat": 4, - "nbformat_minor": 4 + "nbformat_minor": 5 } diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html index 93fc20e67..aeb83f7e6 100644 --- a/doc/LectureNotes/_build/html/chapter3.html +++ b/doc/LectureNotes/_build/html/chapter3.html @@ -499,7 +499,8 @@ const thebe_selector_output = ".output, .cell_output"
-
+

5. Resampling Methods

5.1. Introduction

@@ -625,9 +626,8 @@ mean value \(\boldsymbol{X}\boldsymbo \[ \mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. \]
-

This means that the estimator of the regression parameters is unbiased. -v -We can also calculate the variance

+

This means that the estimator of the regression parameters is unbiased.

+

We can also calculate the variance

The variance of \(\boldsymbol{\beta}\) is

\[\begin{split} @@ -700,7 +700,7 @@ we discuss the

  • prediction error or simply the test error \(\mathrm{Err_{Test}}\), where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the

  • training error \(\mathrm{Err_{Train}}\), which is the average loss over the training data.

    • -

    As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. +

    As our model becomes more and more complex, more of the training data tends to be used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. For a certain level of complexity the test error will reach minimum, before starting to increase again. The training error reaches a saturation.

    Two famous @@ -760,10 +760,10 @@ number \(i\) is left out. Usin

    -
    Runtime: 0.0903549 sec
+
    Runtime: 0.0893679 sec
 Jackknife Statistics :
 original           bias      std. error
- 100.107        100.097        0.150184
+ 99.9524        99.9424        0.148854
    @@ -828,7 +828,7 @@ estimate any statistics thereof. In principle you never draw the histogram of the relative frequency of \(\widehat{\beta}^*\). Instead you use the estimators corresponding to the statistic of interest. For example, if you are interested in estimating the variance of \(\widehat -\beta\), apply the etsimator \(\widehat \sigma^2\) to the values +\beta\), apply the estimator \(\widehat \sigma^2\) to the values \(\widehat \beta^*\).

    Before we proceed however, we need to remind ourselves about a central theorem in statistics, namely the so-called central limit theorem. @@ -982,7 +982,7 @@ theorem.

    Bootstrap Statistics :
 original           bias      std. error
- 99.9033  14.9678         99.904        0.151348
+ 100.188  15.1133         100.19        0.149655
    @@ -1004,7 +1004,7 @@ original bias std. error
    -_images/chapter3_47_0.png +_images/chapter3_51_0.png
    @@ -1036,12 +1036,12 @@ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_ \[ \mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2. \] -

    The three terms represent the square of the bias of the learning +

    The first term represents the square of the bias of the learning method, which can be thought of as the error caused by the simplifying assumptions built into the method. The second term represents the variance of the chosen model and finally the last terms is variance of the error \(\boldsymbol{\epsilon}\).

    -

    To derive this equation, we need to recall that the variance of \(\boldsymbol{y}\) and \(\boldsymbol{\epsilon}\) are both equal to \(\sigma^2\). The mean value of \(\boldsymbol{\epsilon}\) is by definition equal to zero. Furthermore, the function \(f\) is not a stochastics variable, idem for \(\boldsymbol{\tilde{y}}\). +

    To derive this equation, we need to recall that the variance of \(\boldsymbol{y}\) and \(\boldsymbol{\epsilon}\) are both equal to \(\sigma^2\). The mean value of \(\boldsymbol{\epsilon}\) is by definition equal to zero. Furthermore, the function \(f\) is not a stochastic variable, idem for \(\boldsymbol{\tilde{y}}\). We use a more compact notation in terms of the expectation value

    \[ @@ -1124,7 +1124,7 @@ Var: 0.0010479245926411787 0.013121574062587286 >= 0.012073649469946107 + 0.0010479245926411787 = 0.013121574062587286
    -_images/chapter3_61_1.png +_images/chapter3_65_1.png
    @@ -1216,7 +1216,9 @@ Error: 0.03781367141738902 Bias^2: 0.03365768507152769 Var: 0.0041559863458613296 0.03781367141738902 >= 0.03365768507152769 + 0.0041559863458613296 = 0.03781367141738902 -Polynomial degree: 7 +
    + +
    Polynomial degree: 7
 Error: 0.027609773491022394
 Bias^2: 0.022999498260366198
 Var: 0.004610275230656182
@@ -1253,7 +1255,7 @@ Var: 0.20867052175003364
 0.2284246870217162 >= 0.01975416527168255 + 0.20867052175003364 = 0.2284246870217162
    -_images/chapter3_62_2.png +_images/chapter3_66_3.png

    The bias-variance tradeoff summarizes the fundamental tension in @@ -1378,7 +1380,7 @@ set, the higher, the less likely the model generalizes correctly from the training data. -_images/chapter3_64_1.png +_images/chapter3_68_1.png

    @@ -1466,9 +1468,7 @@ training data. 
    Degree of polynomial:   1
-
    -
    -
    Mean squared error on training data: 439230.69504801
+Mean squared error on training data: 439230.69504801
 Mean squared error on test data: 481979.17861098
 Degree of polynomial:   2
 Mean squared error on training data: 115822.95008046
@@ -1508,12 +1508,12 @@ Mean squared error on test data: 0.17446471
 Degree of polynomial:  13
 Mean squared error on training data: 0.00759119
 Mean squared error on test data: 1.08131003
-Degree of polynomial:  14
-Mean squared error on training data: 0.00472199
-Mean squared error on test data: 0.81333804
    -
    Degree of polynomial:  15
+
    Degree of polynomial:  14
+Mean squared error on training data: 0.00472199
+Mean squared error on test data: 0.81333804
+Degree of polynomial:  15
 Mean squared error on training data: 0.00410478
 Mean squared error on test data: 92.09172409
 Degree of polynomial:  16
@@ -1528,12 +1528,12 @@ Mean squared error on test data: 108.27092910
 Degree of polynomial:  19
 Mean squared error on training data: 0.00156376
 Mean squared error on test data: 1371.99051150
-Degree of polynomial:  20
-Mean squared error on training data: 0.00137818
-Mean squared error on test data: 1887.86252988
 
    
 
    
-
    Degree of polynomial:  21
+
    Degree of polynomial:  20
+Mean squared error on training data: 0.00137818
+Mean squared error on test data: 1887.86252988
+Degree of polynomial:  21
 Mean squared error on training data: 0.00118508
 Mean squared error on test data: 14859.69908626
 Degree of polynomial:  22
@@ -1548,12 +1548,12 @@ Mean squared error on test data: 1277.61702282
 Degree of polynomial:  25
 Mean squared error on training data: 0.00079129
 Mean squared error on test data: 128664.31650694
-Degree of polynomial:  26
-Mean squared error on training data: 0.00076905
-Mean squared error on test data: 19003.94822514
 
    
 
    
-
    Degree of polynomial:  27
+
    Degree of polynomial:  26
+Mean squared error on training data: 0.00076905
+Mean squared error on test data: 19003.94822514
+Degree of polynomial:  27
 Mean squared error on training data: 0.00068946
 Mean squared error on test data: 2379.66219404
 Degree of polynomial:  28
@@ -1564,13 +1564,13 @@ Mean squared error on training data: 0.00060705
 Mean squared error on test data: 3250.17647619
 
    
 
    
-
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label='Test Error')
 
    
 
    
-_images/chapter3_65_7.png
+_images/chapter3_69_6.png
 
    
 
    
 
    
@@ -1723,7 +1723,7 @@ cross-validation (LOOCV).
    
 
    
 
    
 
    
-_images/chapter3_71_0.png
+_images/chapter3_75_0.png
 
    
 
    
 
    More examples of the application of cross-validation follow here.
    
@@ -1800,14 +1800,14 @@ cross-validation (LOOCV).
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
 
    
 
    
-_images/chapter3_73_1.png
+_images/chapter3_77_1.png
    -

    Note that we have kept the intercept in the first column of design matrix \(\boldsymbol{X}\). When we call the corresponding Scikit-Learn function we need thus to set the intercept to False. Libraries like Scikit-Learn normally scale the design matrix and does not fit intercept. See the discussions below.

    +

    Note that we have kept the intercept in the first column of design matrix \(\boldsymbol{X}\). When we call the corresponding Scikit-Learn function we need thus to set the intercept to False. Libraries like Scikit-Learn normally scale the design matrix and do not fit intercept. See the discussions below.

    5.6. More on Rescaling data

    @@ -2077,7 +2077,7 @@ MSE with Sklearn intercept 0.004113634617443131
    -_images/chapter3_107_1.png +_images/chapter3_112_1.png

    The intercept is the value of our output/target variable @@ -2088,7 +2088,7 @@ the way we treat the intercept may give a larger or smaller MSE, meaning that the MSE can be penalized by the value of the intercept. Not including the intercept in the fit, means that the regularization term does not include \(\beta_0\). For different values -of \(\lambda\), this may lead to differeing MSE values.

    +of \(\lambda\), this may lead to different MSE values.

    To remind the reader, the regularization term, with the intercept in Ridge regression, is given by

    \[ @@ -2272,7 +2272,7 @@ MSE values for Scikit-Learn Ridge implementation 0.26409315307910025
    -_images/chapter3_115_1.png +_images/chapter3_120_1.png

    The results here agree when we force Scikit-Learn’s Ridge function to include the first column in our design matrix. @@ -2475,7 +2475,7 @@ MSE values for Scikit-Learn Ridge implementation 0.002381316302584886 -_images/chapter3_117_1.png +_images/chapter3_122_1.png

    We see here, when compared to the code which includes explicitely the @@ -2689,13 +2689,13 @@ linear system as an equation would reduce this down to

    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
-_images/chapter3_148_1.png
+_images/chapter3_154_1.png

    It is interesting to note that OLS @@ -2835,13 +2835,13 @@ with the form utilized in linear regression, viz.

    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
-_images/chapter3_166_1.png
+_images/chapter3_172_1.png

    The results agree perfectly with our previous discussion where we used our own code.

    @@ -2850,23 +2850,15 @@ regression. In ridge regression we include a regularizer. This involves a new cost function which leads to a new estimate for the weights \(\boldsymbol{\beta}\). This results in a penalized regression problem. The cost function is given by

    -

    6 -0

    -

    < -< -< -! -! -M -A -T -H -_ -B -L -O -C -K

    + +
    +
    +\[ +\begin{equation} + C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}. +\label{_auto11} \tag{11} +\end{equation} +\]
    _lambda = 0.1
@@ -2885,13 +2877,13 @@ K
    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
-_images/chapter3_169_1.png
+_images/chapter3_175_1.png

    In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.

    @@ -2922,13 +2914,13 @@ K

    -
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   cb = fig.colorbar(im)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
    
 
    
-_images/chapter3_173_1.png
+_images/chapter3_179_1.png

    It is quite striking how LASSO breaks the symmetry of the coupling @@ -2977,49 +2969,49 @@ constant as opposed to ridge and OLS. We get a sparse solution with

    -
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+
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    /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:647: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.924e+00, tolerance: 1.797e+00
   model = cd_fast.enet_coordinate_descent(
 
- 10%|███████████▊                                                                                                          | 1/10 [00:00<00:06,  1.32it/s]
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-_images/chapter3_175_13.png
+_images/chapter3_181_13.png
 
    
 
    
 
    We see that LASSO reaches a good solution for low
@@ -3068,7 +3060,7 @@ testing set that is close to the accuracy of the training set.
    
 
    
 
    
 
    
-_images/chapter3_177_0.png
+_images/chapter3_183_0.png
 
    
 
    
 
    From the above figure we can see that LASSO with \(\lambda = 10^{-2}\)
@@ -3160,13 +3152,13 @@ which polynomial fits the data best.
    
 
    
 
    
 
    
-
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3980313467.py:9: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
+
    /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3980313467.py:9: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
   ax = fig.gca(projection='3d')
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3980313467.py:37: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3980313467.py:37: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.
   fig.colorbar(surf, shrink=0.5, aspect=5)
 
    
 
    
-_images/chapter3_181_1.png
+_images/chapter3_188_1.png
 
    
 
    
 
    
diff --git a/doc/LectureNotes/_build/html/chapter4.html b/doc/LectureNotes/_build/html/chapter4.html
index c72d38972..d06f34eed 100644
--- a/doc/LectureNotes/_build/html/chapter4.html
+++ b/doc/LectureNotes/_build/html/chapter4.html
@@ -328,8 +328,8 @@ const thebe_selector_output = ".output, .cell_output"
   
  
  
  • 
-  
-   6.4. Examples of likelihood functions used in logistic regression and nueral networks
+  
+   6.4. Examples of likelihood functions used in logistic regression and neural networks
   
  
    • 
  
  • 
@@ -373,8 +373,8 @@ const thebe_selector_output = ".output, .cell_output"
   
  
    • 
  
  • 
-  
-   6.4. Examples of likelihood functions used in logistic regression and nueral networks
+  
+   6.4. Examples of likelihood functions used in logistic regression and neural networks
   
  
    • 
  
  • 
@@ -391,7 +391,8 @@ const thebe_selector_output = ".output, .cell_output"
             
               
    
                 
-  
    
+  
    
 
    6. Logistic Regression
    
 
    
 
    6.1. Logistic Regression
    
@@ -647,7 +648,7 @@ the probability of a given category. This leads us to the logistic function.
    
   
 
 
    100 rows × 4 columns
    
-
    _images/chapter4_5_1.png
+
    _images/chapter4_8_1.png
 
    • 
 
    
 
    What we could attempt however is to plot the mean value for each group.
    
@@ -665,7 +666,7 @@ the probability of a given category. This leads us to the logistic function.
    
 
    
 
    
 
    
-_images/chapter4_7_0.png
+_images/chapter4_10_0.png
 
    
 
    
 
    We are now trying to find a function \(f(y\vert x)\), that is a function which gives us an expected value for the output \(y\) with a given input \(x\).
@@ -704,8 +705,8 @@ p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
 \]
    
 
    Note that \(1-p(t)= p(-t)\).
    
 
    
-
    
-
    6.4. Examples of likelihood functions used in logistic regression and nueral networks
    
+
    
+
    6.4. Examples of likelihood functions used in logistic regression and neural networks
    
 
    The following code plots the logistic function, the step function and other functions we will encounter from here and on.
    
 
    
 
    
@@ -767,9 +768,9 @@ p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
 
    
 
    
 
    
-_images/chapter4_13_0.png
-_images/chapter4_13_1.png
-_images/chapter4_13_2.png
+_images/chapter4_18_0.png
+_images/chapter4_18_1.png
+_images/chapter4_18_2.png
 
    
 
    
 
    We assume now that we have two classes with \(y_i\) either \(0\) or \(1\). Furthermore we assume also that we have only two parameters \(\beta\) in our fitting of the Sigmoid function, that is we define probabilities
    
@@ -996,8 +997,8 @@ We use Pandas to compute the correlation matrix.
    
 
    
 
    
 
    
-_images/chapter4_51_0.png
-_images/chapter4_51_1.png
+_images/chapter4_57_0.png
+_images/chapter4_57_1.png
 
    
 
    
 
    In the above example we note two things. In the first plot we display
@@ -1030,6 +1031,20 @@ matrix.
    
 features are of relevance and which are not. This leads  us to
 the classical Principal Component Analysis (PCA) theorem with
 applications. This will be discussed later this semester (week 43).
    
+
    Here we present a further way to present our results in terms of a so-called confusion matrix, the cumulative gain and the ROC curve.
+This way of displaying our data are based upon different ways to classify our possible outcomes. Before we proceed we need some definitions.
    
