From 25b62ff6183164b006cf454ed5750b4af6e36832 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Wed, 16 Sep 2020 10:57:03 +0200 Subject: [PATCH] update on logreg --- doc/pub/week38/html/._week38-bs000.html | 86 +++++----- doc/pub/week38/html/._week38-bs001.html | 94 +++++----- doc/pub/week38/html/._week38-bs002.html | 103 +++++------ doc/pub/week38/html/._week38-bs003.html | 108 +++++------- doc/pub/week38/html/._week38-bs004.html | 115 ++++++------- doc/pub/week38/html/._week38-bs005.html | 120 ++++++------- doc/pub/week38/html/._week38-bs006.html | 115 ++++++------- doc/pub/week38/html/._week38-bs007.html | 121 +++++++------ doc/pub/week38/html/._week38-bs008.html | 114 +++++++------ doc/pub/week38/html/._week38-bs009.html | 158 +++++++---------- doc/pub/week38/html/._week38-bs010.html | 112 ++++++------ doc/pub/week38/html/._week38-bs011.html | 163 +++++++++++------- doc/pub/week38/html/._week38-bs012.html | 106 ++++++------ doc/pub/week38/html/._week38-bs013.html | 111 ++++++------ doc/pub/week38/html/._week38-bs014.html | 108 ++++++------ doc/pub/week38/html/._week38-bs015.html | 102 ++++++----- doc/pub/week38/html/._week38-bs016.html | 115 ++++++------- doc/pub/week38/html/._week38-bs017.html | 120 ++++++------- doc/pub/week38/html/._week38-bs018.html | 148 +++++++--------- doc/pub/week38/html/._week38-bs019.html | 145 +++++++++------- doc/pub/week38/html/._week38-bs020.html | 170 ++++++++++--------- doc/pub/week38/html/week38-bs.html | 86 +++++----- doc/pub/week38/html/week38-reveal.html | 56 +++--- doc/pub/week38/html/week38-solarized.html | 98 ++++++----- doc/pub/week38/html/week38.html | 98 ++++++----- doc/pub/week38/ipynb/ipynb-week38-src.tar.gz | Bin 197 -> 197 bytes doc/pub/week38/ipynb/week38.ipynb | 11 +- 27 files changed, 1472 insertions(+), 1411 deletions(-) diff --git a/doc/pub/week38/html/._week38-bs000.html b/doc/pub/week38/html/._week38-bs000.html index 5a7692ec7..5b21e45a1 100644 --- a/doc/pub/week38/html/._week38-bs000.html +++ b/doc/pub/week38/html/._week38-bs000.html @@ -41,36 +41,38 @@ Automatically generated HTML file from DocOnce source @@ -108,26 +110,28 @@ MathJax.Hub.Config({
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    To do for log reg

    +

    Plans for week 38

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    - + -

    Logistic Regression

    - -

    -In linear regression our main interest was centered on learning the -coefficients of a functional fit (say a polynomial) in order to be -able to predict the response of a continuous variable on some unseen -data. The fit to the continuous variable \( y_i \) is based on some -independent variables \( \hat{x}_i \). Linear regression resulted in -analytical expressions for standard ordinary Least Squares or Ridge -regression (in terms of matrices to invert) for several quantities, -ranging from the variance and thereby the confidence intervals of the -parameters \( \hat{\beta} \) to the mean squared error. If we can invert -the product of the design matrices, linear regression gives then a -simple recipe for fitting our data. +

    Thursday:

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    - + -

    Classification problems

    - -

    -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 -out which mutations are important for a certain disease; or based on -scans of various patients' brains, figure out if there is a tumor or -not; or given a specific physical system, we'd like to identify its -state, say whether it is an ordered or disordered system (typical -situation in solid state physics); or classify the status of a -patient, whether she/he has a stroke or not and many other similar -situations. - -

    -The most common situation we encounter when we apply logistic -regression is that of two possible outcomes, normally denoted as a -binary outcome, true or false, positive or negative, success or -failure etc. +

    Friday: Intro to Logistic Regression

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    - + -

    Optimization and Deep learning

    +

    Logistic Regression

    -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 -equation in the parameters \( \hat{\beta} \). The optimization of the -problem calls therefore for minimization algorithms. This forms the -bottle neck of all machine learning algorithms, namely how to find -reliable minima of a multi-variable function. This leads us to the -family of gradient descent methods. The latter are the working horses -of basically all modern machine learning algorithms. - -

    -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. +In linear regression our main interest was centered on learning the +coefficients of a functional fit (say a polynomial) in order to be +able to predict the response of a continuous variable on some unseen +data. The fit to the continuous variable \( y_i \) is based on some +independent variables \( \hat{x}_i \). Linear regression resulted in +analytical expressions for standard ordinary Least Squares or Ridge +regression (in terms of matrices to invert) for several quantities, +ranging from the variance and thereby the confidence intervals of the +parameters \( \hat{\beta} \) to the mean squared error. If we can invert +the product of the design matrices, linear regression gives then a +simple recipe for fitting our data.

