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Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
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+ + +
+ +

 

 

 

+ + + + + + +
+

Data Analysis and Machine Learning: Logistic Regression

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 16, 2020

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+

+ + +

Read »

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+ +

+ + +
+ + + + + + + +
+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs001.html b/doc/pub/week38/html/._week38-bs001.html new file mode 100644 index 000000000..11627073b --- /dev/null +++ b/doc/pub/week38/html/._week38-bs001.html @@ -0,0 +1,186 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Logistic Regression +
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+ + + + +

To do for log reg

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+ + Data Analysis and Machine Learning: Logistic Regression +
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+ + + + +

Logistic Regression

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+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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+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs003.html b/doc/pub/week38/html/._week38-bs003.html new file mode 100644 index 000000000..7a20afd9e --- /dev/null +++ b/doc/pub/week38/html/._week38-bs003.html @@ -0,0 +1,200 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Logistic Regression +
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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. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs004.html b/doc/pub/week38/html/._week38-bs004.html new file mode 100644 index 000000000..298c19e6e --- /dev/null +++ b/doc/pub/week38/html/._week38-bs004.html @@ -0,0 +1,199 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Optimization and Deep learning

+ +

+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. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs005.html b/doc/pub/week38/html/._week38-bs005.html new file mode 100644 index 000000000..0625d52da --- /dev/null +++ b/doc/pub/week38/html/._week38-bs005.html @@ -0,0 +1,206 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Basics

+ +

+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). + +

+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 + +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. +$$ + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs006.html b/doc/pub/week38/html/._week38-bs006.html new file mode 100644 index 000000000..c3742d046 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs006.html @@ -0,0 +1,204 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Linear classifier

+ +

+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 +$$ +\begin{equation} +\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, +\tag{1} +\end{equation} +$$ + +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. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs007.html b/doc/pub/week38/html/._week38-bs007.html new file mode 100644 index 000000000..5c4d49656 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs007.html @@ -0,0 +1,203 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Some selected properties

+ +

+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. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs008.html b/doc/pub/week38/html/._week38-bs008.html new file mode 100644 index 000000000..35d593d40 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs008.html @@ -0,0 +1,204 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

The logistic function

+ +

+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, +$$ +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. +$$ + +Note that \( 1-p(t)= p(-t) \). + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs009.html b/doc/pub/week38/html/._week38-bs009.html new file mode 100644 index 000000000..fbb3a245b --- /dev/null +++ b/doc/pub/week38/html/._week38-bs009.html @@ -0,0 +1,247 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

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. + +

+ + +

"""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()
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs010.html b/doc/pub/week38/html/._week38-bs010.html new file mode 100644 index 000000000..37598fe3a --- /dev/null +++ b/doc/pub/week38/html/._week38-bs010.html @@ -0,0 +1,203 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

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 +$$ +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ + +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). + +

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

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs011.html b/doc/pub/week38/html/._week38-bs011.html new file mode 100644 index 000000000..16b319b9a --- /dev/null +++ b/doc/pub/week38/html/._week38-bs011.html @@ -0,0 +1,203 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Maximum likelihood

+ +

+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 +$$ +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} +$$ + +from which we obtain the log-likelihood and our cost/loss function +$$ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). +$$ + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs012.html b/doc/pub/week38/html/._week38-bs012.html new file mode 100644 index 000000000..cadf8fee8 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs012.html @@ -0,0 +1,200 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

The cost function rewritten

+ +

+Reordering the logarithms, we can rewrite the cost/loss function as +$$ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +

+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 +$$ +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +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. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs013.html b/doc/pub/week38/html/._week38-bs013.html new file mode 100644 index 000000000..abe1ad6a1 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs013.html @@ -0,0 +1,200 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Minimizing the cross entropy

+ +

+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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), +$$ + +and +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +$$ + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs014.html b/doc/pub/week38/html/._week38-bs014.html new file mode 100644 index 000000000..eb7603a2b --- /dev/null +++ b/doc/pub/week38/html/._week38-bs014.html @@ -0,0 +1,200 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

A more compact expression

+ +

+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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ + +

+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 + +$$ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +$$ + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs015.html b/doc/pub/week38/html/._week38-bs015.html new file mode 100644 index 000000000..89c9ea022 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs015.html @@ -0,0 +1,192 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

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 +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ + +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 +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs016.html b/doc/pub/week38/html/._week38-bs016.html new file mode 100644 index 000000000..9b51accbd --- /dev/null +++ b/doc/pub/week38/html/._week38-bs016.html @@ -0,0 +1,203 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

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 + +$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ + +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ + +and so on till the class \( C=K-1 \) class +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ + +

+and the model is specified in term of \( K-1 \) so-called log-odds or +logit transformations. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs017.html b/doc/pub/week38/html/._week38-bs017.html new file mode 100644 index 000000000..c025008c3 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs017.html @@ -0,0 +1,214 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

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. + +

+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): + +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ + +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. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week38/html/._week38-bs018.html b/doc/pub/week38/html/._week38-bs018.html new file mode 100644 index 000000000..955b2c7e7 --- /dev/null +++ b/doc/pub/week38/html/._week38-bs018.html @@ -0,0 +1,225 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Logistic Regression +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