+
      
+
    1. TP: true positive or in other words, something equivalent with a proper classification
      2. 
+
    2. TN: true negative, which is equivalent with a correct rejection
      3. 
+
    3. FP: false positive, or in simpler words something that is equivalent with a false alarm
      4. 
+
    4. FN: false negative, which is mean to be equivalent with a miss.
      5. 
+
    
+
    The total data set is then the sum of the true positive and true negative targets or outputs, labeled by \(n\).
+Based on this we can then define the accuracy score as the sum of correctly predicted TP and TN cases divided by the sum of true positive and treue negative events in our data set, or as
    
+
    
+\[
+\mathrm{Accuracy} = \frac{\sum_{i=0}^{n-1}I(y_i=\tilde{y}_i)}{n}.
+\]
    
 
    
 
    
 
    import matplotlib.pyplot as plt
@@ -1099,18 +1114,9 @@ Please also refer to the documentation for alternative solver options:
   n_iter_i = _check_optimize_result(
 
    
 
    
-
    ---------------------------------------------------------------------------
-ModuleNotFoundError                       Traceback (most recent call last)
-Input In [8], in <cell line: 36>()
-     32 print(accuracy)
-     33 print("Test set accuracy with Logistic Regression  and scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
----> 36 import scikitplot as skplt
-     37 y_pred = logreg.predict(X_test_scaled)
-     38 skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-
-ModuleNotFoundError: No module named 'scikitplot'
-
    
-
    
+_images/chapter4_64_2.png
+_images/chapter4_64_3.png
+_images/chapter4_64_4.png
 
    
 
    
 