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    Basics

    +

    Classification problems

    -We consider the case where the dependent variables, also called the -responses or the outcomes, \( y_i \) are discrete and only take values -from \( k=0,\dots,K-1 \) (i.e. \( K \) classes). +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 +out which mutations are important for a certain disease; or based on +scans of various patients' brains, figure out if there is a tumor or +not; or given a specific physical system, we'd like to identify its +state, say whether it is an ordered or disordered system (typical +situation in solid state physics); or classify the status of a +patient, whether she/he has a stroke or not and many other similar +situations.

    -The goal is to predict the -output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \) -made of \( n \) samples, each of which carries \( p \) features or predictors. The -primary goal is to identify the classes to which new unseen samples -belong. - -

    -Let us specialize to the case of two classes only, with outputs -\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a -credit card user that could default or not on her/his credit card -debt. That is - -$$ -y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. -$$ +The most common situation we encounter when we apply logistic +regression is that of two possible outcomes, normally denoted as a +binary outcome, true or false, positive or negative, success or +failure etc.

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    Linear classifier

    +

    Optimization and Deep learning

    -Before moving to the logistic model, let us try to use our linear -regression model to classify these two outcomes. We could for example -fit a linear model to the default case if \( y_i > 0.5 \) and the no -default case \( y_i \leq 0.5 \). +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 +equation in the parameters \( \hat{\beta} \). The optimization of the +problem calls therefore for minimization algorithms. This forms the +bottle neck of all machine learning algorithms, namely how to find +reliable minima of a multi-variable function. This leads us to the +family of gradient descent methods. The latter are the working horses +of basically all modern machine learning algorithms.

    -We would then have our -weighted linear combination, namely -$$ -\begin{equation} -\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, -\tag{1} -\end{equation} -$$ - -where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our -\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors. +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.

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    - + -

    Some selected properties

    +

    Basics

    -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 -card data discussed below here, where we can set the state of -defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons -in the data set (see the full example below). +We consider the case where the dependent variables, also called the +responses or the outcomes, \( y_i \) are discrete and only take values +from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).

    -One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to values \( \{0,1\} \), -\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. -We will encounter this model in our first demonstration of neural networks. Historically it is called the "perceptron" model in the machine learning -literature. This model is extremely simple. However, in many cases it is more -favorable to use a ``soft" classifier that outputs -the probability of a given category. This leads us to the logistic function. +The goal is to predict the +output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \) +made of \( n \) samples, each of which carries \( p \) features or predictors. The +primary goal is to identify the classes to which new unseen samples +belong. + +

    +Let us specialize to the case of two classes only, with outputs +\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a +credit card user that could default or not on her/his credit card +debt. That is + +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. +$$

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    The logistic function

    +

    Linear classifier

    -The perceptron is an example of a ``hard classification" model. We -will encounter this model when we discuss neural networks as -well. Each datapoint is deterministically assigned to a category (i.e -\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft" -classifier that outputs the probability of a given category rather -than a single value. For example, given \( x_i \), the classifier -outputs the probability of being in a category \( k \). Logistic regression -is the most common example of a so-called soft classifier. In logistic -regression, the probability that a data point \( x_i \) -belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, +Before moving to the logistic model, let us try to use our linear +regression model to classify these two outcomes. We could for example +fit a linear model to the default case if \( y_i > 0.5 \) and the no +default case \( y_i \leq 0.5 \). + +

    +We would then have our +weighted linear combination, namely $$ -p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. +\begin{equation} +\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, +\tag{1} +\end{equation} $$ -Note that \( 1-p(t)= p(-t) \). +where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our +\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors.

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    Examples of likelihood functions used in logistic regression and nueral networks

    +

    Some selected properties

    -The following code plots the logistic function, the step function and other functions we will encounter from here and on. +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 +card data discussed below here, where we can set the state of +defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons +in the data set (see the full example below).