A simple classification problem

+

+ + +

import numpy as np
+from sklearn import datasets, linear_model
+import matplotlib.pyplot as plt
+
+
+def generate_data():
+    np.random.seed(0)
+    X, y = datasets.make_moons(200, noise=0.20)
+    return X, y
+
+
+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/reveal.js/.gitignore b/doc/pub/week38/html/reveal.js/.gitignore new file mode 100644 index 000000000..a5df3133d --- /dev/null +++ b/doc/pub/week38/html/reveal.js/.gitignore @@ -0,0 +1,8 @@ +.DS_Store +.svn +log/*.log +tmp/** +node_modules/ +.sass-cache +css/reveal.min.css +js/reveal.min.js diff --git a/doc/pub/week38/html/reveal.js/.travis.yml b/doc/pub/week38/html/reveal.js/.travis.yml new file mode 100644 index 000000000..165d9ae9f --- /dev/null +++ b/doc/pub/week38/html/reveal.js/.travis.yml @@ -0,0 +1,5 @@ +language: node_js +node_js: + - 0.10 +before_script: + - npm install -g grunt-cli \ No newline at end of file diff --git a/doc/pub/week38/html/reveal.js/CONTRIBUTING.md b/doc/pub/week38/html/reveal.js/CONTRIBUTING.md new file mode 100644 index 000000000..c2091e88f --- /dev/null +++ b/doc/pub/week38/html/reveal.js/CONTRIBUTING.md @@ -0,0 +1,23 @@ +## Contributing + +Please keep the [issue tracker](http://github.com/hakimel/reveal.js/issues) limited to **bug reports**, **feature requests** and **pull requests**. + + +### Personal Support +If you have personal support or setup questions the best place to ask those are [StackOverflow](http://stackoverflow.com/questions/tagged/reveal.js). + + +### Bug Reports +When reporting a bug make sure to include information about which browser and operating system you are on as well as the necessary steps to reproduce the issue. If possible please include a link to a sample presentation where the bug can be tested. + + +### Pull Requests +- Should follow the coding style of the file you work in, most importantly: + - Tabs to indent + - Single-quoted strings +- Should be made towards the **dev branch** +- Should be submitted from a feature/topic branch (not your master) + + +### Plugins +Please do not submit plugins as pull requests. They should be maintained in their own separate repository. More information here: https://github.com/hakimel/reveal.js/wiki/Plugin-Guidelines diff --git a/doc/pub/week38/html/reveal.js/Gruntfile.js b/doc/pub/week38/html/reveal.js/Gruntfile.js new file mode 100644 index 000000000..b257e8f32 --- /dev/null +++ b/doc/pub/week38/html/reveal.js/Gruntfile.js @@ -0,0 +1,140 @@ +/* global module:false */ +module.exports = function(grunt) { + var port = grunt.option('port') || 8000; + // Project configuration + grunt.initConfig({ + pkg: grunt.file.readJSON('package.json'), + meta: { + banner: + '/*!\n' + + ' * reveal.js <%= pkg.version %> (<%= grunt.template.today("yyyy-mm-dd, HH:MM") %>)\n' + + ' * http://lab.hakim.se/reveal-js\n' + + ' * MIT licensed\n' + + ' *\n' + + ' * Copyright (C) 2014 Hakim El Hattab, http://hakim.se\n' + + ' */' + }, + + qunit: { + files: [ 'test/*.html' ] + }, + + uglify: { + options: { + banner: '<%= meta.banner %>\n' + }, + build: { + src: 'js/reveal.js', + dest: 'js/reveal.min.js' + } + }, + + cssmin: { + compress: { + files: { + 'css/reveal.min.css': [ 'css/reveal.css' ] + } + } + }, + + sass: { + main: { + files: { + 'css/theme/darkgray.css': 'css/theme/source/darkgray.scss', + 'css/theme/beigesmall.css': 'css/theme/source/beigesmall.scss', + 'css/theme/cbc.css': 'css/theme/source/cbc.scss', + 'css/theme/default.css': 'css/theme/source/default.scss', + 'css/theme/beige.css': 'css/theme/source/beige.scss', + 'css/theme/night.css': 'css/theme/source/night.scss', + 'css/theme/serif.css': 'css/theme/source/serif.scss', + 'css/theme/simple.css': 'css/theme/source/simple.scss', + 'css/theme/sky.css': 'css/theme/source/sky.scss', + 'css/theme/moon.css': 'css/theme/source/moon.scss', + 'css/theme/solarized.css': 'css/theme/source/solarized.scss', + 'css/theme/blood.css': 'css/theme/source/blood.scss' + } + } + }, + + jshint: { + options: { + curly: false, + eqeqeq: true, + immed: true, + latedef: true, + newcap: true, + noarg: true, + sub: true, + undef: true, + eqnull: true, + browser: true, + expr: true, + globals: { + head: false, + module: false, + console: false, + unescape: false + } + }, + files: [ 'Gruntfile.js', 'js/reveal.js' ] + }, + + connect: { + server: { + options: { + port: port, + base: '.' + } + } + }, + + zip: { + 'reveal-js-presentation.zip': [ + 'index.html', + 'css/**', + 'js/**', + 'lib/**', + 'images/**', + 'plugin/**' + ] + }, + + watch: { + main: { + files: [ 'Gruntfile.js', 'js/reveal.js', 'css/reveal.css' ], + tasks: 'default' + }, + theme: { + files: [ 'css/theme/source/*.scss', 'css/theme/template/*.scss' ], + tasks: 'themes' + } + } + + }); + + // Dependencies + grunt.loadNpmTasks( 'grunt-contrib-qunit' ); + grunt.loadNpmTasks( 'grunt-contrib-jshint' ); + grunt.loadNpmTasks( 'grunt-contrib-cssmin' ); + grunt.loadNpmTasks( 'grunt-contrib-uglify' ); + grunt.loadNpmTasks( 'grunt-contrib-watch' ); + grunt.loadNpmTasks( 'grunt-contrib-sass' ); + grunt.loadNpmTasks( 'grunt-contrib-connect' ); + grunt.loadNpmTasks( 'grunt-zip' ); + + // Default task + grunt.registerTask( 'default', [ 'jshint', 'cssmin', 'uglify', 'qunit' ] ); + + // Theme task + grunt.registerTask( 'themes', [ 'sass' ] ); + + // Package presentation to archive + grunt.registerTask( 'package', [ 'default', 'zip' ] ); + + // Serve presentation locally + grunt.registerTask( 'serve', [ 'connect', 'watch' ] ); + + // Run tests + grunt.registerTask( 'test', [ 'jshint', 'qunit' ] ); + +}; diff --git a/doc/pub/week38/html/reveal.js/LICENSE b/doc/pub/week38/html/reveal.js/LICENSE new file mode 100644 index 000000000..09623076f --- /dev/null +++ b/doc/pub/week38/html/reveal.js/LICENSE @@ -0,0 +1,19 @@ +Copyright (C) 2015 Hakim El Hattab, http://hakim.se + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in +all copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN +THE SOFTWARE. \ No newline at end of file diff --git a/doc/pub/week38/html/reveal.js/README.md b/doc/pub/week38/html/reveal.js/README.md new file mode 100644 index 000000000..573b19597 --- /dev/null +++ b/doc/pub/week38/html/reveal.js/README.md @@ -0,0 +1,1052 @@ +# reveal.js [![Build Status](https://travis-ci.org/hakimel/reveal.js.svg?branch=master)](https://travis-ci.org/hakimel/reveal.js) + +A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://lab.hakim.se/reveal-js/). + +reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). It's