mhjensen
217 lines
9.0 KiB
HTML
217 lines
9.0 KiB
HTML
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<!-- navigation toc: --> <li><a href="._LogReg-bs001.html#___sec0" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs002.html#___sec1" style="font-size: 80%;">Optimization and Deep learning</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs003.html#___sec2" style="font-size: 80%;">Basics</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs004.html#___sec3" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs005.html#___sec4" style="font-size: 80%;">Some selected properties</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs006.html#___sec5" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs007.html#___sec6" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs012.html#___sec11" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="#___sec14" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">A simple classification problem</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs017.html#___sec16" style="font-size: 80%;">The Credit Card example</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs018.html#___sec17" style="font-size: 80%;">How to read the Credit Card data</a></li>
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<h2 id="___sec14" class="anchor">More classes </h2>
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<p>
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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 <b>Softmax</b> function.
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<p>
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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 \( \hat{x} \) and a weighting vector \( \hat{\beta} \) 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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<p>
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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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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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