From 6e4633d19f713fd41c7ebc03f4f3e55225a96c73 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Tue, 25 Sep 2018 22:20:28 +0200 Subject: [PATCH] updated log reg --- doc/pub/LogReg/html/._LogReg-bs000.html | 47 ++- doc/pub/LogReg/html/._LogReg-bs001.html | 85 ++-- doc/pub/LogReg/html/._LogReg-bs002.html | 66 +-- doc/pub/LogReg/html/._LogReg-bs003.html | 76 ++-- doc/pub/LogReg/html/._LogReg-bs004.html | 75 ++-- doc/pub/LogReg/html/._LogReg-bs005.html | 77 ++-- doc/pub/LogReg/html/._LogReg-bs006.html | 73 ++-- doc/pub/LogReg/html/._LogReg-bs007.html | 74 ++-- doc/pub/LogReg/html/._LogReg-bs008.html | 77 ++-- doc/pub/LogReg/html/._LogReg-bs009.html | 89 ++-- doc/pub/LogReg/html/._LogReg-bs010.html | 175 ++++++++ doc/pub/LogReg/html/._LogReg-bs011.html | 175 ++++++++ doc/pub/LogReg/html/._LogReg-bs012.html | 185 +++++++++ doc/pub/LogReg/html/._LogReg-bs013.html | 185 +++++++++ doc/pub/LogReg/html/._LogReg-bs014.html | 188 +++++++++ doc/pub/LogReg/html/._LogReg-bs015.html | 217 ++++++++++ doc/pub/LogReg/html/LogReg-bs.html | 47 ++- doc/pub/LogReg/html/LogReg-reveal.html | 344 +++++++++++----- doc/pub/LogReg/html/LogReg-solarized.html | 353 +++++++++++----- doc/pub/LogReg/html/LogReg.html | 353 +++++++++++----- doc/pub/LogReg/ipynb/LogReg.ipynb | 382 ++++++++++++------ doc/pub/LogReg/ipynb/ipynb-LogReg-src.tar.gz | Bin 209 -> 210 bytes .../LogReg/pdf/LogReg-beamer-handouts2x3.pdf | Bin 293044 -> 255477 bytes doc/pub/LogReg/pdf/LogReg-beamer.pdf | Bin 285359 -> 245185 bytes doc/pub/LogReg/pdf/LogReg-minted.pdf | Bin 326366 -> 287260 bytes doc/src/LogisticRegression/LogReg.do.txt | 189 ++++++--- 26 files changed, 2726 insertions(+), 806 deletions(-) create mode 100644 doc/pub/LogReg/html/._LogReg-bs010.html create mode 100644 doc/pub/LogReg/html/._LogReg-bs011.html create mode 100644 doc/pub/LogReg/html/._LogReg-bs012.html create mode 100644 doc/pub/LogReg/html/._LogReg-bs013.html create mode 100644 doc/pub/LogReg/html/._LogReg-bs014.html create mode 100644 doc/pub/LogReg/html/._LogReg-bs015.html diff --git a/doc/pub/LogReg/html/._LogReg-bs000.html b/doc/pub/LogReg/html/._LogReg-bs000.html index d4f17bb42..ae97688d0 100644 --- a/doc/pub/LogReg/html/._LogReg-bs000.html +++ b/doc/pub/LogReg/html/._LogReg-bs000.html @@ -41,16 +41,19 @@ Automatically generated HTML file from DocOnce source @@ -89,13 +92,19 @@ MathJax.Hub.Config({ Contents @@ -130,7 +139,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Sep 20, 2018

+

Sep 25, 2018


@@ -146,6 +155,8 @@ MathJax.Hub.Config({

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    Logistic Regression

    -So far we have focused on learning from datasets for which there is a -continuous output. In linear regression we have been -concerned with learning the coefficients of a polynomial to predict -the response of a continuous variable \( y_i \) on unseen data based on -its independent variables \( {\bf x}_i \). +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 (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, -however, are concerned with outcomes taking the form of discrete -variables (i.e. categories). For example, we may want to detect if -there's a cat or a dog in an image. Or given a specific system, -we'd like to identify its state, say whether it is an ordered or disordered system (typical situation in solid state physics). -(e.g. ordered/disordered). +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.

    -Logistic regression deals with binary, dichotomous outcomes (e.g. True or -False, Success or Failure, etc.). It is worth noting that logistic -regression is also commonly used in modern supervised Deep Learning -models, as we will see later. +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.

    @@ -152,6 +170,9 @@ models, as we will see later.

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    - + -

    Basics

    +

    Optimization and Deep learning

    -We consider the case where the dependent variables \( y_i\in\mathbb{Z} \) -are discrete and only take values from \( m=0,\dots,M-1 \) (i.e. \( M \) -classes). +Logistic regression will also serve as our stepping 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 optmization 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.

    -The goal is to predict the -output classes from the design matrix \( X\in\mathbb{R}^{n\times p} \) -made of \( n \) samples, each of which bears \( p \) features. The -primary goal is to identify the classes to which new unseen samples -belong. +We note also that many of the topics discussed here +regression are also commonly used in modern supervised Deep Learning +models, as we will see later.

    @@ -140,6 +148,10 @@ belong.

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    - + -

    Linear classifier

    +

    Basics

    -Let us start by considering a slightly simpler classifier: a linear classifier that categorizes examples using a weighted linear-combination of the features and an additive offset -$$ -\begin{equation} -s_i = \boldsymbol{x}_i^T\boldsymbol{w} + b_0 \equiv \mathbf{x}_i^T\mathbf{w}, -\tag{1} -\end{equation} -$$ +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). -where we use the short-hand notation -\( \mathbf{x}_i = (1,\boldsymbol{x}_i) \) and \( \mathbf{w}_i = (b_0,\boldsymbol{w}_i) \). +

    +The goal is to predict the +output classes from the design matrix \( X\in\mathbb{R}^{n\times p} \) +made of \( n \) samples, each of which carries \( p \) features. 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 use who could default or not on her/his debt. That is +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}. +$$

    @@ -140,6 +155,11 @@ where we use the short-hand notation

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    Some selected properties

    +

    Linear classifier

    -This function takes values on the entire real axis. In the case of -logistic regression, however, the labels \( y_i \) are discrete -variables. One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to \( \{0,1\} \), -\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, -this is commonly known as the "perceptron" in the machine learning -literature. This model is extremely simple, and it is favorable in -many cases (e.g. noisy data) to have a ``soft" classifier that outputs -the probability of a given category. For example, given -\( \mathbf{x}_i \), the classifier outputs the probability of being in -category \( m \). One such function is the logistic (or sigmoid) function: +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 $$ \begin{equation} -f(s) = \frac{1}{1+\mathrm e^{-s}}. -\tag{2} -\end{equation} +\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon}, +\tag{1} +\end{equation} $$ -Note that \( 1-f(s)= f(-s) \), which will be useful shortly. +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.