    +One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to values \( \{0,1\} \), +\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. +We will encounter this model in our first demonstration of neural networks. Historically it is called the "perceptron" model in the machine learning +literature. This model is extremely simple. However, in many cases it is more +favorable to use a ``soft" classifier that outputs +the probability of a given category. This leads us to the logistic function. - -

    """The sigmoid function (or the logistic curve) is a
-function that takes any real number, z, and outputs a number (0,1).
-It is useful in neural networks for assigning weights on a relative scale.
-The value z is the weighted sum of parameters involved in the learning algorithm."""
-
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
-
-z = numpy.arange(-5, 5, .1)
-sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
-sigma = sigma_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, sigma)
-ax.set_ylim([-0.1, 1.1])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sigmoid function')
-
-plt.show()
-
-"""Step Function"""
-z = numpy.arange(-5, 5, .02)
-step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
-step = step_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, step)
-ax.set_ylim([-0.5, 1.5])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('step function')
-
-plt.show()
-
-"""tanh Function"""
-z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
-t = numpy.tanh(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, t)
-ax.set_ylim([-1.0, 1.0])
-ax.set_xlim([-2*mt.pi,2*mt.pi])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('tanh function')
-
-plt.show()
-

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    Two parameters

    +

    The logistic function

    -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 +The perceptron is an example of a ``hard classification" model. We +will encounter this model when we discuss neural networks as +well. Each datapoint is deterministically assigned to a category (i.e +\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft" +classifier that outputs the probability of a given category rather +than a single value. For example, given \( x_i \), the classifier +outputs the probability of being in a category \( k \). Logistic regression +is the most common example of a so-called soft classifier. In logistic +regression, the probability that a data point \( x_i \) +belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, $$ -\begin{align*} -p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ -p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), -\end{align*} +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. $$ -where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). - -

    -Note that we used -$$ -p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). -$$ +Note that \( 1-p(t)= p(-t) \).

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    - + -

    Maximum likelihood

    +

    Examples of likelihood functions used in logistic regression and nueral networks

    -In order to define the total likelihood for all possible outcomes from a -dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels -\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. -We aim thus at maximizing -the probability of seeing the observed data. We can then approximate the -likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is -$$ -\begin{align*} -P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ -\end{align*} -$$ +The following code plots the logistic function, the step function and other functions we will encounter from here and on. -from which we obtain the log-likelihood and our cost/loss function -$$ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). -$$ +

    + +

    """The sigmoid function (or the logistic curve) is a
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
+
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""tanh Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.tanh(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('tanh function')
+
+plt.show()
+

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    The cost function rewritten

    +

    Two parameters

    -Reordering the logarithms, we can rewrite the cost/loss function as +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 $$ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} $$ +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). +

    -The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \). -Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that +Note that we used $$ -\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). $$ -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. -

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    - + -

    Minimizing the cross entropy

    +

    Maximum likelihood

    -The cross entropy is a convex function of the weights \( \hat{\beta} \) 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 - +In order to define the total likelihood for all possible outcomes from a +dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels +\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. +We aim thus at maximizing +the probability of seeing the observed data. We can then approximate the +likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is $$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} $$ -and +from which we obtain the log-likelihood and our cost/loss function $$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). $$

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    A more compact expression

    +

    The cost function rewritten

    -Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an -\( n\times p \) matrix \( \hat{X} \) which contains the \( x_i \) values and a -vector \( \hat{p} \) of fitted probabilities \( p(y_i\vert x_i,\hat{\beta}) \). We can rewrite in a more compact form the first -derivative of cost function as - +Reordering the logarithms, we can rewrite the cost/loss function as $$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). $$

    -If we in addition define a diagonal matrix \( \hat{W} \) with elements -\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as +The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \). +Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that +$$ +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ -$$ -\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. -$$ +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.

