Morten Hjorth-Jensen
423 lines
14 KiB
Python
423 lines
14 KiB
Python
#!/usr/bin/env python
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# coding: utf-8
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# <!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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# doconce format html additionweek42.do.txt --no_mako -->
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# <!-- dom:TITLE: Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40 -->
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# # Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40
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# **Morten Hjorth-Jensen**, Department of Physics and Center for Computing in Science Education, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University
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#
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# Date: **October 14-18, 2024**
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# ## The logistic function
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#
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# A widely studied model, is the
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# perceptron model, which is an example of a "hard classification" model. We
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# have used this model when we discussed 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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# ## Examples of likelihood functions used in logistic regression and nueral networks
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#
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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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# In[1]:
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get_ipython().run_line_magic('matplotlib', 'inline')
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"""The sigmoid function (or the logistic curve) is a
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function that takes any real number, z, and outputs a number (0,1).
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It is useful in neural networks for assigning weights on a relative scale.
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The value z is the weighted sum of parameters involved in the learning algorithm."""
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import numpy
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import matplotlib.pyplot as plt
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import math as mt
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z = numpy.arange(-5, 5, .1)
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sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
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sigma = sigma_fn(z)
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fig = plt.figure()
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ax = fig.add_subplot(111)
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ax.plot(z, sigma)
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ax.set_ylim([-0.1, 1.1])
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ax.set_xlim([-5,5])
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ax.grid(True)
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ax.set_xlabel('z')
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ax.set_title('sigmoid function')
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plt.show()
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"""Step Function"""
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z = numpy.arange(-5, 5, .02)
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step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
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step = step_fn(z)
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fig = plt.figure()
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ax = fig.add_subplot(111)
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ax.plot(z, step)
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ax.set_ylim([-0.5, 1.5])
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ax.set_xlim([-5,5])
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ax.grid(True)
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ax.set_xlabel('z')
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ax.set_title('step function')
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plt.show()
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"""tanh Function"""
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z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
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t = numpy.tanh(z)
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fig = plt.figure()
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ax = fig.add_subplot(111)
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ax.plot(z, t)
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ax.set_ylim([-1.0, 1.0])
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ax.set_xlim([-2*mt.pi,2*mt.pi])
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ax.grid(True)
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ax.set_xlabel('z')
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ax.set_title('tanh function')
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plt.show()
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# ## Two parameters
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#
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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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# $$
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# \begin{align*}
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# p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
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# p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
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# \end{align*}
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# $$
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# where $\boldsymbol{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$.
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#
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# Note that we used
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# $$
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# p(y_i=0\vert x_i, \boldsymbol{\beta}) = 1-p(y_i=1\vert x_i, \boldsymbol{\beta}).
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# $$
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# ## The cost function
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#
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# Reordering the logarithms, we can rewrite the **cost/loss** function as
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# $$
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# \mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
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# $$
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# The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\beta$.
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# Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
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# $$
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# \mathcal{C}(\boldsymbol{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
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# $$
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# This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression,
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# in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.
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# ## Minimizing the cross entropy
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#
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# The cross entropy is a convex function of the weights $\boldsymbol{\beta}$ and,
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# therefore, any local minimizer is a global minimizer.
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#
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# Minimizing this
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# cost function with respect to the two parameters $\beta_0$ and $\beta_1$ we obtain
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# $$
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# \frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
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# $$
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# and
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# $$
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# \frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
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# $$
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# ## A more compact expression
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#
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# Let us now define a vector $\boldsymbol{y}$ with $n$ elements $y_i$, an
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# $n\times p$ matrix $\boldsymbol{X}$ which contains the $x_i$ values and a
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# vector $\boldsymbol{p}$ of fitted probabilities $p(y_i\vert x_i,\boldsymbol{\beta})$. We can rewrite in a more compact form the first
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# derivative of cost function as
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# $$
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# \frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
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# $$
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# If we in addition define a diagonal matrix $\boldsymbol{W}$ with elements
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# $p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})$, we can obtain a compact expression of the second derivative as
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# $$
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# \frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
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# $$
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# ## Extending to more predictors
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#
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# Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors
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# $$
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# \log{ \frac{p(\boldsymbol{\beta}\boldsymbol{x})}{1-p(\boldsymbol{\beta}\boldsymbol{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
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# $$
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# Here we defined $\boldsymbol{x}=[1,x_1,x_2,\dots,x_p]$ and $\boldsymbol{\beta}=[\beta_0, \beta_1, \dots, \beta_p]$ leading to
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# $$
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# p(\boldsymbol{\beta}\boldsymbol{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
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# $$
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# ## Including more classes
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#
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# Till now we have mainly focused on two classes, the so-called binary
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# system. Suppose we wish to extend to $K$ classes. Let us for the sake
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# of simplicity assume we have only two predictors. We have then following model
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# $$
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# \log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
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# $$
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# and
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# $$
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# \log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
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# $$
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# and so on till the class $C=K-1$ class
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# $$
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# \log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
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# $$
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# and the model is specified in term of $K-1$ so-called log-odds or
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# **logit** transformations.
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# ## More classes
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#
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# In our discussion of neural networks we will encounter the above again
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# in terms of a slightly modified function, the so-called **Softmax** function.
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#
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# The softmax function is used in various multiclass classification
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# methods, such as multinomial logistic regression (also known as
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# softmax regression), multiclass linear discriminant analysis, naive
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# Bayes classifiers, and artificial neural networks. Specifically, in
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# multinomial logistic regression and linear discriminant analysis, the
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# input to the function is the result of $K$ distinct linear functions,
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# and the predicted probability for the $k$-th class given a sample
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# vector $\boldsymbol{x}$ and a weighting vector $\boldsymbol{\beta}$ is (with two
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# predictors):
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# $$
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# p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
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# $$
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# It is easy to extend to more predictors. The final class is
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# $$
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# p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
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# $$
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# and they sum to one.