    @@ -150,6 +153,12 @@ Note that \( 1-f(s)= f(-s) \), which will be useful shortly.

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    The cross-entropy as a cost function for logistic regression

    +

    Some selected properties

    -The perceptron is an example of a ``hard classification": 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 \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). -Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point \( \boldsymbol{x}_i \) belongs to a category \( y_i=\{0,1\} \) is is given by -$$ -\begin{eqnarray} -P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& \frac{1}{1+\mathrm{e}^{-\mathbf{x}^T_i\mathbf{w}}},\nonumber\\ -P(y_i=0|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& 1 - P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)}, -\end{eqnarray} -$$ - -where \( \boldsymbol{\theta}=\mathbf{w} \) are the weights we wish to learn from the data. +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.

    -Notice that in terms of the logistic function, we can write -$$ -P(y_i=1) =f(\mathbf{x}_i^T\mathbf{w})=1-P(y_i=0). -$$ +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.

    @@ -146,6 +152,13 @@ $$

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    - + -

    Maximum likelihood

    +

    The logistic function

    -We now define the cost function for logistic regression using Maximum -Likelihood Estimation (MLE). Recall, that in MLE we choose parameters -to maximize the probability of seeing the observed data. Consider a -dataset \( \mathcal{D}=\{(y_i,\boldsymbol{x}_i)\} \) with binary labels -\( y_i\in\{0,1\} \) where the data points are drawn independently. The -likelihood of the seeing the data under our model is just: +A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \), $$ -\begin{align} -P(\mathcal{D}|\mathbf{w})& = \prod_{i=1}^n \left[f(\mathbf{x}_i^T\mathbf{w})\right]^{y_i}\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]^{1-y_i}\nonumber \\ -\tag{3} -\end{align} +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. $$ -from which we can readily compute the log-likelihood: -$$ -\begin{equation} -l(\mathbf{w}) = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. -\tag{4} -\end{equation} -$$ +Note that \( 1-p(t)= p(-t) \).

    @@ -150,6 +145,12 @@ $$

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  • diff --git a/doc/pub/LogReg/html/._LogReg-bs007.html b/doc/pub/LogReg/html/._LogReg-bs007.html index 58dd903f9..7eb3a129b 100644 --- a/doc/pub/LogReg/html/._LogReg-bs007.html +++ b/doc/pub/LogReg/html/._LogReg-bs007.html @@ -41,16 +41,19 @@ Automatically generated HTML file from DocOnce source @@ -89,13 +92,19 @@ MathJax.Hub.Config({ Contents @@ -110,21 +119,20 @@ MathJax.Hub.Config({ -The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \theta \) -$$ -\hat{\mathbf{w}} = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. -$$ -Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that -$$ -\begin{eqnarray} -\mathcal{C}(\mathbf{w}) &=& - l(\mathbf{w}) \\ -&=& \sum_{i=1}^n -y_i\log f(\mathbf{x}_i^T\mathbf{w}) - (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].\nonumber -\end{eqnarray} -$$ +

    Probabilistic classifiers

    -This equation is known in statistics as the \emph{cross entropy}. Finally, we note that just as in linear regression, -in practice we usually supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression. +

    +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)

    @@ -141,6 +149,12 @@ in practice we usually supplement the cross-entropy with additional regularizati

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    Minimizing the cross entropy

    +

    Two parameters

    -The cross entropy is a convex function of the weights \( \mathbf{w} \) and, -therefore, any local minimizer is a global minimizer. Minimizing this -cost function leads to the following equation +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 +$$ +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ -$$ -\begin{equation} -\boldsymbol{0}=\boldsymbol{\nabla} \mathcal{C}(\mathbf{w}) = \sum_{i=1}^n\left[f(\mathbf{x}_i^T\mathbf{w})-y_i\right]\mathbf{x}_i, -\tag{5} -\end{equation} -$$ +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).

    -where we made use of the logistic function identity \( \partial_z f(z) = -f(z)[1-f(z)] \). This equation defines a transcendental equation for -\( \mathbf{w} \), the solution of which, unlike linear regression, cannot -be written in a closed form. -Here we need gradient descent methods! +Note that we used +$$ +p(y_i=1) = 1-p(y_i=0). +$$

    @@ -147,6 +154,12 @@ Here we need gradient descent methods!

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    - + -

    A scikit-learn example

    +

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

    import numpy as np
    -import matplotlib.pyplot as plt
    -from sklearn import datasets
    -iris = datasets.load_iris()
    -list(iris.keys())
    -['data', 'target_names', 'feature_names', 'target', 'DESCR']
    -X = iris["data"][:, 3:] # petal width
    -y = (iris["target"] == 2).astype(np.int) # 1 if Iris-Virginica, else 0
    +from which we obtain the log-likelihood and our cost/loss function
    +$$
    +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n  y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right].
    +$$
     
    -from sklearn.linear_model import LogisticRegression
    -log_reg = LogisticRegression()
    -log_reg.fit(X, y)
    -
    -X_new = np.linspace(0, 3, 1000).reshape(-1, 1)
    -y_proba = log_reg.predict_proba(X_new)
    -plt.plot(X_new, y_proba[:, 1], "g-", label="Iris-Virginica")
    -plt.plot(X_new, y_proba[:, 0], "b--", label="Not Iris-Virginica")
    -plt.show()
    -

