Morten Hjorth-Jensen
565 lines
38 KiB
HTML
565 lines
38 KiB
HTML
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('Why Linear Regression (aka Ordinary Least Squares and family), '
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'repeat from last week',
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<a class="navbar-brand" href="week35-bs.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#plans-for-week-35" style="font-size: 80%;"><b>Plans for week 35</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#reading-recommendations" style="font-size: 80%;"> Reading recommendations:</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs002.html#why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week" style="font-size: 80%;"><b>Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs003.html#the-equations-for-ordinary-least-squares" style="font-size: 80%;"><b>The equations for ordinary least squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs004.html#the-cost-loss-function" style="font-size: 80%;"><b>The cost/loss function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#some-useful-matrix-and-vector-expressions" style="font-size: 80%;"><b>Some useful matrix and vector expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs008.html#the-jacobian" style="font-size: 80%;"><b>The Jacobian</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs009.html#derivatives-example-1" style="font-size: 80%;"><b>Derivatives, example 1</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs010.html#example-2" style="font-size: 80%;"><b>Example 2</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs011.html#example-3" style="font-size: 80%;"><b>Example 3</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#example-4" style="font-size: 80%;"><b>Example 4</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#the-mean-squared-error-and-its-derivative" style="font-size: 80%;"><b>The mean squared error and its derivative</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs014.html#other-useful-relations" style="font-size: 80%;"><b>Other useful relations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs015.html#meet-the-hessian-matrix" style="font-size: 80%;"><b>Meet the Hessian Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs017.html#example-relevant-for-the-exercises" style="font-size: 80%;"><b>Example relevant for the exercises</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#own-code-for-ordinary-least-squares" style="font-size: 80%;"><b>Own code for Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs019.html#adding-error-analysis-and-training-set-up" style="font-size: 80%;"><b>Adding error analysis and training set up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs020.html#splitting-our-data-in-training-and-test-data" style="font-size: 80%;"><b>Splitting our Data in Training and Test data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs021.html#the-complete-code-with-a-simple-data-set" style="font-size: 80%;"><b>The complete code with a simple data set</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs022.html#making-your-own-test-train-splitting" style="font-size: 80%;"><b>Making your own test-train splitting</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#reducing-the-number-of-degrees-of-freedom-overarching-view" style="font-size: 80%;"><b>Reducing the number of degrees of freedom, overarching view</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs024.html#preprocessing-our-data" style="font-size: 80%;"><b>Preprocessing our data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs025.html#functionality-in-scikit-learn" style="font-size: 80%;"><b>Functionality in Scikit-Learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs026.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.html#frequently-used-scaling-functions" style="font-size: 80%;"><b>Frequently used scaling functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs028.html#example-of-own-standard-scaling" style="font-size: 80%;"><b>Example of own Standard scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs029.html#min-max-scaling" style="font-size: 80%;"><b>Min-Max Scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs030.html#testing-the-means-squared-error-as-function-of-complexity" style="font-size: 80%;"><b>Testing the Means Squared Error as function of Complexity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs031.html#more-preprocessing-examples-two-dimensional-example-the-franke-function" style="font-size: 80%;"><b>More preprocessing examples, two-dimensional example, the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs032.html#to-think-about-first-part" style="font-size: 80%;"><b>To think about, first part</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs033.html#more-thinking" style="font-size: 80%;"><b>More thinking</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs034.html#still-thinking" style="font-size: 80%;"><b>Still thinking</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs035.html#what-does-centering-subtracting-the-mean-values-mean-mathematically" style="font-size: 80%;"><b>What does centering (subtracting the mean values) mean mathematically?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.html#further-manipulations" style="font-size: 80%;"><b>Further Manipulations</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs037.html#wrapping-it-up" style="font-size: 80%;"><b>Wrapping it up</b></a></li>
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|
<!-- navigation toc: --> <li><a href="#linear-regression-code-intercept-handling-first" style="font-size: 80%;"><b>Linear Regression code, Intercept handling first</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs039.html#the-boston-housing-data-example" style="font-size: 80%;"><b>The Boston housing data example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs040.html#housing-data-the-code" style="font-size: 80%;"><b>Housing data, the code</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs041.html#material-for-lecture-thursday-august-31" style="font-size: 80%;"><b>Material for lecture Thursday, August 31</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs042.html#mathematical-interpretation-of-ordinary-least-squares" style="font-size: 80%;"><b>Mathematical Interpretation of Ordinary Least Squares</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs043.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs044.