best viewed in a modern browser but [fallbacks](https://github.com/hakimel/reveal.js/wiki/Browser-Support) are available to make sure your presentation can still be viewed elsewhere. + + +#### More reading: +- [Installation](#installation): Step-by-step instructions for getting reveal.js running on your computer. +- [Changelog](https://github.com/hakimel/reveal.js/releases): Up-to-date version history. +- [Examples](https://github.com/hakimel/reveal.js/wiki/Example-Presentations): Presentations created with reveal.js, add your own! +- [Browser Support](https://github.com/hakimel/reveal.js/wiki/Browser-Support): Explanation of browser support and fallbacks. +- [Plugins](https://github.com/hakimel/reveal.js/wiki/Plugins,-Tools-and-Hardware): A list of plugins that can be used to extend reveal.js. + +## Online Editor + +Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [http://slides.com](http://slides.com). + + +## Instructions + +### Markup + +Markup hierarchy needs to be ``
`` where the ``
`` represents one slide and can be repeated indefinitely. If you place multiple ``
``'s inside of another ``
`` they will be shown as vertical slides. The first of the vertical slides is the "root" of the others (at the top), and it will be included in the horizontal sequence. For example: + +```html +
+
+
Single Horizontal Slide
+
+
Vertical Slide 1
+
Vertical Slide 2
+
+
+
+``` + +### Markdown + +It's possible to write your slides using Markdown. To enable Markdown, add the ```data-markdown``` attribute to your ```
``` elements and wrap the contents in a ``` +
+``` + +#### External Markdown + +You can write your content as a separate file and have reveal.js load it at runtime. Note the separator arguments which determine how slides are delimited in the external file. The ```data-charset``` attribute is optional and specifies which charset to use when loading the external file. + +When used locally, this feature requires that reveal.js [runs from a local web server](#full-setup). + +```html +
+
+``` + +#### Element Attributes + +Special syntax (in html comment) is available for adding attributes to Markdown elements. This is useful for fragments, amongst other things. + +```html +
+ +
+``` + +#### Slide Attributes + +Special syntax (in html comment) is available for adding attributes to the slide `
` elements generated by your Markdown. + +```html +
+ +
+``` + + +### Configuration + +At the end of your page you need to initialize reveal by running the following code. Note that all config values are optional and will default as specified below. + +```javascript +Reveal.initialize({ + + // Display controls in the bottom right corner + controls: true, + + // Display a presentation progress bar + progress: true, + + // Display the page number of the current slide + slideNumber: false, + + // Push each slide change to the browser history + history: false, + + // Enable keyboard shortcuts for navigation + keyboard: true, + + // Enable the slide overview mode + overview: true, + + // Vertical centering of slides + center: true, + + // Enables touch navigation on devices with touch input + touch: true, + + // Loop the presentation + loop: false, + + // Change the presentation direction to be RTL + rtl: false, + + // Turns fragments on and off globally + fragments: true, + + // Flags if the presentation is running in an embedded mode, + // i.e. contained within a limited portion of the screen + embedded: false, + + // Flags if we should show a help overlay when the questionmark + // key is pressed + help: true, + + // Number of milliseconds between automatically proceeding to the + // next slide, disabled when set to 0, this value can be overwritten + // by using a data-autoslide attribute on your slides + autoSlide: 0, + + // Stop auto-sliding after user input + autoSlideStoppable: true, + + // Enable slide navigation via mouse wheel + mouseWheel: false, + + // Hides the address bar on mobile devices + hideAddressBar: true, + + // Opens links in an iframe preview overlay + previewLinks: false, + + // Transition style + transition: 'default', // none/fade/slide/convex/concave/zoom + + // Transition speed + transitionSpeed: 'default', // default/fast/slow + + // Transition style for full page slide backgrounds + backgroundTransition: 'default', // none/fade/slide/convex/concave/zoom + + // Number of slides away from the current that are visible + viewDistance: 3, + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "'https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg'" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" + + // Amount to move parallax background (horizontal and vertical) on slide change + // Number, e.g. 100 + parallaxBackgroundHorizontal: '', + parallaxBackgroundVertical: '' + +}); +``` + + +The configuration can be updated after initialization using the ```configure``` method: + +```javascript +// Turn autoSlide off +Reveal.configure({ autoSlide: 0 }); + +// Start auto-sliding every 5s +Reveal.configure({ autoSlide: 5000 }); +``` + + +### Dependencies + +Reveal.js doesn't _rely_ on any third party scripts to work but a few optional libraries are included by default. These libraries are loaded as dependencies in the order they appear, for example: + +```javascript +Reveal.initialize({ + dependencies: [ + // Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/ + { src: 'lib/js/classList.js', condition: function() { return !document.body.classList; } }, + + // Interpret Markdown in
elements + { src: 'plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + { src: 'plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + + // Syntax highlight for elements + { src: 'plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } }, + + // Zoom in and out with Alt+click + { src: 'plugin/zoom-js/zoom.js', async: true }, + + // Speaker notes + { src: 'plugin/notes/notes.js', async: true }, + + // Remote control your reveal.js presentation using a touch device + { src: 'plugin/remotes/remotes.js', async: true }, + + // MathJax + { src: 'plugin/math/math.js', async: true } + ] +}); +``` + +You can add your own extensions using the same syntax. The following properties are available for each dependency object: +- **src**: Path to the script to load +- **async**: [optional] Flags if the script should load after reveal.js has started, defaults to false +- **callback**: [optional] Function to execute when the script has loaded +- **condition**: [optional] Function