    -

    diff --git a/doc/pub/LogReg/html/._LogReg-bs010.html b/doc/pub/LogReg/html/._LogReg-bs010.html new file mode 100644 index 000000000..d0ee85632 --- /dev/null +++ b/doc/pub/LogReg/html/._LogReg-bs010.html @@ -0,0 +1,175 @@ + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +

     

     

     

    + + + + +

    The cost function rewritten

    + +

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

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/LogReg/html/._LogReg-bs011.html b/doc/pub/LogReg/html/._LogReg-bs011.html new file mode 100644 index 000000000..0a94cd42d --- /dev/null +++ b/doc/pub/LogReg/html/._LogReg-bs011.html @@ -0,0 +1,175 @@ + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +

     

     

     

    + + + +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/LogReg/html/._LogReg-bs012.html b/doc/pub/LogReg/html/._LogReg-bs012.html new file mode 100644 index 000000000..8747675c5 --- /dev/null +++ b/doc/pub/LogReg/html/._LogReg-bs012.html @@ -0,0 +1,185 @@ + + + + + + + +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/LogReg/html/._LogReg-bs013.html b/doc/pub/LogReg/html/._LogReg-bs013.html new file mode 100644 index 000000000..61ea427f6 --- /dev/null +++ b/doc/pub/LogReg/html/._LogReg-bs013.html @@ -0,0 +1,185 @@ + + + + + + + +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/LogReg/html/._LogReg-bs014.html b/doc/pub/LogReg/html/._LogReg-bs014.html new file mode 100644 index 000000000..b7a8dcb07 --- /dev/null +++ b/doc/pub/LogReg/html/._LogReg-bs014.html @@ -0,0 +1,188 @@ + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +

     

     

     

    + + + + +

    A scikit-learn example

    + +

    + + +

    import numpy as np
    +import matplotlib.pyplot as plt
    +from sklearn import datasets
    +iris = datasets.load_iris()
    +list(iris.keys())
    +['data', 'target_names', 'feature_names', 'target', 'DESCR']
    +X = iris["data"][:, 3:] # petal width
    +y = (iris["target"] == 2).astype(np.int) # 1 if Iris-Virginica, else 0
    +
    +from sklearn.linear_model import LogisticRegression
    +log_reg = LogisticRegression()
    +log_reg.fit(X, y)
    +
    +X_new = np.linspace(0, 3, 1000).reshape(-1, 1)
    +y_proba = log_reg.predict_proba(X_new)
    +plt.plot(X_new, y_proba[:, 1], "g-", label="Iris-Virginica")
    +plt.plot(X_new, y_proba[:, 0], "b--", label="Not Iris-Virginica")
    +plt.show()
    +
    +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/LogReg/html/._LogReg-bs015.html b/doc/pub/LogReg/html/._LogReg-bs015.html new file mode 100644 index 000000000..b9cf8ae1f --- /dev/null +++ b/doc/pub/LogReg/html/._LogReg-bs015.html @@ -0,0 +1,217 @@ + + + + + + + +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):
    +    # plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
    +    # plt.show()
    +    plot_decision_boundary(lambda x: clf.predict(x), X, y)
    +    plt.title("Logistic Regression")
    +
    +
    +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/LogReg/html/LogReg-bs.html b/doc/pub/LogReg/html/LogReg-bs.html index d4f17bb42..ae97688d0 100644 --- a/doc/pub/LogReg/html/LogReg-bs.html +++ b/doc/pub/LogReg/html/LogReg-bs.html @@ -41,16 +41,19 @@ Automatically generated HTML file from DocOnce source @@ -89,13 +92,19 @@ MathJax.Hub.Config({ Contents @@ -130,7 +139,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 20, 2018

    +

    Sep 25, 2018


    @@ -146,6 +155,8 @@ MathJax.Hub.Config({

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

     
    -

    Sep 20, 2018

    +

    Sep 25, 2018


    @@ -162,202 +162,288 @@ MathJax.Hub.Config({

    Logistic Regression

    -So far we have focused on learning from datasets for which there is a -continuous output. In linear regression we have been -concerned with learning the coefficients of a polynomial to predict -the response of a continuous variable \( y_i \) on unseen data based on -its independent variables \( {\bf x}_i \). +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 (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, -however, are concerned with outcomes taking the form of discrete -variables (i.e. categories). For example, we may want to detect if -there's a cat or a dog in an image. Or given a specific system, -we'd like to identify its state, say whether it is an ordered or disordered system (typical situation in solid state physics). -(e.g. ordered/disordered). +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.

    -Logistic regression deals with binary, dichotomous outcomes (e.g. True or -False, Success or Failure, etc.). It is worth noting that logistic -regression is also commonly used in modern supervised Deep Learning -models, as we will see later. +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.

    -

    Basics

    +

    Optimization and Deep learning

    -We consider the case where the dependent variables \( y_i\in\mathbb{Z} \) -are discrete and only take values from \( m=0,\dots,M-1 \) (i.e. \( M \) -classes). +Logistic regression will also serve as our stepping 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 optmization 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 +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 \( X\in\mathbb{R}^{n\times p} \) -made of \( n \) samples, each of which bears \( p \) features. The +made of \( n \) samples, each of which carries \( p \) features. 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 use who could default or not on her/his debt. That is +

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

     

    -

    Linear classifier

    +

    Linear classifier

    -Let us start by considering a slightly simpler classifier: a linear classifier that categorizes examples using a weighted linear-combination of the features and an additive offset +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

     
    $$ \begin{equation} -s_i = \boldsymbol{x}_i^T\boldsymbol{w} + b_0 \equiv \mathbf{x}_i^T\mathbf{w}, +\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon}, \tag{1} \end{equation} $$

     
    -where we use the short-hand notation -\( \mathbf{x}_i = (1,\boldsymbol{x}_i) \) and \( \mathbf{w}_i = (b_0,\boldsymbol{w}_i) \). +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.