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs045.html#the-singular-value-decomposition" style="font-size: 80%;"><b>The singular value decomposition</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs046.html#linear-regression-problems" style="font-size: 80%;"><b>Linear Regression Problems</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs047.html#fixing-the-singularity" style="font-size: 80%;"><b>Fixing the singularity</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs048.html#basic-math-of-the-svd" style="font-size: 80%;"><b>Basic math of the SVD</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs049.html#the-svd-a-fantastic-algorithm" style="font-size: 80%;"><b>The SVD, a Fantastic Algorithm</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs050.html#economy-size-svd" style="font-size: 80%;"><b>Economy-size SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs051.html#codes-for-the-svd" style="font-size: 80%;"><b>Codes for the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs056.html#further-properties-important-for-our-analyses-later" style="font-size: 80%;"><b>Further properties (important for our analyses later)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs057.html#meet-the-covariance-matrix" style="font-size: 80%;"><b>Meet the Covariance Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs058.html#introducing-the-covariance-and-correlation-functions" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs059.html#covariance-and-correlation-matrix" style="font-size: 80%;"><b>Covariance and Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs060.html#correlation-function-and-design-feature-matrix" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs063.html#correlation-matrix-with-pandas" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs064.html#correlation-matrix-with-pandas-and-the-franke-function" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs065.html#rewriting-the-covariance-and-or-correlation-matrix" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs066.html#linking-with-the-svd" style="font-size: 80%;"><b>Linking with the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs067.html#what-does-it-mean" style="font-size: 80%;"><b>What does it mean?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs068.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;"><b>And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs069.html#ridge-and-lasso-regression" style="font-size: 80%;"><b>Ridge and LASSO Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs070.html#deriving-the-ridge-regression-equations" style="font-size: 80%;"><b>Deriving the Ridge Regression Equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs071.html#interpreting-the-ridge-results" style="font-size: 80%;"><b>Interpreting the Ridge results</b></a></li>
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<a name="part0038"></a>
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<h2 id="linear-regression-code-intercept-handling-first" class="anchor">Linear Regression code, Intercept handling first </h2>
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<p>This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (<em>code example thanks to Øyvind Sigmundson Schøyen</em>). Here our scaling of the data is done by subtracting the mean values only.
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Note also that we do not split the data into training and test.
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</p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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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">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.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2021</span>)
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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: #008000; font-weight: bold">def</span> <span style="color: #0000FF">fit_beta</span>(X, y):
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
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true_beta <span style="color: #666666">=</span> [<span style="color: #666666">2</span>, <span style="color: #666666">0.5</span>, <span style="color: #666666">3.7</span>]
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">11</span>)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(
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np<span style="color: #666666">.</span>asarray([x <span style="color: #666666">**</span> p <span style="color: #666666">*</span> b <span style="color: #008000; font-weight: bold">for</span> p, b <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(true_beta)]), axis<span style="color: #666666">=0</span>
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) <span style="color: #666666">+</span> <span style="color: #666666">0.1</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span><span style="color: #008000">len</span>(x))
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degree <span style="color: #666666">=</span> <span style="color: #666666">3</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x), degree))
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<span style="color: #408080; font-style: italic"># Include the intercept in the design matrix</span>
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<span style="color: #008000; font-weight: bold">for</span> p <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(degree):
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X[:, p] <span style="color: #666666">=</span> x <span style="color: #666666">**</span> p
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beta <span style="color: #666666">=</span> fit_beta(X, y)
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<span style="color: #408080; font-style: italic"># Intercept is included in the design matrix</span>
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skl <span style="color: #666666">=</span> LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)<span style="color: #666666">.</span>fit(X, y)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"True beta: </span><span style="color: #BB6688; font-weight: bold">{</span>true_beta<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Fitted beta: </span><span style="color: #BB6688; font-weight: bold">{</span>beta<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Sklearn fitted beta: </span><span style="color: #BB6688; font-weight: bold">{</span>skl<span style="color: #666666">.</span>coef_<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
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ypredictOwn <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
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ypredictSKL <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>predict(X)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with intercept column"</span>)
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<span style="color: #008000">print</span>(MSE(y,ypredictOwn))
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with intercept column from SKL"</span>)
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<span style="color: #008000">print</span>(MSE(y,ypredictSKL))
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plt<span style="color: #666666">.</span>figure()
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plt<span style="color: #666666">.</span>scatter(x, y, label<span style="color: #666666">=</span><span style="color: #BA2121">"Data"</span>)
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plt<span style="color: #666666">.</span>plot(x, X <span style="color: #666666">@</span> beta, label<span style="color: #666666">=</span><span style="color: #BA2121">"Fit"</span>)
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plt<span style="color: #666666">.</span>plot(x, skl<span style="color: #666666">.</span>predict(X), label<span style="color: #666666">=</span><span style="color: #BA2121">"Sklearn (fit_intercept=False)"</span>)
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<span style="color: #408080; font-style: italic"># Do not include the intercept in the design matrix</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x), degree <span style="color: #666666">-</span> <span style="color: #666666">1</span>))
