Morten Hjorth-Jensen
463 lines
31 KiB
HTML
463 lines
31 KiB
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<!-- navigation toc: --> <li><a href="._week38-bs001.html#plans-for-week-38" style="font-size: 80%;">Plans for week 38</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs002.html#thursday-september-23" style="font-size: 80%;">Thursday September 23</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs003.html#ridge-and-lasso-regression-reminder" style="font-size: 80%;">Ridge and LASSO Regression, reminder</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs004.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs005.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" 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#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs007.html#code-example-for-cross-validation-and-k-fold-cross-validation" 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#to-think-about-first-part" style="font-size: 80%;">To think about, first part</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs009.html#more-thinking" style="font-size: 80%;">More thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs010.html#still-thinking" style="font-size: 80%;">Still thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs011.html#what-does-centering-subtracting-the-mean-values-mean-mathematically" style="font-size: 80%;">What does centering (subtracting the mean values) mean mathematically?</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs012.html#further-manipulations" style="font-size: 80%;">Further Manipulations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs013.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs014.html#linear-regression-code-intercept-handling-first" style="font-size: 80%;">Linear Regression code, Intercept handling first</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs015.html#code-examples" style="font-size: 80%;">Code Examples</a></li>
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<!-- navigation toc: --> <li><a href="#taking-out-the-mean" style="font-size: 80%;">Taking out the mean</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs017.html#more-complicated-example-the-ising-model" style="font-size: 80%;">More complicated Example: The Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs018.html#reformulating-the-problem-to-suit-regression" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs019.html#linear-regression" style="font-size: 80%;">Linear regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs020.html#singular-value-decomposition" style="font-size: 80%;">Singular Value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs021.html#the-one-dimensional-ising-model" style="font-size: 80%;">The one-dimensional Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#ridge-regression" style="font-size: 80%;">Ridge regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs023.html#lasso-regression" style="font-size: 80%;">LASSO regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs024.html#performance-as-function-of-the-regularization-parameter" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs025.html#finding-the-optimal-value-of-lambda" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs026.html#logistic-regression" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#classification-problems" style="font-size: 80%;">Classification problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs028.html#optimization-and-deep-learning" style="font-size: 80%;">Optimization and Deep learning</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#basics" style="font-size: 80%;">Basics</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs030.html#linear-classifier" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs031.html#some-selected-properties" style="font-size: 80%;">Some selected properties</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs032.html#simple-example" style="font-size: 80%;">Simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs033.html#plotting-the-mean-value-for-each-group" style="font-size: 80%;">Plotting the mean value for each group</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs034.html#the-logistic-function" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks" 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-bs036.html#two-parameters" style="font-size: 80%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#maximum-likelihood" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#the-cost-function-rewritten" style="font-size: 80%;">The cost function rewritten</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#minimizing-the-cross-entropy" style="font-size: 80%;">Minimizing the cross entropy</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs040.html#a-more-compact-expression" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs041.html#extending-to-more-predictors" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs042.html#including-more-classes" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs043.html#more-classes" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs044.html#friday-september-24" style="font-size: 80%;">Friday September 24</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs045.html#wisconsin-cancer-data" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs046.html#using-the-correlation-matrix" style="font-size: 80%;">Using the correlation matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs047.html#discussing-the-correlation-data" style="font-size: 80%;">Discussing the correlation data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs048.html#other-measures-in-classification-studies-cancer-data-again" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs049.html#friday-september-25" style="font-size: 80%;">Friday September 25</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs050.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs051.html#revisiting-our-logistic-regression-case" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs052.html#the-equations-to-solve" style="font-size: 80%;">The equations to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs053.html#solving-using-newton-raphson-s-method" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs054.html#brief-reminder-on-newton-raphson-s-method" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs055.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs056.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs057.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs058.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs059.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs060.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs061.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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</ul>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0016"></a>
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<!-- !split -->
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<h2 id="taking-out-the-mean" class="anchor">Taking out the mean </h2>
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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><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> linear_model
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
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n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
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<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
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<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">315</span>)
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n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)
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Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">20</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n,Maxpolydegree<span style="color: #666666">-1</span>))
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<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,Maxpolydegree): <span style="color: #408080; font-style: italic">#No intercept column</span>