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    Extending to more predictors

    +

    Minimizing the cross entropy

    -Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors +The cross entropy is a convex function of the weights \( \hat{\beta} \) 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 + $$ -\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), $$ -Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to +and $$ -p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). $$

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    Including more classes

    +

    A more compact expression

    -Till now we have mainly focused on two classes, the so-called binary -system. Suppose we wish to extend to \( K \) classes. Let us for the sake -of simplicity assume we have only two predictors. We have then -following model +Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an +\( n\times p \) matrix \( \hat{X} \) which contains the \( x_i \) values and a +vector \( \hat{p} \) of fitted probabilities \( p(y_i\vert x_i,\hat{\beta}) \). We can rewrite in a more compact form the first +derivative of cost function as $$ -\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, -$$ - -$$ -\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, -$$ - -and so on till the class \( C=K-1 \) class -$$ -\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). $$

    -and the model is specified in term of \( K-1 \) so-called log-odds or -logit transformations. +If we in addition define a diagonal matrix \( \hat{W} \) with elements +\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as + +$$ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +$$

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    More classes

    +

    Extending to more predictors

    -In our discussion of neural networks we will encounter the above again -in terms of a slightly modified function, the so-called Softmax function. - -

    -The softmax function is used in various multiclass classification -methods, such as multinomial logistic regression (also known as -softmax regression), multiclass linear discriminant analysis, naive -Bayes classifiers, and artificial neural networks. Specifically, in -multinomial logistic regression and linear discriminant analysis, the -input to the function is the result of \( K \) distinct linear functions, -and the predicted probability for the \( k \)-th class given a sample -vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two -predictors): - +Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors $$ -p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. $$ -It is easy to extend to more predictors. The final class is +Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to $$ -p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. $$ -

    -and they sum to one. Our earlier discussions were all specialized to -the case with two classes only. It is easy to see from the above that -what we derived earlier is compatible with these equations. - -

    -To find the optimal parameters we would typically use a gradient -descent method. Newton's method and gradient descent methods are -discussed in the material on optimization -methods. -

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    A simple classification problem

    +

    Including more classes

    +

    +Till now we have mainly focused on two classes, the so-called binary +system. Suppose we wish to extend to \( K \) classes. Let us for the sake +of simplicity assume we have only two predictors. We have then +following model - -

    import numpy as np
-from sklearn import datasets, linear_model
-import matplotlib.pyplot as plt
+$$
+\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
+$$
 
+$$
+\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
+$$
 
-def generate_data():
-    np.random.seed(0)
-    X, y = datasets.make_moons(200, noise=0.20)
-    return X, y
+and so on till the class \( C=K-1 \) class
+$$
+\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
+$$
 
+
    
+and the model is specified in term of \( K-1 \) so-called log-odds or
+logit transformations.
 
-def visualize(X, y, clf):
-    plot_decision_boundary(lambda x: clf.predict(x), X, y)
-
-def plot_decision_boundary(pred_func, X, y):
-    # Set min and max values and give it some padding
-    x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
-    y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
-    h = 0.01
-    # Generate a grid of points with distance h between them
-    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
-    # Predict the function value for the whole gid
-    Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
-    Z = Z.reshape(xx.shape)
-    # Plot the contour and training examples
-    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
-    plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)
-    plt.show()
-
-
-def classify(X, y):
-    clf = linear_model.LogisticRegressionCV()
-    clf.fit(X, y)
-    return clf
-
-
-def main():
-    X, y = generate_data()
-    # visualize(X, y)
-    clf = classify(X, y)
-    visualize(X, y, clf)
-
-if __name__ == "__main__":
-    main()
-

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    Cancer Data again now with Decision Trees and other Methods

    +

    More classes

    +

    +In our discussion of neural networks we will encounter the above again +in terms of a slightly modified function, the so-called Softmax function. - -

    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
+
    
+The softmax function is used in various multiclass classification
+methods, such as multinomial logistic regression (also known as
+softmax regression), multiclass linear discriminant analysis, naive
+Bayes classifiers, and artificial neural networks.  Specifically, in
+multinomial logistic regression and linear discriminant analysis, the
+input to the function is the result of \( K \) distinct linear functions,
+and the predicted probability for the \( k \)-th class given a sample
+vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two
+predictors):
 
-# Load the data
-cancer = load_breast_cancer()
+$$
+p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
+$$
+
+It is easy to extend to more predictors. The final class is 
+$$
+p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
+$$
+
+
    
+and they sum to one. Our earlier discussions were all specialized to
+the case with two classes only. It is easy to see from the above that
+what we derived earlier is compatible with these equations.
+
+
    
+To find the optimal parameters we would typically use a gradient
+descent method.  Newton's method and gradient descent methods are
+discussed in the material on optimization
+methods.
 