which must return true for the script to be loaded + + +### Ready Event + +A 'ready' event is fired when reveal.js has loaded all non-async dependencies and is ready to start navigating. To check if reveal.js is already 'ready' you can call `Reveal.isReady()`. + +```javascript +Reveal.addEventListener( 'ready', function( event ) { + // event.currentSlide, event.indexh, event.indexv +} ); +``` + + +### Presentation Size + +All presentations have a normal size, that is the resolution at which they are authored. The framework will automatically scale presentations uniformly based on this size to ensure that everything fits on any given display or viewport. + +See below for a list of configuration options related to sizing, including default values: + +```javascript +Reveal.initialize({ + + ... + + // The "normal" size of the presentation, aspect ratio will be preserved + // when the presentation is scaled to fit different resolutions. Can be + // specified using percentage units. + width: 960, + height: 700, + + // Factor of the display size that should remain empty around the content + margin: 0.1, + + // Bounds for smallest/largest possible scale to apply to content + minScale: 0.2, + maxScale: 1.5 + +}); +``` + + +### Auto-sliding + +Presentations can be configured to progress through slides automatically, without any user input. To enable this you will need to tell the framework how many milliseconds it should wait between slides: + +```javascript +// Slide every five seconds +Reveal.configure({ + autoSlide: 5000 +}); +``` +When this is turned on a control element will appear that enables users to pause and resume auto-sliding. Alternatively, sliding can be paused or resumed by pressing »a« on the keyboard. Sliding is paused automatically as soon as the user starts navigating. You can disable these controls by specifying ```autoSlideStoppable: false``` in your reveal.js config. + +You can also override the slide duration for individual slides and fragments by using the ```data-autoslide``` attribute: + +```html +
+

After 2 seconds the first fragment will be shown.

+

After 10 seconds the next fragment will be shown.

+

Now, the fragment is displayed for 2 seconds before the next slide is shown.

+
+``` + +Whenever the auto-slide mode is resumed or paused the ```autoslideresumed``` and ```autoslidepaused``` events are fired. + + +### Keyboard Bindings + +If you're unhappy with any of the default keyboard bindings you can override them using the ```keyboard``` config option: + +```javascript +Reveal.configure({ + keyboard: { + 13: 'next', // go to the next slide when the ENTER key is pressed + 27: function() {}, // do something custom when ESC is pressed + 32: null // don't do anything when SPACE is pressed (i.e. disable a reveal.js default binding) + } +}); +``` + +### Lazy Loading + +When working on presentation with a lot of media or iframe content it's important to load lazily. Lazy loading means that reveal.js will only load content for the few slides nearest to the current slide. The number of slides that are preloaded is determined by the `viewDistance` configuration option. + +To enable lazy loading all you need to do is change your "src" attributes to "data-src" as shown below. This is supported for image, video, audio and iframe elements. Lazy loaded iframes will also unload when the containing slide is no longer visible. + +```html +
+ + + +
+``` + + +### API + +The ``Reveal`` object exposes a JavaScript API for controlling navigation and reading state: + +```javascript +// Navigation +Reveal.slide( indexh, indexv, indexf ); +Reveal.left(); +Reveal.right(); +Reveal.up(); +Reveal.down(); +Reveal.prev(); +Reveal.next(); +Reveal.prevFragment(); +Reveal.nextFragment(); + +// Toggle presentation states, optionally pass true/false to force on/off +Reveal.toggleOverview(); +Reveal.togglePause(); +Reveal.toggleAutoSlide(); + +// Change a config value at runtime +Reveal.configure({ controls: true }); + +// Returns the present configuration options +Reveal.getConfig(); + +// Fetch the current scale of the presentation +Reveal.getScale(); + +// Retrieves the previous and current slide elements +Reveal.getPreviousSlide(); +Reveal.getCurrentSlide(); + +Reveal.getIndices(); // { h: 0, v: 0 } } +Reveal.getProgress(); // 0-1 +Reveal.getTotalSlides(); + +// State checks +Reveal.isFirstSlide(); +Reveal.isLastSlide(); +Reveal.isOverview(); +Reveal.isPaused(); +Reveal.isAutoSliding(); +``` + +### Slide Changed Event + +A 'slidechanged' event is fired each time the slide is changed (regardless of state). The event object holds the index values of the current slide as well as a reference to the previous and current slide HTML nodes. + +Some libraries, like MathJax (see [#226](https://github.com/hakimel/reveal.js/issues/226#issuecomment-10261609)), get confused by the transforms and display states of slides. Often times, this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'slidechanged', function( event ) { + // event.previousSlide, event.currentSlide, event.indexh, event.indexv +} ); +``` + +### Presentation State + +The presentation's current state can be fetched by using the `getState` method. A state object contains all of the information required to put the presentation back as it was when `getState` was first called. Sort of like a snapshot. It's a simple object that can easily be stringified and persisted or sent over the wire. + +```javascript +Reveal.slide( 1 ); +// we're on slide 1 + +var state = Reveal.getState(); + +Reveal.slide( 3 ); +// we're on slide 3 + +Reveal.setState( state ); +// we're back on slide 1 +``` + +### Slide States + +If you set ``data-state="somestate"`` on a slide ``
``, "somestate" will be applied as a class on the document element when that slide is opened. This allows you to apply broad style changes to the page based on the active slide. + +Furthermore you can also listen to these changes in state via JavaScript: + +```javascript +Reveal.addEventListener( 'somestate', function() { + // TODO: Sprinkle magic +}, false ); +``` + +### Slide Backgrounds + +Slides are contained within a limited portion of the screen by default to allow them to fit any display and scale uniformly. You can apply full page backgrounds outside of the slide area by adding a ```data-background``` attribute to your ```
``` elements. Four different types of backgrounds are supported: color, image, video and iframe. Below are a few examples. + +```html +
+