    -

    Some selected properties

    +

    Some selected properties

    -This function takes values on the entire real axis. In the case of +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. One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to \( \{0,1\} \), -\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, -this is commonly known as the "perceptron" in the machine learning -literature. This model is extremely simple, and it is favorable in -many cases (e.g. noisy data) to have a ``soft" classifier that outputs -the probability of a given category. For example, given -\( \mathbf{x}_i \), the classifier outputs the probability of being in -category \( m \). One such function is the logistic (or sigmoid) function: +variables. -

     
    -$$ -\begin{equation} -f(s) = \frac{1}{1+\mathrm e^{-s}}. -\tag{2} -\end{equation} -$$ -

     
    - -Note that \( 1-f(s)= f(-s) \), which will be useful shortly. +

    +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 cross-entropy as a cost function for logistic regression

    +

    The logistic function

    -The perceptron is an example of a ``hard classification": 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 \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). -Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point \( \boldsymbol{x}_i \) belongs to a category \( y_i=\{0,1\} \) is is given by +A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \),

     
    $$ -\begin{eqnarray} -P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& \frac{1}{1+\mathrm{e}^{-\mathbf{x}^T_i\mathbf{w}}},\nonumber\\ -P(y_i=0|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& 1 - P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)}, -\end{eqnarray} +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. $$

     
    -where \( \boldsymbol{\theta}=\mathbf{w} \) are the weights we wish to learn from the data. +Note that \( 1-p(t)= p(-t) \). +

    + + +
    +

    Probabilistic classifiers

    -Notice that in terms of the logistic function, we can write +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) +

    + + +
    +

    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

     
    $$ -P(y_i=1) =f(\mathbf{x}_i^T\mathbf{w})=1-P(y_i=0). +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ +

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

    +Note that we used +

     
    +$$ +p(y_i=1) = 1-p(y_i=0). $$

     

    -

    Maximum likelihood

    +

    Maximum likelihood

    -We now define the cost function for logistic regression using Maximum -Likelihood Estimation (MLE). Recall, that in MLE we choose parameters -to maximize the probability of seeing the observed data. Consider a -dataset \( \mathcal{D}=\{(y_i,\boldsymbol{x}_i)\} \) with binary labels -\( y_i\in\{0,1\} \) where the data points are drawn independently. The -likelihood of the seeing the data under our model is just: +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

     
    $$ -\begin{align} -P(\mathcal{D}|\mathbf{w})& = \prod_{i=1}^n \left[f(\mathbf{x}_i^T\mathbf{w})\right]^{y_i}\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]^{1-y_i}\nonumber \\ -\tag{3} -\end{align} +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} $$

     
    -from which we can readily compute the log-likelihood: +from which we obtain the log-likelihood and our cost/loss function

     
    $$ -\begin{equation} -l(\mathbf{w}) = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. -\tag{4} -\end{equation} +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]. $$

     

    -The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \theta \) -

     
    -$$ -\hat{\mathbf{w}} = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. -$$ -

     
    +

    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

     
    $$ -\begin{eqnarray} -\mathcal{C}(\mathbf{w}) &=& - l(\mathbf{w}) \\ -&=& \sum_{i=1}^n -y_i\log f(\mathbf{x}_i^T\mathbf{w}) - (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].\nonumber -\end{eqnarray} +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right). $$

     
    -This equation is known in statistics as the \emph{cross entropy}. Finally, we note that just as in linear regression, -in practice we usually supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression. +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

    +

    Minimizing the cross entropy

    -The cross entropy is a convex function of the weights \( \mathbf{w} \) and, -therefore, any local minimizer is a global minimizer. Minimizing this -cost function leads to the following equation +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

     
    $$ -\begin{equation} -\boldsymbol{0}=\boldsymbol{\nabla} \mathcal{C}(\mathbf{w}) = \sum_{i=1}^n\left[f(\mathbf{x}_i^T\mathbf{w})-y_i\right]\mathbf{x}_i, -\tag{5} -\end{equation} +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), $$

     
    -

    -where we made use of the logistic function identity \( \partial_z f(z) = -f(z)[1-f(z)] \). This equation defines a transcendental equation for -\( \mathbf{w} \), the solution of which, unlike linear regression, cannot -be written in a closed form. -Here we need gradient descent methods! +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 scikit-learn example

    +

    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 + +

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

     
    +

    + + +
    +

    A scikit-learn example

    @@ -384,6 +470,64 @@ plt.show()

    +
    +

    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):
    +    # plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
    +    # plt.show()
    +    plot_decision_boundary(lambda x: clf.predict(x), X, y)
    +    plt.title("Logistic Regression")
    +
    +
    +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/LogReg/html/LogReg-solarized.html b/doc/pub/LogReg/html/LogReg-solarized.html index e482c3596..f37054fab 100644 --- a/doc/pub/LogReg/html/LogReg-solarized.html +++ b/doc/pub/LogReg/html/LogReg-solarized.html @@ -35,16 +35,19 @@ div { text-align: justify; text-justify: inter-word; } @@ -86,7 +89,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 20, 2018

    +

    Sep 25, 2018


    @@ -94,183 +97,260 @@ MathJax.Hub.Config({

    Logistic Regression

    -So far we have focused on learning from datasets for which there is a -continuous output. In linear regression we have been -concerned with learning the coefficients of a polynomial to predict -the response of a continuous variable \( y_i \) on unseen data based on -its independent variables \( {\bf x}_i \). +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 (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, -however, are concerned with outcomes taking the form of discrete -variables (i.e. categories). For example, we may want to detect if -there's a cat or a dog in an image. Or given a specific system, -we'd like to identify its state, say whether it is an ordered or disordered system (typical situation in solid state physics). -(e.g. ordered/disordered). +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.

    -Logistic regression deals with binary, dichotomous outcomes (e.g. True or -False, Success or Failure, etc.). It is worth noting that logistic -regression is also commonly used in modern supervised Deep Learning -models, as we will see later. +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 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 optmization 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 +regression are also commonly used in modern supervised Deep Learning +models, as we will see later.

