mhjensen
235 lines
9.7 KiB
HTML
235 lines
9.7 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%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs008.html#___sec7" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs009.html#___sec8" style="font-size: 80%;">The cost function rewritten</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs010.html#___sec9" style="font-size: 80%;">Minimizing the cross entropy</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs011.html#___sec10" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs012.html#___sec11" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A <b>scikit-learn</b> example</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 two-dimensional Ising model, Predicting phase transition of the two-dimensional Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs018.html#___sec17" style="font-size: 80%;">Reading in the data</a></li>
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<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Logistic regression</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs020.html#___sec19" style="font-size: 80%;">Exploring the logistic regression</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs021.html#___sec20" style="font-size: 80%;">Accuracy of a classification model</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs022.html#___sec21" style="font-size: 80%;">Analyzing the results</a></li>
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<a name="part0019"></a>
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<!-- !split -->
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<h2 id="___sec18" class="anchor">Logistic regression </h2>
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<p>
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Logistic regression is a linear model for classification. Recalling
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the cost function for ordinary least squares with both L2 (ridge) and
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L1 (LASSO) penalties we will see that the logistic cost function is
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very similar. In OLS we wish to predict a continuous variable
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\( \hat{y} \) using
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$$
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\begin{align}
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\hat{y} = X\omega,
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\tag{4}
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\end{align}
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$$
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<p>
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where \( X \in \mathbb{R}^{n \times p} \) is the input data and \( \omega^{p
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\times d} \) are the weights of the regression. In a classification
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setting (binary classification in our situation) we are interested in
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a positive or negative answer. We can thus define either answer to be
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above or below some threshold. But, in order to limit the size of the
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answer and also to get a probability interpretation on how sure we are
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for either answer we can compute the sigmoid function of OLS. That is,
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$$
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\begin{align}
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f(X\omega) = \frac{1}{1 + \exp(-X\omega)}.
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\tag{5}
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\end{align}
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$$
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We are thus interested in minizming the following cost function
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$$
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\begin{align}
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C(X, \omega) = \sum_{i = 1}^n \left\{
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- y_i\log\left( f(x_i^T\omega) \right)
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- (1 - y_i)\log\left[1 - f(x_i^T\omega)\right]
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\right\},
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\tag{6}
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\end{align}
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$$
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<p>
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where we will restrict ourselves to a value for \( f(z) \) as the sigmoid
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described above. We can also tack on a L2 (Ridge) or L1 (LASSO)
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penalization to this cost function in the same manner we did for
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linear regression.
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<p>
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<p>
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