All CSS color formats are supported, like rgba() or hsl().

+
+
+

This slide will have a full-size background image.

+
+
+

This background image will be sized to 100px and repeated.

+
+
+

Video. Multiple sources can be defined using a comma separated list. Video will loop when the data-background-video-loop attribute is provided.

+
+
+

Embeds a web page as a background. Note that the page won't be interactive.

+
+``` + +Backgrounds transition using a fade animation by default. This can be changed to a linear sliding transition by passing ```backgroundTransition: 'slide'``` to the ```Reveal.initialize()``` call. Alternatively you can set ```data-background-transition``` on any section with a background to override that specific transition. + + +### Parallax Background + +If you want to use a parallax scrolling background, set the first two config properties below when initializing reveal.js (the other two are optional). + +```javascript +Reveal.initialize({ + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" - currently only pixels are supported (don't use % or auto) + + // Amount of pixels to move the parallax background per slide step, + // a value of 0 disables movement along the given axis + // These are optional, if they aren't specified they'll be calculated automatically + parallaxBackgroundHorizontal: 200, + parallaxBackgroundVertical: 50 + +}); +``` + +Make sure that the background size is much bigger than screen size to allow for some scrolling. [View example](http://lab.hakim.se/reveal-js/?parallaxBackgroundImage=https%3A%2F%2Fs3.amazonaws.com%2Fhakim-static%2Freveal-js%2Freveal-parallax-1.jpg¶llaxBackgroundSize=2100px%20900px). + + + +### Slide Transitions +The global presentation transition is set using the ```transition``` config value. You can override the global transition for a specific slide by using the ```data-transition``` attribute: + +```html +
+

This slide will override the presentation transition and zoom!

+
+ +
+

Choose from three transition speeds: default, fast or slow!

