Morten Hjorth-Jensen
763 lines
43 KiB
HTML
763 lines
43 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>
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|
<!-- 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="#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>
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|
<!-- 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>
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|
<!-- 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>
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|
<!-- 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="._week35-bs049.html#example-matrix" style="font-size: 80%;"><b>Example Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs050.html#setting-up-the-matrix-to-be-inverted" style="font-size: 80%;"><b>Setting up the Matrix to be inverted</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs051.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-bs052.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-bs053.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-bs055.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-bs056.html#covariance-matrix-examples" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs057.html#correlation-matrix" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs058.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-bs059.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-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-bs061.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-bs062.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-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-bs064.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-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-bs067.html#more-interpretations" style="font-size: 80%;"><b>More interpretations</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#exercises-for-week-36-september-6-10" style="font-size: 80%;"><b>Exercises for week 36, September 6-10</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="housing-data-the-code" class="anchor">Housing data, the code </h2>
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<p>We start by importing the libraries</p>
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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">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">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
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<p>and load the Boston Housing DataSet from <b>Scikit-Learn</b></p>
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_boston
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boston_dataset <span style="color: #666666">=</span> load_boston()
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<span style="color: #408080; font-style: italic"># boston_dataset is a dictionary</span>
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boston_dataset<span style="color: #666666">.</span>keys()
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<p>Then we invoke Pandas</p>
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<pre style="line-height: 125%;">boston <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(boston_dataset<span style="color: #666666">.</span>data, columns<span style="color: #666666">=</span>boston_dataset<span style="color: #666666">.</span>feature_names)
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boston<span style="color: #666666">.</span>head()
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boston[<span style="color: #BA2121">'MEDV'</span>] <span style="color: #666666">=</span> boston_dataset<span style="color: #666666">.</span>target
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<p>and preprocess the data</p>
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># check for missing values in all the columns</span>
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boston<span style="color: #666666">.</span>isnull()<span style="color: #666666">.</span>sum()
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<p>We can then visualize the data</p>
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># set the size of the figure</span>
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sns<span style="color: #666666">.</span>set(rc<span style="color: #666666">=</span>{<span style="color: #BA2121">'figure.figsize'</span>:(<span style="color: #666666">11.7</span>,<span style="color: #666666">8.27</span>)})
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<span style="color: #408080; font-style: italic"># plot a histogram showing the distribution of the target values</span>
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sns<span style="color: #666666">.</span>distplot(boston[<span style="color: #BA2121">'MEDV'</span>], bins<span style="color: #666666">=30</span>)
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plt<span style="color: #666666">.</span>show()
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<p>It is now useful to look at the correlation matrix</p>
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># compute the pair wise correlation for all columns </span>
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correlation_matrix <span style="color: #666666">=</span> boston<span style="color: #666666">.</span>corr()<span style="color: #666666">.</span>round(<span style="color: #666666">2</span>)
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<span style="color: #408080; font-style: italic"># use the heatmap function from seaborn to plot the correlation matrix</span>
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<span style="color: #408080; font-style: italic"># annot = True to print the values inside the square</span>
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sns<span style="color: #666666">.</span>heatmap(data<span style="color: #666666">=</span>correlation_matrix, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
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<p>From the above coorelation plot we can see that <b>MEDV</b> is strongly correlated to <b>LSTAT</b> and <b>RM</b>. We see also that <b>RAD</b> and <b>TAX</b> are stronly correlated, but we don't include this in our features together to avoid multi-colinearity</p>
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<pre style="line-height: 125%;">plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">5</span>))
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features <span style="color: #666666">=</span> [<span style="color: #BA2121">'LSTAT'</span>, <span style="color: #BA2121">'RM'</span>]
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target <span style="color: #666666">=</span> boston[<span style="color: #BA2121">'MEDV'</span>]
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<span style="color: #008000; font-weight: bold">for</span> i, col <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(features):
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plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(features) , i<span style="color: #666666">+1</span>)
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x <span style="color: #666666">=</span> boston[col]
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y <span style="color: #666666">=</span> target
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plt<span style="color: #666666">.</span>scatter(x, y, marker<span style="color: #666666">=</span><span style="color: #BA2121">'o'</span>)
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plt<span style="color: #666666">.</span>title(col)
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plt<span style="color: #666666">.</span>xlabel(col)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MEDV'</span>)
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<p>Now we start training our model</p>
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<pre style="line-height: 125%;">X <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(np<span style="color: #666666">.</span>c_[boston[<span style="color: #BA2121">'LSTAT'</span>], boston[<span style="color: #BA2121">'RM'</span>]], columns <span style="color: #666666">=</span> [<span style="color: #BA2121">'LSTAT'</span>,<span style="color: #BA2121">'RM'</span>])
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Y <span style="color: #666666">=</span> boston[<span style="color: #BA2121">'MEDV'</span>]
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<p>We split the data into training and test sets</p>
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<pre style="line-height: 125%;"><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: #408080; font-style: italic"># splits the training and test data set in 80% : 20%</span>
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<span style="color: #408080; font-style: italic"># assign random_state to any value.This ensures consistency.</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">=</span> <span style="color: #666666">0.2</span>, random_state<span style="color: #666666">=5</span>)
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<span style="color: #008000">print</span>(X_train<span style="color: #666666">.</span>shape)
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<span style="color: #008000">print</span>(X_test<span style="color: #666666">.</span>shape)
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<span style="color: #008000">print</span>(Y_train<span style="color: #666666">.</span>shape)
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<span style="color: #008000">print</span>(Y_test<span style="color: #666666">.</span>shape)
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<p>Then we use the linear regression functionality from <b>Scikit-Learn</b></p>
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<pre style="line-height: 125%;"><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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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score
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lin_model <span style="color: #666666">=</span> LinearRegression()
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y_train_predict <span style="color: #666666">=</span> lin_model<span style="color: #666666">.</span>predict(X_train)
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rmse <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>sqrt(mean_squared_error(Y_train, y_train_predict)))
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r2 <span style="color: #666666">=</span> r2_score(Y_train, y_train_predict)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"The model performance for training set"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"--------------------------------------"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'RMSE is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(rmse))
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'R2 score is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(r2))
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">"</span>)
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<span style="color: #408080; font-style: italic"># model evaluation for testing set</span>
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y_test_predict <span style="color: #666666">=</span> lin_model<span style="color: #666666">.</span>predict(X_test)
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<span style="color: #408080; font-style: italic"># root mean square error of the model</span>
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rmse <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>sqrt(mean_squared_error(Y_test, y_test_predict)))
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<span style="color: #408080; font-style: italic"># r-squared score of the model</span>
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r2 <span style="color: #666666">=</span> r2_score(Y_test, y_test_predict)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"The model performance for testing set"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"--------------------------------------"</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'RMSE is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(rmse))
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'R2 score is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(r2))
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</pre>
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</div>
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</div>
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</div>
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</div>
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<div class="output_wrapper">
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<div class="output">
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<div class="output_area">
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<div class="output_subarea output_stream output_stdout output_text">
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</div>
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</div>
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</div>
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</div>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># plotting the y_test vs y_pred</span>
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<span style="color: #408080; font-style: italic"># ideally should have been a straight line</span>
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plt<span style="color: #666666">.</span>scatter(Y_test, y_test_predict)
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plt<span style="color: #666666">.</span>show()
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</pre>
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