    
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 7e5e8f82a..2cb77ef5a 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations  with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks"],titleterms:{"1":0,"10":17,"11":17,"12":17,"13":[],"14":17,"15":[],"16":17,"17":17,"18":17,"19":17,"2":[0,17],"20":[],"2021":[],"2022":19,"21":17,"22":17,"23":17,"24":17,"25":17,"26":17,"27":[],"28":17,"29":17,"3":[0,17],"30":17,"31":17,"34":17,"35":17,"36":17,"37":17,"38":17,"39":17,"4":[0,17],"40":17,"41":17,"4155":[],"42":17,"43":17,"44":17,"45":17,"46":17,"47":17,"5":[0,17],"6":[],"7":17,"8":[],"9":17,"case":[8,10,18],"do":1,"final":12,"function":[0,1,6,7,8,10,11,12,13,18],"import":[5,16],"new":4,A:[0,1,4,8,9],And:[],Ising:6,The:[0,1,2,3,5,6,7,8,9,11,12,15],With:4,activ:[1,12],actual:[],ad:[0,6],adaboost:10,adagrad:13,adam:13,adapt:10,adjust:1,adversari:4,again:[3,9],aim:[8,9],algebra:16,algorithm:[9,10,11,12],algortithm:13,all:8,an:[0,4,10],analys:5,analysi:[0,5,6,11,15,18],analyt:0,ani:13,anoth:9,appli:15,approach:[0,8,14],approxim:12,architectur:1,arrai:16,assist:19,august:17,autocorrel:18,autograd:[2,13],automat:13,back:[1,11,12],background:15,bag:10,base:13,basic:[0,5,7,9,10,11,16],batch:1,bay:5,befor:11,better:8,bia:6,binari:1,binomi:[],bird:10,block:[],boost:10,bootstrap:[6,10],boston:0,breast:1,bring:12,build:[1,3,9],calcul:[],cancer:[1,7,9,11],cart:9,central:[13,15,18],chain:12,chang:10,chi:0,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],collect:[1,3],compar:[2,10],complex:[0,6],complic:6,compon:11,comput:9,con:9,concept:18,conjug:13,continu:[],convex:[8,13],convolut:[3,12],correl:11,cost:[1,10],cours:[15,20],covari:[5,11,18],cross:6,cumul:[],data:[0,1,3,6,7,9,11,15,18],dataset:[1,3],decai:2,decis:[9,10],decomposit:[5,11,16],deep:[1,2],defin:1,definit:[],degre:0,demonstr:[],dens:0,deriv:[5,12],descent:[2,10,13],detail:3,develop:1,deviat:[],diagon:11,dice:[],differ:8,differenti:[2,13],diffus:2,dimension:[2,3,8],disadvantag:9,discret:18,disguis:[],distribut:[5,18],domain:18,down:1,dropout:1,element:[0,18],elimin:16,ensembl:10,entropi:9,environ:0,equat:[0,2,12],error:[0,10],euler:2,evalu:1,event:[],exampl:[0,1,2,3,4,6,7,8,9,10],exercis:[0,6],expect:18,experi:18,explor:0,exponenti:2,extrapol:4,extrem:10,ey:10,fall:19,famili:1,famou:16,featur:[9,16],feed:[1,12],fine:1,first:[4,12],fit:[0,10],forc:3,forest:10,forward:[1,2,12],fourier:3,frank:6,freedom:0,frequentist:0,from:[5,10,12],full:2,further:[3,5],fy:[],gan:4,gaussian:16,gd:13,gener:[4,9],geometr:11,gini:9,good:0,grade:19,gradient:[1,2,10,13],growth:2,ha:15,handl:16,hidden:2,hous:0,how:[],hyperparamet:1,hyperplan:8,i:1,id3:9,idea:11,implement:1,implic:5,improv:1,includ:13,increment:11,index:9,inform:19,input:2,instal:15,instructor:19,interpret:[5,11],introduc:[11,13],introduct:[0,6,15,16],invers:[5,16],iter:10,its:[],jackknif:[],jax:13,jungl:10,kera:[1,3],kernel:[8,11],lagrangian:8,lasso:[5,6],later:5,layer:[1,2,3,12],learn:[0,1,2,11,13,14,15],least:[5,6],level:10,librari:15,likelihood:7,limit:[1,13,18],linear:[0,8,13,16],link:[5,11,17,20],logist:7,lu:16,machin:[0,8,13,15],main:18,make:[0,9,10],mani:[10,12],materi:17,math:5,mathemat:[3,5,8],matric:[5,16],matrix:[1,5,11,12,16],matter:0,mean:0,meet:[5,10,18],mercer:8,mersenn:[],method:[6,9,10,13],mlp:12,mnist:[3,4],model:[0,1,4,6,12],moment:[],momentum:13,moon:[8,9],more:[3,6,16],multilay:12,multipl:[1,3],multipli:8,name:[],network:[1,2,3,4,7,12],neural:[1,2,3,4,7,12],non:8,normal:[0,1],norwai:[],notat:12,novemb:17,now:[1,9,13],nuclear:0,nueral:[],number:[0,2,18],numer:[2,18],numpi:16,object:3,observ:[],obtain:11,octob:17,od:2,off:6,ol:[5,6],one:[2,12],oper:16,optim:[1,8,13,15],order:13,ordinari:[5,6],organ:0,oslo:20,other:[4,9,11,12,16],our:[0,4,5,11,13],outcom:15,output:2,overarch:[0,4,8,9],overview:10,own:[0,10,11],packag:16,part:[13,15],partial:2,pass:1,pca:11,pdf:18,perceptron:12,perform:[1,9],period:3,perspect:1,point:4,poisson:2,polynomi:3,popul:2,practic:13,pre:[1,3],predict:4,prerequisit:[3,15],princip:11,principl:3,pro:9,probabl:[5,18],problem:[1,2,13],procedur:9,process:[1,3],program:[2,13],project:6,prop:13,propag:[1,12],properti:[5,18],pseudo:[],python:[0,9,15,16],quick:8,ran0:[],random:[10,11,18],read:9,real:6,recip:[],recurr:[4,12],reduc:0,reduct:3,reformul:2,regress:[0,5,6,7,9,10,13],regular:1,relev:20,relu:1,remark:3,remind:[6,8],replac:13,requir:[2,15],resampl:6,rescal:6,resourc:2,revisit:13,ridg:[0,5,6],rm:13,rng:[],rule:12,s:[8,10],same:13,sampl:11,schedul:17,schemat:9,scheme:2,scikit:[0,1,11],second:13,select:[],semest:19,septemb:17,set:[0,2,3,9,12],sgd:13,should:1,similar:13,simpl:[0,4,9,13],singl:10,singular:[5,11],situat:[],soft:8,softmax:1,solv:2,solver:13,some:[13,16],specifi:2,split:0,squar:[0,5,6,10],standard:13,state:0,statist:[5,6,15,18],steepest:[10,13],stk3155:[],stochast:[13,18],superposit:3,supervis:1,support:8,svd:5,systemat:3,teach:[17,19],teacher:19,techniqu:[6,11],technolog:15,tensorflow:[1,3],test:[0,1],textbook:20,theorem:[5,8,11,12,18],theori:18,three:[],tip:13,togeth:12,top:1,toss:[],toward:11,trade:6,tradeoff:6,train:[0,1,4],transform:3,tree:[9,10],tune:1,two:[3,8,15],type:[2,4,12],uncorrel:[],uniform:[],univers:[12,20],unsupervis:14,up:[0,2,9,12],us:[0,1,2,3,7,13,15],valid:6,valu:[5,11,18],variabl:18,varianc:6,variou:0,vector:[8,12,16],view:[0,4,10],visual:[1,9],vs:3,wai:9,wave:2,week:17,weekli:17,what:0,which:1,why:[],wisconsin:7,write:[4,11],xgboost:10,your:[0,10]}})
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
index 3a25f8c1a..e347cd688 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
@@ -2,10 +2,32 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "51186d57",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    ""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "79a63fb4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Resampling Methods"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f3d916b4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "# Resampling Methods\n",
-    "\n",
     "## Introduction\n",
     "\n",
     "Resampling methods are an indispensable tool in modern\n",
@@ -27,7 +49,6 @@
     "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n",
     "cross-validation and the bootstrap method. \n",
     "\n",
-    "\n",
     "Resampling approaches can be computationally expensive, because they\n",
     "involve fitting the same statistical method multiple times using\n",
     "different subsets of the training data. However, due to recent\n",
@@ -44,23 +65,28 @@
     "level of flexibility for a model is known as model selection. The\n",
     "bootstrap is widely used.\n",
     "\n",
-    "\n",
     "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n",
     "\n",
     "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n",
     "\n",
-    "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n",
-    "\n",
+    "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "46cb3279",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Reminder on Statistics\n",
     "\n",
-    "\n",
     "* As in other experiments, many numerical  experiments have two classes of errors:\n",
     "\n",
     "  * Statistical errors\n",
     "\n",
     "  * Systematical errors\n",
     "\n",
-    "\n",
     "* Statistical errors can be estimated using standard tools from statistics\n",
     "\n",
     "* Systematical errors are method specific and must be treated differently from case to case. \n",
@@ -72,7 +98,6 @@
     "derive quantities like the variance and other expectation values in a\n",
     "rather straightforward way.\n",
     "\n",
-    "\n",
     "It is assumed that $\\varepsilon_i\n",
     "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n",
     "independent, i.e.:"
@@ -80,7 +105,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0fe38e07",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \n",
@@ -93,7 +121,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d9d6955b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The randomness of $\\varepsilon_i$ implies that\n",
     "$\\mathbf{y}_i$ is also a random variable. In particular,\n",
@@ -106,7 +137,6 @@
     "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n",
     "row number $i$ and perform a sum over all values $p$.\n",
     "\n",
-    "\n",
     "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n",
     "that there exists a function $f(\\boldsymbol{x})$ and  a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n",
     "which describe our data"
@@ -114,7 +144,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f41f7049",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
@@ -123,7 +156,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2ee172f4",
+   "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"
@@ -131,7 +167,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1aecc768",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
@@ -140,14 +179,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "12e9bdea",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8f9db9db",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \n",
@@ -160,7 +205,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b10abe89",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "while\n",
     "its variance is"
@@ -168,7 +216,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bec51521",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
@@ -188,18 +239,23 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4859640c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n",
     "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD). \n",
     "\n",
-    "\n",
     "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we can evaluate the expectation value"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "69978823",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}(\\boldsymbol{\\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",
@@ -208,10 +264,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5caee9d9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This means that the estimator of the regression parameters is unbiased.\n",
-    "v\n",
+    "\n",
     "We can also calculate the variance\n",
     "\n",
     "The variance of $\\boldsymbol{\\beta}$ is"
@@ -219,7 +278,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6791e5b4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{eqnarray*}\n",
@@ -247,7 +309,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4f747992",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have used  that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n",
     "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n",
@@ -258,7 +323,6 @@
     "[(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} }$. This may be used to\n",
     "construct a confidence interval for the estimates.\n",
     "\n",
-    "\n",
     "In a similar way, we can obtain analytical expressions for say the\n",
     "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n",
     "when we employ Ridge regression, allowing us again to define a confidence interval. \n",
@@ -268,7 +332,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d4679496",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
@@ -277,7 +344,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bd3534e1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see clearly that \n",
     "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n",
@@ -287,7 +357,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "125b139e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
@@ -296,7 +369,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "98b59ea0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n",
     "\n",
@@ -305,7 +381,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "dcfd7a6c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
@@ -314,14 +393,23 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "30f43be3",
+   "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",
-    "\n",
-    "\n",
+    "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3c1bf3c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Resampling methods\n",
     "\n",
     "With all these analytical equations for both the OLS and Ridge\n",
@@ -339,12 +427,10 @@
     "\n",
     "2. training error $\\mathrm{Err_{Train}}$, which is the average loss over the training data.\n",
     "\n",
-    "As our model becomes more and more complex, more of the training data tends to  used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n",
+    "As our model becomes more and more complex, more of the training data tends to  be used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n",
     "For a certain level of complexity the test error will reach minimum, before starting to increase again. The\n",
     "training error reaches a saturation.\n",
     "\n",
-    "\n",
-    "\n",
     "Two famous\n",
     "resampling methods are the **independent bootstrap** and **the jackknife**. \n",
     "\n",
@@ -360,7 +446,6 @@
     "variance of $\\overline{X}$ (which often is the case), then there is no\n",
     "need for bootstrapping. \n",
     "\n",
-    "\n",
     "The Jackknife works by making many replicas of the estimator $\\widehat{\\beta}$. \n",
     "The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\\boldsymbol{x} = (x_1,x_2,\\cdots,X_n)$. \n",
     "Let $\\boldsymbol{x}_i$ denote the vector"
@@ -368,7 +453,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3925f435",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{x}_i = (x_1,x_2,\\cdots,x_{i-1},x_{i+1},\\cdots,x_n),\n",
@@ -377,7 +465,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "cc6328ee",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which equals the vector $\\boldsymbol{x}$ with the exception that observation\n",
     "number $i$ is left out. Using this notation, define\n",
@@ -388,6 +479,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
+   "id": "b751a941",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -397,10 +489,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Runtime: 0.0903549 sec\n",
+      "Runtime: 0.0893679 sec\n",
       "Jackknife Statistics :\n",
       "original           bias      std. error\n",
-      " 100.107        100.097        0.150184\n"
+      " 99.9524        99.9424        0.148854\n"
      ]
     }
    ],
@@ -437,7 +529,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "25ff562a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "### Bootstrap\n",
     "\n",
@@ -463,8 +558,6 @@
     "estimate all the interesting parameters of $p(\\boldsymbol{t})$ using point\n",
     "estimators.  \n",
     "\n",
-    "\n",
-    "\n",
     "In the case that $\\widehat{\\beta}$ has\n",
     "more than one component, and the components are independent, we use the\n",
     "same estimator on each component separately.  If the probability\n",
@@ -479,7 +572,6 @@
     "idea is to use the relative frequency of $\\widehat{\\beta}^*$\n",
     "(think of a histogram) as an estimate of $p(\\boldsymbol{t})$.\n",
     "\n",
-    "\n",
     "But\n",
     "unless there is enough information available about the process that\n",
     "generated $X_1,X_2,\\cdots,X_n$, $p(x)$ is in general\n",
@@ -489,13 +581,11 @@
     "the relative frequency of the observations, will we obtain the same\n",
     "result in some asymptotic sense? The answer is yes.\n",
     "\n",
-    "\n",
     "Instead of generating the histogram for the relative\n",
     "frequency of the observation $X_i$, just draw the values\n",
     "$(X_1^*,X_2^*,\\cdots,X_n^*)$ with replacement from the vector\n",
     "$\\boldsymbol{X}$. \n",
     "\n",
-    "\n",
     "The independent bootstrap works like this: \n",
     "\n",
     "1. Draw with replacement $n$ numbers for the observed variables $\\boldsymbol{x} = (x_1,x_2,\\cdots,x_n)$. \n",
@@ -513,7 +603,7 @@
     "histogram of the relative frequency of $\\widehat{\\beta}^*$. Instead\n",
     "you use the estimators corresponding to the statistic of interest. For\n",
     "example, if you are interested in estimating the variance of $\\widehat\n",
-    "\\beta$, apply the etsimator $\\widehat \\sigma^2$ to the values\n",
+    "\\beta$, apply the estimator $\\widehat \\sigma^2$ to the values\n",
     "$\\widehat \\beta^*$.\n",
     "\n",
     "Before we proceed however, we need to remind ourselves about a central\n",
@@ -522,7 +612,6 @@
     "(and other resampling methods) work so well on independent and\n",
     "identically distributed variables.\n",
     "\n",
-    "\n",
     "Suppose we have a PDF $p(x)$ from which we generate  a series $N$\n",
     "of averages $\\langle x_i \\rangle$. Each mean value $\\langle x_i \\rangle$\n",
     "is viewed as the average of a specific measurement, e.g., throwing \n",
@@ -536,7 +625,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fa55ab5a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "z=\\frac{x_1+x_2+\\dots+x_m}{m},\n",
@@ -545,11 +637,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "697c0c94",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "the question we pose is which is the PDF of the new variable $z$.\n",
     "\n",
-    "\n",
     "The probability of obtaining an average value $z$ is the product of the \n",
     "probabilities of obtaining arbitrary individual mean values $x_i$,\n",
     "but with the constraint that the average is $z$. We can express this through\n",
@@ -558,7 +652,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bd26bbd1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{p}(z)=\\int dx_1p(x_1)\\int dx_2p(x_2)\\dots\\int dx_mp(x_m)\n",
@@ -568,21 +665,25 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "68664e4f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where the $\\delta$-function enbodies the constraint that the mean is $z$.\n",
     "All measurements that lead to each individual $x_i$ are expected to\n",
     "be independent, which in turn means that we can express $\\tilde{p}$ as the \n",
     "product of individual $p(x_i)$.  The independence assumption is important in the derivation of the central limit theorem.\n",
     "\n",
-    "\n",
-    "\n",
     "If we use the integral expression for the $\\delta$-function"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "742a107a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\delta(z-\\frac{x_1+x_2+\\dots+x_m}{m})=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n",
@@ -592,7 +693,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b6d624c1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and inserting $e^{i\\mu q-i\\mu q}$ where $\\mu$ is the mean value\n",
     "we arrive at"
@@ -600,7 +704,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "46458586",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{p}(z)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n",
@@ -611,14 +718,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ca119461",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "with the integral over $x$ resulting in"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f75b40fc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}=\n",
@@ -629,7 +742,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a4b64e20",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The second term on the rhs disappears since this is just the mean and \n",
     "employing the definition of $\\sigma^2$ we have"
@@ -637,7 +753,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fc76951f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\int_{-\\infty}^{\\infty}dxp(x)e^{\\left(iq(\\mu-x)/m\\right)}=\n",
@@ -647,14 +766,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "76979572",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "resulting in"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3edff7d2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\left[\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}\\right]^m\\approx\n",
@@ -664,14 +789,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "73a9341f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and in the limit $m\\rightarrow \\infty$ we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "88f1cc30",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{p}(z)=\\frac{1}{\\sqrt{2\\pi}(\\sigma/\\sqrt{m})}\n",
@@ -681,13 +812,15 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1aec913e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which is the normal distribution with variance\n",
     "$\\sigma^2_m=\\sigma^2/m$, where $\\sigma$ is the variance of the PDF $p(x)$\n",
     "and $\\mu$ is also the mean of the PDF $p(x)$. \n",
     "\n",
-    "\n",
     "Thus, the central limit theorem states that the PDF $\\tilde{p}(z)$ of\n",
     "the average of $m$ random values corresponding to a PDF $p(x)$ \n",
     "is a normal distribution whose mean is the \n",
@@ -700,7 +833,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "42b317e7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sigma_m=\n",
@@ -710,7 +846,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e5baf71d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The latter is true only if the average value is known exactly. This is obtained in the limit\n",
     "$m\\rightarrow \\infty$  only. Because the mean and the variance are measured quantities we obtain \n",
@@ -719,7 +858,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e2b39e5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sigma_m\\approx \n",
@@ -729,7 +871,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "472e7c2c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In many cases however the above estimate for the standard deviation,\n",
     "in particular if correlations are strong, may be too simplistic. Keep\n",
@@ -739,13 +884,10 @@
     "we generate in various calculations do always exhibit some\n",
     "correlations.\n",
     "\n",
-    "\n",
-    "\n",
     "The theorem is satisfied by a large class of PDFs. Note however that for a\n",
     "finite $m$, it is not always possible to find a closed form /analytic expression for\n",
     "$\\tilde{p}(x)$.\n",
     "\n",
-    "\n",
     "The following code starts with a Gaussian distribution with mean value\n",
     "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n",
     "used in the bootstrap analysis. The bootstrap analysis returns a data\n",
@@ -763,6 +905,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
+   "id": "0ff7b796",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -774,7 +917,7 @@
      "text": [
       "Bootstrap Statistics :\n",
       "original           bias      std. error\n",
-      " 99.9033  14.9678         99.904        0.151348\n"
+      " 100.188  15.1133         100.19        0.149655\n"
      ]
     }
    ],
@@ -811,7 +954,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "22680159",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see that our new variance and from that the standard deviation, agrees with the central limit theorem.\n",
     "\n",
@@ -821,6 +967,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
+   "id": "fe018bb3",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -828,14 +975,14 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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\n",
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\n",
       "text/plain": [
        "
    "
       ]
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_47_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_51_0.png"
       }
      },
      "output_type": "display_data"
@@ -855,11 +1002,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5bbd613c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The bias-variance tradeoff\n",
     "\n",
-    "\n",
     "We will discuss the bias-variance tradeoff in the context of\n",
     "continuous predictions such as regression. However, many of the\n",
     "intuitions and ideas discussed here also carry over to classification\n",
@@ -871,7 +1020,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ca3fde4a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}\n",
@@ -880,7 +1032,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e11f84b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\epsilon$ is normally distributed with mean zero and standard deviation $\\sigma^2$.\n",
     "\n",
@@ -894,7 +1049,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "026a65c8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
@@ -903,14 +1061,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e59918c7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can rewrite this as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2fd3f73c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n",
@@ -919,21 +1083,27 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7daf46c9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "The three terms represent the square of the bias of the learning\n",
+    "The first term represents the square of the bias of the learning\n",
     "method, which can be thought of as the error caused by the simplifying\n",
     "assumptions built into the method. The second term represents the\n",
     "variance of the chosen model and finally the last terms is variance of\n",
     "the error $\\boldsymbol{\\epsilon}$.\n",
     "\n",
-    "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\\boldsymbol{\\tilde{y}}$.\n",
+    "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastic variable, idem for $\\boldsymbol{\\tilde{y}}$.\n",
     "We use a more compact notation in terms of the expectation value"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6094266b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}})^2\\right],\n",
@@ -942,14 +1112,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "643e0047",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and adding and subtracting $\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]$ we get"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1319bde5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}}+\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right],\n",
@@ -958,14 +1134,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9c6d6da1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which, using the abovementioned expectation values can be rewritten as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "855756ef",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right]+\\mathrm{Var}\\left[\\boldsymbol{\\tilde{y}}\\right]+\\sigma^2,\n",
@@ -974,7 +1156,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "34d24717",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "that is the rewriting in terms of the so-called bias, the variance of the model $\\boldsymbol{\\tilde{y}}$ and the variance of $\\boldsymbol{\\epsilon}$."
    ]
@@ -982,6 +1167,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
+   "id": "d51b6100",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1006,7 +1192,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_61_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_65_1.png"
       }
      },
      "output_type": "display_data"
@@ -1072,6 +1258,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
+   "id": "bd636def",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1121,7 +1308,13 @@
       "Error: 0.03781367141738902\n",
       "Bias^2: 0.03365768507152769\n",
       "Var: 0.0041559863458613296\n",
-      "0.03781367141738902 >= 0.03365768507152769 + 0.0041559863458613296 = 0.03781367141738902\n",
+      "0.03781367141738902 >= 0.03365768507152769 + 0.0041559863458613296 = 0.03781367141738902\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
       "Polynomial degree: 7\n",
       "Error: 0.027609773491022394\n",
       "Bias^2: 0.022999498260366198\n",
@@ -1168,7 +1361,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_62_2.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_66_3.png"
       }
      },
      "output_type": "display_data"
@@ -1225,7 +1418,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8cf88b3a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The bias-variance tradeoff summarizes the fundamental tension in\n",
     "machine learning, particularly supervised learning, between the\n",
@@ -1236,8 +1432,6 @@
     "train and less sensitive to sampling noise arising from having a\n",
     "finite-sized training dataset (smaller variance). \n",
     "\n",
-    "\n",
-    "\n",
     "The above equations tell us that in\n",
     "order to minimize the expected test error, we need to select a\n",
     "statistical learning method that simultaneously achieves low variance\n",
@@ -1245,7 +1439,6 @@
     "and squared bias is also nonnegative. Hence, we see that the expected\n",
     "test MSE can never lie below $Var(\\epsilon)$, the irreducible error.\n",
     "\n",
-    "\n",
     "What do we mean by the variance and bias of a statistical learning\n",
     "method? The variance refers to the amount by which our model would change if we\n",
     "estimated it using a different training data set. Since the training\n",
@@ -1256,13 +1449,13 @@
     "the training data can result in large changes in the model. In general, more\n",
     "flexible statistical methods have higher variance.\n",
     "\n",
-    "\n",
     "You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 6,
+   "id": "86bfc49a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1304,7 +1497,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_64_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_68_1.png"
       }
      },
      "output_type": "display_data"
@@ -1387,6 +1580,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
+   "id": "0c2a183a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1396,14 +1590,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Degree of polynomial:   1"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "\n",
+      "Degree of polynomial:   1\n",
       "Mean squared error on training data: 439230.69504801\n",
       "Mean squared error on test data: 481979.17861098\n",
       "Degree of polynomial:   2\n",
@@ -1447,16 +1634,16 @@
       "Mean squared error on test data: 0.17446471\n",
       "Degree of polynomial:  13\n",
       "Mean squared error on training data: 0.00759119\n",
-      "Mean squared error on test data: 1.08131003\n",
-      "Degree of polynomial:  14\n",
-      "Mean squared error on training data: 0.00472199\n",
-      "Mean squared error on test data: 0.81333804\n"
+      "Mean squared error on test data: 1.08131003\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Degree of polynomial:  14\n",
+      "Mean squared error on training data: 0.00472199\n",
+      "Mean squared error on test data: 0.81333804\n",
       "Degree of polynomial:  15\n",
       "Mean squared error on training data: 0.00410478\n",
       "Mean squared error on test data: 92.09172409\n",
@@ -1471,16 +1658,16 @@
       "Mean squared error on test data: 108.27092910\n",
       "Degree of polynomial:  19\n",
       "Mean squared error on training data: 0.00156376\n",
-      "Mean squared error on test data: 1371.99051150\n",
-      "Degree of polynomial:  20\n",
-      "Mean squared error on training data: 0.00137818\n",
-      "Mean squared error on test data: 1887.86252988\n"
+      "Mean squared error on test data: 1371.99051150\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Degree of polynomial:  20\n",
+      "Mean squared error on training data: 0.00137818\n",
+      "Mean squared error on test data: 1887.86252988\n",
       "Degree of polynomial:  21\n",
       "Mean squared error on training data: 0.00118508\n",
       "Mean squared error on test data: 14859.69908626\n",
@@ -1495,16 +1682,16 @@
       "Mean squared error on test data: 1277.61702282\n",
       "Degree of polynomial:  25\n",
       "Mean squared error on training data: 0.00079129\n",
-      "Mean squared error on test data: 128664.31650694\n",
-      "Degree of polynomial:  26\n",
-      "Mean squared error on training data: 0.00076905\n",
-      "Mean squared error on test data: 19003.94822514\n"
+      "Mean squared error on test data: 128664.31650694\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "Degree of polynomial:  26\n",
+      "Mean squared error on training data: 0.00076905\n",
+      "Mean squared error on test data: 19003.94822514\n",
       "Degree of polynomial:  27\n",
       "Mean squared error on training data: 0.00068946\n",
       "Mean squared error on test data: 2379.66219404\n",
@@ -1520,9 +1707,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/626635268.py:73: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/626635268.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_94529/626635268.py:74: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/626635268.py:74: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(testerror), label='Test Error')\n"
      ]
     },
@@ -1535,7 +1722,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_65_7.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_69_6.png"
       }
      },
      "output_type": "display_data"
@@ -1624,7 +1811,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2c6c9e89",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Cross-validation\n",
     "\n",
@@ -1643,7 +1833,6 @@
     "choosing $k=n$. This particular case is referred to as leave-one-out\n",
     "cross-validation (LOOCV). \n",
     "\n",
-    "\n",
     "* Define a range of interest for the penalty parameter.\n",
     "\n",
     "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n",
@@ -1653,7 +1842,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "71738b2a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -1666,7 +1858,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "14db46b6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n",
     "\n",
@@ -1677,7 +1872,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "95e5c8e4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -1688,7 +1886,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7e60f51d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "For the various values of $k$\n",
     "\n",
@@ -1706,7 +1907,6 @@
     "\n",
     "d. Retain the evaluation score and discard the model\n",
     "\n",
-    "\n",
     "5. Summarize the model using the sample of model evaluation scores\n",
     "\n",
     "The code here uses Ridge regression with cross-validation (CV)  resampling and $k$-fold CV in order to fit a specific polynomial."
@@ -1715,6 +1915,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
+   "id": "2cef0eb7",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1729,7 +1930,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_71_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_75_0.png"
       }
      },
      "output_type": "display_data"
@@ -1829,7 +2030,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f501c9cf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "More examples of the application of cross-validation follow here."
    ]
@@ -1837,6 +2041,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
+   "id": "30e1e320",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1846,7 +2051,7 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10\n",
       "  plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n"
      ]
     },
@@ -1859,7 +2064,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_73_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_77_1.png"
       }
      },
      "output_type": "display_data"
@@ -1937,10 +2142,21 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "383e5c2a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Note that we have kept the intercept in the first column of design matrix $\\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and do not fit intercept. See the discussions below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1f9cd409",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "Note that we have kept the intercept in the first column of design matrix $\\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and does not fit intercept. See the discussions below.\n",
-    "\n",
     "## More on Rescaling data\n",
     "\n",
     "We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.\n",
@@ -1963,7 +2179,6 @@
     "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n",
     "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n",
     "\n",
-    "\n",
     "If our predictors represent different scales, then it is important to\n",
     "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n",
     "column from the corresponding column and dividing the column with its\n",
@@ -1989,8 +2204,6 @@
     "coefficient term, than if measured in millimeters.\n",
     "This can clearly lead to problems in evaluating the cost/loss functions.\n",
     "\n",
-    "\n",
-    "\n",
     "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n",
     "on your eventual new data set  before making a prediction. If we translate this into a Python code, it would could be implemented as follows"
    ]
@@ -1998,6 +2211,7 @@
   {
    "cell_type": "code",
    "execution_count": 10,
+   "id": "eab81633",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2035,7 +2249,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0ca51a54",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us try to understand what this may imply mathematically when we\n",
     "subtract the mean values, also known as *zero centering*. For\n",
@@ -2046,7 +2263,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fa3a4110",
+   "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",
@@ -2055,7 +2275,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9f499302",
+   "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",
@@ -2069,7 +2292,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b1f7ba52",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
@@ -2078,14 +2304,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7ff75b7f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "for all $j$. For $\\beta_0$ we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "eac37d3c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\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",
@@ -2094,14 +2326,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6e653464",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Multiplying away the constant $2/n$, we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8afa60a1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
@@ -2110,7 +2348,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "055b7975",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
     "Our result for $\\beta_0$ simplifies then to"
@@ -2118,7 +2359,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a2bf6e5d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
@@ -2127,14 +2371,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e15afbd5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We obtain then"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d2f9a064",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
@@ -2143,14 +2393,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e8035d8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we define"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0bbcef44",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n",
@@ -2159,14 +2415,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "919462c2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and the mean value of the outputs as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3a0899ec",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
@@ -2175,14 +2437,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9b255ec5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7716aca1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
@@ -2191,14 +2459,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "24863f48",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6c10f940",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
@@ -2207,14 +2481,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7afefe95",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can rewrite the latter equation as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a2e7bc1a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\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",
@@ -2223,14 +2503,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ded7d78",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have defined"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bcac7366",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
@@ -2239,18 +2525,22 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bafe8156",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
     "\n",
-    "\n",
-    "\n",
     "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7c91d280",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
@@ -2259,14 +2549,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1b81d6de",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7d6d497e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
@@ -2275,7 +2571,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "772593b2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
     "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
@@ -2285,7 +2584,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3edbdb56",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
@@ -2294,11 +2596,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7de9723e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
     "\n",
-    "\n",
     "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to  Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n",
     "Note also that we do not split the data into training and test."
    ]
@@ -2306,6 +2610,7 @@
   {
    "cell_type": "code",
    "execution_count": 11,
+   "id": "2e4a0363",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2341,7 +2646,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_107_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_112_1.png"
       }
      },
      "output_type": "display_data"
@@ -2438,7 +2743,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c6d3a071",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The intercept is the value of our output/target variable\n",
     "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
@@ -2449,14 +2757,17 @@
     "meaning that the MSE can be penalized by the value of the\n",
     "intercept. Not including the intercept in the fit, means that the\n",
     "regularization term does not include $\\beta_0$. For different values\n",
-    "of $\\lambda$, this may lead to differeing MSE values. \n",
+    "of $\\lambda$, this may lead to different MSE values. \n",
     "\n",
     "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "db803b77",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
@@ -2465,14 +2776,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d7a9c9d8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "but when we take out the intercept, this equation becomes"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e08af763",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
@@ -2481,14 +2798,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "10fe24e4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "For Lasso regression we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6feb3bdd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
@@ -2497,7 +2820,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "657a0777",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It means that, when scaling the design matrix and the outputs/targets,\n",
     "by subtracting the mean values, we have an optimization problem which\n",
@@ -2506,13 +2832,13 @@
     "back the intercept, we will get a MSE which then contains the\n",
     "intercept.\n",
     "\n",
-    "\n",
     "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known  vanilla data set."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 12,
+   "id": "9f21317d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2621,7 +2947,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_115_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_120_1.png"
       }
      },
      "output_type": "display_data"
@@ -2697,7 +3023,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "20de82fd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
     "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
@@ -2708,6 +3037,7 @@
   {
    "cell_type": "code",
    "execution_count": 13,
+   "id": "4138ed50",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2838,7 +3168,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_117_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_122_1.png"
       }
      },
      "output_type": "display_data"
@@ -2927,7 +3257,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7e9dfda5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see here, when compared to the code which includes explicitely the\n",
     "intercept column, that our MSE value is actually smaller. This is\n",
@@ -2938,8 +3271,16 @@
     "also that the problem with the intercept occurs mainly in these type\n",
     "of polynomial fitting problem.\n",
     "\n",
-    "The next example is indeed an example where all these discussions about the role of intercept are not present.\n",
-    "\n",
+    "The next example is indeed an example where all these discussions about the role of intercept are not present."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a7eb252c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## More complicated Example: The Ising model\n",
     "\n",
     "The one-dimensional Ising model with nearest neighbor interaction, no\n",
@@ -2948,7 +3289,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "990ff280",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2963,7 +3307,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ea46b31",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n",
     "in the system is determined by $L$. For the one-dimensional system\n",
@@ -2977,6 +3324,7 @@
   {
    "cell_type": "code",
    "execution_count": 14,
+   "id": "fa60aaab",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3007,7 +3355,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b9e8214e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we use ordinary least squares\n",
     "regression to predict the energy for the nearest neighbor\n",
@@ -3020,7 +3371,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a578cd63",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3035,7 +3389,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d90dcfee",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we allow for interactions beyond the nearest neighbors and a state dependent\n",
     "coupling constant. This latter expression can be formulated as\n",
@@ -3044,7 +3401,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c4eec69a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3059,7 +3419,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ae46a11b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n",
     "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
@@ -3068,7 +3431,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "87ebb9b8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3083,7 +3449,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "856efd3f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We split the data in training and test data as discussed in the previous example"
    ]
@@ -3091,6 +3460,7 @@
   {
    "cell_type": "code",
    "execution_count": 15,
+   "id": "5ae2828f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3106,14 +3476,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0d6201d9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the ordinary least squares method we choose the cost function"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4af380a7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3128,7 +3504,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e8eefa3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n",
     "This yields the expression for $\\boldsymbol{\\beta}$ to be"
@@ -3136,7 +3515,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a1c0765c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n",
@@ -3145,7 +3527,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c754e28b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n",
     "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n",
@@ -3156,6 +3541,7 @@
   {
    "cell_type": "code",
    "execution_count": 16,
+   "id": "ff056cf9",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3174,7 +3560,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "524f259e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Doing the inversion directly turns out to be a bad idea since the matrix\n",
     "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n",
@@ -3184,7 +3573,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "52be4553",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n",
@@ -3193,14 +3585,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c37ff09a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where the pseudoinverse of $\\boldsymbol{X}$ is given by"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ea49a105",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n",
@@ -3209,7 +3607,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bf31067a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Using singular value decomposition we can decompose the matrix  $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n",
     "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n",
@@ -3219,7 +3620,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ef2c839",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3234,7 +3638,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "96bbf134",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Note that solving this equation by actually doing the pseudoinverse\n",
     "(which is what we will do) is not a good idea as this operation scales\n",
@@ -3247,6 +3654,7 @@
   {
    "cell_type": "code",
    "execution_count": 17,
+   "id": "5e729b1e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3261,6 +3669,7 @@
   {
    "cell_type": "code",
    "execution_count": 18,
+   "id": "80b923d1",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3272,7 +3681,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b4cf2f1a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "When extracting the $J$-matrix  we need to make sure that we remove the intercept, as is done here"
    ]
@@ -3280,6 +3692,7 @@
   {
    "cell_type": "code",
    "execution_count": 19,
+   "id": "93be2c0d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3291,7 +3704,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ae869089",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "A way of looking at the coefficients in $J$ is to plot the matrices as images."
    ]
@@ -3299,6 +3715,7 @@
   {
    "cell_type": "code",
    "execution_count": 20,
+   "id": "8d6d8152",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3308,9 +3725,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/4162706317.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/4162706317.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -3323,7 +3740,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_148_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_154_1.png"
       }
      },
      "output_type": "display_data"
@@ -3342,7 +3759,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8092ac84",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It is interesting to note that OLS\n",
     "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n",
@@ -3352,10 +3772,6 @@
     "\n",
     "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n",
     "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
     "Let us now \n",
     "focus on Ridge and Lasso regression as well. We repeat some of the\n",
     "basic parts of the Ising model and the setup of the training and test\n",
@@ -3366,7 +3782,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d9251cde",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3381,7 +3800,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "abe454c9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n",
     "\n",
@@ -3391,6 +3813,7 @@
   {
    "cell_type": "code",
    "execution_count": 21,
+   "id": "89f8fbd0",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3422,14 +3845,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e06639d4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "A more general form for the one-dimensional Ising model is"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "43ef0f40",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3444,7 +3873,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "523dc03b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we allow for interactions beyond the nearest neighbors and a more\n",
     "adaptive coupling matrix. This latter expression can be formulated as\n",
@@ -3453,7 +3885,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0f389eef",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3468,7 +3903,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b6e00f24",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n",
     "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
@@ -3477,7 +3915,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ea1084d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3492,7 +3933,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5551bb05",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We organize the data as we did above"
    ]
@@ -3500,6 +3944,7 @@
   {
    "cell_type": "code",
    "execution_count": 22,
+   "id": "f5dd7795",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3525,7 +3970,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f36c807b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We will do all fitting with **Scikit-Learn**,"
    ]
@@ -3533,6 +3981,7 @@
   {
    "cell_type": "code",
    "execution_count": 23,
+   "id": "eb701dd4",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3544,7 +3993,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bd356ada",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "When  extracting the $J$-matrix we make sure to remove the intercept"
    ]
@@ -3552,6 +4004,7 @@
   {
    "cell_type": "code",
    "execution_count": 24,
+   "id": "ae737db8",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3563,7 +4016,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d152a32a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "And then we plot the results"
    ]
@@ -3571,6 +4027,7 @@
   {
    "cell_type": "code",
    "execution_count": 25,
+   "id": "8d713977",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3580,9 +4037,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3777801602.py:6: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3777801602.py:7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -3595,7 +4052,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_166_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_172_1.png"
       }
      },
      "output_type": "display_data"
@@ -3614,11 +4071,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ede30c18",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The results agree perfectly with our previous discussion where we used our own code.\n",
     "\n",
-    "\n",
     "Having explored the ordinary least squares we move on to ridge\n",
     "regression. In ridge regression we include a **regularizer**. This\n",
     "involves a new cost function which leads to a new estimate for the\n",
@@ -3628,31 +4087,26 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "eac772e9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "6\n",
-    "0\n",
-    " \n",
-    "<\n",
-    "<\n",
-    "<\n",
-    "!\n",
-    "!\n",
-    "M\n",
-    "A\n",
-    "T\n",
-    "H\n",
-    "_\n",
-    "B\n",
-    "L\n",
-    "O\n",
-    "C\n",
-    "K"
+    "\n",
+    "
    \n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}.\n",
+    "\\label{_auto11} \\tag{11}\n",
+    "\\end{equation}\n",
+    "$$"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 26,
+   "id": "b66675f0",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3662,9 +4116,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/438060758.py:9: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/438060758.py:10: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -3677,7 +4131,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_169_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_175_1.png"
       }
      },
      "output_type": "display_data"
@@ -3700,14 +4154,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "71aa37b5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8b70ef66",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -3722,7 +4182,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "079981ff",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**."
    ]
@@ -3730,6 +4193,7 @@
   {
    "cell_type": "code",
    "execution_count": 27,
+   "id": "c8d4a180",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3739,9 +4203,9 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3544313922.py:8: MatplotlibDeprecationWarning: Auto-removal of grids by pcolor() and pcolormesh() is deprecated since 3.5 and will be removed two minor releases later; please call grid(False) first.\n",
       "  cb = fig.colorbar(im)\n",
-      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94529/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+      "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_41811/3544313922.py:9: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
       "  cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
      ]
     },
@@ -3754,7 +4218,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_173_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_179_1.png"
       }
      },
      "output_type": "display_data"
@@ -3776,21 +4240,22 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "57839941",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It is quite striking how LASSO breaks the symmetry of the coupling\n",
     "constant as opposed to ridge and OLS. We get a sparse solution with\n",
     "$J_{j, j + 1} = -1$.\n",
     "\n",
-    "\n",
-    "\n",
-    "\n",
     "We see how the different models perform for a different set of values for $\\lambda$."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 28,
+   "id": "0e3bbb4e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3801,7 +4266,7 @@
      "output_type": "stream",
      "text": [
       "\r",
-      "  0%|                                                                                                                              | 0/10 [00:00
+
 # # Resampling Methods
-# 
+
 # ## Introduction
 # 
 # Resampling methods are an indispensable tool in modern
@@ -24,7 +27,6 @@
 # In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
 # cross-validation and the bootstrap method. 
 # 
-# 
 # Resampling approaches can be computationally expensive, because they
 # involve fitting the same statistical method multiple times using
 # different subsets of the training data. However, due to recent
@@ -41,23 +43,20 @@
 # level of flexibility for a model is known as model selection. The
 # bootstrap is widely used.
 # 
-# 
 # * Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods
 # 
 # * The results can be analysed with the same statistical tools as we would use analysing experimental data.
 # 
 # * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
-# 
+
 # ## Reminder on Statistics
 # 
-# 
 # * As in other experiments, many numerical  experiments have two classes of errors:
 # 
 #   * Statistical errors
 # 
 #   * Systematical errors
 # 
-# 
 # * Statistical errors can be estimated using standard tools from statistics
 # 
 # * Systematical errors are method specific and must be treated differently from case to case. 
@@ -69,7 +68,6 @@
 # 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.:
@@ -93,7 +91,6 @@
 # notation above $\mathbf{X}_{i,\ast}$ means that we are looking at the
 # row number $i$ and perform a sum over all values $p$.
 # 
-# 
 # 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 $f(\boldsymbol{x})$ and  a normal distributed error $\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)$
 # which describe our data
@@ -140,7 +137,6 @@
 # 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). 
 # 
-# 
 # With the OLS expressions for the parameters $\boldsymbol{\beta}$ we can evaluate the expectation value
 