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-

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    Other measures in classification studies: Cancer Data again

    +

    A simple classification problem

    -

    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+
    import numpy as np
+from sklearn import datasets, linear_model
+import matplotlib.pyplot as plt
 
 
-from sklearn.preprocessing import LabelEncoder
-from sklearn.model_selection import cross_validate
-#Cross validation
-accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Logistic Regression  and scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+def generate_data():
+    np.random.seed(0)
+    X, y = datasets.make_moons(200, noise=0.20)
+    return X, y
 
 
-import scikitplot as skplt
-y_pred = logreg.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-plt.show()
-y_probas = logreg.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-plt.show()
+def visualize(X, y, clf):
+    plot_decision_boundary(lambda x: clf.predict(x), X, y)
+
+def plot_decision_boundary(pred_func, X, y):
+    # Set min and max values and give it some padding
+    x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
+    y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
+    h = 0.01
+    # Generate a grid of points with distance h between them
+    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
+    # Predict the function value for the whole gid
+    Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
+    Z = Z.reshape(xx.shape)
+    # Plot the contour and training examples
+    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
+    plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)
+    plt.show()
+
+
+def classify(X, y):
+    clf = linear_model.LogisticRegressionCV()
+    clf.fit(X, y)
+    return clf
+
+
+def main():
+    X, y = generate_data()
+    # visualize(X, y)
+    clf = classify(X, y)
+    visualize(X, y, clf)
+
+if __name__ == "__main__":
+    main()
 
    
 
    
-
 
    
 
 
    
 
 
diff --git a/doc/pub/week38/html/week38-bs.html b/doc/pub/week38/html/week38-bs.html
index 5a7692ec7..5b21e45a1 100644
--- a/doc/pub/week38/html/week38-bs.html
+++ b/doc/pub/week38/html/week38-bs.html
@@ -41,36 +41,38 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -108,26 +110,28 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -186,7 +190,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 21
    • 
+  
  • 23
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/week38-reveal.html b/doc/pub/week38/html/week38-reveal.html
index 04a81430a..10c4f13ae 100644
--- a/doc/pub/week38/html/week38-reveal.html
+++ b/doc/pub/week38/html/week38-reveal.html
@@ -159,19 +159,27 @@ MathJax.Hub.Config({
 
 
 
    
-
    To do for log reg 
    
+
    Plans for week 38 
    
 
 
      
-
    •  Develop code for log reg step by step, with link to gradient descent part
      • 
-
    •  show how to read and set up design matrix
      • 
-
    •  use breast cancer data as example
      • 
-
    •  develop other classification examples, pulsar example
      • 
+
    •  Thursday: Summary of regression methods and discussion of project 1. We revisit also cross-validation and bootstrap as resampling techniques with examples
      • 
+
    •  Friday: Logistic Regression
      • 
 
    
 
    
 
 
 
    
-
    Logistic Regression 
    
+
    Thursday: 
    
+
    
+
+
+
    
+
    Friday: Intro to Logistic Regression 
    
+
    
+
+
+
    
+
    Logistic Regression 
    
 
 
    
 In linear regression our main interest was centered on learning the
@@ -189,7 +197,7 @@ simple recipe for fitting our data.
 
 
 
    
-
    Classification problems 
    
+
    Classification problems 
    
 
 
    
 Classification problems, however, are concerned with outcomes taking
@@ -212,7 +220,7 @@ failure etc.
 
 
 
    
-
    Optimization and Deep learning 
    
+
    Optimization and Deep learning 
    
 
 
    
 Logistic regression will also serve as our stepping stone towards
@@ -233,7 +241,7 @@ models, as we will see later.
 
 
 
    
-
    Basics 
    
+
    Basics 
    
 
 
    
 We consider the case where the dependent variables, also called the
@@ -262,7 +270,7 @@ $$
 
 
 
    
-
    Linear classifier 
    
+
    Linear classifier 
    
 
 
    
 Before moving to the logistic model, let us try to use our linear
@@ -288,7 +296,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
 
 
 
    
-
    Some selected properties 
    
+
    Some selected properties 
    
 
 
    
 The main problem with our function is that it takes values on the
@@ -310,7 +318,7 @@ the probability of a given category. This leads us to the logistic function.
 