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# ## Wisconsin Cancer Data
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#
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# We show here how we can use a simple regression case on the breast
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# cancer data using Logistic regression as our algorithm for
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# classification.
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# In[2]:
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.model_selection import train_test_split
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from sklearn.datasets import load_breast_cancer
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from sklearn.linear_model import LogisticRegression
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# Load the data
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cancer = load_breast_cancer()
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X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
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print(X_train.shape)
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print(X_test.shape)
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# Logistic Regression
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logreg = LogisticRegression(solver='lbfgs')
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logreg.fit(X_train, y_train)
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print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
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# ## Using the correlation matrix
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#
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# In addition to the above scores, we could also study the covariance (and the correlation matrix).
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# We use **Pandas** to compute the correlation matrix.
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# In[3]:
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.model_selection import train_test_split
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from sklearn.datasets import load_breast_cancer
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from sklearn.linear_model import LogisticRegression
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cancer = load_breast_cancer()
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import pandas as pd
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# Making a data frame
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cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
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fig, axes = plt.subplots(15,2,figsize=(10,20))
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malignant = cancer.data[cancer.target == 0]
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benign = cancer.data[cancer.target == 1]
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ax = axes.ravel()
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for i in range(30):
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_, bins = np.histogram(cancer.data[:,i], bins =50)
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ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)
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ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)
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ax[i].set_title(cancer.feature_names[i])
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ax[i].set_yticks(())
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ax[0].set_xlabel("Feature magnitude")
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ax[0].set_ylabel("Frequency")
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ax[0].legend(["Malignant", "Benign"], loc ="best")
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fig.tight_layout()
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plt.show()
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import seaborn as sns
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correlation_matrix = cancerpd.corr().round(1)
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# use the heatmap function from seaborn to plot the correlation matrix
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# annot = True to print the values inside the square
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plt.figure(figsize=(15,8))
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sns.heatmap(data=correlation_matrix, annot=True)
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plt.show()
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# ## Discussing the correlation data
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#
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# In the above example we note two things. In the first plot we display
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# the overlap of benign and malignant tumors as functions of the various
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# features in the Wisconsing breast cancer data set. We see that for
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# some of the features we can distinguish clearly the benign and
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# malignant cases while for other features we cannot. This can point to
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# us which features may be of greater interest when we wish to classify
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# a benign or not benign tumour.
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#
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# In the second figure we have computed the so-called correlation
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# matrix, which in our case with thirty features becomes a $30\times 30$
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# matrix.
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#
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# We constructed this matrix using **pandas** via the statements
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# In[4]:
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cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
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# and then
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# In[5]:
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correlation_matrix = cancerpd.corr().round(1)
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# Diagonalizing this matrix we can in turn say something about which
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# features are of relevance and which are not. This leads us to
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# the classical Principal Component Analysis (PCA) theorem with
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# applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)).
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# ## Other measures in classification studies: Cancer Data again
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# In[6]:
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.model_selection import train_test_split
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from sklearn.datasets import load_breast_cancer
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from sklearn.linear_model import LogisticRegression
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# Load the data
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cancer = load_breast_cancer()
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X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
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print(X_train.shape)
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print(X_test.shape)
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# Logistic Regression
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logreg = LogisticRegression(solver='lbfgs')
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logreg.fit(X_train, y_train)
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from sklearn.preprocessing import LabelEncoder
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from sklearn.model_selection import cross_validate
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#Cross validation
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accuracy = cross_validate(logreg,X_test,y_test,cv=10)['test_score']
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print(accuracy)
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print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
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import scikitplot as skplt
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y_pred = logreg.predict(X_test)
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skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
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plt.show()
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y_probas = logreg.predict_proba(X_test)
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skplt.metrics.plot_roc(y_test, y_probas)
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plt.show()
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skplt.metrics.plot_cumulative_gain(y_test, y_probas)
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plt.show()
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# ## Gradient descent and Logistic regression
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#
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# We complete these examples by adding a simple code for
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# Logistic regression. Note the more general approach with a class for
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# the method. Here we use a so-called **AND** gate for our data set.
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# In[7]:
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import numpy as np
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class LogisticRegression:
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def __init__(self, learning_rate=0.01, num_iterations=1000):
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self.learning_rate = learning_rate
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self.num_iterations = num_iterations
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self.beta_logreg = None
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def sigmoid(self, z):
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return 1 / (1 + np.exp(-z))
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def GDfit(self, X, y):
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n_data, num_features = X.shape
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self.beta_logreg = np.zeros(num_features)
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for _ in range(self.num_iterations):
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linear_model = X @ self.beta_logreg
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y_predicted = self.sigmoid(linear_model)
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# Gradient calculation
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gradient = (X.T @ (y_predicted - y))/n_data
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# Update beta_logreg
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self.beta_logreg -= self.learning_rate*gradient
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def predict(self, X):
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linear_model = X @ self.beta_logreg
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y_predicted = self.sigmoid(linear_model)
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return [1 if i >= 0.5 else 0 for i in y_predicted]
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# Example usage
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if __name__ == "__main__":
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# Sample data
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X = np.array([[0, 0], [1, 0], [0, 1], [1, 1]])
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y = np.array([0, 0, 0, 1]) # This is an AND gate
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model = LogisticRegression(learning_rate=0.01, num_iterations=1000)
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model.GDfit(X, y)
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predictions = model.predict(X)
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print("Predictions:", predictions)
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