 # $$
@@ -148,7 +144,7 @@
 # $$
 
 # This means that the estimator of the regression parameters is unbiased.
-# v
+# 
 # We can also calculate the variance
 # 
 # The variance of $\boldsymbol{\beta}$ is
@@ -185,7 +181,6 @@
 # [(\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. 
@@ -215,10 +210,8 @@
 
 # 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. 
-# 
-# 
-# 
+# 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.
+
 # ## Resampling methods
 # 
 # With all these analytical equations for both the OLS and Ridge
@@ -236,12 +229,10 @@
 # 
 # 2. training error $\mathrm{Err_{Train}}$, which is the average loss over the training data.
 # 
-# As our model becomes more and more complex, more of the training data tends to  used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.
+# As our model becomes more and more complex, more of the training data tends to  be used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.
 # For a certain level of complexity the test error will reach minimum, before starting to increase again. The
 # training error reaches a saturation.
 # 
-# 
-# 
 # Two famous
 # resampling methods are the **independent bootstrap** and **the jackknife**. 
 # 
@@ -257,7 +248,6 @@
 # variance of $\overline{X}$ (which often is the case), then there is no
 # need for bootstrapping. 
 # 
-# 
 # The Jackknife works by making many replicas of the estimator $\widehat{\beta}$. 
 # The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\boldsymbol{x} = (x_1,x_2,\cdots,X_n)$. 
 # Let $\boldsymbol{x}_i$ denote the vector
@@ -328,8 +318,6 @@ t = jackknife(x, stat)
 # estimate all the interesting parameters of $p(\boldsymbol{t})$ using point
 # estimators.  
 # 
-# 
-# 
 # In the case that $\widehat{\beta}$ has
 # more than one component, and the components are independent, we use the
 # same estimator on each component separately.  If the probability
@@ -344,7 +332,6 @@ t = jackknife(x, stat)
 # idea is to use the relative frequency of $\widehat{\beta}^*$
 # (think of a histogram) as an estimate of $p(\boldsymbol{t})$.
 # 
-# 
 # But
 # unless there is enough information available about the process that
 # generated $X_1,X_2,\cdots,X_n$, $p(x)$ is in general
@@ -354,13 +341,11 @@ t = jackknife(x, stat)
 # the relative frequency of the observations, will we obtain the same
 # result in some asymptotic sense? The answer is yes.
 # 
-# 
 # Instead of generating the histogram for the relative
 # frequency of the observation $X_i$, just draw the values
 # $(X_1^*,X_2^*,\cdots,X_n^*)$ with replacement from the vector
 # $\boldsymbol{X}$. 
 # 
-# 
 # The independent bootstrap works like this: 
 # 
 # 1. Draw with replacement $n$ numbers for the observed variables $\boldsymbol{x} = (x_1,x_2,\cdots,x_n)$. 
@@ -378,7 +363,7 @@ t = jackknife(x, stat)
 # histogram of the relative frequency of $\widehat{\beta}^*$. Instead
 # you use the estimators corresponding to the statistic of interest. For
 # example, if you are interested in estimating the variance of $\widehat
-# \beta$, apply the etsimator $\widehat \sigma^2$ to the values
+# \beta$, apply the estimator $\widehat \sigma^2$ to the values
 # $\widehat \beta^*$.
 # 
 # Before we proceed however, we need to remind ourselves about a central
@@ -387,7 +372,6 @@ t = jackknife(x, stat)
 # (and other resampling methods) work so well on independent and
 # identically distributed variables.
 # 
-# 
 # Suppose we have a PDF $p(x)$ from which we generate  a series $N$
 # of averages $\langle x_i \rangle$. Each mean value $\langle x_i \rangle$
 # is viewed as the average of a specific measurement, e.g., throwing 
@@ -404,7 +388,6 @@ t = jackknife(x, stat)
 
 # the question we pose is which is the PDF of the new variable $z$.
 # 
-# 
 # The probability of obtaining an average value $z$ is the product of the 
 # probabilities of obtaining arbitrary individual mean values $x_i$,
 # but with the constraint that the average is $z$. We can express this through
@@ -420,8 +403,6 @@ t = jackknife(x, stat)
 # be independent, which in turn means that we can express $\tilde{p}$ as the 
 # product of individual $p(x_i)$.  The independence assumption is important in the derivation of the central limit theorem.
 # 
-# 
-# 
 # If we use the integral expression for the $\delta$-function
 
 # $$
@@ -472,7 +453,6 @@ t = jackknife(x, stat)
 # $\sigma^2_m=\sigma^2/m$, where $\sigma$ is the variance of the PDF $p(x)$
 # and $\mu$ is also the mean of the PDF $p(x)$. 
 # 
-# 
 # Thus, the central limit theorem states that the PDF $\tilde{p}(z)$ of
 # the average of $m$ random values corresponding to a PDF $p(x)$ 
 # is a normal distribution whose mean is the 
@@ -504,13 +484,10 @@ t = jackknife(x, stat)
 # we generate in various calculations do always exhibit some
 # correlations.
 # 
-# 
-# 
 # The theorem is satisfied by a large class of PDFs. Note however that for a
 # finite $m$, it is not always possible to find a closed form /analytic expression for
 # $\tilde{p}(x)$.
 # 
-# 
 # The following code starts with a Gaussian distribution with mean value
 # $\mu =100$ and variance $\sigma=15$. We use this to generate the data
 # used in the bootstrap analysis. The bootstrap analysis returns a data
@@ -577,7 +554,6 @@ plt.show()
 
 # ## The bias-variance tradeoff
 # 
-# 
 # We will discuss the bias-variance tradeoff in the context of
 # continuous predictions such as regression. However, many of the
 # intuitions and ideas discussed here also carry over to classification
@@ -609,13 +585,13 @@ plt.show()
 # \mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
 # $$
 
-# The three terms represent the square of the bias of the learning
+# The first term represents the square of the bias of the learning
 # method, which can be thought of as the error caused by the simplifying
 # assumptions built into the method. The second term represents the
 # variance of the chosen model and finally the last terms is variance of
 # the error $\boldsymbol{\epsilon}$.
 # 
-# To derive this equation, we need to recall that the variance of $\boldsymbol{y}$ and $\boldsymbol{\epsilon}$ are both equal to $\sigma^2$. The mean value of $\boldsymbol{\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\boldsymbol{\tilde{y}}$.
+# To derive this equation, we need to recall that the variance of $\boldsymbol{y}$ and $\boldsymbol{\epsilon}$ are both equal to $\sigma^2$. The mean value of $\boldsymbol{\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastic variable, idem for $\boldsymbol{\tilde{y}}$.
 # We use a more compact notation in terms of the expectation value
 
 # $$
@@ -755,8 +731,6 @@ plt.show()
 # train and less sensitive to sampling noise arising from having a
 # finite-sized training dataset (smaller variance). 
 # 
-# 
-# 
 # The above equations tell us that in
 # order to minimize the expected test error, we need to select a
 # statistical learning method that simultaneously achieves low variance
@@ -764,7 +738,6 @@ plt.show()
 # and squared bias is also nonnegative. Hence, we see that the expected
 # test MSE can never lie below $Var(\epsilon)$, the irreducible error.
 # 
-# 
 # What do we mean by the variance and bias of a statistical learning
 # method? The variance refers to the amount by which our model would change if we
 # estimated it using a different training data set. Since the training
@@ -775,7 +748,6 @@ plt.show()
 # the training data can result in large changes in the model. In general, more
 # flexible statistical methods have higher variance.
 # 
-# 
 # You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest.
 
 # In[6]:
@@ -954,7 +926,6 @@ plt.show()
 # choosing $k=n$. This particular case is referred to as leave-one-out
 # cross-validation (LOOCV). 
 # 
-# 
 # * Define a range of interest for the penalty parameter.
 # 
 # * Divide the data set into training and test set comprising samples $\{1, \ldots, n\} \setminus i$ and $\{ i \}$, respectively.
@@ -997,7 +968,6 @@ plt.show()
 # 
 # d. Retain the evaluation score and discard the model
 # 
-# 
 # 5. Summarize the model using the sample of model evaluation scores
 # 
 # The code here uses Ridge regression with cross-validation (CV)  resampling and $k$-fold CV in order to fit a specific polynomial.
@@ -1170,8 +1140,8 @@ plt.legend()
 plt.show()
 
 
-# Note that we have kept the intercept in the first column of design matrix $\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and does not fit intercept. See the discussions below.
-# 
+# Note that we have kept the intercept in the first column of design matrix $\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and do not fit intercept. See the discussions below.
+
 # ## More on Rescaling data
 # 
 # We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.
@@ -1194,7 +1164,6 @@ plt.show()
 # $\boldsymbol{\beta}$, are derived under the assumption that both $\boldsymbol{y}$ and
 # $\boldsymbol{X}$ are zero centered, that is we subtract the mean values.
 # 
-# 
 # If our predictors represent different scales, then it is important to
 # standardize the design matrix $\boldsymbol{X}$ by subtracting the mean of each
 # column from the corresponding column and dividing the column with its
@@ -1220,8 +1189,6 @@ plt.show()
 # coefficient term, than if measured in millimeters.
 # This can clearly lead to problems in evaluating the cost/loss functions.
 # 
-# 
-# 
 # Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
 # on your eventual new data set  before making a prediction. If we translate this into a Python code, it would could be implemented as follows
 
@@ -1332,8 +1299,6 @@ y_pred = y_pred + y_train_mean
 
 # the mean value for all elements of the column vector $\boldsymbol{x}_j$.
 # 
-# 
-# 
 # 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)
 
 # $$
@@ -1357,7 +1322,6 @@ y_pred = y_pred + y_train_mean
 
 # What does this mean? And why do we insist on all this? Let us look at some examples.
 # 
-# 
 # This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to  Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.
 # Note also that we do not split the data into training and test.
 
@@ -1461,7 +1425,7 @@ plt.show()
 # meaning that the MSE can be penalized by the value of the
 # intercept. Not including the intercept in the fit, means that the
 # regularization term does not include $\beta_0$. For different values
-# of $\lambda$, this may lead to differeing MSE values. 
+# of $\lambda$, this may lead to different MSE values. 
 # 
 # To remind the reader, the regularization term, with the intercept in Ridge regression, is given by
 
@@ -1488,7 +1452,6 @@ plt.show()
 # back the intercept, we will get a MSE which then contains the
 # intercept.
 # 
-# 
 # Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known  vanilla data set.
 