+
+``` + +You can also use different in and out transitions for the same slide: + +```html +
+ The train goes on … +
+
+ and on … +
+
+ and stops. +
+
+ (Passengers entering and leaving) +
+
+ And it starts again. +
+``` + + +Note that this does not work with the page and cube transitions. + + +### Internal links + +It's easy to link between slides. The first example below targets the index of another slide whereas the second targets a slide with an ID attribute (```
```): + +```html +Link +Link +``` + +You can also add relative navigation links, similar to the built in reveal.js controls, by appending one of the following classes on any element. Note that each element is automatically given an ```enabled``` class when it's a valid navigation route based on the current slide. + +```html + + + + + + +``` + + +### Fragments +Fragments are used to highlight individual elements on a slide. Every element with the class ```fragment``` will be stepped through before moving on to the next slide. Here's an example: http://lab.hakim.se/reveal-js/#/fragments + +The default fragment style is to start out invisible and fade in. This style can be changed by appending a different class to the fragment: + +```html +
+

grow

+

shrink

+

fade-out

+

visible only once

+

blue only once

+

highlight-red

+

highlight-green

+

highlight-blue

+
+``` + +Multiple fragments can be applied to the same element sequentially by wrapping it, this will fade in the text on the first step and fade it back out on the second. + +```html +
+ + I'll fade in, then out + +
+``` + +The display order of fragments can be controlled using the ```data-fragment-index``` attribute. + +```html +
+

Appears last

+

Appears first

+

Appears second

+
+``` + +### Fragment events + +When a slide fragment is either shown or hidden reveal.js will dispatch an event. + +Some libraries, like MathJax (see #505), get confused by the initially hidden fragment elements. Often times this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'fragmentshown', function( event ) { + // event.fragment = the fragment DOM element +} ); +Reveal.addEventListener( 'fragmenthidden', function( event ) { + // event.fragment = the fragment DOM element +} ); +``` + +### Code syntax highlighting + +By default, Reveal is configured with [highlight.js](http://softwaremaniacs.org/soft/highlight/en/) for code syntax highlighting. Below is an example with clojure code that will be syntax highlighted. When the `data-trim` attribute is present surrounding whitespace is automatically removed. + +```html +
+ 

+(def lazy-fib
+  (concat
+   [0 1]
+   ((fn rfib [a b]
+        (lazy-cons (+ a b) (rfib b (+ a b)))) 0 1)))
+
+
+``` + +### Slide number +If you would like to display the page number of the current slide you can do so using the ```slideNumber``` configuration value. + +```javascript +// Shows the slide number using default formatting +Reveal.configure({ slideNumber: true }); + +// Slide number formatting can be configured using these variables: +// h: current slide's horizontal index +// v: current slide's vertical index +// c: current slide index (flattened) +// t: total number of slides (flattened) +Reveal.configure({ slideNumber: 'c / t' }); + +``` + + +### Overview mode + +Press "Esc" or "o" keys to toggle the overview mode on and off. While you're in this mode, you can still navigate between slides, +as if you were at 1,000 feet above your presentation. The overview mode comes with a few API hooks: + +```javascript +Reveal.addEventListener( 'overviewshown', function( event ) { /* ... */ } ); +Reveal.addEventListener( 'overviewhidden', function( event ) { /* ... */ } ); + +// Toggle the overview mode programmatically +Reveal.toggleOverview(); +``` + +### Fullscreen mode +Just press »F« on your keyboard to show your presentation in fullscreen mode. Press the »ESC« key to exit fullscreen mode. + + +### Embedded media +Embedded HTML5 `
+ +
+ +

 

 

 

+ + + + + + +
+

Data Analysis and Machine Learning: Logistic Regression

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 16, 2020

+
+

+ + +

Read »

+ + +
+ +

+ +

+ + +
+ + + + + + + +
+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/week38/html/week38-reveal.html b/doc/pub/week38/html/week38-reveal.html new file mode 100644 index 000000000..cc3db2e84 --- /dev/null +++ b/doc/pub/week38/html/week38-reveal.html @@ -0,0 +1,828 @@ + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + + + + + + + + + + +
+ + + + +

Data Analysis and Machine Learning: Logistic Regression

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

 
+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

 
+

Sep 16, 2020

+
+

+ +

+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+
+ + +
+

To do for log reg

+ +
    +

  • 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
    • +
+
+ + +
+

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. +

+ + +
+

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. +

+ + +
+

Optimization and Deep learning

+ +

+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. +

+ + +
+

Basics

+ +

+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). + +

+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 + +

 
+$$ +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. +$$ +

 
+

+ + +
+

Linear classifier

+ +

+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 +

 
+$$ +\begin{equation} +\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, +\tag{1} +\end{equation} +$$ +

 
+ +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. +

+ + +
+

Some selected properties

+ +

+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 logistic function

+ +

+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, +

 
+$$ +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. +$$ +

 
+ +Note that \( 1-p(t)= p(-t) \). +

+ + +
+

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. + +

+ + +

"""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()
+
+
+ + +
+

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 +

 
+$$ +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ +

 
+ +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). + +

+Note that we used +

 
+$$ +p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). +$$ +

 
+

+ + +
+

Maximum likelihood

+ +

+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 +

 
+$$ +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} +$$ +

 
+ +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 cost function rewritten

+ +

+Reordering the logarithms, we can rewrite the cost/loss function as +

 
+$$ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ +

 
+ +

+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 +

 
+$$ +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ +

 
+ +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. +

+ + +
+

Minimizing the cross entropy

+ +

+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 + +

 
+$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), +$$ +

 
+ +and +

 
+$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +$$ +

 
+

+ + +
+

A more compact expression

+ +

+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 + +

 
+$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ +

 
+ +

+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 + +

 
+$$ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +$$ +

 
+

+ + +
+

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 +

 
+$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ +

 
+ +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 +

 
+$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ +

 
+

+ + +
+

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 + +

 
+$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ +

 
+ +

 
+$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ +

 
+ +and so on till the class \( C=K-1 \) class +

 
+$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ +

 
+ +

+and the model is specified in term of \( K-1 \) so-called log-odds or +logit transformations. +

+ + +
+

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. + +

+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): + +

 
+$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ +

 
+ +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. +

+ + +
+

A simple classification problem

+

+ + +

import numpy as np
+from sklearn import datasets, linear_model
+import matplotlib.pyplot as plt
+
+
+def generate_data():
+    np.random.seed(0)
+    X, y = datasets.make_moons(200, noise=0.20)
+    return X, y
+
+
+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-solarized.html b/doc/pub/week38/html/week38-solarized.html new file mode 100644 index 000000000..b0cebc0bb --- /dev/null +++ b/doc/pub/week38/html/week38-solarized.html @@ -0,0 +1,597 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Data Analysis and Machine Learning: Logistic Regression

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 16, 2020

+
+

+









+ +

To do for log reg

+ +
    +
  • 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
    • +
+ + + +

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. + +

+ + +

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. + +

+









+ +

Optimization and Deep learning

+ +

+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. + +

+ + +

Basics

+ +

+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). + +

+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 + +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. +$$ + +

+









+ +

Linear classifier

+ +

+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 +$$ +\begin{equation} +\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, +\label{_auto1} +\end{equation} +$$ + +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. + +

+









+ +

Some selected properties

+ +

+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 logistic function

+ +

+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, +$$ +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. +$$ + +Note that \( 1-p(t)= p(-t) \). + +

+









+ +

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. + +

+ + +

"""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()
+
+

+









+ +

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 +$$ +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ + +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). + +

