mhjensen
294 lines
16 KiB
HTML
294 lines
16 KiB
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'___sec19'),
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('Logistic Regression', 2, None, '___sec20'),
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('Optimization and Deep learning', 2, None, '___sec22'),
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('Some selected properties', 2, None, '___sec25'),
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('Simple example', 2, None, '___sec26'),
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('Plotting the mean value for each group', 2, None, '___sec27'),
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('Examples of likelihood functions used in logistic regression '
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('Two parameters', 2, None, '___sec30'),
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<!-- navigation toc: --> <li><a href="._week38-bs001.html#___sec0" style="font-size: 80%;">Plans for week 38</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs002.html#___sec1" style="font-size: 80%;">Thursday September 17</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs003.html#___sec2" style="font-size: 80%;">Ridge and LASSO Regression, reminder</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs004.html#___sec3" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs005.html#___sec4" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs006.html#___sec5" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs007.html#___sec6" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs008.html#___sec7" style="font-size: 80%;">Bias-Variance tradeoff with Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs009.html#___sec8" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs010.html#___sec9" style="font-size: 80%;">Cross-validation with Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs011.html#___sec10" style="font-size: 80%;">The Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs012.html#___sec11" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs013.html#___sec12" style="font-size: 80%;">Linear regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs014.html#___sec13" style="font-size: 80%;">Singular Value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs015.html#___sec14" style="font-size: 80%;">The one-dimensional Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs016.html#___sec15" style="font-size: 80%;">Ridge regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs017.html#___sec16" style="font-size: 80%;">LASSO regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs018.html#___sec17" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs019.html#___sec18" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs020.html#___sec19" style="font-size: 80%;">Friday September 18: Intro to Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs021.html#___sec20" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#___sec21" style="font-size: 80%;">Classification problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs023.html#___sec22" style="font-size: 80%;">Optimization and Deep learning</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs024.html#___sec23" style="font-size: 80%;">Basics</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs025.html#___sec24" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs026.html#___sec25" style="font-size: 80%;">Some selected properties</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#___sec26" style="font-size: 80%;">Simple example</a></li>
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<!-- navigation toc: --> <li><a href="#___sec27" style="font-size: 80%;">Plotting the mean value for each group</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#___sec28" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs030.html#___sec29" 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="._week38-bs031.html#___sec30" style="font-size: 80%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs032.html#___sec31" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs033.html#___sec32" style="font-size: 80%;">The cost function rewritten</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs034.html#___sec33" style="font-size: 80%;">Minimizing the cross entropy</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#___sec34" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Using the correlation matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs041.html#___sec40" style="font-size: 80%;">Discussing the correlation data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs042.html#___sec41" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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</li>
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<!-- !split -->
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<h2 id="___sec27" class="anchor">Plotting the mean value for each group </h2>
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<p>
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What we could attempt however is to plot the mean value for each group.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>agegroupmean <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0.1</span>, <span style="color: #666666">0.133</span>, <span style="color: #666666">0.250</span>, <span style="color: #666666">0.333</span>, <span style="color: #666666">0.462</span>, <span style="color: #666666">0.625</span>, <span style="color: #666666">0.765</span>, <span style="color: #666666">0.800</span>])
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group <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1</span>, <span style="color: #666666">2</span>, <span style="color: #666666">3</span>, <span style="color: #666666">4</span>, <span style="color: #666666">5</span>, <span style="color: #666666">6</span>, <span style="color: #666666">7</span>, <span style="color: #666666">8</span>])
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plt<span style="color: #666666">.</span>plot(group, agegroupmean, <span style="color: #BA2121">"r-"</span>)
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plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">9</span>,<span style="color: #666666">0</span>, <span style="color: #666666">1.0</span>])
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'Age group'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'CHD mean values'</span>)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Mean values for each age group'</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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We are now trying to find a function \( f(y\vert x) \), that is a function which gives us an expected value for the output \( y \) with a given input \( x \).
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In standard linear regression with a linear dependence on \( x \), we would write this in terms of our model
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$$
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f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
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$$
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<p>
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This expression implies however that \( f(y_i\vert x_i) \) could take any
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value from minus infinity to plus infinity. If we however let
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\( f(y\vert y) \) be represented by the mean value, the above example
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shows us that we can constrain the function to take values between
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zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
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at our last curve we see also that it has an S-shaped form. This leads
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us to a very popular model for the function \( f \), namely the so-called
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Sigmoid function or logistic model. We will consider this function as
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representing the probability for finding a value of \( y_i \) with a given
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\( x_i \).
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<p>
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