 # In[12]:
@@ -1659,7 +1622,7 @@ plt.show()
 # of polynomial fitting problem.
 # 
 # The next example is indeed an example where all these discussions about the role of intercept are not present.
-# 
+
 # ## More complicated Example: The Ising model
 # 
 # The one-dimensional Ising model with nearest neighbor interaction, no
@@ -1885,10 +1848,6 @@ plt.show()
 # 
 # In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
 # 
-# 
-# 
-# 
-# 
 # Let us now 
 # focus on Ridge and Lasso regression as well. We repeat some of the
 # basic parts of the Ising model and the setup of the training and test
@@ -2031,31 +1990,21 @@ plt.show()
 
 # The results agree perfectly with our previous discussion where we used our own code.
 # 
-# 
 # Having explored the ordinary least squares we move on to ridge
 # regression. In ridge regression we include a **regularizer**. This
 # involves a new cost function which leads to a new estimate for the
 # weights $\boldsymbol{\beta}$. This results in a penalized regression problem. The
 # cost function is given by
 
-# 6
-# 0
-#  
-# <
-# <
-# <
-# !
-# !
-# M
-# A
-# T
-# H
-# _
-# B
-# L
-# O
-# C
-# K
+# 
+# 
    
+# 
+# $$
+# \begin{equation}
+#     C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
+# \label{_auto11} \tag{11}
+# \end{equation}
+# $$
 
 # In[26]:
 
@@ -2108,9 +2057,6 @@ plt.show()
 # constant as opposed to ridge and OLS. We get a sparse solution with
 # $J_{j, j + 1} = -1$.
 # 
-# 
-# 
-# 
 # We see how the different models perform for a different set of values for $\lambda$.
 
 # In[28]:
@@ -2159,7 +2105,6 @@ plt.show()
 # much. Ridge is more stable over a larger range of values for
 # $\lambda$, but eventually also fades away.
 # 
-# 
 # To determine which value of $\lambda$ is best we plot the accuracy of
 # the models when predicting the training and the testing set. We expect
 # the accuracy of the training set to be quite good, but if the accuracy
@@ -2205,21 +2150,13 @@ plt.show()
 # From the above figure we can see that LASSO with $\lambda = 10^{-2}$
 # achieves a very good accuracy on the test set. This by far surpasses the
 # other models for all values of $\lambda$.
-# 
-# 
-# 
-# 
-# 
-# 
+
 # ## Exercises and Projects
 # 
-# 
-# 
 # The main aim of this project is to study in more detail various
 # regression methods, including the Ordinary Least Squares (OLS) method,
 # The total score is **100** points. Each subtask has its own final score.
 # 
-# 
 # We will first study how to fit polynomials to a specific
 # two-dimensional function called [Franke's
 # function](http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf).  This
@@ -2230,7 +2167,6 @@ plt.show()
 # proper assessment of our models. We will also study in detail the
 # so-called Bias-Variance trade off.
 # 
-# 
 # The Franke function, which is a weighted sum of four exponentials  reads as follows
 
 # $$
@@ -2259,7 +2195,6 @@ plt.show()
 # beyond the second-order polynomials metioned above and explore 
 # which polynomial fits the data best.
 # 
-# 
 # The Python code for the Franke function is included here (it performs also a three-dimensional plot of it)
 
 # In[30]:
@@ -2355,11 +2290,8 @@ plt.show()
 # data and say test data.  An accepted rule of thumb is to use
 # approximately $2/3$ to $4/5$ of the data as training data.
 # 
-# 
 # You can easily reuse the solutions to your exercises from week 35 and week 36.
-# 
-# 
-# 
+
 # ### Exercise: Bias-variance trade-off and resampling techniques
 # 
 # Our aim here is to study the bias-variance trade-off by implementing the **bootstrap** resampling technique.
@@ -2422,8 +2354,7 @@ plt.show()
 # of data points, and possibly also your training and test data using the **bootstrap** resampling method.
 # 
 # Note also that when you calculate the bias, in all applications you don't know the function values $f_i$. You would hence replace them with the actual data points $y_i$.
-# 
-# 
+
 # ### Exercise:  Cross-validation as resampling techniques, adding more complexity
 # 
 # The aim here is to write your own code for another widely popular
@@ -2440,8 +2371,7 @@ plt.show()
 # you got from your **bootstrap** code. Comment your results. Try $5-10$
 # folds.  You can also compare your own cross-validation code with the
 # one provided by **Scikit-Learn**.
-# 
-# 
+
 # ### Exercise: Ridge Regression on the Franke function  with resampling
 # 
 # Write your own code for the Ridge method, either using matrix
@@ -2452,8 +2382,8 @@ plt.show()
 # dependence on $\lambda$.
 # 
 # Study also the bias-variance trade-off as function of various values of
-# the parameter $\lambda$. For the bias-variance trade-off, use the **bootstrap** resampling method. Comment your results. 
-# 
+# the parameter $\lambda$. For the bias-variance trade-off, use the **bootstrap** resampling method. Comment your results.
+
 # ### Exercise: Lasso Regression on the Franke function  with resampling
 # 
 # This exercise is essentially a repeat of the previous two ones, but now
@@ -2461,8 +2391,8 @@ plt.show()
 # you can also use the functionalities of **Scikit-Learn** (recommended). 
 # Give a
 # critical discussion of the three methods and a judgement of which
-# model fits the data best.  Perform here as well an analysis of the bias-variance trade-off using the **bootstrap** resampling technique and an analysis of the mean squared error using cross-validation. 
-# 
+# model fits the data best.  Perform here as well an analysis of the bias-variance trade-off using the **bootstrap** resampling technique and an analysis of the mean squared error using cross-validation.
+
 # ### Exercise: Analysis of real data
 # 
 # With our codes functioning and having been tested properly on a
@@ -2491,7 +2421,7 @@ scipy.misc.imread
 # Here is a simple part of a Python code which reads and plots the data
 # from such files
 
-# In[ ]:
+# In[32]:
 
 
 """
@@ -2519,10 +2449,8 @@ plt.show()
 # Austfjell, again in Norway.
 # Feel free to produce your own terrain data.
 # 
-# 
 # Alternatively, if you would like to use another data set, feel free to do so. This could be data close to your reseach area or simply a data set you found interesting. See for example [kaggle.com](https://www.kaggle.com/datasets) for examples.
 # 
-# 
 # Our final part deals with the parameterization of your digital terrain
 # data (or your own data).  We will apply all three methods for linear regression, the same type (or higher order) of polynomial
 # approximation and cross-validation as resampling technique to evaluate which
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diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter4.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter4.ipynb
index d9e89993b..1c1376439 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter4.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter4.ipynb
@@ -2,12 +2,32 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f63b8c1a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    ""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9a284cf5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Logistic Regression"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d78b73e2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "# Logistic Regression\n",
-    "\n",
-    "\n",
-    "\n",
     "## Logistic Regression\n",
     "\n",
     "In linear regression our main interest was centered on learning the\n",
@@ -22,7 +42,6 @@
     "the product of the design matrices, linear regression gives then a\n",
     "simple recipe for fitting our data.\n",
     "\n",
-    "\n",
     "Classification problems, however, are concerned with outcomes taking\n",
     "the form of discrete variables (i.e. categories). We may for example,\n",
     "on the basis of DNA sequencing for a number of patients, like to find\n",
@@ -39,7 +58,6 @@
     "binary outcome, true or false, positive or negative, success or\n",
     "failure etc.\n",
     "\n",
-    "\n",
     "Logistic regression will also serve as our stepping stone towards\n",
     "neural network algorithms and supervised deep learning. For logistic\n",
     "learning, the minimization of the cost function leads to a non-linear\n",
@@ -52,10 +70,16 @@
     "\n",
     "We note also that many of the topics discussed here on logistic \n",
     "regression are also commonly used in modern supervised Deep Learning\n",
-    "models, as we will see later.\n",
-    "\n",
-    "\n",
-    "\n",
+    "models, as we will see later."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d0c829b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Basics\n",
     "\n",
     "We consider the case where the dependent variables, also called the\n",
@@ -76,7 +100,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f1f4d5f2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\  1 & \\mathrm{yes} \\end{bmatrix}.\n",
@@ -85,7 +112,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e2f836b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Before moving to the logistic model, let us try to use our linear\n",
     "regression model to classify these two outcomes. We could for example\n",
@@ -98,7 +128,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6b58f1ba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -113,12 +146,14 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "54e3604b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{y}$ is a vector representing the possible outcomes, $\\boldsymbol{X}$ is our\n",
     "$n\\times p$ design matrix and $\\boldsymbol{\\beta}$ represents our estimators/predictors.\n",
     "\n",
-    "\n",
     "The main problem with our function is that it takes values on the\n",
     "entire real axis. In the case of logistic regression, however, the\n",
     "labels $y_i$ are discrete variables. A typical example is the credit\n",
@@ -134,13 +169,13 @@
     "favorable to use a ``soft\" classifier that outputs\n",
     "the probability of a given category. This leads us to the logistic function.\n",
     "\n",
-    "\n",
     "The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput  is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
+   "id": "cd1dd1c5",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -285,7 +320,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_5_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_8_1.png"
       }
      },
      "output_type": "display_data"
@@ -352,7 +387,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2e06ee82",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "What we could attempt however is to plot the mean value for each group."
    ]
@@ -360,6 +398,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
+   "id": "a774f001",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -374,7 +413,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_7_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_10_0.png"
       }
      },
      "output_type": "display_data"
@@ -393,7 +432,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e102fa56",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We are now trying to find a function $f(y\\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.\n",
     "In standard linear regression with a linear dependence on $x$, we would write this in terms of our model"
@@ -401,7 +443,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "75d995ba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "f(y_i\\vert x_i)=\\beta_0+\\beta_1 x_i.\n",
@@ -410,7 +455,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e2cacca5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This expression implies however that $f(y_i\\vert x_i)$ could take any\n",
     "value from minus infinity to plus infinity. If we however let\n",
@@ -421,9 +469,16 @@
     "us to a very popular model for the function $f$, namely the so-called\n",
     "Sigmoid function or logistic model. We will consider this function as\n",
     "representing the probability for finding a value of $y_i$ with a given\n",
-    "$x_i$.\n",
-    "\n",
-    "\n",
+    "$x_i$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af9ac586",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## The logistic function\n",
     "\n",
     "Another widely studied model, is the so-called \n",
@@ -441,7 +496,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "80aa3cbe",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n",
@@ -450,12 +508,22 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "28d49915",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "Note that $1-p(t)= p(-t)$.\n",
-    "\n",
-    "## Examples of likelihood functions used in logistic regression and nueral networks\n",
-    "\n",
+    "Note that $1-p(t)= p(-t)$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eeff6cd9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Examples of likelihood functions used in logistic regression and neural networks\n",
     "\n",
     "The following code plots the logistic function, the step function and other functions we will encounter from here and on."
    ]
@@ -463,6 +531,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
+   "id": "5ecec05e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -477,7 +546,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_13_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_18_0.png"
       }
      },
      "output_type": "display_data"
@@ -491,7 +560,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_13_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_18_1.png"
       }
      },
      "output_type": "display_data"
@@ -505,7 +574,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_13_2.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_18_2.png"
       }
      },
      "output_type": "display_data"
@@ -570,14 +639,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5802efe3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f85d14ea",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -589,7 +664,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6c4e0334",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n",
     "\n",
@@ -598,7 +676,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "df7facc9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(y_i=0\\vert x_i, \\boldsymbol{\\beta}) = 1-p(y_i=1\\vert x_i, \\boldsymbol{\\beta}).\n",
@@ -607,7 +688,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a920e6d8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In order to define the total likelihood for all possible outcomes from a  \n",
     "dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n",
@@ -619,7 +703,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a780bfe1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -630,14 +717,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9f810a44",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "from which we obtain the log-likelihood and our **cost/loss** function"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3fac4ef4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathcal{C}(\\boldsymbol{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\boldsymbol{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\boldsymbol{\\beta}))\\right]\\right).\n",
@@ -646,14 +739,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "75aa2c14",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Reordering the logarithms, we can rewrite the **cost/loss** function as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "780f2038",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathcal{C}(\\boldsymbol{\\beta}) = \\sum_{i=1}^n  \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
@@ -662,7 +761,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c8c940aa",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\beta$.\n",
     "Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that"
@@ -670,7 +772,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9d4a527b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathcal{C}(\\boldsymbol{\\beta})=-\\sum_{i=1}^n  \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
@@ -679,23 +784,27 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c1706606",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n",
     "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n",
     "\n",
-    "\n",
     "The cross entropy is a convex function of the weights $\\boldsymbol{\\beta}$ and,\n",
     "therefore, any local minimizer is a global minimizer. \n",
     "\n",
-    "\n",
     "Minimizing this\n",
     "cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fe8fb387",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n  \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n",
@@ -704,14 +813,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bb564e81",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "700c5443",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n  \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n",
@@ -720,7 +835,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f442c956",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us now define a vector $\\boldsymbol{y}$ with $n$ elements $y_i$, an\n",
     "$n\\times p$ matrix $\\boldsymbol{X}$ which contains the $x_i$ values and a\n",
@@ -730,7 +848,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "83937f6f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = -\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{p}\\right).\n",
@@ -739,7 +860,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "71819c1d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we in addition define a diagonal matrix $\\boldsymbol{W}$ with elements \n",
     "$p(y_i\\vert x_i,\\boldsymbol{\\beta})(1-p(y_i\\vert x_i,\\boldsymbol{\\beta})$, we can obtain a compact expression of the second derivative as"
@@ -747,7 +871,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "23dfd975",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T} = \\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X}.\n",
@@ -756,14 +883,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6ca9f74b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ddb3d93a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{ \\frac{p(\\boldsymbol{\\beta}\\boldsymbol{x})}{1-p(\\boldsymbol{\\beta}\\boldsymbol{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n",
@@ -772,14 +905,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0cfae560",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we defined $\\boldsymbol{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\boldsymbol{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7c427fc3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta}\\boldsymbol{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n",
@@ -788,7 +927,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6a6814bf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Till now we have mainly focused on two classes, the so-called binary\n",
     "system. Suppose we wish to extend to $K$ classes.  Let us for the sake\n",
@@ -797,7 +939,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "85282137",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{\\frac{p(C=1\\vert x)}{p(K\\vert x)}} = \\beta_{10}+\\beta_{11}x_1,\n",
@@ -806,14 +951,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4864f76d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "390e9a55",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n",
@@ -822,14 +973,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c3f39c44",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and so on till the class $C=K-1$ class"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "92f32a03",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n",
@@ -838,13 +995,14 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "677a5c4d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and the model is specified in term of $K-1$ so-called log-odds or\n",
     "**logit** transformations.\n",
     "\n",
-    "\n",
-    "\n",
     "In our discussion of neural networks we will encounter the above again\n",
     "in terms of a slightly modified function, the so-called **Softmax** function.\n",
     "\n",
@@ -861,7 +1019,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fa70e3bf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n",
@@ -870,14 +1031,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4c59400c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It is easy to extend to more predictors. The final class is"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f4073652",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n",
@@ -886,7 +1053,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "810e872a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and they sum to one. Our earlier discussions were all specialized to\n",
     "the case with two classes only. It is easy to see from the above that\n",
@@ -895,8 +1065,16 @@
     "To find the optimal parameters we would typically use a gradient\n",
     "descent method.  Newton's method and gradient descent methods are\n",
     "discussed in the material on [optimization\n",
-    "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n",
-    "\n",
+    "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f4ebc1da",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Wisconsin Cancer Data\n",
     "\n",
     "We show here how we can use a simple regression case on the breast\n",
@@ -907,6 +1085,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
+   "id": "b7a8cbac",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -967,7 +1146,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "10f74b93",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In addition to the above scores, we could also study the covariance (and the correlation matrix).\n",
     "We use **Pandas** to compute the correlation matrix."
@@ -976,6 +1158,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
+   "id": "d0b8025d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -990,7 +1173,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_51_0.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_57_0.png"
       }
      },
      "output_type": "display_data"
@@ -1004,7 +1187,7 @@
      },
      "metadata": {
       "filenames": {
-       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_51_1.png"
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_57_1.png"
       }
      },
      "output_type": "display_data"
@@ -1049,7 +1232,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9d84a2aa",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the above example we note two things. In the first plot we display\n",
     "the overlap of benign and malignant tumors as functions of the various\n",
@@ -1069,6 +1255,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
+   "id": "6144ea0a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1080,7 +1267,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5627f5ae",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and then"
    ]
@@ -1088,6 +1278,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
+   "id": "a7c8662a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1099,17 +1290,46 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d03ee2a7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Diagonalizing this matrix we can in turn say something about which\n",
     "features are of relevance and which are not. This leads  us to\n",
     "the classical Principal Component Analysis (PCA) theorem with\n",
-    "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html))."
+    "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)).\n",
+    "\n",
+    "Here we present a further way to present our results in terms of a so-called **confusion matrix**, the cumulative gain and the **ROC** curve.\n",
+    "This way of displaying our data are based upon different ways to classify our possible outcomes. Before we proceed we need some definitions.\n",
+    "1. **TP**: true positive or in other words, something equivalent with a proper classification\n",
+    "\n",
+    "2. **TN**: true negative, which is equivalent with a correct rejection\n",
+    "\n",
+    "3. **FP**: false positive, or in simpler words something that is equivalent with a false alarm\n",
+    "\n",
+    "4. **FN**: false negative, which is mean to be equivalent with a miss.\n",
+    "\n",
+    "The total data set is then the sum of the true positive and true negative targets or outputs, labeled by $n$.\n",
+    "Based on this we can then define the accuracy score as the sum of correctly predicted **TP** and **TN** cases divided by the sum of true positive and treue negative events in our data set, or as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "44ce5095",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathrm{Accuracy} = \\frac{\\sum_{i=0}^{n-1}I(y_i=\\tilde{y}_i)}{n}.\n",
+    "$$"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 8,
+   "id": "24ff3dd3",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1143,15 +1363,46 @@
      ]
     },
     {
-     "ename": "ModuleNotFoundError",
-     "evalue": "No module named 'scikitplot'",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mModuleNotFoundError\u001b[0m                       Traceback (most recent call last)",
-      "Input \u001b[0;32mIn [8]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m     32\u001b[0m \u001b[38;5;28mprint\u001b[39m(accuracy)\n\u001b[1;32m     33\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mTest set accuracy with Logistic Regression  and scaled data: \u001b[39m\u001b[38;5;132;01m{:.2f}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;241m.\u001b[39mformat(logreg\u001b[38;5;241m.\u001b[39mscore(X_test_scaled,y_test)))\n\u001b[0;32m---> 36\u001b[0m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;21;01mscikitplot\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m \u001b[38;5;21;01mskplt\u001b[39;00m\n\u001b[1;32m     37\u001b[0m y_pred \u001b[38;5;241m=\u001b[39m logreg\u001b[38;5;241m.\u001b[39mpredict(X_test_scaled)\n\u001b[1;32m     38\u001b[0m skplt\u001b[38;5;241m.\u001b[39mmetrics\u001b[38;5;241m.\u001b[39mplot_confusion_matrix(y_test, y_pred, normalize\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mTrue\u001b[39;00m)\n",
-      "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'scikitplot'"
-     ]
+     "data": {
+      "image/png": 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+       "
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+      ]
+     },
+     "metadata": {
+      "filenames": {
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_64_2.png"
+      }
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
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\n",
+      "text/plain": [
+       "
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+      ]
+     },
+     "metadata": {
+      "filenames": {
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_64_3.png"
+      }
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
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\n",
+      "text/plain": [
+       "
    "
+      ]
+     },
+     "metadata": {
+      "filenames": {
+       "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter4_64_4.png"
+      }
+     },
+     "output_type": "display_data"
     }
    ],
    "source": [
@@ -1217,5 +1468,5 @@
   }
  },
  "nbformat": 4,
- "nbformat_minor": 4
+ "nbformat_minor": 5
 }
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter4.py b/doc/LectureNotes/_build/jupyter_execute/chapter4.py
index f3e017492..529cfd821 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter4.py
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter4.py
@@ -1,10 +1,11 @@
 #!/usr/bin/env python
 # coding: utf-8
 