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

+ + +

Maximum likelihood

+ +

+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 +$$ +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} +$$ + +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 cost function rewritten

+ +

+Reordering the logarithms, we can rewrite the cost/loss function as +$$ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +

+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 +$$ +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +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. + +

+









+ +

Minimizing the cross entropy

+ +

+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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), +$$ + +and +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +$$ + +

+









+ +

A more compact expression

+ +

+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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ + +

+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 + +$$ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +$$ + +

+









+ +

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 +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ + +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 +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ + +

+









+ +

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 + +$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ + +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ + +and so on till the class \( C=K-1 \) class +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ + +

+and the model is specified in term of \( K-1 \) so-called log-odds or +logit transformations. + +

+









+ +

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. + +

+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): + +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ + +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. + +

+









+ +

A simple classification problem

+

+ + +

import numpy as np
+from sklearn import datasets, linear_model
+import matplotlib.pyplot as plt
+
+
+def generate_data():
+    np.random.seed(0)
+    X, y = datasets.make_moons(200, noise=0.20)
+    return X, y
+
+
+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()
+
+

+ + + + +

+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/week38/html/week38.html b/doc/pub/week38/html/week38.html new file mode 100644 index 000000000..faa233a10 --- /dev/null +++ b/doc/pub/week38/html/week38.html @@ -0,0 +1,602 @@ + + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + +

Data Analysis and Machine Learning: Logistic Regression

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 16, 2020

+
+

+









+ +

To do for log reg

+ +
    +
  • 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
    • +
+ + + +

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. + +

+ + +

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. + +

+









+ +

Optimization and Deep learning

+ +

+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. + +

+ + +

Basics

+ +

+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). + +

+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 + +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. +$$ + +

+









+ +

Linear classifier

+ +

+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 +$$ +\begin{equation} +\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, +\label{_auto1} +\end{equation} +$$ + +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. + +

+









+ +

Some selected properties

+ +

+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 logistic function

+ +

+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, +$$ +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. +$$ + +Note that \( 1-p(t)= p(-t) \). + +

+









+ +

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. + +

+ + +

"""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()
+
+

+









+ +

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 +$$ +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ + +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \). + +

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

+ + +

Maximum likelihood

+ +

+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 +$$ +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} +$$ + +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 cost function rewritten

+ +

+Reordering the logarithms, we can rewrite the cost/loss function as +$$ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +

+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 +$$ +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +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. + +

+









+ +

Minimizing the cross entropy

+ +

+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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), +$$ + +and +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +$$ + +

+









+ +

A more compact expression

+ +

+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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ + +

+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 + +$$ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +$$ + +

+









+ +

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 +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ + +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 +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ + +

+









+ +

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 + +$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ + +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ + +and so on till the class \( C=K-1 \) class +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ + +

+and the model is specified in term of \( K-1 \) so-called log-odds or +logit transformations. + +

+









+ +

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. + +

+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): + +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ + +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. + +

+









+ +

A simple classification problem

+

+ + +

import numpy as np
+from sklearn import datasets, linear_model
+import matplotlib.pyplot as plt
+
+
+def generate_data():
+    np.random.seed(0)
+    X, y = datasets.make_moons(200, noise=0.20)
+    return X, y
+
+
+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()
+
+

+ + + + +

+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz b/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz new file mode 100644 index 0000000000000000000000000000000000000000..b704baad8e9691f66d12ed193cf6605f87b40dfb GIT binary patch literal 199 zcmb2|=3t08nHbN&{Pw(U)?o*Mw#2J;N9QNG-}b&kL&hCO?>ff zna@MLHXStWI@_FedzDq~DcHZi> zTC;t+zWVK&bu49(M;VV{WPSbn<^TGtUf04TFBmi!7yzH5U047B literal 0 HcmV?d00001 diff --git a/doc/pub/week38/ipynb/week38.ipynb b/doc/pub/week38/ipynb/week38.ipynb new file mode 100644 index 000000000..624f17304 --- /dev/null +++ b/doc/pub/week38/ipynb/week38.ipynb @@ -0,0 +1,683 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Data Analysis and Machine Learning: Logistic Regression\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Sep 16, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## To do for log reg\n", + "\n", + "* Develop code for log reg step by step, with link to gradient descent part\n", + "\n", + "* show how to read and set up design matrix\n", + "\n", + "* use breast cancer data as example\n", + "\n", + "* develop other classification examples, pulsar example\n", + "\n", + "\n", + "## Logistic Regression\n", + "\n", + "In linear regression our main interest was centered on learning the\n", + "coefficients of a functional fit (say a polynomial) in order to be\n", + "able to predict the response of a continuous variable on some unseen\n", + "data. The fit to the continuous variable $y_i$ is based on some\n", + "independent variables $\\hat{x}_i$. Linear regression resulted in\n", + "analytical expressions for standard ordinary Least Squares or Ridge\n", + "regression (in terms of matrices to invert) for several quantities,\n", + "ranging from the variance and thereby the confidence intervals of the\n", + "parameters $\\hat{\\beta}$ to the mean squared error. If we can invert\n", + "the product of the design matrices, linear regression gives then a\n", + "simple recipe for fitting our data.\n", + "\n", + "\n", + "## Classification problems\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", + "out which mutations are important for a certain disease; or based on\n", + "scans of various patients' brains, figure out if there is a tumor or\n", + "not; or given a specific physical system, we'd like to identify its\n", + "state, say whether it is an ordered or disordered system (typical\n", + "situation in solid state physics); or classify the status of a\n", + "patient, whether she/he has a stroke or not and many other similar\n", + "situations.\n", + "\n", + "The most common situation we encounter when we apply logistic\n", + "regression is that of two possible outcomes, normally denoted as a\n", + "binary outcome, true or false, positive or negative, success or\n", + "failure etc.\n", + "\n", + "## Optimization and Deep learning\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", + "equation in the parameters $\\hat{\\beta}$. The optimization of the\n", + "problem calls therefore for minimization algorithms. This forms the\n", + "bottle neck of all machine learning algorithms, namely how to find\n", + "reliable minima of a multi-variable function. This leads us to the\n", + "family of gradient descent methods. The latter are the working horses\n", + "of basically all modern machine learning algorithms.\n", + "\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", + "## Basics\n", + "\n", + "We consider the case where the dependent variables, also called the\n", + "responses or the outcomes, $y_i$ are discrete and only take values\n", + "from $k=0,\\dots,K-1$ (i.e. $K$ classes).\n", + "\n", + "The goal is to predict the\n", + "output classes from the design matrix $\\hat{X}\\in\\mathbb{R}^{n\\times p}$\n", + "made of $n$ samples, each of which carries $p$ features or predictors. The\n", + "primary goal is to identify the classes to which new unseen samples\n", + "belong.