    -

    Basics

    +

    Basics

    -We consider the case where the dependent variables \( y_i\in\mathbb{Z} \) -are discrete and only take values from \( m=0,\dots,M-1 \) (i.e. \( M \) -classes). +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 \( X\in\mathbb{R}^{n\times p} \) -made of \( n \) samples, each of which bears \( p \) features. The +made of \( n \) samples, each of which carries \( p \) features. The primary goal is to identify the classes to which new unseen samples belong.

    -









    - -

    Linear classifier

    +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 use who could default or not on her/his debt. That is +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}. +$$

    -Let us start by considering a slightly simpler classifier: a linear classifier that categorizes examples using a weighted linear-combination of the features and an additive offset +









    + +

    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 $$ \begin{equation} -s_i = \boldsymbol{x}_i^T\boldsymbol{w} + b_0 \equiv \mathbf{x}_i^T\mathbf{w}, +\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon}, \label{_auto1} \end{equation} $$ -where we use the short-hand notation -\( \mathbf{x}_i = (1,\boldsymbol{x}_i) \) and \( \mathbf{w}_i = (b_0,\boldsymbol{w}_i) \). +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.











    -

    Some selected properties

    +

    Some selected properties

    -This function takes values on the entire real axis. In the case of +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. One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to \( \{0,1\} \), -\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, -this is commonly known as the "perceptron" in the machine learning -literature. This model is extremely simple, and it is favorable in -many cases (e.g. noisy data) to have a ``soft" classifier that outputs -the probability of a given category. For example, given -\( \mathbf{x}_i \), the classifier outputs the probability of being in -category \( m \). One such function is the logistic (or sigmoid) function: +variables. -$$ -\begin{equation} -f(s) = \frac{1}{1+\mathrm e^{-s}}. -\label{eq:log_fun} -\end{equation} -$$ - -Note that \( 1-f(s)= f(-s) \), which will be useful shortly. +

    +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 cross-entropy as a cost function for logistic regression

    +

    The logistic function

    -The perceptron is an example of a ``hard classification": 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 \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). -Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point \( \boldsymbol{x}_i \) belongs to a category \( y_i=\{0,1\} \) is is given by +A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \), $$ -\begin{eqnarray} -P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& \frac{1}{1+\mathrm{e}^{-\mathbf{x}^T_i\mathbf{w}}},\nonumber\\ -P(y_i=0|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& 1 - P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)}, -\end{eqnarray} +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. $$ -where \( \boldsymbol{\theta}=\mathbf{w} \) are the weights we wish to learn from the data. +Note that \( 1-p(t)= p(-t) \).

    -Notice that in terms of the logistic function, we can write +









    + +

    Probabilistic classifiers

    + +

    +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) + +

    +









    + +

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

    +Note that we used +$$ +p(y_i=1) = 1-p(y_i=0). $$

    -

    Maximum likelihood

    +

    Maximum likelihood

    -We now define the cost function for logistic regression using Maximum -Likelihood Estimation (MLE). Recall, that in MLE we choose parameters -to maximize the probability of seeing the observed data. Consider a -dataset \( \mathcal{D}=\{(y_i,\boldsymbol{x}_i)\} \) with binary labels -\( y_i\in\{0,1\} \) where the data points are drawn independently. The -likelihood of the seeing the data under our model is just: +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 $$ -\begin{align} -P(\mathcal{D}|\mathbf{w})& = \prod_{i=1}^n \left[f(\mathbf{x}_i^T\mathbf{w})\right]^{y_i}\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]^{1-y_i}\nonumber \\ -\label{_auto2} -\end{align} +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} $$ -from which we can readily compute the log-likelihood: +from which we obtain the log-likelihood and our cost/loss function $$ -\begin{equation} -l(\mathbf{w}) = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. -\label{_auto3} -\end{equation} +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]. $$











    -The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \theta \) + +

    The cost function rewritten

    + +

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









    +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 $$ -\begin{eqnarray} -\mathcal{C}(\mathbf{w}) &=& - l(\mathbf{w}) \\ -&=& \sum_{i=1}^n -y_i\log f(\mathbf{x}_i^T\mathbf{w}) - (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].\nonumber -\end{eqnarray} +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right). $$ -This equation is known in statistics as the \emph{cross entropy}. Finally, we note that just as in linear regression, -in practice we usually supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression. +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

    +

    Minimizing the cross entropy

    -The cross entropy is a convex function of the weights \( \mathbf{w} \) and, -therefore, any local minimizer is a global minimizer. Minimizing this -cost function leads to the following equation - -$$ -\begin{equation} -\boldsymbol{0}=\boldsymbol{\nabla} \mathcal{C}(\mathbf{w}) = \sum_{i=1}^n\left[f(\mathbf{x}_i^T\mathbf{w})-y_i\right]\mathbf{x}_i, -\label{_auto4} -\end{equation} -$$ +The cross entropy is a convex function of the weights \( \hat{\beta} \) and, +therefore, any local minimizer is a global minimizer.

    -where we made use of the logistic function identity \( \partial_z f(z) = -f(z)[1-f(z)] \). This equation defines a transcendental equation for -\( \mathbf{w} \), the solution of which, unlike linear regression, cannot -be written in a closed form. -Here we need gradient descent methods! +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 scikit-learn example

    +

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

    +









    + +

    A scikit-learn example

    @@ -295,6 +375,63 @@ plt.plot(X_new, y_proba[:, 0], 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):
    +    # plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
    +    # plt.show()
    +    plot_decision_boundary(lambda x: clf.predict(x), X, y)
    +    plt.title("Logistic Regression")
    +
    +
    +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/LogReg/html/LogReg.html b/doc/pub/LogReg/html/LogReg.html index 889e00fb5..1dab7dca5 100644 --- a/doc/pub/LogReg/html/LogReg.html +++ b/doc/pub/LogReg/html/LogReg.html @@ -40,16 +40,19 @@ div { text-align: justify; text-justify: inter-word; } @@ -91,7 +94,7 @@ MathJax.Hub.Config({

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

    -

    Sep 20, 2018

    +

    Sep 25, 2018


    @@ -99,183 +102,260 @@ MathJax.Hub.Config({

    Logistic Regression

    -So far we have focused on learning from datasets for which there is a -continuous output. In linear regression we have been -concerned with learning the coefficients of a polynomial to predict -the response of a continuous variable \( y_i \) on unseen data based on -its independent variables \( {\bf x}_i \). +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 (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, -however, are concerned with outcomes taking the form of discrete -variables (i.e. categories). For example, we may want to detect if -there's a cat or a dog in an image. Or given a specific system, -we'd like to identify its state, say whether it is an ordered or disordered system (typical situation in solid state physics). -(e.g. ordered/disordered). +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.

