Morten Hjorth-Jensen
500 lines
33 KiB
HTML
500 lines
33 KiB
HTML
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2,
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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-august-30-september-3" style="font-size: 80%;"><b>Plans for week 35, August 30 -September 3</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs002.html#thursday-september-2" style="font-size: 80%;"><b>Thursday September 2</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs003.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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs004.html#regression-analysis-overarching-aims" style="font-size: 80%;"><b>Regression analysis, overarching aims</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs005.html#regression-analysis-overarching-aims-ii" style="font-size: 80%;"><b>Regression analysis, overarching aims II</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#general-linear-models" style="font-size: 80%;"><b>General linear models</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs008.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs009.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs011.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs011.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#optimizing-our-parameters" style="font-size: 80%;"><b>Optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#our-model-for-the-nuclear-binding-energies" style="font-size: 80%;"><b>Our model for the nuclear binding energies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs014.html#optimizing-our-parameters-more-details" style="font-size: 80%;"><b>Optimizing our parameters, more details</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs019.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-bs019.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#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-bs018.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-bs019.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-bs020.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-bs021.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-bs022.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-bs023.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs024.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-bs025.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-bs026.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-bs027.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-bs028.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-bs029.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-bs030.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs031.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-bs032.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-bs033.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-bs034.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-bs035.html#more-preprocessing-examples-franke-function-and-regression" style="font-size: 80%;"><b>More preprocessing examples, Franke function and regression</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs036.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-bs037.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs038.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs039.html#the-singular-value-decomposition" style="font-size: 80%;"><b>The singular value decomposition</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs040.html#linear-regression-problems" style="font-size: 80%;"><b>Linear Regression Problems</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs041.html#fixing-the-singularity" style="font-size: 80%;"><b>Fixing the singularity</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs042.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-bs043.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-bs044.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-bs045.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-bs046.html#note-about-svd-calculations" style="font-size: 80%;"><b>Note about SVD Calculations</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs047.html#friday-september-3" style="font-size: 80%;"><b>Friday September 3</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs048.html#mathematics-of-the-svd-and-implications" style="font-size: 80%;"><b>Mathematics of the SVD and implications</b></a></li>
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<!-- navigation toc: --> <li><a href="#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-bs060.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-bs063.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-bs065.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-bs066.html#interpreting-the-ridge-results" style="font-size: 80%;"><b>Interpreting the Ridge results</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs068.html#deriving-the-lasso-regression-equations" style="font-size: 80%;"><b>Deriving the Lasso Regression Equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs069.html#exercise-1-adding-ridge-and-lasso-regression" style="font-size: 80%;"><b>Exercise 1: Adding Ridge and Lasso Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs069.html#exercise-linear-regression-for-a-two-dimensional-function" style="font-size: 80%;"> Exercise: Linear Regression for a two-dimensional function</a></li>
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<h2 id="correlation-matrix-with-pandas-and-the-franke-function" class="anchor">Correlation Matrix with Pandas and the Franke function </h2>
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Common imports</span>
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<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">def</span> <span style="color: #0000FF">FrankeFunction</span>(x,y):
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term1 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">0.25*</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>))
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term2 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>((<span style="color: #666666">9*</span>x<span style="color: #666666">+1</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">/49.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.1*</span>(<span style="color: #666666">9*</span>y<span style="color: #666666">+1</span>))
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term3 <span style="color: #666666">=</span> <span style="color: #666666">0.5*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-7</span>)<span style="color: #666666">**2/4.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-3</span>)<span style="color: #666666">**2</span>))
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term4 <span style="color: #666666">=</span> <span style="color: #666666">-0.2*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-4</span>)<span style="color: #666666">**2</span> <span style="color: #666666">-</span> (<span style="color: #666666">9*</span>y<span style="color: #666666">-7</span>)<span style="color: #666666">**2</span>)
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<span style="color: #008000; font-weight: bold">return</span> term1 <span style="color: #666666">+</span> term2 <span style="color: #666666">+</span> term3 <span style="color: #666666">+</span> term4
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">create_X</span>(x, y, n ):
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<span style="color: #008000; font-weight: bold">if</span> <span style="color: #008000">len</span>(x<span style="color: #666666">.</span>shape) <span style="color: #666666">></span> <span style="color: #666666">1</span>:
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(x)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(y)
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N <span style="color: #666666">=</span> <span style="color: #008000">len</span>(x)
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l <span style="color: #666666">=</span> <span style="color: #008000">int</span>((n<span style="color: #666666">+1</span>)<span style="color: #666666">*</span>(n<span style="color: #666666">+2</span>)<span style="color: #666666">/2</span>) <span style="color: #408080; font-style: italic"># Number of elements in beta</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones((N,l))
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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>(<span style="color: #666666">1</span>,n<span style="color: #666666">+1</span>):
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q <span style="color: #666666">=</span> <span style="color: #008000">int</span>((i)<span style="color: #666666">*</span>(i<span style="color: #666666">+1</span>)<span style="color: #666666">/2</span>)
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<span style="color: #008000; font-weight: bold">for</span> k <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(i<span style="color: #666666">+1</span>):
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X[:,q<span style="color: #666666">+</span>k] <span style="color: #666666">=</span> (x<span style="color: #666666">**</span>(i<span style="color: #666666">-</span>k))<span style="color: #666666">*</span>(y<span style="color: #666666">**</span>k)
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<span style="color: #008000; font-weight: bold">return</span> X
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<span style="color: #408080; font-style: italic"># Making meshgrid of datapoints and compute Franke's function</span>
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n <span style="color: #666666">=</span> <span style="color: #666666">4</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>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
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z <span style="color: #666666">=</span> FrankeFunction(x, y)
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X <span style="color: #666666">=</span> create_X(x, y, n<span style="color: #666666">=</span>n)
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Xpd <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
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<span style="color: #408080; font-style: italic"># subtract the mean values and set up the covariance matrix</span>
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Xpd <span style="color: #666666">=</span> Xpd <span style="color: #666666">-</span> Xpd<span style="color: #666666">.</span>mean()
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covariance_matrix <span style="color: #666666">=</span> Xpd<span style="color: #666666">.</span>cov()
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<span style="color: #008000">print</span>(covariance_matrix)
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</pre>
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<p>We note here that the covariance is zero for the first rows and
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columns since all matrix elements in the design matrix were set to one
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(we are fitting the function in terms of a polynomial of degree \( n \)).
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</p>
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<p>This means that the variance for these elements will be zero and will
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cause problems when we set up the correlation matrix. We can simply
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drop these elements and construct a correlation
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matrix without these elements.
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</p>
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