+# 
+
 # # Logistic Regression
-# 
-# 
-# 
+
 # ## Logistic Regression
 # 
 # In linear regression our main interest was centered on learning the
@@ -19,7 +20,6 @@
 # the product of the design matrices, linear regression gives then a
 # simple recipe for fitting our data.
 # 
-# 
 # Classification problems, however, are concerned with outcomes taking
 # the form of discrete variables (i.e. categories). We may for example,
 # on the basis of DNA sequencing for a number of patients, like to find
@@ -36,7 +36,6 @@
 # binary outcome, true or false, positive or negative, success or
 # failure etc.
 # 
-# 
 # Logistic regression will also serve as our stepping stone towards
 # neural network algorithms and supervised deep learning. For logistic
 # learning, the minimization of the cost function leads to a non-linear
@@ -50,9 +49,7 @@
 # We note also that many of the topics discussed here on logistic 
 # regression are also commonly used in modern supervised Deep Learning
 # models, as we will see later.
-# 
-# 
-# 
+
 # ## Basics
 # 
 # We consider the case where the dependent variables, also called the
@@ -95,7 +92,6 @@
 # where $\boldsymbol{y}$ is a vector representing the possible outcomes, $\boldsymbol{X}$ is our
 # $n\times p$ design matrix and $\boldsymbol{\beta}$ represents our estimators/predictors.
 # 
-# 
 # The main problem with our function is that it takes values on the
 # entire real axis. In the case of logistic regression, however, the
 # labels $y_i$ are discrete variables. A typical example is the credit
@@ -111,7 +107,6 @@
 # favorable to use a ``soft" classifier that outputs
 # the probability of a given category. This leads us to the logistic function.
 # 
-# 
 # The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput  is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.
 
 # In[1]:
@@ -207,8 +202,7 @@ plt.show()
 # Sigmoid function or logistic model. We will consider this function as
 # representing the probability for finding a value of $y_i$ with a given
 # $x_i$.
-# 
-# 
+
 # ## The logistic function
 # 
 # Another widely studied model, is the so-called 
@@ -228,9 +222,8 @@ plt.show()
 # $$
 
 # Note that $1-p(t)= p(-t)$.
-# 
-# ## Examples of likelihood functions used in logistic regression and nueral networks
-# 
+
+# ## Examples of likelihood functions used in logistic regression and neural networks
 # 
 # The following code plots the logistic function, the step function and other functions we will encounter from here and on.
 
@@ -345,11 +338,9 @@ plt.show()
 # This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, 
 # in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.
 # 
-# 
 # The cross entropy is a convex function of the weights $\boldsymbol{\beta}$ and,
 # therefore, any local minimizer is a global minimizer. 
 # 
-# 
 # Minimizing this
 # cost function with respect to the two parameters $\beta_0$ and $\beta_1$ we obtain
 
@@ -414,8 +405,6 @@ plt.show()
 # and the model is specified in term of $K-1$ so-called log-odds or
 # **logit** transformations.
 # 
-# 
-# 
 # In our discussion of neural networks we will encounter the above again
 # in terms of a slightly modified function, the so-called **Softmax** function.
 # 
@@ -447,7 +436,7 @@ plt.show()
 # descent method.  Newton's method and gradient descent methods are
 # discussed in the material on [optimization
 # methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).
-# 
+
 # ## Wisconsin Cancer Data
 # 
 # We show here how we can use a simple regression case on the breast
@@ -558,6 +547,23 @@ correlation_matrix = cancerpd.corr().round(1)
 # features are of relevance and which are not. This leads  us to
 # the classical Principal Component Analysis (PCA) theorem with
 # applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)).
+# 
+# Here we present a further way to present our results in terms of a so-called **confusion matrix**, the cumulative gain and the **ROC** curve.
+# This way of displaying our data are based upon different ways to classify our possible outcomes. Before we proceed we need some definitions.
+# 1. **TP**: true positive or in other words, something equivalent with a proper classification
+# 
+# 2. **TN**: true negative, which is equivalent with a correct rejection
+# 
+# 3. **FP**: false positive, or in simpler words something that is equivalent with a false alarm
+# 
+# 4. **FN**: false negative, which is mean to be equivalent with a miss.
+# 
+# The total data set is then the sum of the true positive and true negative targets or outputs, labeled by $n$.
+# Based on this we can then define the accuracy score as the sum of correctly predicted **TP** and **TN** cases divided by the sum of true positive and treue negative events in our data set, or as
+
+# $$
+# \mathrm{Accuracy} = \frac{\sum_{i=0}^{n-1}I(y_i=\tilde{y}_i)}{n}.
+# $$
 