    -Logistic regression deals with binary, dichotomous outcomes (e.g. True or -False, Success or Failure, etc.). It is worth noting that logistic -regression is also commonly used in modern supervised Deep Learning -models, as we will see later. +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 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 optmization 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 +regression are also commonly used in modern supervised Deep Learning +models, as we will see later.

    -

    Basics

    +

    Basics

    -We consider the case where the dependent variables \( y_i\in\mathbb{Z} \) -are discrete and only take values from \( m=0,\dots,M-1 \) (i.e. \( M \) -classes). +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 \( X\in\mathbb{R}^{n\times p} \) -made of \( n \) samples, each of which bears \( p \) features. The +made of \( n \) samples, each of which carries \( p \) features. The primary goal is to identify the classes to which new unseen samples belong.

    -









    - -

    Linear classifier

    +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 use who could default or not on her/his debt. That is +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no default}\\ 1 & \mathrm{default} \end{bmatrix}. +$$

    -Let us start by considering a slightly simpler classifier: a linear classifier that categorizes examples using a weighted linear-combination of the features and an additive offset +









    + +

    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 $$ \begin{equation} -s_i = \boldsymbol{x}_i^T\boldsymbol{w} + b_0 \equiv \mathbf{x}_i^T\mathbf{w}, +\hat{y} = \hat{X}_i^T\hat{\beta} + \hat{\epsilon}, \label{_auto1} \end{equation} $$ -where we use the short-hand notation -\( \mathbf{x}_i = (1,\boldsymbol{x}_i) \) and \( \mathbf{w}_i = (b_0,\boldsymbol{w}_i) \). +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.











    -

    Some selected properties

    +

    Some selected properties

    -This function takes values on the entire real axis. In the case of +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. One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to \( \{0,1\} \), -\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, -this is commonly known as the "perceptron" in the machine learning -literature. This model is extremely simple, and it is favorable in -many cases (e.g. noisy data) to have a ``soft" classifier that outputs -the probability of a given category. For example, given -\( \mathbf{x}_i \), the classifier outputs the probability of being in -category \( m \). One such function is the logistic (or sigmoid) function: +variables. -$$ -\begin{equation} -f(s) = \frac{1}{1+\mathrm e^{-s}}. -\label{eq:log_fun} -\end{equation} -$$ - -Note that \( 1-f(s)= f(-s) \), which will be useful shortly. +

    +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 cross-entropy as a cost function for logistic regression

    +

    The logistic function

    -The perceptron is an example of a ``hard classification": 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 \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). -Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point \( \boldsymbol{x}_i \) belongs to a category \( y_i=\{0,1\} \) is is given by +A widely used model is the so-called logistic (or sigmoid) function which ranges from \( 0 \) to \( 1 \), $$ -\begin{eqnarray} -P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& \frac{1}{1+\mathrm{e}^{-\mathbf{x}^T_i\mathbf{w}}},\nonumber\\ -P(y_i=0|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& 1 - P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)}, -\end{eqnarray} +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. $$ -where \( \boldsymbol{\theta}=\mathbf{w} \) are the weights we wish to learn from the data. +Note that \( 1-p(t)= p(-t) \).

    -Notice that in terms of the logistic function, we can write +









    + +

    Probabilistic classifiers

    + +

    +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) + +

    +









    + +

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

    +Note that we used +$$ +p(y_i=1) = 1-p(y_i=0). $$

    -

    Maximum likelihood

    +

    Maximum likelihood

    -We now define the cost function for logistic regression using Maximum -Likelihood Estimation (MLE). Recall, that in MLE we choose parameters -to maximize the probability of seeing the observed data. Consider a -dataset \( \mathcal{D}=\{(y_i,\boldsymbol{x}_i)\} \) with binary labels -\( y_i\in\{0,1\} \) where the data points are drawn independently. The -likelihood of the seeing the data under our model is just: +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 $$ -\begin{align} -P(\mathcal{D}|\mathbf{w})& = \prod_{i=1}^n \left[f(\mathbf{x}_i^T\mathbf{w})\right]^{y_i}\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]^{1-y_i}\nonumber \\ -\label{_auto2} -\end{align} +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} $$ -from which we can readily compute the log-likelihood: +from which we obtain the log-likelihood and our cost/loss function $$ -\begin{equation} -l(\mathbf{w}) = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. -\label{_auto3} -\end{equation} +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]. $$











    -The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \theta \) + +

    The cost function rewritten

    + +

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









    +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 $$ -\begin{eqnarray} -\mathcal{C}(\mathbf{w}) &=& - l(\mathbf{w}) \\ -&=& \sum_{i=1}^n -y_i\log f(\mathbf{x}_i^T\mathbf{w}) - (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].\nonumber -\end{eqnarray} +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{1+\exp{(\beta_0+\beta_1x_i)}}\right). $$ -This equation is known in statistics as the \emph{cross entropy}. Finally, we note that just as in linear regression, -in practice we usually supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression. +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

    +

    Minimizing the cross entropy

    -The cross entropy is a convex function of the weights \( \mathbf{w} \) and, -therefore, any local minimizer is a global minimizer. Minimizing this -cost function leads to the following equation - -$$ -\begin{equation} -\boldsymbol{0}=\boldsymbol{\nabla} \mathcal{C}(\mathbf{w}) = \sum_{i=1}^n\left[f(\mathbf{x}_i^T\mathbf{w})-y_i\right]\mathbf{x}_i, -\label{_auto4} -\end{equation} -$$ +The cross entropy is a convex function of the weights \( \hat{\beta} \) and, +therefore, any local minimizer is a global minimizer.