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<span style="color: #008000; font-weight: bold">for</span> p <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(degree <span style="color: #666666">-</span> <span style="color: #666666">1</span>):
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X[:, p] <span style="color: #666666">=</span> x <span style="color: #666666">**</span> (p <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
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<span style="color: #408080; font-style: italic"># Intercept is not included in the design matrix</span>
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skl <span style="color: #666666">=</span> LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)<span style="color: #666666">.</span>fit(X, y)
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<span style="color: #408080; font-style: italic"># Use centered values for X and y when computing coefficients</span>
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y_offset <span style="color: #666666">=</span> np<span style="color: #666666">.</span>average(y, axis<span style="color: #666666">=0</span>)
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X_offset <span style="color: #666666">=</span> np<span style="color: #666666">.</span>average(X, axis<span style="color: #666666">=0</span>)
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beta <span style="color: #666666">=</span> fit_beta(X <span style="color: #666666">-</span> X_offset, y <span style="color: #666666">-</span> y_offset)
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intercept <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(y_offset <span style="color: #666666">-</span> X_offset <span style="color: #666666">@</span> beta)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Manual intercept: </span><span style="color: #BB6688; font-weight: bold">{</span>intercept<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Fitted beta (wiothout intercept): </span><span style="color: #BB6688; font-weight: bold">{</span>beta<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Sklearn intercept: </span><span style="color: #BB6688; font-weight: bold">{</span>skl<span style="color: #666666">.</span>intercept_<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Sklearn fitted beta (without intercept): </span><span style="color: #BB6688; font-weight: bold">{</span>skl<span style="color: #666666">.</span>coef_<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
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ypredictOwn <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
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ypredictSKL <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>predict(X)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with Manual intercept"</span>)
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<span style="color: #008000">print</span>(MSE(y,ypredictOwn<span style="color: #666666">+</span>intercept))
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<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with Sklearn intercept"</span>)
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<span style="color: #008000">print</span>(MSE(y,ypredictSKL))
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plt<span style="color: #666666">.</span>plot(x, X <span style="color: #666666">@</span> beta <span style="color: #666666">+</span> intercept, <span style="color: #BA2121">"--"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Fit (manual intercept)"</span>)
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plt<span style="color: #666666">.</span>plot(x, skl<span style="color: #666666">.</span>predict(X), <span style="color: #BA2121">"--"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Sklearn (fit_intercept=True)"</span>)
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plt<span style="color: #666666">.</span>grid()
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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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</div>
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</div>
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<div class="output_wrapper">
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<p>The intercept is the value of our output/target variable
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when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case).
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</p>
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<p>Printing the MSE, we see first that both methods give the same MSE, as
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they should. However, when we move to for example Ridge regression (discussed next week),
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the way we treat the intercept may give a larger or smaller MSE,
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meaning that the MSE can be penalized by the value of the
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intercept. Not including the intercept in the fit, means that the
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regularization term does not include \( \beta_0 \). For different values
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of \( \lambda \), this may lead to differing MSE values.
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</p>
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<p>To remind the reader, the regularization term, with the intercept in Ridge regression is given by</p>
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$$
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\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2,
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$$
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<p>but when we take out the intercept, this equation becomes</p>
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$$
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\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2.
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$$
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<p>For Lasso regression we have</p>
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$$
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\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert.
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$$
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<p>It means that, when scaling the design matrix and the outputs/targets,
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by subtracting the mean values, we have an optimization problem which
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is not penalized by the intercept. The MSE value can then be smaller
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since it focuses only on the remaining quantities. If we however bring
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back the intercept, we will get an MSE which then contains the
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intercept. This becomes more important when we discuss Ridge and Lasso
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regression next week.
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</p>
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<p>
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