 # In[8]:
 
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diff --git a/doc/LectureNotes/chapter3.ipynb b/doc/LectureNotes/chapter3.ipynb
index 7cb02906e..6e1a008cf 100644
--- a/doc/LectureNotes/chapter3.ipynb
+++ b/doc/LectureNotes/chapter3.ipynb
@@ -2,10 +2,32 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "51186d57",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    ""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "79a63fb4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Resampling Methods"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f3d916b4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "# Resampling Methods\n",
-    "\n",
     "## Introduction\n",
     "\n",
     "Resampling methods are an indispensable tool in modern\n",
@@ -27,7 +49,6 @@
     "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n",
     "cross-validation and the bootstrap method. \n",
     "\n",
-    "\n",
     "Resampling approaches can be computationally expensive, because they\n",
     "involve fitting the same statistical method multiple times using\n",
     "different subsets of the training data. However, due to recent\n",
@@ -44,23 +65,28 @@
     "level of flexibility for a model is known as model selection. The\n",
     "bootstrap is widely used.\n",
     "\n",
-    "\n",
     "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n",
     "\n",
     "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n",
     "\n",
-    "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n",
-    "\n",
+    "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "46cb3279",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Reminder on Statistics\n",
     "\n",
-    "\n",
     "* As in other experiments, many numerical  experiments have two classes of errors:\n",
     "\n",
     "  * Statistical errors\n",
     "\n",
     "  * Systematical errors\n",
     "\n",
-    "\n",
     "* Statistical errors can be estimated using standard tools from statistics\n",
     "\n",
     "* Systematical errors are method specific and must be treated differently from case to case. \n",
@@ -72,7 +98,6 @@
     "derive quantities like the variance and other expectation values in a\n",
     "rather straightforward way.\n",
     "\n",
-    "\n",
     "It is assumed that $\\varepsilon_i\n",
     "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n",
     "independent, i.e.:"
@@ -80,7 +105,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0fe38e07",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \n",
@@ -93,7 +121,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d9d6955b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The randomness of $\\varepsilon_i$ implies that\n",
     "$\\mathbf{y}_i$ is also a random variable. In particular,\n",
@@ -106,7 +137,6 @@
     "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n",
     "row number $i$ and perform a sum over all values $p$.\n",
     "\n",
-    "\n",
     "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n",
     "that there exists a function $f(\\boldsymbol{x})$ and  a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n",
     "which describe our data"
@@ -114,7 +144,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f41f7049",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n",
@@ -123,7 +156,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2ee172f4",
+   "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"
@@ -131,7 +167,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1aecc768",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n",
@@ -140,14 +179,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "12e9bdea",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8f9db9db",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \n",
@@ -160,7 +205,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b10abe89",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "while\n",
     "its variance is"
@@ -168,7 +216,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bec51521",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n",
@@ -188,18 +239,23 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4859640c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n",
     "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD). \n",
     "\n",
-    "\n",
     "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we can evaluate the expectation value"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "69978823",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}(\\boldsymbol{\\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",
@@ -208,10 +264,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5caee9d9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This means that the estimator of the regression parameters is unbiased.\n",
-    "v\n",
+    "\n",
     "We can also calculate the variance\n",
     "\n",
     "The variance of $\\boldsymbol{\\beta}$ is"
@@ -219,7 +278,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6791e5b4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{eqnarray*}\n",
@@ -247,7 +309,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4f747992",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have used  that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n",
     "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n",
@@ -258,7 +323,6 @@
     "[(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} }$. This may be used to\n",
     "construct a confidence interval for the estimates.\n",
     "\n",
-    "\n",
     "In a similar way, we can obtain analytical expressions for say the\n",
     "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n",
     "when we employ Ridge regression, allowing us again to define a confidence interval. \n",
@@ -268,7 +332,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d4679496",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
@@ -277,7 +344,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bd3534e1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see clearly that \n",
     "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n",
@@ -287,7 +357,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "125b139e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}  \\mathbf{X}^{T} \\mathbf{X} \\{ [  \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
@@ -296,7 +369,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "98b59ea0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n",
     "\n",
@@ -305,7 +381,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "dcfd7a6c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [  \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n",
@@ -314,14 +393,23 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "30f43be3",
+   "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",
-    "\n",
-    "\n",
+    "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e3c1bf3c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Resampling methods\n",
     "\n",
     "With all these analytical equations for both the OLS and Ridge\n",
@@ -339,12 +427,10 @@
     "\n",
     "2. training error $\\mathrm{Err_{Train}}$, which is the average loss over the training data.\n",
     "\n",
-    "As our model becomes more and more complex, more of the training data tends to  used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n",
+    "As our model becomes more and more complex, more of the training data tends to  be used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n",
     "For a certain level of complexity the test error will reach minimum, before starting to increase again. The\n",
     "training error reaches a saturation.\n",
     "\n",
-    "\n",
-    "\n",
     "Two famous\n",
     "resampling methods are the **independent bootstrap** and **the jackknife**. \n",
     "\n",
@@ -360,7 +446,6 @@
     "variance of $\\overline{X}$ (which often is the case), then there is no\n",
     "need for bootstrapping. \n",
     "\n",
-    "\n",
     "The Jackknife works by making many replicas of the estimator $\\widehat{\\beta}$. \n",
     "The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\\boldsymbol{x} = (x_1,x_2,\\cdots,X_n)$. \n",
     "Let $\\boldsymbol{x}_i$ denote the vector"
@@ -368,7 +453,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3925f435",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{x}_i = (x_1,x_2,\\cdots,x_{i-1},x_{i+1},\\cdots,x_n),\n",
@@ -377,7 +465,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "cc6328ee",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which equals the vector $\\boldsymbol{x}$ with the exception that observation\n",
     "number $i$ is left out. Using this notation, define\n",
@@ -387,7 +478,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 1,
+   "id": "b751a941",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -426,7 +518,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "25ff562a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "### Bootstrap\n",
     "\n",
@@ -452,8 +547,6 @@
     "estimate all the interesting parameters of $p(\\boldsymbol{t})$ using point\n",
     "estimators.  \n",
     "\n",
-    "\n",
-    "\n",
     "In the case that $\\widehat{\\beta}$ has\n",
     "more than one component, and the components are independent, we use the\n",
     "same estimator on each component separately.  If the probability\n",
@@ -468,7 +561,6 @@
     "idea is to use the relative frequency of $\\widehat{\\beta}^*$\n",
     "(think of a histogram) as an estimate of $p(\\boldsymbol{t})$.\n",
     "\n",
-    "\n",
     "But\n",
     "unless there is enough information available about the process that\n",
     "generated $X_1,X_2,\\cdots,X_n$, $p(x)$ is in general\n",
@@ -478,13 +570,11 @@
     "the relative frequency of the observations, will we obtain the same\n",
     "result in some asymptotic sense? The answer is yes.\n",
     "\n",
-    "\n",
     "Instead of generating the histogram for the relative\n",
     "frequency of the observation $X_i$, just draw the values\n",
     "$(X_1^*,X_2^*,\\cdots,X_n^*)$ with replacement from the vector\n",
     "$\\boldsymbol{X}$. \n",
     "\n",
-    "\n",
     "The independent bootstrap works like this: \n",
     "\n",
     "1. Draw with replacement $n$ numbers for the observed variables $\\boldsymbol{x} = (x_1,x_2,\\cdots,x_n)$. \n",
@@ -502,7 +592,7 @@
     "histogram of the relative frequency of $\\widehat{\\beta}^*$. Instead\n",
     "you use the estimators corresponding to the statistic of interest. For\n",
     "example, if you are interested in estimating the variance of $\\widehat\n",
-    "\\beta$, apply the etsimator $\\widehat \\sigma^2$ to the values\n",
+    "\\beta$, apply the estimator $\\widehat \\sigma^2$ to the values\n",
     "$\\widehat \\beta^*$.\n",
     "\n",
     "Before we proceed however, we need to remind ourselves about a central\n",
@@ -511,7 +601,6 @@
     "(and other resampling methods) work so well on independent and\n",
     "identically distributed variables.\n",
     "\n",
-    "\n",
     "Suppose we have a PDF $p(x)$ from which we generate  a series $N$\n",
     "of averages $\\langle x_i \\rangle$. Each mean value $\\langle x_i \\rangle$\n",
     "is viewed as the average of a specific measurement, e.g., throwing \n",
@@ -525,7 +614,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fa55ab5a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "z=\\frac{x_1+x_2+\\dots+x_m}{m},\n",
@@ -534,11 +626,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "697c0c94",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "the question we pose is which is the PDF of the new variable $z$.\n",
     "\n",
-    "\n",
     "The probability of obtaining an average value $z$ is the product of the \n",
     "probabilities of obtaining arbitrary individual mean values $x_i$,\n",
     "but with the constraint that the average is $z$. We can express this through\n",
@@ -547,7 +641,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bd26bbd1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{p}(z)=\\int dx_1p(x_1)\\int dx_2p(x_2)\\dots\\int dx_mp(x_m)\n",
@@ -557,21 +654,25 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "68664e4f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where the $\\delta$-function enbodies the constraint that the mean is $z$.\n",
     "All measurements that lead to each individual $x_i$ are expected to\n",
     "be independent, which in turn means that we can express $\\tilde{p}$ as the \n",
     "product of individual $p(x_i)$.  The independence assumption is important in the derivation of the central limit theorem.\n",
     "\n",
-    "\n",
-    "\n",
     "If we use the integral expression for the $\\delta$-function"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "742a107a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\delta(z-\\frac{x_1+x_2+\\dots+x_m}{m})=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n",
@@ -581,7 +682,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b6d624c1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and inserting $e^{i\\mu q-i\\mu q}$ where $\\mu$ is the mean value\n",
     "we arrive at"
@@ -589,7 +693,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "46458586",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{p}(z)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n",
@@ -600,14 +707,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ca119461",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "with the integral over $x$ resulting in"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f75b40fc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}=\n",
@@ -618,7 +731,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a4b64e20",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The second term on the rhs disappears since this is just the mean and \n",
     "employing the definition of $\\sigma^2$ we have"
@@ -626,7 +742,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fc76951f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\int_{-\\infty}^{\\infty}dxp(x)e^{\\left(iq(\\mu-x)/m\\right)}=\n",
@@ -636,14 +755,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "76979572",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "resulting in"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3edff7d2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\left[\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}\\right]^m\\approx\n",
@@ -653,14 +778,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "73a9341f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and in the limit $m\\rightarrow \\infty$ we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "88f1cc30",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\tilde{p}(z)=\\frac{1}{\\sqrt{2\\pi}(\\sigma/\\sqrt{m})}\n",
@@ -670,13 +801,15 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1aec913e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which is the normal distribution with variance\n",
     "$\\sigma^2_m=\\sigma^2/m$, where $\\sigma$ is the variance of the PDF $p(x)$\n",
     "and $\\mu$ is also the mean of the PDF $p(x)$. \n",
     "\n",
-    "\n",
     "Thus, the central limit theorem states that the PDF $\\tilde{p}(z)$ of\n",
     "the average of $m$ random values corresponding to a PDF $p(x)$ \n",
     "is a normal distribution whose mean is the \n",
@@ -689,7 +822,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "42b317e7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sigma_m=\n",
@@ -699,7 +835,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e5baf71d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The latter is true only if the average value is known exactly. This is obtained in the limit\n",
     "$m\\rightarrow \\infty$  only. Because the mean and the variance are measured quantities we obtain \n",
@@ -708,7 +847,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e2b39e5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sigma_m\\approx \n",
@@ -718,7 +860,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "472e7c2c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In many cases however the above estimate for the standard deviation,\n",
     "in particular if correlations are strong, may be too simplistic. Keep\n",
@@ -728,13 +873,10 @@
     "we generate in various calculations do always exhibit some\n",
     "correlations.\n",
     "\n",
-    "\n",
-    "\n",
     "The theorem is satisfied by a large class of PDFs. Note however that for a\n",
     "finite $m$, it is not always possible to find a closed form /analytic expression for\n",
     "$\\tilde{p}(x)$.\n",
     "\n",
-    "\n",
     "The following code starts with a Gaussian distribution with mean value\n",
     "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n",
     "used in the bootstrap analysis. The bootstrap analysis returns a data\n",
@@ -751,7 +893,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 2,
+   "id": "0ff7b796",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -790,7 +933,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "22680159",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see that our new variance and from that the standard deviation, agrees with the central limit theorem.\n",
     "\n",
@@ -799,7 +945,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 3,
+   "id": "fe018bb3",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -819,11 +966,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5bbd613c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The bias-variance tradeoff\n",
     "\n",
-    "\n",
     "We will discuss the bias-variance tradeoff in the context of\n",
     "continuous predictions such as regression. However, many of the\n",
     "intuitions and ideas discussed here also carry over to classification\n",
@@ -835,7 +984,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ca3fde4a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}\n",
@@ -844,7 +996,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e11f84b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\epsilon$ is normally distributed with mean zero and standard deviation $\\sigma^2$.\n",
     "\n",
@@ -858,7 +1013,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "026a65c8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
@@ -867,14 +1025,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e59918c7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can rewrite this as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2fd3f73c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n",
@@ -883,21 +1047,27 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7daf46c9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "The three terms represent the square of the bias of the learning\n",
+    "The first term represents the square of the bias of the learning\n",
     "method, which can be thought of as the error caused by the simplifying\n",
     "assumptions built into the method. The second term represents the\n",
     "variance of the chosen model and finally the last terms is variance of\n",
     "the error $\\boldsymbol{\\epsilon}$.\n",
     "\n",
-    "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\\boldsymbol{\\tilde{y}}$.\n",
+    "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastic variable, idem for $\\boldsymbol{\\tilde{y}}$.\n",
     "We use a more compact notation in terms of the expectation value"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6094266b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}})^2\\right],\n",
@@ -906,14 +1076,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "643e0047",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and adding and subtracting $\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]$ we get"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1319bde5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}}+\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right],\n",
@@ -922,14 +1098,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9c6d6da1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which, using the abovementioned expectation values can be rewritten as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "855756ef",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right]+\\mathrm{Var}\\left[\\boldsymbol{\\tilde{y}}\\right]+\\sigma^2,\n",
@@ -938,14 +1120,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "34d24717",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "that is the rewriting in terms of the so-called bias, the variance of the model $\\boldsymbol{\\tilde{y}}$ and the variance of $\\boldsymbol{\\epsilon}$."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 4,
+   "id": "d51b6100",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1010,7 +1196,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 5,
+   "id": "bd636def",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1067,7 +1254,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8cf88b3a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The bias-variance tradeoff summarizes the fundamental tension in\n",
     "machine learning, particularly supervised learning, between the\n",
@@ -1078,8 +1268,6 @@
     "train and less sensitive to sampling noise arising from having a\n",
     "finite-sized training dataset (smaller variance). \n",
     "\n",
-    "\n",
-    "\n",
     "The above equations tell us that in\n",
     "order to minimize the expected test error, we need to select a\n",
     "statistical learning method that simultaneously achieves low variance\n",
@@ -1087,7 +1275,6 @@
     "and squared bias is also nonnegative. Hence, we see that the expected\n",
     "test MSE can never lie below $Var(\\epsilon)$, the irreducible error.\n",
     "\n",
-    "\n",
     "What do we mean by the variance and bias of a statistical learning\n",
     "method? The variance refers to the amount by which our model would change if we\n",
     "estimated it using a different training data set. Since the training\n",
@@ -1098,13 +1285,13 @@
     "the training data can result in large changes in the model. In general, more\n",
     "flexible statistical methods have higher variance.\n",
     "\n",
-    "\n",
     "You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 6,
+   "id": "86bfc49a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1186,7 +1373,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 7,
+   "id": "0c2a183a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1275,7 +1463,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2c6c9e89",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Cross-validation\n",
     "\n",
@@ -1294,7 +1485,6 @@
     "choosing $k=n$. This particular case is referred to as leave-one-out\n",
     "cross-validation (LOOCV). \n",
     "\n",
-    "\n",
     "* Define a range of interest for the penalty parameter.\n",
     "\n",
     "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n",
@@ -1304,7 +1494,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "71738b2a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -1317,7 +1510,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "14db46b6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n",
     "\n",
@@ -1328,7 +1524,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "95e5c8e4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -1339,7 +1538,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7e60f51d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "For the various values of $k$\n",
     "\n",
@@ -1357,7 +1559,6 @@
     "\n",
     "d. Retain the evaluation score and discard the model\n",
     "\n",
-    "\n",
     "5. Summarize the model using the sample of model evaluation scores\n",
     "\n",
     "The code here uses Ridge regression with cross-validation (CV)  resampling and $k$-fold CV in order to fit a specific polynomial."
@@ -1365,7 +1566,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 8,
+   "id": "2cef0eb7",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1465,14 +1667,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f501c9cf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "More examples of the application of cross-validation follow here."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 9,
+   "id": "30e1e320",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1550,10 +1756,21 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "383e5c2a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Note that we have kept the intercept in the first column of design matrix $\\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and do not fit intercept. See the discussions below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1f9cd409",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "Note that we have kept the intercept in the first column of design matrix $\\boldsymbol{X}$. When we call the corresponding **Scikit-Learn** function we need thus to set the intercept to **False**. Libraries like **Scikit-Learn** normally scale the design matrix and does not fit intercept. See the discussions below.\n",
-    "\n",
     "## More on Rescaling data\n",
     "\n",
     "We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.\n",
@@ -1576,7 +1793,6 @@
     "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n",
     "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n",
     "\n",
-    "\n",
     "If our predictors represent different scales, then it is important to\n",
     "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n",
     "column from the corresponding column and dividing the column with its\n",
@@ -1602,15 +1818,14 @@
     "coefficient term, than if measured in millimeters.\n",
     "This can clearly lead to problems in evaluating the cost/loss functions.\n",
     "\n",
-    "\n",
-    "\n",
     "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n",
     "on your eventual new data set  before making a prediction. If we translate this into a Python code, it would could be implemented as follows"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 10,
+   "id": "eab81633",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1637,7 +1852,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0ca51a54",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us try to understand what this may imply mathematically when we\n",
     "subtract the mean values, also known as *zero centering*. For\n",
@@ -1648,7 +1866,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fa3a4110",
+   "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",
@@ -1657,7 +1878,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9f499302",
+   "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",
@@ -1671,7 +1895,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b1f7ba52",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
@@ -1680,14 +1907,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7ff75b7f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "for all $j$. For $\\beta_0$ we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "eac37d3c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\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",
@@ -1696,14 +1929,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6e653464",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Multiplying away the constant $2/n$, we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8afa60a1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
@@ -1712,7 +1951,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "055b7975",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
     "Our result for $\\beta_0$ simplifies then to"
@@ -1720,7 +1962,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a2bf6e5d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
@@ -1729,14 +1974,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e15afbd5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We obtain then"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d2f9a064",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n",
@@ -1745,14 +1996,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e8035d8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we define"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0bbcef44",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n",
@@ -1761,14 +2018,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "919462c2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and the mean value of the outputs as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3a0899ec",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
@@ -1777,14 +2040,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9b255ec5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7716aca1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n",
@@ -1793,14 +2062,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "24863f48",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6c10f940",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n",
@@ -1809,14 +2084,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7afefe95",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can rewrite the latter equation as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a2e7bc1a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\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",
@@ -1825,14 +2106,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ded7d78",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have defined"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bcac7366",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n",
@@ -1841,18 +2128,22 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bafe8156",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n",
     "\n",
-    "\n",
-    "\n",
     "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7c91d280",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
@@ -1861,14 +2152,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1b81d6de",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7d6d497e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
@@ -1877,7 +2174,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "772593b2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
     "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
@@ -1887,7 +2187,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3edbdb56",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
@@ -1896,18 +2199,21 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7de9723e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
     "\n",
-    "\n",
     "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to  Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n",
     "Note also that we do not split the data into training and test."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 11,
+   "id": "2e4a0363",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2004,7 +2310,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c6d3a071",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The intercept is the value of our output/target variable\n",
     "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
@@ -2015,14 +2324,17 @@
     "meaning that the MSE can be penalized by the value of the\n",
     "intercept. Not including the intercept in the fit, means that the\n",
     "regularization term does not include $\\beta_0$. For different values\n",
-    "of $\\lambda$, this may lead to differeing MSE values. \n",
+    "of $\\lambda$, this may lead to different MSE values. \n",
     "\n",
     "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "db803b77",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
@@ -2031,14 +2343,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d7a9c9d8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "but when we take out the intercept, this equation becomes"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e08af763",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
@@ -2047,14 +2365,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "10fe24e4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "For Lasso regression we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6feb3bdd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
@@ -2063,7 +2387,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "657a0777",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It means that, when scaling the design matrix and the outputs/targets,\n",
     "by subtracting the mean values, we have an optimization problem which\n",
@@ -2072,13 +2399,13 @@
     "back the intercept, we will get a MSE which then contains the\n",
     "intercept.\n",
     "\n",
-    "\n",
     "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known  vanilla data set."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 12,
+   "id": "9f21317d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2154,7 +2481,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "20de82fd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
     "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
@@ -2164,7 +2494,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 13,
+   "id": "4138ed50",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2253,7 +2584,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7e9dfda5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see here, when compared to the code which includes explicitely the\n",
     "intercept column, that our MSE value is actually smaller. This is\n",
@@ -2264,8 +2598,16 @@
     "also that the problem with the intercept occurs mainly in these type\n",
     "of polynomial fitting problem.\n",
     "\n",
-    "The next example is indeed an example where all these discussions about the role of intercept are not present.\n",
-    "\n",
+    "The next example is indeed an example where all these discussions about the role of intercept are not present."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a7eb252c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## More complicated Example: The Ising model\n",
     "\n",
     "The one-dimensional Ising model with nearest neighbor interaction, no\n",
@@ -2274,7 +2616,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "990ff280",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2289,7 +2634,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ea46b31",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n",
     "in the system is determined by $L$. For the one-dimensional system\n",
@@ -2302,7 +2650,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 14,
+   "id": "fa60aaab",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2333,7 +2682,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b9e8214e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we use ordinary least squares\n",
     "regression to predict the energy for the nearest neighbor\n",
@@ -2346,7 +2698,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a578cd63",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2361,7 +2716,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d90dcfee",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we allow for interactions beyond the nearest neighbors and a state dependent\n",
     "coupling constant. This latter expression can be formulated as\n",
@@ -2370,7 +2728,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c4eec69a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2385,7 +2746,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ae46a11b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n",
     "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
@@ -2394,7 +2758,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "87ebb9b8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2409,14 +2776,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "856efd3f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We split the data in training and test data as discussed in the previous example"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 15,
+   "id": "5ae2828f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2432,14 +2803,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0d6201d9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the ordinary least squares method we choose the cost function"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4af380a7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2454,7 +2831,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e8eefa3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n",