    -where we made use of the logistic function identity \( \partial_z f(z) = -f(z)[1-f(z)] \). This equation defines a transcendental equation for -\( \mathbf{w} \), the solution of which, unlike linear regression, cannot -be written in a closed form. -Here we need gradient descent methods! +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 scikit-learn example

    +

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

    +









    + +

    A scikit-learn example

    @@ -300,6 +380,63 @@ plt.plot(X_new, y_proba[:, .show()

    +









    + +

    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):
    +    # plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
    +    # plt.show()
    +    plot_decision_boundary(lambda x: clf.predict(x), X, y)
    +    plt.title("Logistic Regression")
    +
    +
    +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/LogReg/ipynb/LogReg.ipynb b/doc/pub/LogReg/ipynb/LogReg.ipynb index 90c236b5c..b6532eb40 100644 --- a/doc/pub/LogReg/ipynb/LogReg.ipynb +++ b/doc/pub/LogReg/ipynb/LogReg.ipynb @@ -10,7 +10,7 @@ " \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 20, 2018**\n", + "Date: **Sep 25, 2018**\n", "\n", "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -20,42 +20,81 @@ "\n", "## Logistic Regression\n", "\n", - "So far we have focused on learning from datasets for which there is a\n", - "**continuous** output. In linear regression we have been \n", - "concerned with learning the coefficients of a polynomial to predict\n", - "the response of a continuous variable $y_i$ on unseen data based on\n", - "its independent variables ${\\bf x}_i$. \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 (in terms of matrices to invert) for several\n", + "quantities, ranging from the variance and thereby the confidence\n", + "intervals of the parameters $\\hat{\\beta}$ to the mean squared\n", + "error. If we can invert the product of the design matrices, linear\n", + "regression gives then a simple recipe for fitting our data.\n", "\n", - "Classification problems,\n", - "however, are concerned with outcomes taking the form of discrete\n", - "variables (i.e. categories). For example, we may want to detect if\n", - "there's a cat or a dog in an image. Or given a specific system,\n", - "we'd like to identify its state, say whether it is an ordered or disordered system (typical situation in solid state physics).\n", - "(e.g. ordered/disordered). \n", "\n", - "**Logistic regression deals with binary, dichotomous outcomes (e.g. True or\n", - "False, Success or Failure, etc.). It is worth noting that logistic\n", - "regression is also commonly used in modern supervised Deep Learning\n", - "models**, as we will see later.\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 towards neural\n", + "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 optmization 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. \n", + "\n", + "We note also that many of the topics discussed here \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 $y_i\\in\\mathbb{Z}$\n", - "are discrete and only take values from $m=0,\\dots,M-1$ (i.e. $M$\n", - "classes).\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 $X\\in\\mathbb{R}^{n\\times p}$\n", - "made of $n$ samples, each of which bears $p$ features. The\n", + "made of $n$ samples, each of which carries $p$ features. The\n", "primary goal is to identify the classes to which new unseen samples\n", "belong.\n", "\n", - "\n", + "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 use who could default or not on her/his debt. That is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y_i = \\begin{bmatrix} 0 & \\mathrm{no default}\\\\ 1 & \\mathrm{default} \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "## Linear classifier\n", "\n", - "Let us start by considering a slightly simpler classifier: a linear classifier that categorizes examples using a weighted linear-combination of the features and an additive offset" + "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 $y_i \\leq 0.5$. \n", + "\n", + "We would then have our \n", + "weighted linear combination, namely" ] }, { @@ -67,7 +106,7 @@ "\n", "$$\n", "\\begin{equation}\n", - "s_i = \\boldsymbol{x}_i^T\\boldsymbol{w} + b_0 \\equiv \\mathbf{x}_i^T\\mathbf{w},\n", + "\\hat{y} = \\hat{X}_i^T\\hat{\\beta} + \\hat{\\epsilon},\n", "\\label{_auto1} \\tag{1}\n", "\\end{equation}\n", "$$" @@ -77,36 +116,35 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "where we use the short-hand notation \n", - "$\\mathbf{x}_i = (1,\\boldsymbol{x}_i)$ and $\\mathbf{w}_i = (b_0,\\boldsymbol{w}_i)$. \n", + "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.\n", "\n", "## Some selected properties\n", "\n", - "This function takes values on the entire real axis. In the case of\n", + "The main problem with our function is that it \n", + "takes values on the entire real axis. In the case of\n", "logistic regression, however, the labels $y_i$ are discrete\n", - "variables. One simple way to get a discrete output is to have sign\n", - "functions that map the output of a linear regressor to $\\{0,1\\}$,\n", - "$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. Indeed,\n", - "this is commonly known as the \"perceptron\" in the machine learning\n", - "literature. This model is extremely simple, and it is favorable in\n", - "many cases (e.g. noisy data) to have a ``soft\" classifier that outputs\n", - "the probability of a given category. For example, given\n", - "$\\mathbf{x}_i$, the classifier outputs the probability of being in\n", - "category $m$. One such function is the logistic (or sigmoid) function:" + "variables. \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", + "## The logistic function\n", + "\n", + "A widely used model is the so-called logistic (or sigmoid) function which ranges from $0$ to $1$," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "\n", - "