 
 
    
-
    The logistic function 
    
+
    The logistic function 
    
 
 
    
 The perceptron is an example of a ``hard classification" model. We
@@ -334,7 +342,7 @@ 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 nueral networks 
    
 
 
    
 The following code plots the logistic function, the step function and other functions we will encounter from here and on.
@@ -401,7 +409,7 @@ plt.show()
 
 
 
    
-
    Two parameters 
    
+
    Two parameters 
    
 
 
    
 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 
@@ -427,7 +435,7 @@ $$
 
 
 
    
-
    Maximum likelihood 
    
+
    Maximum likelihood 
    
 
 
    
 In order to define the total likelihood for all possible outcomes from a  
@@ -454,7 +462,7 @@ $$
 
 
 
    
-
    The cost function rewritten 
    
+
    The cost function rewritten 
    
 
 
    
 Reordering the logarithms, we can rewrite the cost/loss function as
@@ -479,7 +487,7 @@ in practice we often supplement the cross-entropy with additional regularization
 
 
 
    
-
    Minimizing the cross entropy 
    
+
    Minimizing the cross entropy 
    
 
 
    
 The cross entropy is a convex function of the weights \( \hat{\beta} \) and,
@@ -505,7 +513,7 @@ $$
 
 
 
    
-
    A more compact expression 
    
+
    A more compact expression 
    
 
 
    
 Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
@@ -532,7 +540,7 @@ $$
 
 
 
    
-
    Extending to more predictors 
    
+
    Extending to more predictors 
    
 
 
    
 Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
@@ -552,7 +560,7 @@ $$
 
 
 
    
-
    Including more classes 
    
+
    Including more classes 
    
 
 
    
 Till now we have mainly focused on two classes, the so-called binary
@@ -586,7 +594,7 @@ and the model is specified in term of \( K-1 \) so-called log-odds or
 
 
 
    
-
    More classes 
    
+
    More classes 
    
 
 
    
 In our discussion of neural networks we will encounter the above again
@@ -630,7 +638,7 @@ methods.
 
 
 
    
-
    A simple classification problem 
    
+
    A simple classification problem 
    
 
    
 
 
@@ -683,7 +691,7 @@ methods.
 
 
 
    
-
    Cancer Data again now with Decision Trees and other Methods 
    
+
    Cancer Data again now with Decision Trees and other Methods 
    
 
    
 
 
@@ -717,7 +725,7 @@ logreg.fit(X_train_scaled, y_train)
 
 
 
    
-
    Other measures in classification studies: Cancer Data  again 
    
+
    Other measures in classification studies: Cancer Data  again 
    
 
    
 
 
diff --git a/doc/pub/week38/html/week38-solarized.html b/doc/pub/week38/html/week38-solarized.html
index a21c3811f..bbd654b0b 100644
--- a/doc/pub/week38/html/week38-solarized.html
+++ b/doc/pub/week38/html/week38-solarized.html
@@ -35,36 +35,38 @@ div { text-align: justify; text-justify: inter-word; }
 
 
 
 
@@ -111,18 +113,26 @@ MathJax.Hub.Config({
 
    
 









    
 
-
    To do for log reg 
    
+
    Plans for week 38 
    
 
 
      
-
    •  Develop code for log reg step by step, with link to gradient descent part
      • 
-
    •  show how to read and set up design matrix
      • 
-
    •  use breast cancer data as example
      • 
-
    •  develop other classification examples, pulsar example
      • 
+
    •  Thursday: Summary of regression methods and discussion of project 1. We revisit also cross-validation and bootstrap as resampling techniques with examples
      • 
+
    •  Friday: Logistic Regression
      • 
 
    
 
+









    
+
+
    Thursday: 
    
+
+
    
+









    
+
+
    Friday: Intro to Logistic Regression 
    
+
+
    
 
 
-
    Logistic Regression 
    
+
    Logistic Regression 
    
 
 
    
 In linear regression our main interest was centered on learning the
@@ -140,7 +150,7 @@ simple recipe for fitting our data.
 
    
 
 
-
    Classification problems 
    
+
    Classification problems 
    
 
 
    
 Classification problems, however, are concerned with outcomes taking
@@ -163,7 +173,7 @@ failure etc.
 
    
 









    
 
-
    Optimization and Deep learning 
    
+
    Optimization and Deep learning 
    
 
 
    
 Logistic regression will also serve as our stepping stone towards
@@ -184,7 +194,7 @@ models, as we will see later.
 
    
 
 
-
    Basics 
    
+
    Basics 
    
 
 
    
 We consider the case where the dependent variables, also called the
@@ -211,7 +221,7 @@ $$
 
    
 









    
 
-
    Linear classifier 
    
+
    Linear classifier 
    
 
 
    
 Before moving to the logistic model, let us try to use our linear
@@ -235,7 +245,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
 
    
 









    
 
-
    Some selected properties 
    
+
    Some selected properties 
    
 
 
    
 The main problem with our function is that it takes values on the
@@ -257,7 +267,7 @@ the probability of a given category. This leads us to the logistic function.
 
