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@@ -281,116 +282,117 @@ MathJax.Hub.Config({
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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%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</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%;">Optimizing our parameters</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</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%;">Some useful matrix and vector expressions</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</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%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">The code</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Codes for the SVD</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">A better understanding of regularization</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Spectral Decomposition of the OLS</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Jackknife code example</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Various steps in cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Cross-validation in brief</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</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%;">Example code for Bias-Variance tradeoff</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">The same example but now with cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Cross-validation with Ridge</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">The Ising model</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Linear regression</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">Singular Value decomposition</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">The one-dimensional Ising model</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Ridge regression</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">LASSO regression</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="#___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>
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more variance</a></li>
<!-- 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%;">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="._Regression-bs111.html#___sec110" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
</ul>
</li>
@@ -406,105 +408,27 @@ MathJax.Hub.Config({
<a name="part0026"></a>
<!-- !split -->
<h2 id="___sec25" class="anchor">The code </h2>
<h2 id="___sec25" class="anchor">Fitting an Equation of State for Dense Nuclear Matter </h2>
<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: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<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>
<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>
<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>
<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>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
<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, mean_absolute_error
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, format<span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">&quot;EoS.csv&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</span>)
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">&#39;Density&#39;</span>, <span style="color: #BA2121">&#39;Energy&#39;</span>))
EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</span>)
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>]
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Density&#39;</span>]
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),<span style="color: #666666">4</span>))
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">4.0/3.0</span>)
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> Density
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
<span style="color: #408080; font-style: italic"># We use now Scikit-Learn&#39;s linear regressor and ridge regressor</span>
<span style="color: #408080; font-style: italic"># OLS part</span>
clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X, Energies)
ytilde <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)
EoS[<span style="color: #BA2121">&#39;Eols&#39;</span>] <span style="color: #666666">=</span> ytilde
<span style="color: #408080; font-style: italic"># The mean squared error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> mean_squared_error(Energies, ytilde))
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> r2_score(Energies, ytilde))
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, ytilde))
<span style="color: #008000; font-weight: bold">print</span>(clf<span style="color: #666666">.</span>coef_, clf<span style="color: #666666">.</span>intercept_)
<span style="color: #408080; font-style: italic"># The Ridge regression with a hyperparameter lambda = 0.1</span>
_lambda <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
clf_ridge <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X, Energies)
yridge <span style="color: #666666">=</span> clf_ridge<span style="color: #666666">.</span>predict(X)
EoS[<span style="color: #BA2121">&#39;Eridge&#39;</span>] <span style="color: #666666">=</span> yridge
<span style="color: #408080; font-style: italic"># The mean squared error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> mean_squared_error(Energies, yridge))
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> r2_score(Energies, yridge))
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, yridge))
<span style="color: #008000; font-weight: bold">print</span>(clf_ridge<span style="color: #666666">.</span>coef_, clf_ridge<span style="color: #666666">.</span>intercept_)
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r&#39;$\rho[\mathrm{fm}^{-3}]$&#39;</span>)
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r&#39;Energy per particle&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Theoretical data&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Eols&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">&#39;m&#39;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;OLS&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Eridge&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">&#39;g&#39;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Ridge $\lambda = 0.1$&#39;</span>)
ax<span style="color: #666666">.</span>legend()
save_fig(<span style="color: #BA2121">&quot;EoSfitting&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
The above simple polynomial in density \( \rho \) gives an excellent fit
to the data.
Before we continue, let us introduce yet another example. We are going to fit the
nuclear equation of state using results from many-body calculations.
The equation of state we have made available here, as function of
density, has been derived using modern nucleon-nucleon potentials with
<a href="https://www.sciencedirect.com/science/article/pii/S0370157399001106" target="_self">the addition of three-body
forces</a>. This
time the file is presented as a standard <b>csv</b> file.
<p>
We note also that there is a small deviation between the
standard OLS and the Ridge regression at higher densities. We discuss this in more detail
below.
The beginning of the Python code here is similar to what you have seen
before, with the same initializations and declarations. We use also
<b>pandas</b> again, rather extensively in order to organize our data.
<p>
The difference now is that we use <b>Scikit-Learn's</b> regression tools
instead of our own matrix inversion implementation. Furthermore, we
sneak in <b>Ridge</b> regression (to be discussed below) which includes a
hyperparameter \( \lambda \), also to be explained below.
<p>
<p>
@@ -532,7 +456,7 @@ below.
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