mhjensen
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704 lines
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Logistic Regression', 2, None, '___sec0'),
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('Optimization and Deep learning', 2, None, '___sec1'),
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('Basics', 2, None, '___sec2'),
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('Linear classifier', 2, None, '___sec3'),
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('Some selected properties', 2, None, '___sec4'),
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('The logistic function', 2, None, '___sec5'),
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('Examples of likelihood functions used in logistic regression '
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'and nueral networks',
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2,
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None,
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'___sec6'),
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('Two parameters', 2, None, '___sec7'),
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('Maximum likelihood', 2, None, '___sec8'),
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('The cost function rewritten', 2, None, '___sec9'),
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('Minimizing the cross entropy', 2, None, '___sec10'),
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('A more compact expression', 2, None, '___sec11'),
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('Extending to more predictors', 2, None, '___sec12'),
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('Including more classes', 2, None, '___sec13'),
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('More classes', 2, None, '___sec14'),
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('A simple classification problem', 2, None, '___sec15'),
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('The Credit Card example', 2, None, '___sec16'),
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('How to read the Credit Card data', 2, None, '___sec17')]}
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end of tocinfo -->
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<!-- ------------------- main content ---------------------- -->
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<center><h1>Data Analysis and Machine Learning: Logistic Regression</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Oct 17, 2019</h4></center> <!-- date -->
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<br>
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<p>
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<!-- !split -->
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<h2 id="___sec0">Logistic Regression </h2>
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<p>
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In linear regression our main interest was centered on learning the
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coefficients of a functional fit (say a polynomial) in order to be
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able to predict the response of a continuous variable on some unseen
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data. The fit to the continuous variable \( y_i \) is based on some
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independent variables \( \hat{x}_i \). Linear regression resulted in
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analytical expressions for standard ordinary Least Squares or Ridge
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regression (in terms of matrices to invert) for several quantities,
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ranging from the variance and thereby the confidence intervals of the
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parameters \( \hat{\beta} \) to the mean squared error. If we can invert
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the product of the design matrices, linear regression gives then a
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simple recipe for fitting our data.
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<p>
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Classification problems, however, are concerned with outcomes taking
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the form of discrete variables (i.e. categories). We may for example,
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on the basis of DNA sequencing for a number of patients, like to find
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out which mutations are important for a certain disease; or based on
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scans of various patients' brains, figure out if there is a tumor or
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not; or given a specific physical system, we'd like to identify its
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state, say whether it is an ordered or disordered system (typical
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situation in solid state physics); or classify the status of a
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patient, whether she/he has a stroke or not and many other similar
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situations.
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<p>
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The most common situation we encounter when we apply logistic
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regression is that of two possible outcomes, normally denoted as a
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binary outcome, true or false, positive or negative, success or
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failure etc.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec1">Optimization and Deep learning </h2>
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<p>
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Logistic regression will also serve as our stepping stone towards
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neural network algorithms and supervised deep learning. For logistic
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learning, the minimization of the cost function leads to a non-linear
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equation in the parameters \( \hat{\beta} \). The optimization of the
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problem calls therefore for minimization algorithms. This forms the
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bottle neck of all machine learning algorithms, namely how to find
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reliable minima of a multi-variable function. This leads us to the
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family of gradient descent methods. The latter are the working horses
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of basically all modern machine learning algorithms.
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<p>
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We note also that many of the topics discussed here on logistic
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regression are also commonly used in modern supervised Deep Learning
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models, as we will see later.
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<p>
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<!-- !split -->
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<h2 id="___sec2">Basics </h2>
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<p>
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We consider the case where the dependent variables, also called the
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responses or the outcomes, \( y_i \) are discrete and only take values
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from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
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<p>
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The goal is to predict the
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output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \)
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made of \( n \) samples, each of which carries \( p \) features or predictors. The
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primary goal is to identify the classes to which new unseen samples
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belong.
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<p>
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Let us specialize to the case of two classes only, with outputs
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\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
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credit card user that could default or not on her/his credit card
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debt. That is
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$$
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y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
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$$
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec3">Linear classifier </h2>
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<p>
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Before moving to the logistic model, let us try to use our linear
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regression model to classify these two outcomes. We could for example
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fit a linear model to the default case if \( y_i > 0.5 \) and the no
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default case \( y_i \leq 0.5 \).
