mhjensen
202 lines
8.6 KiB
HTML
202 lines
8.6 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="#___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-bs008.html#___sec7" style="font-size: 80%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs009.html#___sec8" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs010.html#___sec9" style="font-size: 80%;">The cost function rewritten</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs011.html#___sec10" style="font-size: 80%;">Minimizing the cross entropy</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="._LogReg-bs015.html#___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="___sec4" class="anchor">Some selected properties </h2>
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<p>
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The main problem with our function is that it takes values on the
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entire real axis. In the case of logistic regression, however, the
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labels \( y_i \) are discrete variables. A typical example is the credit
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card data discussed below here, where we can set the state of
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defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons
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in the data set (see the full example below).
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<p>
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One simple way to get a discrete output is to have sign
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functions that map the output of a linear regressor to values \( \{0,1\} \),
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\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise.
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We will encounter this model in our first demonstration of neural networks. Historically it is called the "perceptron" model in the machine learning
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literature. This model is extremely simple. However, in many cases it is more
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favorable to use a ``soft" classifier that outputs
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the probability of a given category. This leads us to the logistic function.
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<p>
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<li><a href="._LogReg-bs004.html">«</a></li>
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<li><a href="._LogReg-bs000.html">1</a></li>
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<li><a href="._LogReg-bs001.html">2</a></li>
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<li><a href="._LogReg-bs002.html">3</a></li>
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<li><a href="._LogReg-bs003.html">4</a></li>
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<li><a href="._LogReg-bs004.html">5</a></li>
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<li><a href="._LogReg-bs014.html">15</a></li>
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<li><a href="">...</a></li>
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<li><a href="._LogReg-bs018.html">19</a></li>
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<li><a href="._LogReg-bs006.html">»</a></li>
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