Morten Hjorth-Jensen
330 lines
18 KiB
HTML
330 lines
18 KiB
HTML
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<!-- navigation toc: --> <li><a href="._week39-bs001.html#plan-for-week-39-september-22-26-2025" style="font-size: 80%;"><b>Plan for week 39, September 22-26, 2025</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs002.html#readings-and-videos-resampling-methods" style="font-size: 80%;"><b>Readings and Videos, resampling methods</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs003.html#readings-and-videos-logistic-regression" style="font-size: 80%;"><b>Readings and Videos, logistic regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs004.html#lab-sessions-week-39" style="font-size: 80%;"><b>Lab sessions week 39</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs005.html#lecture-material" style="font-size: 80%;"><b>Lecture material</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs010.html#resampling-methods" style="font-size: 80%;"><b>Resampling methods</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs007.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;"><b>Resampling approaches can be computationally expensive</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs008.html#why-resampling-methods" style="font-size: 80%;"><b>Why resampling methods ?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs009.html#statistical-analysis" style="font-size: 80%;"><b>Statistical analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs010.html#resampling-methods" style="font-size: 80%;"><b>Resampling methods</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs011.html#resampling-methods-bootstrap" style="font-size: 80%;"><b>Resampling methods: Bootstrap</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs012.html#the-bias-variance-tradeoff" style="font-size: 80%;"><b>The bias-variance tradeoff</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs013.html#a-way-to-read-the-bias-variance-tradeoff" style="font-size: 80%;"><b>A way to Read the Bias-Variance Tradeoff</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs014.html#understanding-what-happens" style="font-size: 80%;"><b>Understanding what happens</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs015.html#summing-up" style="font-size: 80%;"><b>Summing up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs016.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;"><b>Another Example from Scikit-Learn's Repository</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs017.html#various-steps-in-cross-validation" style="font-size: 80%;"><b>Various steps in cross-validation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs018.html#cross-validation-in-brief" style="font-size: 80%;"><b>Cross-validation in brief</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs019.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;"><b>Code Example for Cross-validation and \( k \)-fold Cross-validation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs020.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;"><b>More examples on bootstrap and cross-validation and errors</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs021.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;"><b>The same example but now with cross-validation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs022.html#logistic-regression" style="font-size: 80%;"><b>Logistic Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs023.html#classification-problems" style="font-size: 80%;"><b>Classification problems</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs024.html#optimization-and-deep-learning" style="font-size: 80%;"><b>Optimization and Deep learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs025.html#basics" style="font-size: 80%;"><b>Basics</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs026.html#linear-classifier" style="font-size: 80%;"><b>Linear classifier</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs027.html#some-selected-properties" style="font-size: 80%;"><b>Some selected properties</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs028.html#simple-example" style="font-size: 80%;"><b>Simple example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs029.html#plotting-the-mean-value-for-each-group" style="font-size: 80%;"><b>Plotting the mean value for each group</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs030.html#the-logistic-function" style="font-size: 80%;"><b>The logistic function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs031.html#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks" style="font-size: 80%;"><b>Examples of likelihood functions used in logistic regression and nueral networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs032.html#two-parameters" style="font-size: 80%;"><b>Two parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs033.html#maximum-likelihood" style="font-size: 80%;"><b>Maximum likelihood</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs034.html#the-cost-function-rewritten" style="font-size: 80%;"><b>The cost function rewritten</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs035.html#minimizing-the-cross-entropy" style="font-size: 80%;"><b>Minimizing the cross entropy</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs036.html#a-more-compact-expression" style="font-size: 80%;"><b>A more compact expression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs037.html#extending-to-more-predictors" style="font-size: 80%;"><b>Extending to more predictors</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs038.html#including-more-classes" style="font-size: 80%;"><b>Including more classes</b></a></li>
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<!-- navigation toc: --> <li><a href="#more-classes" style="font-size: 80%;"><b>More classes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs040.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;"><b>Optimization, the central part of any Machine Learning algortithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs041.html#revisiting-our-logistic-regression-case" style="font-size: 80%;"><b>Revisiting our Logistic Regression case</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs042.html#the-equations-to-solve" style="font-size: 80%;"><b>The equations to solve</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs043.html#solving-using-newton-raphson-s-method" style="font-size: 80%;"><b>Solving using Newton-Raphson's method</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs044.html#example-code-for-logistic-regression" style="font-size: 80%;"><b>Example code for Logistic Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs044.html#synthetic-data-generation" style="font-size: 80%;"> Synthetic data generation</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0039"></a>
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<!-- !split -->
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<h2 id="more-classes" class="anchor">More classes </h2>
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<p>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 <b>Softmax</b> function.
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</p>
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<p>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 \( \boldsymbol{\theta} \) is (with two
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predictors):
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</p>
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$$
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p(C=k\vert \mathbf {x} )=\frac{\exp{(\theta_{k0}+\theta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\theta_{l0}+\theta_{l1}x_1)}}.
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$$
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<p>It is easy to extend to more predictors. The final class is </p>
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$$
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p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\theta_{l0}+\theta_{l1}x_1)}},
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$$
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<p>and they sum to one. Our earlier discussions were all specialized to
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the case with two classes only. It is easy to see from the above that
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what we derived earlier is compatible with these equations.
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</p>
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<p>To find the optimal parameters we would typically use a gradient
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descent method. Newton's method and gradient descent methods are
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discussed in the material on <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_self">optimization
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methods</a>.
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</p>
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