     "This yields the expression for $\\boldsymbol{\\beta}$ to be"
@@ -2462,7 +2842,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a1c0765c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n",
@@ -2471,7 +2854,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c754e28b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n",
     "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n",
@@ -2481,7 +2867,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 16,
+   "id": "ff056cf9",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2500,7 +2887,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "524f259e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Doing the inversion directly turns out to be a bad idea since the matrix\n",
     "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n",
@@ -2510,7 +2900,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "52be4553",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n",
@@ -2519,14 +2912,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c37ff09a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where the pseudoinverse of $\\boldsymbol{X}$ is given by"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ea49a105",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n",
@@ -2535,7 +2934,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bf31067a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Using singular value decomposition we can decompose the matrix  $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n",
     "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n",
@@ -2545,7 +2947,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ef2c839",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2560,7 +2965,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "96bbf134",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Note that solving this equation by actually doing the pseudoinverse\n",
     "(which is what we will do) is not a good idea as this operation scales\n",
@@ -2572,7 +2980,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 17,
+   "id": "5e729b1e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2586,7 +2995,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 18,
+   "id": "80b923d1",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2598,14 +3008,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b4cf2f1a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "When extracting the $J$-matrix  we need to make sure that we remove the intercept, as is done here"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 19,
+   "id": "93be2c0d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2617,14 +3031,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ae869089",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "A way of looking at the coefficients in $J$ is to plot the matrices as images."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 20,
+   "id": "8d6d8152",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2643,7 +3061,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8092ac84",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It is interesting to note that OLS\n",
     "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n",
@@ -2653,10 +3074,6 @@
     "\n",
     "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n",
     "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
     "Let us now \n",
     "focus on Ridge and Lasso regression as well. We repeat some of the\n",
     "basic parts of the Ising model and the setup of the training and test\n",
@@ -2667,7 +3084,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d9251cde",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2682,7 +3102,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "abe454c9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n",
     "\n",
@@ -2691,7 +3114,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 21,
+   "id": "89f8fbd0",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2723,14 +3147,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e06639d4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "A more general form for the one-dimensional Ising model is"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "43ef0f40",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2745,7 +3175,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "523dc03b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we allow for interactions beyond the nearest neighbors and a more\n",
     "adaptive coupling matrix. This latter expression can be formulated as\n",
@@ -2754,7 +3187,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0f389eef",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2769,7 +3205,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b6e00f24",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n",
     "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
@@ -2778,7 +3217,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5ea1084d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2793,14 +3235,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5551bb05",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We organize the data as we did above"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 22,
+   "id": "f5dd7795",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2826,14 +3272,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f36c807b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We will do all fitting with **Scikit-Learn**,"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 23,
+   "id": "eb701dd4",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2845,14 +3295,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bd356ada",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "When  extracting the $J$-matrix we make sure to remove the intercept"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 24,
+   "id": "ae737db8",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2864,14 +3318,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d152a32a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "And then we plot the results"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 25,
+   "id": "8d713977",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2890,11 +3348,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ede30c18",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The results agree perfectly with our previous discussion where we used our own code.\n",
     "\n",
-    "\n",
     "Having explored the ordinary least squares we move on to ridge\n",
     "regression. In ridge regression we include a **regularizer**. This\n",
     "involves a new cost function which leads to a new estimate for the\n",
@@ -2904,31 +3364,26 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "eac772e9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "6\n",
-    "0\n",
-    " \n",
-    "<\n",
-    "<\n",
-    "<\n",
-    "!\n",
-    "!\n",
-    "M\n",
-    "A\n",
-    "T\n",
-    "H\n",
-    "_\n",
-    "B\n",
-    "L\n",
-    "O\n",
-    "C\n",
-    "K"
+    "\n",
+    "
    \n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}.\n",
+    "\\label{_auto11} \\tag{11}\n",
+    "\\end{equation}\n",
+    "$$"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 26,
+   "id": "b66675f0",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2951,14 +3406,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "71aa37b5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8b70ef66",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -2973,14 +3434,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "079981ff",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 27,
+   "id": "c8d4a180",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3002,21 +3467,22 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "57839941",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It is quite striking how LASSO breaks the symmetry of the coupling\n",
     "constant as opposed to ridge and OLS. We get a sparse solution with\n",
     "$J_{j, j + 1} = -1$.\n",
     "\n",
-    "\n",
-    "\n",
-    "\n",
     "We see how the different models perform for a different set of values for $\\lambda$."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 28,
+   "id": "0e3bbb4e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3063,14 +3529,16 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "14711abd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We see that LASSO reaches a good solution for low\n",
     "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n",
     "much. Ridge is more stable over a larger range of values for\n",
     "$\\lambda$, but eventually also fades away.\n",
     "\n",
-    "\n",
     "To determine which value of $\\lambda$ is best we plot the accuracy of\n",
     "the models when predicting the training and the testing set. We expect\n",
     "the accuracy of the training set to be quite good, but if the accuracy\n",
@@ -3081,7 +3549,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 29,
+   "id": "9ff29a72",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3122,26 +3591,29 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9c10be56",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n",
     "achieves a very good accuracy on the test set. This by far surpasses the\n",
-    "other models for all values of $\\lambda$.\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
+    "other models for all values of $\\lambda$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "efce3b63",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Exercises and Projects\n",
     "\n",
-    "\n",
-    "\n",
     "The main aim of this project is to study in more detail various\n",
     "regression methods, including the Ordinary Least Squares (OLS) method,\n",
     "The total score is **100** points. Each subtask has its own final score.\n",
     "\n",
-    "\n",
     "We will first study how to fit polynomials to a specific\n",
     "two-dimensional function called [Franke's\n",
     "function](http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf).  This\n",
@@ -3152,13 +3624,15 @@
     "proper assessment of our models. We will also study in detail the\n",
     "so-called Bias-Variance trade off.\n",
     "\n",
-    "\n",
     "The Franke function, which is a weighted sum of four exponentials  reads as follows"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ef5a4df0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -3170,7 +3644,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "b085eff4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The function will be defined for $x,y\\in [0,1]$.  Our first step will\n",
     "be to perform an OLS regression analysis of this function, trying out\n",
@@ -3191,13 +3668,13 @@
     "beyond the second-order polynomials metioned above and explore \n",
     "which polynomial fits the data best.\n",
     "\n",
-    "\n",
     "The Python code for the Franke function is included here (it performs also a three-dimensional plot of it)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 30,
+   "id": "5056dccb",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3247,7 +3724,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f5e72aef",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "### Exercise: Ordinary Least Square (OLS) on the Franke function\n",
     "\n",
@@ -3267,7 +3747,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0ba33237",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n",
@@ -3277,7 +3760,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1f9af2ef",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and the $R^2$ score function.  If $\\tilde{\\hat{y}}_i$ is the predicted\n",
     "value of the $i-th$ sample and $y_i$ is the corresponding true value,\n",
@@ -3286,7 +3772,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "77d5b0f2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
@@ -3295,14 +3784,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6c1f9df9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have defined the mean value  of $\\hat{y}$ as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1d06102c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\bar{y} =  \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n",
@@ -3311,7 +3806,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8fb40b08",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Your code has to include a scaling of the data (for example by\n",
     "subtracting the mean value), and\n",
@@ -3328,11 +3826,16 @@
     "data and say test data.  An accepted rule of thumb is to use\n",
     "approximately $2/3$ to $4/5$ of the data as training data.\n",
     "\n",
-    "\n",
-    "You can easily reuse the solutions to your exercises from week 35 and week 36.\n",
-    "\n",
-    "\n",
-    "\n",
+    "You can easily reuse the solutions to your exercises from week 35 and week 36."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1701de47",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "### Exercise: Bias-variance trade-off and resampling techniques\n",
     "\n",
     "Our aim here is to study the bias-variance trade-off by implementing the **bootstrap** resampling technique.\n",
@@ -3360,7 +3863,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "15110cdf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}.\n",
@@ -3369,7 +3875,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "33046595",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here $\\epsilon$ is normally distributed with mean zero and standard\n",
     "deviation $\\sigma^2$.\n",
@@ -3385,7 +3894,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "84527747",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
@@ -3394,7 +3906,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "da2f876d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here the expected value $\\mathbb{E}$ is the sample value. \n",
     "\n",
@@ -3403,7 +3918,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "1845a7dc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n",
@@ -3412,7 +3930,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d2f1d7d7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Explain what the terms mean, which one is the bias and which one is\n",
     "the variance and discuss their interpretations.\n",
@@ -3424,9 +3945,16 @@
     "of your model complexity (the degree of the polynomial) and the number\n",
     "of data points, and possibly also your training and test data using the **bootstrap** resampling method.\n",
     "\n",
-    "Note also that when you calculate the bias, in all applications you don't know the function values $f_i$. You would hence replace them with the actual data points $y_i$.\n",
-    "\n",
-    "\n",
+    "Note also that when you calculate the bias, in all applications you don't know the function values $f_i$. You would hence replace them with the actual data points $y_i$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e898c902",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "### Exercise:  Cross-validation as resampling techniques, adding more complexity\n",
     "\n",
     "The aim here is to write your own code for another widely popular\n",
@@ -3442,9 +3970,16 @@
     "Compare the MSE you get from your cross-validation code with the one\n",
     "you got from your **bootstrap** code. Comment your results. Try $5-10$\n",
     "folds.  You can also compare your own cross-validation code with the\n",
-    "one provided by **Scikit-Learn**.\n",
-    "\n",
-    "\n",
+    "one provided by **Scikit-Learn**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d23dc734",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "### Exercise: Ridge Regression on the Franke function  with resampling\n",
     "\n",
     "Write your own code for the Ridge method, either using matrix\n",
@@ -3455,8 +3990,16 @@
     "dependence on $\\lambda$.\n",
     "\n",
     "Study also the bias-variance trade-off as function of various values of\n",
-    "the parameter $\\lambda$. For the bias-variance trade-off, use the **bootstrap** resampling method. Comment your results. \n",
-    "\n",
+    "the parameter $\\lambda$. For the bias-variance trade-off, use the **bootstrap** resampling method. Comment your results."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "39a35330",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "### Exercise: Lasso Regression on the Franke function  with resampling\n",
     "\n",
     "This exercise is essentially a repeat of the previous two ones, but now\n",
@@ -3464,8 +4007,16 @@
     "you can also use the functionalities of **Scikit-Learn** (recommended). \n",
     "Give a\n",
     "critical discussion of the three methods and a judgement of which\n",
-    "model fits the data best.  Perform here as well an analysis of the bias-variance trade-off using the **bootstrap** resampling technique and an analysis of the mean squared error using cross-validation. \n",
-    "\n",
+    "model fits the data best.  Perform here as well an analysis of the bias-variance trade-off using the **bootstrap** resampling technique and an analysis of the mean squared error using cross-validation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "44d9e821",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "### Exercise: Analysis of real data\n",
     "\n",
     "With our codes functioning and having been tested properly on a\n",
@@ -3488,7 +4039,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 31,
+   "id": "0a4e6d7e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3500,7 +4052,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "16a73292",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here is a simple part of a Python code which reads and plots the data\n",
     "from such files"
@@ -3508,7 +4063,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 32,
+   "id": "168356a4",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3536,7 +4092,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e9ec3e68",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If you should have problems in downloading the digital terrain data,\n",
     "we provide two examples under the data folder of project 1. One is\n",
@@ -3544,10 +4103,8 @@
     "Austfjell, again in Norway.\n",
     "Feel free to produce your own terrain data.\n",
     "\n",
-    "\n",
     "Alternatively, if you would like to use another data set, feel free to do so. This could be data close to your reseach area or simply a data set you found interesting. See for example [kaggle.com](https://www.kaggle.com/datasets) for examples.\n",
     "\n",
-    "\n",
     "Our final part deals with the parameterization of your digital terrain\n",
     "data (or your own data).  We will apply all three methods for linear regression, the same type (or higher order) of polynomial\n",
     "approximation and cross-validation as resampling technique to evaluate which\n",
@@ -3561,5 +4118,5 @@
  ],
  "metadata": {},
  "nbformat": 4,
- "nbformat_minor": 4
+ "nbformat_minor": 5
 }
diff --git a/doc/LectureNotes/chapter4.ipynb b/doc/LectureNotes/chapter4.ipynb
index faee62767..bdfb97b2b 100644
--- a/doc/LectureNotes/chapter4.ipynb
+++ b/doc/LectureNotes/chapter4.ipynb
@@ -2,12 +2,32 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f63b8c1a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    ""
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9a284cf5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "# Logistic Regression"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d78b73e2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "# Logistic Regression\n",
-    "\n",
-    "\n",
-    "\n",
     "## Logistic Regression\n",
     "\n",
     "In linear regression our main interest was centered on learning the\n",
@@ -22,7 +42,6 @@
     "the product of the design matrices, linear regression gives then a\n",
     "simple recipe for fitting our data.\n",
     "\n",
-    "\n",
     "Classification problems, however, are concerned with outcomes taking\n",
     "the form of discrete variables (i.e. categories). We may for example,\n",
     "on the basis of DNA sequencing for a number of patients, like to find\n",
@@ -39,7 +58,6 @@
     "binary outcome, true or false, positive or negative, success or\n",
     "failure etc.\n",
     "\n",
-    "\n",
     "Logistic regression will also serve as our stepping stone towards\n",
     "neural network algorithms and supervised deep learning. For logistic\n",
     "learning, the minimization of the cost function leads to a non-linear\n",
@@ -52,10 +70,16 @@
     "\n",
     "We note also that many of the topics discussed here on logistic \n",
     "regression are also commonly used in modern supervised Deep Learning\n",
-    "models, as we will see later.\n",
-    "\n",
-    "\n",
-    "\n",
+    "models, as we will see later."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d0c829b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Basics\n",
     "\n",
     "We consider the case where the dependent variables, also called the\n",
@@ -76,7 +100,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f1f4d5f2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\  1 & \\mathrm{yes} \\end{bmatrix}.\n",
@@ -85,7 +112,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "8e2f836b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Before moving to the logistic model, let us try to use our linear\n",
     "regression model to classify these two outcomes. We could for example\n",
@@ -98,7 +128,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6b58f1ba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "\n",
     "
    \n",
@@ -113,12 +146,14 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "54e3604b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{y}$ is a vector representing the possible outcomes, $\\boldsymbol{X}$ is our\n",
     "$n\\times p$ design matrix and $\\boldsymbol{\\beta}$ represents our estimators/predictors.\n",
     "\n",
-    "\n",
     "The main problem with our function is that it takes values on the\n",
     "entire real axis. In the case of logistic regression, however, the\n",
     "labels $y_i$ are discrete variables. A typical example is the credit\n",
@@ -134,13 +169,13 @@
     "favorable to use a ``soft\" classifier that outputs\n",
     "the probability of a given category. This leads us to the logistic function.\n",
     "\n",
-    "\n",
     "The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput  is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 1,
+   "id": "cd1dd1c5",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -207,14 +242,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "2e06ee82",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "What we could attempt however is to plot the mean value for each group."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 2,
+   "id": "a774f001",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -233,7 +272,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e102fa56",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We are now trying to find a function $f(y\\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.\n",
     "In standard linear regression with a linear dependence on $x$, we would write this in terms of our model"
@@ -241,7 +283,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "75d995ba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "f(y_i\\vert x_i)=\\beta_0+\\beta_1 x_i.\n",
@@ -250,7 +295,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "e2cacca5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This expression implies however that $f(y_i\\vert x_i)$ could take any\n",
     "value from minus infinity to plus infinity. If we however let\n",
@@ -261,9 +309,16 @@
     "us to a very popular model for the function $f$, namely the so-called\n",
     "Sigmoid function or logistic model. We will consider this function as\n",
     "representing the probability for finding a value of $y_i$ with a given\n",
-    "$x_i$.\n",
-    "\n",
-    "\n",
+    "$x_i$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af9ac586",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## The logistic function\n",
     "\n",
     "Another widely studied model, is the so-called \n",
@@ -281,7 +336,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "80aa3cbe",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n",
@@ -290,19 +348,30 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "28d49915",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    "Note that $1-p(t)= p(-t)$.\n",
-    "\n",
-    "## Examples of likelihood functions used in logistic regression and nueral networks\n",
-    "\n",
+    "Note that $1-p(t)= p(-t)$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eeff6cd9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Examples of likelihood functions used in logistic regression and neural networks\n",
     "\n",
     "The following code plots the logistic function, the step function and other functions we will encounter from here and on."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 3,
+   "id": "5ecec05e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -367,14 +436,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5802efe3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f85d14ea",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -386,7 +461,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6c4e0334",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n",
     "\n",
@@ -395,7 +473,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "df7facc9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(y_i=0\\vert x_i, \\boldsymbol{\\beta}) = 1-p(y_i=1\\vert x_i, \\boldsymbol{\\beta}).\n",
@@ -404,7 +485,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a920e6d8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In order to define the total likelihood for all possible outcomes from a  \n",
     "dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n",
@@ -416,7 +500,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "a780bfe1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\begin{align*}\n",
@@ -427,14 +514,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9f810a44",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "from which we obtain the log-likelihood and our **cost/loss** function"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "3fac4ef4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathcal{C}(\\boldsymbol{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\boldsymbol{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\boldsymbol{\\beta}))\\right]\\right).\n",
@@ -443,14 +536,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "75aa2c14",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Reordering the logarithms, we can rewrite the **cost/loss** function as"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "780f2038",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathcal{C}(\\boldsymbol{\\beta}) = \\sum_{i=1}^n  \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
@@ -459,7 +558,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c8c940aa",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\beta$.\n",
     "Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that"
@@ -467,7 +569,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9d4a527b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\mathcal{C}(\\boldsymbol{\\beta})=-\\sum_{i=1}^n  \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
@@ -476,23 +581,27 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c1706606",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n",
     "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n",
     "\n",
-    "\n",
     "The cross entropy is a convex function of the weights $\\boldsymbol{\\beta}$ and,\n",
     "therefore, any local minimizer is a global minimizer. \n",
     "\n",
-    "\n",
     "Minimizing this\n",
     "cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fe8fb387",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n  \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n",
@@ -501,14 +610,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "bb564e81",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "700c5443",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n  \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n",
@@ -517,7 +632,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f442c956",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Let us now define a vector $\\boldsymbol{y}$ with $n$ elements $y_i$, an\n",
     "$n\\times p$ matrix $\\boldsymbol{X}$ which contains the $x_i$ values and a\n",
@@ -527,7 +645,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "83937f6f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = -\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{p}\\right).\n",
@@ -536,7 +657,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "71819c1d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "If we in addition define a diagonal matrix $\\boldsymbol{W}$ with elements \n",
     "$p(y_i\\vert x_i,\\boldsymbol{\\beta})(1-p(y_i\\vert x_i,\\boldsymbol{\\beta})$, we can obtain a compact expression of the second derivative as"
@@ -544,7 +668,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "23dfd975",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T} = \\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X}.\n",
@@ -553,14 +680,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6ca9f74b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "ddb3d93a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{ \\frac{p(\\boldsymbol{\\beta}\\boldsymbol{x})}{1-p(\\boldsymbol{\\beta}\\boldsymbol{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n",
@@ -569,14 +702,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "0cfae560",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Here we defined $\\boldsymbol{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\boldsymbol{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "7c427fc3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(\\boldsymbol{\\beta}\\boldsymbol{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n",
@@ -585,7 +724,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "6a6814bf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Till now we have mainly focused on two classes, the so-called binary\n",
     "system. Suppose we wish to extend to $K$ classes.  Let us for the sake\n",
@@ -594,7 +736,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "85282137",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{\\frac{p(C=1\\vert x)}{p(K\\vert x)}} = \\beta_{10}+\\beta_{11}x_1,\n",
@@ -603,14 +748,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4864f76d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "390e9a55",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n",
@@ -619,14 +770,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "c3f39c44",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and so on till the class $C=K-1$ class"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "92f32a03",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n",
@@ -635,13 +792,14 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "677a5c4d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and the model is specified in term of $K-1$ so-called log-odds or\n",
     "**logit** transformations.\n",
     "\n",
-    "\n",
-    "\n",
     "In our discussion of neural networks we will encounter the above again\n",
     "in terms of a slightly modified function, the so-called **Softmax** function.\n",
     "\n",
@@ -658,7 +816,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "fa70e3bf",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n",
@@ -667,14 +828,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "4c59400c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "It is easy to extend to more predictors. The final class is"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "f4073652",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n",
@@ -683,7 +850,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "810e872a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and they sum to one. Our earlier discussions were all specialized to\n",
     "the case with two classes only. It is easy to see from the above that\n",
@@ -692,8 +862,16 @@
     "To find the optimal parameters we would typically use a gradient\n",
     "descent method.  Newton's method and gradient descent methods are\n",
     "discussed in the material on [optimization\n",
-    "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n",
-    "\n",
+    "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f4ebc1da",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
     "## Wisconsin Cancer Data\n",
     "\n",
     "We show here how we can use a simple regression case on the breast\n",
@@ -703,7 +881,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 4,
+   "id": "b7a8cbac",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -739,7 +918,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "10f74b93",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In addition to the above scores, we could also study the covariance (and the correlation matrix).\n",
     "We use **Pandas** to compute the correlation matrix."
@@ -747,7 +929,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 5,
+   "id": "d0b8025d",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -792,7 +975,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "9d84a2aa",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "In the above example we note two things. In the first plot we display\n",
     "the overlap of benign and malignant tumors as functions of the various\n",
@@ -811,7 +997,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 6,
+   "id": "6144ea0a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -823,14 +1010,18 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "5627f5ae",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and then"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 7,
+   "id": "a7c8662a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -842,17 +1033,46 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "id": "d03ee2a7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Diagonalizing this matrix we can in turn say something about which\n",
     "features are of relevance and which are not. This leads  us to\n",
     "the classical Principal Component Analysis (PCA) theorem with\n",
-    "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html))."
+    "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)).\n",
+    "\n",
+    "Here we present a further way to present our results in terms of a so-called **confusion matrix**, the cumulative gain and the **ROC** curve.\n",
+    "This way of displaying our data are based upon different ways to classify our possible outcomes. Before we proceed we need some definitions.\n",
+    "1. **TP**: true positive or in other words, something equivalent with a proper classification\n",
+    "\n",
+    "2. **TN**: true negative, which is equivalent with a correct rejection\n",
+    "\n",
+    "3. **FP**: false positive, or in simpler words something that is equivalent with a false alarm\n",
+    "\n",
+    "4. **FN**: false negative, which is mean to be equivalent with a miss.\n",
+    "\n",
+    "The total data set is then the sum of the true positive and true negative targets or outputs, labeled by $n$.\n",
+    "Based on this we can then define the accuracy score as the sum of correctly predicted **TP** and **TN** cases divided by the sum of true positive and treue negative events in our data set, or as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "44ce5095",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathrm{Accuracy} = \\frac{\\sum_{i=0}^{n-1}I(y_i=\\tilde{y}_i)}{n}.\n",
+    "$$"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 8,
+   "id": "24ff3dd3",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -908,5 +1128,5 @@
  ],
  "metadata": {},
  "nbformat": 4,
- "nbformat_minor": 4
+ "nbformat_minor": 5
 }