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<!-- navigation toc: --> <li><a href="._week38-bs001.html#___sec0" style="font-size: 80%;">Plans for week 38</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs002.html#___sec1" style="font-size: 80%;">Thursday September 23</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs003.html#___sec2" style="font-size: 80%;">Ridge and LASSO Regression, reminder</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs004.html#___sec3" style="font-size: 80%;">Various steps in cross-validation</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week38-bs006.html#___sec5" style="font-size: 80%;">Cross-validation in brief</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week38-bs008.html#___sec7" style="font-size: 80%;">To think about, first part</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs009.html#___sec8" style="font-size: 80%;">More thinking</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs010.html#___sec9" style="font-size: 80%;">Still thinking</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs011.html#___sec10" style="font-size: 80%;">Linear Regression code, Intercept handling first</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs012.html#___sec11" style="font-size: 80%;">What does centering (subtracting the mean values) mean mathematically?</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs013.html#___sec12" style="font-size: 80%;">Code Examples</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs014.html#___sec13" style="font-size: 80%;">Taking out the mean</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs015.html#___sec14" style="font-size: 80%;">More complicated Example: The Ising model</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs016.html#___sec15" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs017.html#___sec16" style="font-size: 80%;">Linear regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs018.html#___sec17" style="font-size: 80%;">Singular Value decomposition</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs019.html#___sec18" style="font-size: 80%;">The one-dimensional Ising model</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs020.html#___sec19" style="font-size: 80%;">Ridge regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs021.html#___sec20" style="font-size: 80%;">LASSO regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs022.html#___sec21" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs023.html#___sec22" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs024.html#___sec23" style="font-size: 80%;">Logistic Regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs025.html#___sec24" style="font-size: 80%;">Classification problems</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs026.html#___sec25" style="font-size: 80%;">Optimization and Deep learning</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs027.html#___sec26" style="font-size: 80%;">Basics</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs028.html#___sec27" style="font-size: 80%;">Linear classifier</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs029.html#___sec28" style="font-size: 80%;">Some selected properties</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs030.html#___sec29" style="font-size: 80%;">Simple example</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs031.html#___sec30" style="font-size: 80%;">Plotting the mean value for each group</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs032.html#___sec31" style="font-size: 80%;">The logistic function</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs033.html#___sec32" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs034.html#___sec33" style="font-size: 80%;">Two parameters</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs035.html#___sec34" style="font-size: 80%;">Maximum likelihood</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">The cost function rewritten</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Minimizing the cross entropy</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">A more compact expression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Extending to more predictors</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs041.html#___sec40" style="font-size: 80%;">More classes</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs042.html#___sec41" style="font-size: 80%;">Friday September 24</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs043.html#___sec42" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs044.html#___sec43" style="font-size: 80%;">Using the correlation matrix</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs045.html#___sec44" style="font-size: 80%;">Discussing the correlation data</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs046.html#___sec45" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs047.html#___sec46" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs048.html#___sec47" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs049.html#___sec48" style="font-size: 80%;">The equations to solve</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs050.html#___sec49" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs051.html#___sec50" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs052.html#___sec51" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs053.html#___sec52" style="font-size: 80%;">Simple geometric interpretation</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs054.html#___sec53" style="font-size: 80%;">Extending to more than one variable</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs055.html#___sec54" style="font-size: 80%;">Steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs056.html#___sec55" style="font-size: 80%;">More on Steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs057.html#___sec56" style="font-size: 80%;">The ideal</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs058.html#___sec57" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs059.html#___sec58" style="font-size: 80%;">Convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs060.html#___sec59" style="font-size: 80%;">Convex function</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs061.html#___sec60" style="font-size: 80%;">Conditions on convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs062.html#___sec61" style="font-size: 80%;">More on convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs063.html#___sec62" style="font-size: 80%;">Some simple problems</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs064.html#___sec63" style="font-size: 80%;">Friday September 25</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs065.html#___sec64" style="font-size: 80%;">Standard steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs066.html#___sec65" style="font-size: 80%;">Gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs067.html#___sec66" style="font-size: 80%;">Steepest descent method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs068.html#___sec67" style="font-size: 80%;">Steepest descent method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs069.html#___sec68" style="font-size: 80%;">Final expressions</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs070.html#___sec69" style="font-size: 80%;">Steepest descent example</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs071.html#___sec70" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs072.html#___sec71" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs073.html#___sec72" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs074.html#___sec73" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs075.html#___sec74" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs076.html#___sec75" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs077.html#___sec76" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs078.html#___sec77" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs079.html#___sec78" style="font-size: 80%;">Revisiting some of our first Linear Regression Encounters</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs080.html#___sec79" style="font-size: 80%;">Gradient descent example</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs081.html#___sec80" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs082.html#___sec81" style="font-size: 80%;">The Hessian matrix</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs083.html#___sec82" style="font-size: 80%;">Simple program</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs084.html#___sec83" style="font-size: 80%;">Gradient Descent Example</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs085.html#___sec84" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
<!-- navigation toc: --> <li><a href="._week38-bs086.html#___sec85" style="font-size: 80%;">Gradient descent and Ridge</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs087.html#___sec86" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs088.html#___sec87" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
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<center><h1>Data Analysis and Machine Learning: Logistic Regression</h1></center> <!-- document title -->
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<!-- author(s): Morten Hjorth-Jensen -->
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<b>Morten Hjorth-Jensen</b> [1, 2]
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<!-- institution(s) -->
<center>[1] <b>Department of Physics, University of Oslo</b></center>
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<center><h4>Sep 22, 2021</h4></center> <!-- date -->
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