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<p>
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We would then have our
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weighted linear combination, namely
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$$
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\begin{equation}
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\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
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\label{_auto1}
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\end{equation}
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$$
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where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our
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\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec4">Some selected properties </h2>
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<p>
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The main problem with our function is that it takes values on the
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entire real axis. In the case of logistic regression, however, the
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labels \( y_i \) are discrete variables. A typical example is the credit
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card data discussed below here, where we can set the state of
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defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons
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in the data set (see the full example below).
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<p>
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One simple way to get a discrete output is to have sign
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functions that map the output of a linear regressor to values \( \{0,1\} \),
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\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise.
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We will encounter this model in our first demonstration of neural networks. Historically it is called the "perceptron" model in the machine learning
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literature. This model is extremely simple. However, in many cases it is more
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favorable to use a ``soft" classifier that outputs
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the probability of a given category. This leads us to the logistic function.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec5">The logistic function </h2>
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<p>
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The perceptron is an example of a ``hard classification" model. We
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will encounter this model when we discuss neural networks as
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well. Each datapoint is deterministically assigned to a category (i.e
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\( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft"
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classifier that outputs the probability of a given category rather
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than a single value. For example, given \( x_i \), the classifier
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outputs the probability of being in a category \( k \). Logistic regression
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is the most common example of a so-called soft classifier. In logistic
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regression, the probability that a data point \( x_i \)
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belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
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$$
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p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
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$$
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Note that \( 1-p(t)= p(-t) \).
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec6">Examples of likelihood functions used in logistic regression and nueral networks </h2>
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<p>
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The following code plots the logistic function, the step function and other functions we will encounter from here and on.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #BA2121; font-style: italic">"""The sigmoid function (or the logistic curve) is a</span>
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<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
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<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
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<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm."""</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mt</span>
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z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.1</span>)
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sigma_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1/</span>(<span style="color: #666666">1+</span>numpy<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>z)))
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sigma <span style="color: #666666">=</span> sigma_fn(z)
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fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
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ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
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ax<span style="color: #666666">.</span>plot(z, sigma)
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ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
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ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
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ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
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ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
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ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'sigmoid function'</span>)
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plt<span style="color: #666666">.</span>show()
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<span style="color: #BA2121; font-style: italic">"""Step Function"""</span>
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z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.02</span>)
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step_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1.0</span> <span style="color: #008000; font-weight: bold">if</span> z <span style="color: #666666">>=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
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step <span style="color: #666666">=</span> step_fn(z)
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fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
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ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
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ax<span style="color: #666666">.</span>plot(z, step)
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ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
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ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
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ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
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ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
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ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'step function'</span>)
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plt<span style="color: #666666">.</span>show()
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<span style="color: #BA2121; font-style: italic">"""tanh Function"""</span>
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z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">0.1</span>)
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t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>tanh(z)
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fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
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ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
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ax<span style="color: #666666">.</span>plot(z, t)
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ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
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ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi,<span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi])
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ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
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ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
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ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'tanh function'</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec7">Two parameters </h2>
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<p>
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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
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$$
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\begin{align*}
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p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
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p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}),
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\end{align*}
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$$
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where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
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<p>
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Note that we used
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$$
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p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
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$$
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<p>
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<!-- !split -->
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<h2 id="___sec8">Maximum likelihood </h2>
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<p>
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In order to define the total likelihood for all possible outcomes from a
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dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels
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\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called <a href="https://en.wikipedia.org/wiki/Maximum_likelihood_estimation" target="_blank">Maximum Likelihood Estimation</a> (MLE) principle.
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We aim thus at maximizing
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the probability of seeing the observed data. We can then approximate the
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likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is
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$$
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\begin{align*}
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P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\
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\end{align*}
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$$
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from which we obtain the log-likelihood and our <b>cost/loss</b> function
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$$
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\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
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$$
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec9">The cost function rewritten </h2>
|
|
|
|
<p>
|
|
Reordering the logarithms, we can rewrite the <b>cost/loss</b> function as
|
|
$$
|
|
\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
|
|
$$
|
|
|
|
<p>
|
|
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 <b>cross entropy</b>. 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.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec10">Minimizing the cross entropy </h2>
|
|
|
|
<p>
|
|
The cross entropy is a convex function of the weights \( \hat{\beta} \) and,
|
|
therefore, any local minimizer is a global minimizer.
|
|
|
|
<p>
|
|
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
|
|
$$
|
|
\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec11">A more compact expression </h2>
|
|
|
|
<p>
|
|
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
|
|
|
|
$$
|
|
\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right).
|
|
$$
|
|
|
|
<p>
|
|
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
|
|
|
|
$$
|
|
\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}.
