mhjensen
266 lines
13 KiB
HTML
266 lines
13 KiB
HTML
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<title>Week 36: Resampling techniques and Ordinary Least Square</title>
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'sections': [('Plans for week 36', 2, None, '___sec0'),
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('Thursday September 3', 2, None, '___sec1'),
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('Why resampling methods', 2, None, '___sec2'),
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('Resampling methods', 2, None, '___sec3'),
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('Resampling approaches can be computationally expensive',
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('Why resampling methods ?', 2, None, '___sec5'),
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('Statistical analysis', 2, None, '___sec6'),
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('Expectation value and variance', 2, None, '___sec9'),
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('Expectation value and variance for $\\boldsymbol{\\beta}$',
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('Resampling methods', 2, None, '___sec11'),
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('Resampling methods: Jackknife and Bootstrap',
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('Resampling methods: Jackknife', 2, None, '___sec13'),
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('Resampling methods: Bootstrap', 2, None, '___sec15'),
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('Resampling methods: More Bootstrap background',
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('Resampling methods: Bootstrap approach', 2, None, '___sec18'),
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('Resampling methods: Bootstrap steps', 2, None, '___sec19'),
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('Code example for the Bootstrap method', 2, None, '___sec20'),
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('Various steps in cross-validation', 2, None, '___sec21'),
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('How to set up the cross-validation for Ridge and/or Lasso',
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('Cross-validation in brief', 2, None, '___sec23'),
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('The bias-variance tradeoff', 2, None, '___sec25'),
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('Example code for Bias-Variance tradeoff', 2, None, '___sec26'),
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('Understanding what happens', 2, None, '___sec27'),
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("Another Example from Scikit-Learn's Repository",
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<a class="navbar-brand" href="week36-bs.html">Week 36: Resampling techniques and Ordinary Least Square</a>
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<ul class="nav navbar-nav navbar-right">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week36-bs001.html#___sec0" style="font-size: 80%;">Plans for week 36</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs002.html#___sec1" style="font-size: 80%;">Thursday September 3</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs003.html#___sec2" style="font-size: 80%;">Why resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.html#___sec3" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs005.html#___sec4" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs006.html#___sec5" style="font-size: 80%;">Why resampling methods ?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#___sec6" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs008.html#___sec7" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs009.html#___sec8" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs010.html#___sec9" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs011.html#___sec10" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#___sec11" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#___sec12" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#___sec13" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.html#___sec14" style="font-size: 80%;">Jackknife code example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.html#___sec15" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.html#___sec16" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#___sec17" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#___sec18" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#___sec19" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#___sec20" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs023.html#___sec22" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#___sec23" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#___sec24" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#___sec25" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.html#___sec26" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.html#___sec27" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.html#___sec28" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs030.html#___sec29" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#___sec30" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#___sec31" style="font-size: 80%;">The same example but now with cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.html#___sec32" style="font-size: 80%;">Cross-validation with Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#___sec33" style="font-size: 80%;">Friday September 4</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0022"></a>
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<!-- !split -->
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<h2 id="___sec21" class="anchor">Various steps in cross-validation </h2>
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<p>
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When the repetitive splitting of the data set is done randomly,
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samples may accidently end up in a fast majority of the splits in
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either training or test set. Such samples may have an unbalanced
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influence on either model building or prediction evaluation. To avoid
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this \( k \)-fold cross-validation structures the data splitting. The
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samples are divided into \( k \) more or less equally sized exhaustive and
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mutually exclusive subsets. In turn (at each split) one of these
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subsets plays the role of the test set while the union of the
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remaining subsets constitutes the training set. Such a splitting
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warrants a balanced representation of each sample in both training and
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test set over the splits. Still the division into the \( k \) subsets
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involves a degree of randomness. This may be fully excluded when
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choosing \( k=n \). This particular case is referred to as leave-one-out
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cross-validation (LOOCV).
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<p>
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<p>
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