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X[:,degree<span style="color: #666666">-1</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**</span>(degree)
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<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
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X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
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<span style="color: #408080; font-style: italic">#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable</span>
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X_train_mean <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(X_train,axis<span style="color: #666666">=0</span>)
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<span style="color: #408080; font-style: italic">#Center by removing mean from each feature</span>
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X_train_scaled <span style="color: #666666">=</span> X_train <span style="color: #666666">-</span> X_train_mean
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X_test_scaled <span style="color: #666666">=</span> X_test <span style="color: #666666">-</span> X_train_mean
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<span style="color: #408080; font-style: italic">#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)</span>
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<span style="color: #408080; font-style: italic">#Remove the intercept from the training data.</span>
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y_scaler <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(y_train)
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y_train_scaled <span style="color: #666666">=</span> y_train <span style="color: #666666">-</span> y_scaler
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p <span style="color: #666666">=</span> Maxpolydegree<span style="color: #666666">-1</span>
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I <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(p,p)
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<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
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nlambdas <span style="color: #666666">=</span> <span style="color: #666666">6</span>
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MSEOwnRidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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MSERidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">2</span>, nlambdas)
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(nlambdas):
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lmb <span style="color: #666666">=</span> lambdas[i]
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OwnRidgeBeta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X_train_scaled<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X_train_scaled<span style="color: #666666">+</span>lmb<span style="color: #666666">*</span>I) <span style="color: #666666">@</span> X_train_scaled<span style="color: #666666">.</span>T <span style="color: #666666">@</span> (y_train_scaled)
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intercept_ <span style="color: #666666">=</span> y_scaler <span style="color: #666666">-</span> X_train_mean<span style="color: #AA22FF">@OwnRidgeBeta</span> <span style="color: #408080; font-style: italic">#The intercept can be shifted so the model can predict on uncentered data</span>
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<span style="color: #408080; font-style: italic">#Add intercept to prediction</span>
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ypredictOwnRidge <span style="color: #666666">=</span> X_test_scaled <span style="color: #666666">@</span> OwnRidgeBeta <span style="color: #666666">+</span> y_scaler
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RegRidge <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>Ridge(lmb)
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RegRidge<span style="color: #666666">.</span>fit(X_train,y_train)
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ypredictRidge <span style="color: #666666">=</span> RegRidge<span style="color: #666666">.</span>predict(X_test)
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MSEOwnRidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictOwnRidge)
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MSERidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictRidge)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Beta values for own Ridge implementation"</span>)
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<span style="color: #008000">print</span>(OwnRidgeBeta) <span style="color: #408080; font-style: italic">#Intercept is given by mean of target variable</span>
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Beta values for Scikit-Learn Ridge implementation"</span>)
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<span style="color: #008000">print</span>(RegRidge<span style="color: #666666">.</span>coef_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Intercept from own implementation:'</span>)
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<span style="color: #008000">print</span>(intercept_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Intercept from Scikit-Learn Ridge implementation'</span>)
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<span style="color: #008000">print</span>(RegRidge<span style="color: #666666">.</span>intercept_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for own Ridge implementation"</span>)
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<span style="color: #008000">print</span>(MSEOwnRidgePredict[i])
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for Scikit-Learn Ridge implementation"</span>)
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<span style="color: #008000">print</span>(MSERidgePredict[i])
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<span style="color: #408080; font-style: italic"># Now plot the results</span>
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plt<span style="color: #666666">.</span>figure()
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSEOwnRidgePredict, <span style="color: #BA2121">'b--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE own Ridge Test'</span>)
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSERidgePredict, <span style="color: #BA2121">'g--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE SL Ridge Test'</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MSE'</span>)
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plt<span style="color: #666666">.</span>legend()
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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 see here, when compared to the code which includes explicitely the
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intercept column, that our MSE value is actually smaller. This is
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because the regularization term does not include the intercept value
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\( \beta_0 \) in the fitting. This applies to Lasso regularization as
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well. It means that our optimization is now done only with the
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centered matrix and/or vector that enter the fitting procedure. Note
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also that the problem with the intercept occurs mainly in these type
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of polynomial fitting problem.
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<p>
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The next example is indeed an example where all these discussions about the role of intercept are not present.
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<p>
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<p>
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