Morten Hjorth-Jensen
453 lines
31 KiB
HTML
453 lines
31 KiB
HTML
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<a class="navbar-brand" href="week36-bs.html">Week 36: Statistical interpretation of Linear Regression and Resampling techniques</a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week36-bs001.html#plans-for-week-36" style="font-size: 80%;">Plans for week 36</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs002.html#summary-from-last-week-and-discussion-of-svd-ridge-and-lasso-regression-with-examples" style="font-size: 80%;">Summary from last Week and discussion of SVD, Ridge and Lasso regression with examples</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs003.html#linear-regression-and-the-svd" style="font-size: 80%;">Linear Regression and the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.html#what-does-it-mean" style="font-size: 80%;">What does it mean?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs005.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;">And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs006.html#code-for-svd-and-inversion-of-matrices" style="font-size: 80%;">Code for SVD and Inversion of Matrices</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#inverse-of-rectangular-matrix" style="font-size: 80%;">Inverse of Rectangular Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs008.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs009.html#from-ols-to-ridge-and-lasso" style="font-size: 80%;">From OLS to Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs010.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs011.html#note-on-scikit-learn" style="font-size: 80%;">Note on Scikit-Learn</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#comparison-with-ols" style="font-size: 80%;">Comparison with OLS</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.html#simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression" style="font-size: 80%;">Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs023.html#writing-the-cost-function" style="font-size: 80%;">Writing the Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#simple-code-for-solving-the-above-problem" style="font-size: 80%;">Simple code for solving the above problem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.html#another-example-now-with-a-polynomial-fit" style="font-size: 80%;">Another Example, now with a polynomial fit</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.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="._week36-bs030.html#more-thinking" style="font-size: 80%;">More thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#still-thinking" style="font-size: 80%;">Still thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.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="._week36-bs033.html#further-manipulations" style="font-size: 80%;">Further Manipulations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.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="._week36-bs036.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="._week36-bs038.html#friday-september-9" style="font-size: 80%;">Friday September 9</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs039.html#linking-the-regression-analysis-with-a-statistical-interpretation" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs040.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs041.html#expectation-value-and-variance" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs042.html#expectation-value-and-variance-for-boldsymbol-beta" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs043.html#deriving-ols-from-a-probability-distribution" style="font-size: 80%;">Deriving OLS from a probability distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.html#independent-and-identically-distrubuted-iid" style="font-size: 80%;">Independent and Identically Distrubuted (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs045.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs046.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs047.html#more-basic-statistics-and-bayes-theorem" style="font-size: 80%;">More basic Statistics and Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs049.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#interpretations-of-bayes-theorem" style="font-size: 80%;">Interpretations of Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs052.html#example-of-usage-of-bayes-theorem" style="font-size: 80%;">Example of Usage of Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs053.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs054.html#bayes-theorem-and-ridge-and-lasso-regression" style="font-size: 80%;">Bayes' Theorem and Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs055.html#test-function-for-what-happens-with-ols-ridge-and-lasso" style="font-size: 80%;">Test Function for what happens with OLS, Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs056.html#invoking-bayes-theorem" style="font-size: 80%;">Invoking Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs057.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#exercise-1-mean-values-and-variances-in-linear-regression" style="font-size: 80%;">Exercise 1: mean values and variances in linear regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#exercise-2-adding-ridge-and-lasso-regression" style="font-size: 80%;">Exercise 2: Adding Ridge and Lasso Regression</a></li>
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</li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0037"></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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<pre style="line-height: 125%;"><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>
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<p>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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<p>
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