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec12">Extending to more predictors </h2>
|
|
|
|
<p>
|
|
Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
|
|
$$
|
|
\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
|
|
$$
|
|
|
|
Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to
|
|
$$
|
|
p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec13">Including more classes </h2>
|
|
|
|
<p>
|
|
Till now we have mainly focused on two classes, the so-called binary
|
|
system. Suppose we wish to extend to \( K \) classes. Let us for the sake
|
|
of simplicity assume we have only two predictors. We have then
|
|
following model
|
|
|
|
$$
|
|
\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
|
|
$$
|
|
|
|
$$
|
|
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
|
|
$$
|
|
|
|
and so on till the class \( C=K-1 \) class
|
|
$$
|
|
\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
|
|
$$
|
|
|
|
<p>
|
|
and the model is specified in term of \( K-1 \) so-called log-odds or
|
|
<b>logit</b> transformations.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec14">More classes </h2>
|
|
|
|
<p>
|
|
In our discussion of neural networks we will encounter the above again
|
|
in terms of a slightly modified function, the so-called <b>Softmax</b> function.
|
|
|
|
<p>
|
|
The softmax function is used in various multiclass classification
|
|
methods, such as multinomial logistic regression (also known as
|
|
softmax regression), multiclass linear discriminant analysis, naive
|
|
Bayes classifiers, and artificial neural networks. Specifically, in
|
|
multinomial logistic regression and linear discriminant analysis, the
|
|
input to the function is the result of \( K \) distinct linear functions,
|
|
and the predicted probability for the \( k \)-th class given a sample
|
|
vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two
|
|
predictors):
|
|
|
|
$$
|
|
p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
|
|
$$
|
|
|
|
It is easy to extend to more predictors. The final class is
|
|
$$
|
|
p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
|
|
$$
|
|
|
|
<p>
|
|
and they sum to one. Our earlier discussions were all specialized to
|
|
the case with two classes only. It is easy to see from the above that
|
|
what we derived earlier is compatible with these equations.
|
|
|
|
<p>
|
|
To find the optimal parameters we would typically use a gradient
|
|
descent method. Newton's method and gradient descent methods are
|
|
discussed in the material on <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_blank">optimization
|
|
methods</a>.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec15">A simple classification problem </h2>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets, linear_model
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">generate_data</span>():
|
|
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
|
|
X, y <span style="color: #666666">=</span> datasets<span style="color: #666666">.</span>make_moons(<span style="color: #666666">200</span>, noise<span style="color: #666666">=0.20</span>)
|
|
<span style="color: #008000; font-weight: bold">return</span> X, y
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">visualize</span>(X, y, clf):
|
|
plot_decision_boundary(<span style="color: #008000; font-weight: bold">lambda</span> x: clf<span style="color: #666666">.</span>predict(x), X, y)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_decision_boundary</span>(pred_func, X, y):
|
|
<span style="color: #408080; font-style: italic"># Set min and max values and give it some padding</span>
|
|
x_min, x_max <span style="color: #666666">=</span> X[:, <span style="color: #666666">0</span>]<span style="color: #666666">.</span>min() <span style="color: #666666">-</span> <span style="color: #666666">.5</span>, X[:, <span style="color: #666666">0</span>]<span style="color: #666666">.</span>max() <span style="color: #666666">+</span> <span style="color: #666666">.5</span>
|
|
y_min, y_max <span style="color: #666666">=</span> X[:, <span style="color: #666666">1</span>]<span style="color: #666666">.</span>min() <span style="color: #666666">-</span> <span style="color: #666666">.5</span>, X[:, <span style="color: #666666">1</span>]<span style="color: #666666">.</span>max() <span style="color: #666666">+</span> <span style="color: #666666">.5</span>
|
|
h <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
|
|
<span style="color: #408080; font-style: italic"># Generate a grid of points with distance h between them</span>
|
|
xx, yy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(np<span style="color: #666666">.</span>arange(x_min, x_max, h), np<span style="color: #666666">.</span>arange(y_min, y_max, h))
|
|
<span style="color: #408080; font-style: italic"># Predict the function value for the whole gid</span>
|
|
Z <span style="color: #666666">=</span> pred_func(np<span style="color: #666666">.</span>c_[xx<span style="color: #666666">.</span>ravel(), yy<span style="color: #666666">.</span>ravel()])
|
|
Z <span style="color: #666666">=</span> Z<span style="color: #666666">.</span>reshape(xx<span style="color: #666666">.</span>shape)
|
|
