mhjensen
215 lines
9.0 KiB
HTML
215 lines
9.0 KiB
HTML
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<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;">Logistic Regression</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-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-bs016.html#___sec15" style="font-size: 80%;">A simple classification problem</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="___sec0" class="anchor">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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