    
 









    
 
-
    The logistic function 
    
+
    The logistic function 
    
 
 
    
 The perceptron is an example of a ``hard classification" model. We
@@ -279,7 +289,7 @@ 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 nueral networks 
    
 
 
    
 The following code plots the logistic function, the step function and other functions we will encounter from here and on.
@@ -345,7 +355,7 @@ plt.show()
 
    
 









    
 
-
    Two parameters 
    
+
    Two parameters 
    
 
 
    
 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 
@@ -367,7 +377,7 @@ $$
 
    
 
 
-
    Maximum likelihood 
    
+
    Maximum likelihood 
    
 
 
    
 In order to define the total likelihood for all possible outcomes from a  
@@ -390,7 +400,7 @@ $$
 
    
 









    
 
-
    The cost function rewritten 
    
+
    The cost function rewritten 
    
 
 
    
 Reordering the logarithms, we can rewrite the cost/loss function as
@@ -411,7 +421,7 @@ in practice we often supplement the cross-entropy with additional regularization
 
    
 









    
 
-
    Minimizing the cross entropy 
    
+
    Minimizing the cross entropy 
    
 
 
    
 The cross entropy is a convex function of the weights \( \hat{\beta} \) and,
@@ -433,7 +443,7 @@ $$
 
    
 









    
 
-
    A more compact expression 
    
+
    A more compact expression 
    
 
 
    
 Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
@@ -456,7 +466,7 @@ $$
 
    
 









    
 
-
    Extending to more predictors 
    
+
    Extending to more predictors 
    
 
 
    
 Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
@@ -472,7 +482,7 @@ $$
 
    
 









    
 
-
    Including more classes 
    
+
    Including more classes 
    
 
 
    
 Till now we have mainly focused on two classes, the so-called binary
@@ -500,7 +510,7 @@ and the model is specified in term of \( K-1 \) so-called log-odds or
 
    
 









    
 
-
    More classes 
    
+
    More classes 
    
 
 
    
 In our discussion of neural networks we will encounter the above again
@@ -540,7 +550,7 @@ methods.
 
    
 









    
 
-
    A simple classification problem 
    
+
    A simple classification problem 
    
 
    
 
 
@@ -592,7 +602,7 @@ methods.
 
    
 









    
 
-
    Cancer Data again now with Decision Trees and other Methods 
    
+
    Cancer Data again now with Decision Trees and other Methods 
    
 
    
 
 
@@ -625,7 +635,7 @@ logreg.fit(X_train_scaled, y_train)
 
    
 









    
 
-
    Other measures in classification studies: Cancer Data  again 
    
+
    Other measures in classification studies: Cancer Data  again 
    
 
    
 
 
diff --git a/doc/pub/week38/html/week38.html b/doc/pub/week38/html/week38.html
index 0ae1d126a..60b6504f1 100644
--- a/doc/pub/week38/html/week38.html
+++ b/doc/pub/week38/html/week38.html
@@ -40,36 +40,38 @@ div { text-align: justify; text-justify: inter-word; }
 
 
 
 
@@ -116,18 +118,26 @@ MathJax.Hub.Config({
 
    
 









    
 
-
    To do for log reg 
    
+
    Plans for week 38 
    
 
 
      
-
    •  Develop code for log reg step by step, with link to gradient descent part
      • 
-
    •  show how to read and set up design matrix
      • 
-
    •  use breast cancer data as example
      • 
-
    •  develop other classification examples, pulsar example
      • 
+
    •  Thursday: Summary of regression methods and discussion of project 1. We revisit also cross-validation and bootstrap as resampling techniques with examples
      • 
+
    •  Friday: Logistic Regression
      • 
 
    
 
+









    
+
+
    Thursday: 
    
+
+
    
+









    
+
+
    Friday: Intro to Logistic Regression 
    
+
+
    
 
 
-
    Logistic Regression 
    
+
    Logistic Regression 
    
 
 
    
 In linear regression our main interest was centered on learning the
@@ -145,7 +155,7 @@ simple recipe for fitting our data.
 
    
 
 
-
    Classification problems 
    
+
    Classification problems 
    
 
 
    
 Classification problems, however, are concerned with outcomes taking
@@ -168,7 +178,7 @@ failure etc.
 
    
 









    
 
-
    Optimization and Deep learning 
    
+
    Optimization and Deep learning 
    
 
 
    
 Logistic regression will also serve as our stepping stone towards
@@ -189,7 +199,7 @@ models, as we will see later.
 