    \n", - "\n", "$$\n", - "\\begin{equation}\n", - "f(s) = \\frac{1}{1+\\mathrm e^{-s}}.\n", - "\\label{eq:log_fun} \\tag{2}\n", - "\\end{equation}\n", + "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n", "$$" ] }, @@ -114,12 +152,25 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Note that $1-f(s)= f(-s)$, which will be useful shortly. \n", + "Note that $1-p(t)= p(-t)$.\n", "\n", - "## The cross-entropy as a cost function for logistic regression\n", + "## Probabilistic classifiers\n", "\n", - "The perceptron is an example of a ``hard classification\": 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 $\\mathbf{x}_i$, the classifier outputs the probability of being in category $m$. \n", - "Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point $\\boldsymbol{x}_i$ belongs to a category $y_i=\\{0,1\\}$ is is given by" + "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)\n", + "\n", + "\n", + "## 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" ] }, { @@ -127,10 +178,10 @@ "metadata": {}, "source": [ "$$\n", - "\\begin{eqnarray}\n", - "P(y_i=1|\\boldsymbol{x}_i,\\boldsymbol{\\theta)} &=& \\frac{1}{1+\\mathrm{e}^{-\\mathbf{x}^T_i\\mathbf{w}}},\\nonumber\\\\\n", - "P(y_i=0|\\boldsymbol{x}_i,\\boldsymbol{\\theta)} &=& 1 - P(y_i=1|\\boldsymbol{x}_i,\\boldsymbol{\\theta)},\n", - "\\end{eqnarray}\n", + "\\begin{align*}\n", + "p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", + "p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n", + "\\end{align*}\n", "$$" ] }, @@ -138,10 +189,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "where $\\boldsymbol{\\theta}=\\mathbf{w}$ are the weights we wish to learn from the data. \n", + "where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", "\n", - "\n", - "Notice that in terms of the logistic function, we can write" + "Note that we used" ] }, { @@ -149,7 +199,7 @@ "metadata": {}, "source": [ "$$\n", - "P(y_i=1) =f(\\mathbf{x}_i^T\\mathbf{w})=1-P(y_i=0).\n", + "p(y_i=1) = 1-p(y_i=0).\n", "$$" ] }, @@ -160,12 +210,12 @@ "\n", "## Maximum likelihood\n", "\n", - "We now define the cost function for logistic regression using Maximum\n", - "Likelihood Estimation (MLE). Recall, that in MLE we choose parameters\n", - "to maximize the probability of seeing the observed data. Consider a\n", - "dataset $\\mathcal{D}=\\{(y_i,\\boldsymbol{x}_i)\\}$ with binary labels\n", - "$y_i\\in\\{0,1\\}$ where the data points are drawn independently. The\n", - "likelihood of the seeing the data under our model is just:" + "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 (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" ] }, { @@ -173,7 +223,9 @@ "metadata": {}, "source": [ "$$\n", - "P(\\mathcal{D}|\\mathbf{w}) = \\prod_{i=1}^n \\left[f(\\mathbf{x}_i^T\\mathbf{w})\\right]^{y_i}\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right]^{1-y_i}\\nonumber\n", + "\\begin{align*}\n", + "P(\\mathcal{D}|\\hat{\\beta})& = \\prod_{i=1}^n \\left[p(y_i=1|x_i,\\hat{\\beta})\\right]^{y_i}\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]^{1-y_i}\\nonumber \\\\\n", + "\\end{align*}\n", "$$" ] }, @@ -181,43 +233,25 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n", - "
    \n", + "from which we obtain the log-likelihood and our **cost/loss** function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The cost function rewritten\n", "\n", - "$$\n", - "\\begin{equation} \n", - "\\label{_auto2} \\tag{3}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "from which we can readily compute the log-likelihood:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
    \n", - "\n", - "$$\n", - "\\begin{equation}\n", - "l(\\mathbf{w}) = \\sum_{i=1}^n y_i\\log f(\\mathbf{x}_i^T\\mathbf{w}) + (1-y_i)\\log\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right].\n", - "\\label{_auto3} \\tag{4}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "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 $\\theta$" + "Reordering the logarithms, we can rewrite the **cost/loss** function as" ] }, { @@ -225,7 +259,7 @@ "metadata": {}, "source": [ "$$\n", - "\\hat{\\mathbf{w}} = \\sum_{i=1}^n y_i\\log f(\\mathbf{x}_i^T\\mathbf{w}) + (1-y_i)\\log\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right].\n", + "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n", "$$" ] }, @@ -233,6 +267,7 @@ "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" ] }, @@ -241,10 +276,7 @@ "metadata": {}, "source": [ "$$\n", - "\\begin{eqnarray}\n", - "\\mathcal{C}(\\mathbf{w}) &=& - l(\\mathbf{w}) \\\\\n", - "&=& \\sum_{i=1}^n -y_i\\log f(\\mathbf{x}_i^T\\mathbf{w}) - (1-y_i)\\log\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right].\\nonumber\n", - "\\end{eqnarray}\n", + "\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n", "$$" ] }, @@ -252,28 +284,25 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "This equation is known in statistics as the \\emph{cross entropy}. Finally, we note that just as in linear regression, \n", - "in practice we usually supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n", + "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 $\\mathbf{w}$ and,\n", - "therefore, any local minimizer is a global minimizer. Minimizing this\n", - "cost function leads to the following equation" + "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", - "
    \n", - "\n", "$$\n", - "\\begin{equation}\n", - "\\boldsymbol{0}=\\boldsymbol{\\nabla} \\mathcal{C}(\\mathbf{w}) = \\sum_{i=1}^n\\left[f(\\mathbf{x}_i^T\\mathbf{w})-y_i\\right]\\mathbf{x}_i,\n", - "\\label{_auto4} \\tag{5}\n", - "\\end{equation}\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n", "$$" ] }, @@ -281,13 +310,60 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "where we made use of the logistic function identity $\\partial_z f(z) =\n", - "f(z)[1-f(z)]$. This equation defines a transcendental equation for\n", - "$\\mathbf{w}$, the solution of which, unlike linear regression, cannot\n", - "be written in a closed form. \n", - "Here we need gradient descent methods!\n", - "\n", + "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": [ "## A **scikit-learn** example" ] }, @@ -320,6 +396,72 @@ "plt.plot(X_new, y_proba[:, 0], \"b--\", label=\"Not Iris-Virginica\")\n", "plt.show()" ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 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", + " # plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)\n", + " # plt.show()\n", + " plot_decision_boundary(lambda x: clf.predict(x), X, y)\n", + " plt.title(\"Logistic Regression\")\n", + "\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", + "\n", + "if __name__ == \"__main__\":\n", + " main()" + ] } ], "metadata": {}, diff --git a/doc/pub/LogReg/ipynb/ipynb-LogReg-src.tar.gz 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