update
This commit is contained in:
mhjensen
@@ -6,6 +6,7 @@ Automatically generated HTML file from DocOnce source
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<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
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|
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
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|
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|
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||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
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|
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|
||||
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|
||||
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|
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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end of tocinfo -->
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|
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<body>
|
||||
@@ -286,119 +282,117 @@ MathJax.Hub.Config({
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<li class="dropdown">
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||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs001.html#___sec0" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, follows</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">Simple regression model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">Simple regression model, now using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">Simple linear regression model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">Less noise</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">How to study our fits</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Minimizing the cost function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">Relative error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">The richness of <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">Functions in <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Other functions in <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The mean absolute error and other functions in <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Cubic polynomial in <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Polynomial Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">From standard regression to Ridge regressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Fitting vs. predicting when data is in the model class</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Fitting versus predicting when data is not in the model class</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">An example code without the model assessment part</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Generating test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">How can we effectively evaluate the various models?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Code examples for Ridge and Lasso Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">A second-order polynomial with Ridge and Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Log-likelihood</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Predicted Residual Error Sum of Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife estimator</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Blocking</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Blocking Transformations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Blocking Transformations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Blocking Transformations, getting there</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Blocking Transformations, final expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">"Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Training and testing data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Procedure to find a predictor</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">What we want</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">The expected generalization error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Elaborating a little bit more</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">The bias</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">The variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">The one-dimensional Ising model, project 2</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs001.html#___sec0" style="font-size: 80%;">To do list</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Spectral Decomposition of the OLS</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, independent variables</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more variance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Ordinary least squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Fitting with scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec110" style="font-size: 80%;">Performance of the different models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec110" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -414,38 +408,57 @@ MathJax.Hub.Config({
|
||||
<a name="part0111"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec110" class="anchor">Performance of the different models </h2>
|
||||
<h2 id="___sec110" class="anchor">Finding the optimal value of \( \lambda \) </h2>
|
||||
|
||||
<p>
|
||||
In order to judge which model performs best at varying values of \( \lambda \) (for ridge and LASSO) we compute \( R^2 \) which is given by
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
R^2 = 1 - \frac{(y - \hat{y})^2}{(y - \bar{y})^2},
|
||||
\tag{45}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
where \( y \) is a vector with the true values of the energy, \( \hat{y} \) is the predicted values of \( y \) from the models and \( \bar{y} \) is the mean of \( \hat{y} \).
|
||||
To determine which value of \( \lambda \) is best we plot the accuracy of
|
||||
the models when predicting the training and the testing set. We expect
|
||||
the accuracy of the training set to be quite good, but if the accuracy
|
||||
of the testing set is much lower this tells us that we might be
|
||||
subject to an overfit model. The ideal scenario is an accuracy on the
|
||||
testing set that is close to the accuracy of the training set.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">r_squared</span>(y, y_hat):
|
||||
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y <span style="color: #666666">-</span> y_hat) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_hat)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
|
||||
|
||||
colors <span style="color: #666666">=</span> {
|
||||
<span style="color: #BA2121">"ols_sk"</span>: <span style="color: #BA2121">"r"</span>,
|
||||
<span style="color: #BA2121">"ridge_sk"</span>: <span style="color: #BA2121">"y"</span>,
|
||||
<span style="color: #BA2121">"lasso_sk"</span>: <span style="color: #BA2121">"c"</span>
|
||||
}
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> train_errors:
|
||||
plt<span style="color: #666666">.</span>semilogx(
|
||||
lambdas,
|
||||
train_errors[key],
|
||||
colors[key],
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">"Train </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(key),
|
||||
linewidth<span style="color: #666666">=4.0</span>
|
||||
)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> test_errors:
|
||||
plt<span style="color: #666666">.</span>semilogx(
|
||||
lambdas,
|
||||
test_errors[key],
|
||||
colors[key] <span style="color: #666666">+</span> <span style="color: #BA2121">"--"</span>,
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">"Test </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(key),
|
||||
linewidth<span style="color: #666666">=4.0</span>
|
||||
)
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"best"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$\lambda$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$R^2$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>tick_params(labelsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
This is the same metric used by Scikit-learn for their regression models when scoring.
|
||||
From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
|
||||
achieves a very good accuracy on the test set. This by far surpasses the
|
||||
other models for all values of \( \lambda \).
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>y_hat <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X_test)
|
||||
r_test <span style="color: #666666">=</span> r_squared(y_test, y_hat)
|
||||
sk_r_test <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>score(X_test, y_test)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">assert</span> <span style="color: #008000">abs</span>(r_test <span style="color: #666666">-</span> sk_r_test) <span style="color: #666666"><</span> <span style="color: #666666">1e-2</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -461,9 +474,6 @@ sk_r_test <span style="color: #666666">=</span> clf<span style="color: #666666">
|
||||
<li><a href="._Regression-bs109.html">110</a></li>
|
||||
<li><a href="._Regression-bs110.html">111</a></li>
|
||||
<li class="active"><a href="._Regression-bs111.html">112</a></li>
|
||||
<li><a href="._Regression-bs112.html">113</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs112.html">»</a></li>
|
||||
</ul>
|
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|
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|
||||
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