    
 
 
-
    Basics 
    
+
    Basics 
    
 
 
    
 We consider the case where the dependent variables, also called the
@@ -216,7 +226,7 @@ $$
 
    
 









    
 
-
    Linear classifier 
    
+
    Linear classifier 
    
 
 
    
 Before moving to the logistic model, let us try to use our linear
@@ -240,7 +250,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
 
    
 









    
 
-
    Some selected properties 
    
+
    Some selected properties 
    
 
 
    
 The main problem with our function is that it takes values on the
@@ -262,7 +272,7 @@ the probability of a given category. This leads us to the logistic function.
 
    
 









    
 
-
    The logistic function 
    
+
    The logistic function 
    
 
 
    
 The perceptron is an example of a ``hard classification" model. We
@@ -284,7 +294,7 @@ 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 nueral networks 
    
 
 
    
 The following code plots the logistic function, the step function and other functions we will encounter from here and on.
@@ -350,7 +360,7 @@ plt.show()
 
    
 









    
 
-
    Two parameters 
    
+
    Two parameters 
    
 
 
    
 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 
@@ -372,7 +382,7 @@ $$
 
    
 
 
-
    Maximum likelihood 
    
+
    Maximum likelihood 
    
 
 
    
 In order to define the total likelihood for all possible outcomes from a  
@@ -395,7 +405,7 @@ $$
 
    
 









    
 
-
    The cost function rewritten 
    
+
    The cost function rewritten 
    
 
 
    
 Reordering the logarithms, we can rewrite the cost/loss function as
@@ -416,7 +426,7 @@ in practice we often supplement the cross-entropy with additional regularization
 
    
 









    
 
-
    Minimizing the cross entropy 
    
+
    Minimizing the cross entropy 
    
 
 
    
 The cross entropy is a convex function of the weights \( \hat{\beta} \) and,
@@ -438,7 +448,7 @@ $$
 
    
 









    
 
-
    A more compact expression 
    
+
    A more compact expression 
    
 
 
    
 Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
@@ -461,7 +471,7 @@ $$
 
    
 









    
 
-
    Extending to more predictors 
    
+
    Extending to more predictors 
    
 
 
    
 Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
@@ -477,7 +487,7 @@ $$
 
    
 









    
 
-
    Including more classes 
    
+
    Including more classes 
    
 
 
    
 Till now we have mainly focused on two classes, the so-called binary
@@ -505,7 +515,7 @@ and the model is specified in term of \( K-1 \) so-called log-odds or
 
    
 









    
 
-
    More classes 
    
+
    More classes 
    
 
 
    
 In our discussion of neural networks we will encounter the above again
@@ -545,7 +555,7 @@ methods.
 
    
 









    
 
-
    A simple classification problem 
    
+
    A simple classification problem 
    
 
    
 
 
@@ -597,7 +607,7 @@ methods.
 
    
 









    
 
-
    Cancer Data again now with Decision Trees and other Methods 
    
+
    Cancer Data again now with Decision Trees and other Methods 
    
 
    
 
 
@@ -630,7 +640,7 @@ logreg.fit(X_train_scaled, y_train)
 
    
 









    
 
-
    Other measures in classification studies: Cancer Data  again 
    
+
    Other measures in classification studies: Cancer Data  again 
    
 
    
 
 
diff --git a/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz b/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz
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diff --git a/doc/pub/week38/ipynb/week38.ipynb b/doc/pub/week38/ipynb/week38.ipynb
index a45db93d6..54934964e 100644
--- a/doc/pub/week38/ipynb/week38.ipynb
+++ b/doc/pub/week38/ipynb/week38.ipynb
@@ -17,15 +17,16 @@
     "\n",
     "\n",
     "\n",
-    "## To do for log reg\n",
+    "## Plans for week 38\n",
     "\n",
-    "* Develop code for log reg step by step, with link to gradient descent part\n",
+    "* Thursday: Summary of regression methods and discussion of project 1. We revisit also cross-validation and bootstrap as resampling techniques with examples\n",
     "\n",
-    "* show how to read and set up design matrix\n",
+    "* Friday: Logistic Regression\n",
     "\n",
-    "* use breast cancer data as example\n",
+    "## Thursday:\n",
+    "\n",
+    "## Friday: Intro to Logistic Regression\n",
     "\n",
-    "* develop other classification examples, pulsar example\n",
     "\n",
     "\n",
     "## Logistic Regression\n",