<span style="color: #408080; font-style: italic"># Plot the contour and training examples</span>
|
|
plt<span style="color: #666666">.</span>contourf(xx, yy, Z, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>Spectral)
|
|
plt<span style="color: #666666">.</span>scatter(X[:, <span style="color: #666666">0</span>], X[:, <span style="color: #666666">1</span>], c<span style="color: #666666">=</span>y, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>Spectral)
|
|
plt<span style="color: #666666">.</span>show()
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">classify</span>(X, y):
|
|
clf <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>LogisticRegressionCV()
|
|
clf<span style="color: #666666">.</span>fit(X, y)
|
|
<span style="color: #008000; font-weight: bold">return</span> clf
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">main</span>():
|
|
X, y <span style="color: #666666">=</span> generate_data()
|
|
<span style="color: #408080; font-style: italic"># visualize(X, y)</span>
|
|
clf <span style="color: #666666">=</span> classify(X, y)
|
|
visualize(X, y, clf)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #19177C">__name__</span> <span style="color: #666666">==</span> <span style="color: #BA2121">"__main__"</span>:
|
|
main()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec16">The Credit Card example </h2>
|
|
Here we use the the <a href="https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients" target="_blank">credit card data</a>.
|
|
The data are from an extensive database from Taiwan and include more than ten predictors.
|
|
|
|
<p>
|
|
For categorical data -Scikit-Learn- provides a so-called <b>one-hot encoder</b>.
|
|
This is called one-hot
|
|
encoding, because only one attribute will be equal to 1 (hot), while the others will be 0 (cold).
|
|
<b>Scikit-Learn</b> provides a OneHotEncoder encoder to convert integer categorical values into one-hot
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> OneHotEncoder
|
|
encoder <span style="color: #666666">=</span> OneHotEncoder()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec17">How to read the Credit Card data </h2>
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> OneHotEncoder
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.compose</span> <span style="color: #008000; font-weight: bold">import</span> ColumnTransformer
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler, OneHotEncoder
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> confusion_matrix, accuracy_score, roc_auc_score
|
|
|
|
<span style="color: #408080; font-style: italic"># Trying to set the seed</span>
|
|
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">random</span>
|
|
random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Reading file into data frame</span>
|
|
cwd <span style="color: #666666">=</span> os<span style="color: #666666">.</span>getcwd()
|
|
filename <span style="color: #666666">=</span> cwd <span style="color: #666666">+</span> <span style="color: #BA2121">'/default of credit card clients.xls'</span>
|
|
nanDict <span style="color: #666666">=</span> {}
|
|
df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_excel(filename, header<span style="color: #666666">=1</span>, skiprows<span style="color: #666666">=0</span>, index_col<span style="color: #666666">=0</span>, na_values<span style="color: #666666">=</span>nanDict)
|
|
|
|
df<span style="color: #666666">.</span>rename(index<span style="color: #666666">=</span><span style="color: #008000">str</span>, columns<span style="color: #666666">=</span>{<span style="color: #BA2121">"default payment next month"</span>: <span style="color: #BA2121">"defaultPaymentNextMonth"</span>}, inplace<span style="color: #666666">=</span><span style="color: #008000">True</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Features and targets </span>
|
|
X <span style="color: #666666">=</span> df<span style="color: #666666">.</span>loc[:, df<span style="color: #666666">.</span>columns <span style="color: #666666">!=</span> <span style="color: #BA2121">'defaultPaymentNextMonth'</span>]<span style="color: #666666">.</span>values
|
|
y <span style="color: #666666">=</span> df<span style="color: #666666">.</span>loc[:, df<span style="color: #666666">.</span>columns <span style="color: #666666">==</span> <span style="color: #BA2121">'defaultPaymentNextMonth'</span>]<span style="color: #666666">.</span>values
|
|
|
|
<span style="color: #408080; font-style: italic"># Categorical variables to one-hot's</span>
|
|
onehotencoder <span style="color: #666666">=</span> OneHotEncoder(categories<span style="color: #666666">=</span><span style="color: #BA2121">"auto"</span>)
|
|
|
|
X <span style="color: #666666">=</span> ColumnTransformer(
|
|
[(<span style="color: #BA2121">""</span>, onehotencoder, [<span style="color: #666666">3</span>]),],
|
|
remainder<span style="color: #666666">=</span><span style="color: #BA2121">"passthrough"</span>
|
|
)<span style="color: #666666">.</span>fit_transform(X)
|
|
|
|
y<span style="color: #666666">.</span>shape
|
|
|
|
<span style="color: #408080; font-style: italic"># Train-test split</span>
|
|
