mhjensen
294 lines
19 KiB
HTML
294 lines
19 KiB
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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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2,
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None,
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'___sec24'),
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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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2,
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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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<!-- 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="._week36-bs022.html#___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="#___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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<!-- !split -->
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<h2 id="___sec27" class="anchor">Understanding what happens </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></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>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
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n <span style="color: #666666">=</span> <span style="color: #666666">40</span>
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n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
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<span style="color: #408080; font-style: italic"># Make data set.</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
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error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
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<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
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model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000">False</span>))
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y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
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x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
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y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
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polydegree[degree] <span style="color: #666666">=</span> degree
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error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
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bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>))<span style="color: #666666">**2</span> )
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variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Polynomial degree:'</span>, degree)
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Error:'</span>, error[degree])
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias[degree])
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Var:'</span>, variance[degree])
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'{} >= {} + {} = {}'</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
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plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">'Error'</span>)
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plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">'bias'</span>)
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plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">'Variance'</span>)
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plt<span style="color: #666666">.</span>legend()
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<center style="font-size:80%">
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<!-- copyright only on the titlepage -->
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</center>
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</body>
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</html>
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