\n", + "\n", + "Let us specialize to the case of two classes only, with outputs\n", + "$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a\n", + "credit card user that could default or not on her/his credit card\n", + "debt. That is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\ 1 & \\mathrm{yes} \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Linear classifier\n", + "\n", + "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", + "fit a linear model to the default case if $y_i > 0.5$ and the no\n", + "default case $y_i \\leq 0.5$.\n", + "\n", + "We would then have our \n", + "weighted linear combination, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\hat{y} = \\hat{X}^T\\hat{\\beta} + \\hat{\\epsilon},\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\hat{y}$ is a vector representing the possible outcomes, $\\hat{X}$ is our\n", + "$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators/predictors.\n", + "\n", + "## Some selected properties\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", + "card data discussed below here, where we can set the state of\n", + "defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons\n", + "in the data set (see the full example below).\n", + "\n", + "One simple way to get a discrete output is to have sign\n", + "functions that map the output of a linear regressor to values $\\{0,1\\}$,\n", + "$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. \n", + "We will encounter this model in our first demonstration of neural networks. Historically it is called the \"perceptron\" model in the machine learning\n", + "literature. This model is extremely simple. However, in many cases it is more\n", + "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 logistic function\n", + "\n", + "The perceptron is an example of a ``hard classification\" model. We\n", + "will encounter this model when we discuss neural networks as\n", + "well. Each datapoint is deterministically assigned to a category (i.e\n", + "$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a \"soft\"\n", + "classifier that outputs the probability of a given category rather\n", + "than a single value. For example, given $x_i$, the classifier\n", + "outputs the probability of being in a category $k$. Logistic regression\n", + "is the most common example of a so-called soft classifier. In logistic\n", + "regression, the probability that a data point $x_i$\n", + "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," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that $1-p(t)= p(-t)$.\n", + "\n", + "## Examples of likelihood functions used in logistic regression and nueral networks\n", + "\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": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "\"\"\"The sigmoid function (or the logistic curve) is a\n", + "function that takes any real number, z, and outputs a number (0,1).\n", + "It is useful in neural networks for assigning weights on a relative scale.\n", + "The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n", + "\n", + "import numpy\n", + "import matplotlib.pyplot as plt\n", + "import math as mt\n", + "\n", + "z = numpy.arange(-5, 5, .1)\n", + "sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n", + "sigma = sigma_fn(z)\n", + "\n", + "fig = plt.figure()\n", + "ax = fig.add_subplot(111)\n", + "ax.plot(z, sigma)\n", + "ax.set_ylim([-0.1, 1.1])\n", + "ax.set_xlim([-5,5])\n", + "ax.grid(True)\n", + "ax.set_xlabel('z')\n", + "ax.set_title('sigmoid function')\n", + "\n", + "plt.show()\n", + "\n", + "\"\"\"Step Function\"\"\"\n", + "z = numpy.arange(-5, 5, .02)\n", + "step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n", + "step = step_fn(z)\n", + "\n", + "fig = plt.figure()\n", + "ax = fig.add_subplot(111)\n", + "ax.plot(z, step)\n", + "ax.set_ylim([-0.5, 1.5])\n", + "ax.set_xlim([-5,5])\n", + "ax.grid(True)\n", + "ax.set_xlabel('z')\n", + "ax.set_title('step function')\n", + "\n", + "plt.show()\n", + "\n", + "\"\"\"tanh Function\"\"\"\n", + "z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n", + "t = numpy.tanh(z)\n", + "\n", + "fig = plt.figure()\n", + "ax = fig.add_subplot(111)\n", + "ax.plot(z, t)\n", + "ax.set_ylim([-1.0, 1.0])\n", + "ax.set_xlim([-2*mt.pi,2*mt.pi])\n", + "ax.grid(True)\n", + "ax.set_xlabel('z')\n", + "ax.set_title('tanh function')\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Two parameters\n", + "\n", + "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": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", + "p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", + "\n", + "Note that we used" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(y_i=0\\vert x_i, \\hat{\\beta}) = 1-p(y_i=1\\vert x_i, \\hat{\\beta}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Maximum likelihood\n", + "\n", + "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", + "$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. \n", + "We aim thus at maximizing \n", + "the probability of seeing the observed data. We can then approximate the \n", + "likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "P(\\mathcal{D}|\\hat{\\beta})& = \\prod_{i=1}^n \\left[p(y_i=1|x_i,\\hat{\\beta})\\right]^{y_i}\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]^{1-y_i}\\nonumber \\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "from which we obtain the log-likelihood and our **cost/loss** function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The cost function rewritten\n", + "\n", + "Reordering the logarithms, we can rewrite the **cost/loss** function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "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" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "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", + "## Minimizing the cross entropy\n", + "\n", + "The cross entropy is a convex function of the weights $\\hat{\\beta}$ 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": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A more compact expression\n", + "\n", + "Let us now define a vector $\\hat{y}$ with $n$ elements $y_i$, an\n", + "$n\\times p$ matrix $\\hat{X}$ which contains the $x_i$ values and a\n", + "vector $\\hat{p}$ of fitted probabilities $p(y_i\\vert x_i,\\hat{\\beta})$. We can rewrite in a more compact form the first\n", + "derivative of cost function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}} = -\\hat{X}^T\\left(\\hat{y}-\\hat{p}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we in addition define a diagonal matrix $\\hat{W}$ with elements \n", + "$p(y_i\\vert x_i,\\hat{\\beta})(1-p(y_i\\vert x_i,\\hat{\\beta})$, we can obtain a compact expression of the second derivative as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T} = \\hat{X}^T\\hat{W}\\hat{X}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Extending to more predictors\n", + "\n", + "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": {}, + "source": [ + "$$\n", + "\\log{ \\frac{p(\\hat{\\beta}\\hat{x})}{1-p(\\hat{\\beta}\\hat{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(\\hat{\\beta}\\hat{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Including more classes\n", + "\n", + "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", + "of simplicity assume we have only two predictors. We have then\n", + "following model" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1\n", + "5\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" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and so on till the class $C=K-1$ class" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the model is specified in term of $K-1$ so-called log-odds or\n", + "**logit** transformations.\n", + "\n", + "\n", + "## More classes\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", + "The softmax function is used in various multiclass classification\n", + "methods, such as multinomial logistic regression (also known as\n", + "softmax regression), multiclass linear discriminant analysis, naive\n", + "Bayes classifiers, and artificial neural networks. Specifically, in\n", + "multinomial logistic regression and linear discriminant analysis, the\n", + "input to the function is the result of $K$ distinct linear functions,\n", + "and the predicted probability for the $k$-th class given a sample\n", + "vector $\\hat{x}$ and a weighting vector $\\hat{\\beta}$ is (with two\n", + "predictors):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is easy to extend to more predictors. The final class is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "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", + "what we derived earlier is compatible with these equations.\n", + "\n", + "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", + "\n", + "\n", + "\n", + "## A simple classification problem" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "from sklearn import datasets, linear_model\n", + "import matplotlib.pyplot as plt\n", + "\n", + "\n", + "def generate_data():\n", + " np.random.seed(0)\n", + " X, y = datasets.make_moons(200, noise=0.20)\n", + " return X, y\n", + "\n", + "\n", + "def visualize(X, y, clf):\n", + " plot_decision_boundary(lambda x: clf.predict(x), X, y)\n", + "\n", + "def plot_decision_boundary(pred_func, X, y):\n", + " # Set min and max values and give it some padding\n", + " x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5\n", + " y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5\n", + " h = 0.01\n", + " # Generate a grid of points with distance h between them\n", + " xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))\n", + " # Predict the function value for the whole gid\n", + " Z = pred_func(np.c_[xx.ravel(), yy.ravel()])\n", + " Z = Z.reshape(xx.shape)\n", + " # Plot the contour and training examples\n", + " plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)\n", + " plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)\n", + " plt.show()\n", + "\n", + "\n", + "def classify(X, y):\n", + " clf = linear_model.LogisticRegressionCV()\n", + " clf.fit(X, y)\n", + " return clf\n", + "\n", + "\n", + "def main():\n", + " X, y = generate_data()\n", + " # visualize(X, y)\n", + " clf = classify(X, y)\n", + " visualize(X, y, clf)\n", + "\n", + "if __name__ == \"__main__\":\n", + " main()" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 2 +}