trainingShare <span style="color: #666666">=</span> <span style="color: #666666">0.5</span>
|
|
seed <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
|
XTrain, XTest, yTrain, yTest<span style="color: #666666">=</span>train_test_split(X, y, train_size<span style="color: #666666">=</span>trainingShare, \
|
|
test_size <span style="color: #666666">=</span> <span style="color: #666666">1-</span>trainingShare,
|
|
random_state<span style="color: #666666">=</span>seed)
|
|
|
|
<span style="color: #408080; font-style: italic"># Input Scaling</span>
|
|
sc <span style="color: #666666">=</span> StandardScaler()
|
|
XTrain <span style="color: #666666">=</span> sc<span style="color: #666666">.</span>fit_transform(XTrain)
|
|
XTest <span style="color: #666666">=</span> sc<span style="color: #666666">.</span>transform(XTest)
|
|
|
|
<span style="color: #408080; font-style: italic"># One-hot's of the target vector</span>
|
|
Y_train_onehot, Y_test_onehot <span style="color: #666666">=</span> onehotencoder<span style="color: #666666">.</span>fit_transform(yTrain), onehotencoder<span style="color: #666666">.</span>fit_transform(yTest)
|
|
|
|
<span style="color: #408080; font-style: italic"># Remove instances with zeros only for past bill statements or paid amounts</span>
|
|
<span style="color: #BA2121; font-style: italic">'''</span>
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<span style="color: #BA2121; font-style: italic">df = df.drop(df[(df.BILL_AMT1 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.BILL_AMT2 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.BILL_AMT3 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.BILL_AMT4 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.BILL_AMT5 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.BILL_AMT6 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.PAY_AMT1 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.PAY_AMT2 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.PAY_AMT3 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.PAY_AMT4 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.PAY_AMT5 == 0) &</span>
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<span style="color: #BA2121; font-style: italic"> (df.PAY_AMT6 == 0)].index)</span>
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<span style="color: #BA2121; font-style: italic">'''</span>
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df <span style="color: #666666">=</span> df<span style="color: #666666">.</span>drop(df[(df<span style="color: #666666">.</span>BILL_AMT1 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>BILL_AMT2 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>BILL_AMT3 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>BILL_AMT4 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>BILL_AMT5 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>BILL_AMT6 <span style="color: #666666">==</span> <span style="color: #666666">0</span>)]<span style="color: #666666">.</span>index)
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df <span style="color: #666666">=</span> df<span style="color: #666666">.</span>drop(df[(df<span style="color: #666666">.</span>PAY_AMT1 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>PAY_AMT2 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>PAY_AMT3 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>PAY_AMT4 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>PAY_AMT5 <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">&</span>
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(df<span style="color: #666666">.</span>PAY_AMT6 <span style="color: #666666">==</span> <span style="color: #666666">0</span>)]<span style="color: #666666">.</span>index)
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> GridSearchCV
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lambdas<span style="color: #666666">=</span>np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>,<span style="color: #666666">7</span>,<span style="color: #666666">13</span>)
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parameters <span style="color: #666666">=</span> [{<span style="color: #BA2121">'C'</span>: <span style="color: #666666">1./</span>lambdas, <span style="color: #BA2121">"solver"</span>:[<span style="color: #BA2121">"lbfgs"</span>]}]<span style="color: #408080; font-style: italic">#*len(parameters)}]</span>
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scoring <span style="color: #666666">=</span> [<span style="color: #BA2121">'accuracy'</span>, <span style="color: #BA2121">'roc_auc'</span>]
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logReg <span style="color: #666666">=</span> LogisticRegression()
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gridSearch <span style="color: #666666">=</span> GridSearchCV(logReg, parameters, cv<span style="color: #666666">=5</span>, scoring<span style="color: #666666">=</span>scoring, refit<span style="color: #666666">=</span><span style="color: #BA2121">'roc_auc'</span>)
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</pre></div>
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<p>
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