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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="._Regression-bs026.html#___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="#___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>
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<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
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<!-- 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="._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>
<!-- 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="#___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>
@@ -406,22 +408,32 @@ MathJax.Hub.Config({
<a name="part0064"></a>
<!-- !split -->
<h2 id="___sec63" class="anchor">Statistics, more on sample error </h2>
<h2 id="___sec63" class="anchor">Statistics, law of large numbers </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
Let us first take a look at what happens to the sample error as the
size of the sample grows. In a sample, each of the measurements \( x_i \)
can be associated with its own stochastic variable \( X_i \). The
stochastic variable \( \overline X_n \) for the sample mean \( \bar{x}_n \) is
then just a linear combination, already familiar to us:
The law of large numbers
states that as the size of our sample grows to infinity, the sample
mean approaches the true mean \( \mu_X^{\phantom X} \) of the chosen PDF:
$$
\overline X_n = \frac{1}{n}\sum_{i=1}^n X_i
\lim_{n\to\infty}\bar{x}_n = \mu_X^{\phantom X}
$$
All the coefficients are just equal \( 1/n \). The PDF of \( \overline X_n \),
denoted by \( p_{\overline X_n}(x) \) is the desired PDF of the sample
means.
The sample mean \( \bar{x}_n \) works therefore as an estimate of the true
mean \( \mu_X^{\phantom X} \).
<p>
What we need to find out is how good an approximation \( \bar{x}_n \) is to
\( \mu_X^{\phantom X} \). In any stochastic measurement, an estimated
mean is of no use to us without a measure of its error. A quantity
that tells us how well we can reproduce it in another experiment. We
are therefore interested in the PDF of the sample mean itself. Its
standard deviation will be a measure of the spread of sample means,
and we will simply call it the <em>error</em> of the sample mean, or
just sample error, and denote it by \( \mathrm{err}_X^{\phantom X} \). In
practice, we will only be able to produce an <em>estimate</em> of the
sample error since the exact value would require the knowledge of the
true PDFs behind, which we usually do not have.
</div>
</div>
@@ -452,7 +464,7 @@ means.
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