Morten Hjorth-Jensen
374 lines
26 KiB
HTML
374 lines
26 KiB
HTML
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<!-- navigation toc: --> <li><a href="._week38-bs001.html#plans-for-week-38-lecture-monday-september-15" style="font-size: 80%;">Plans for week 38, lecture Monday September 15</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs002.html#readings-and-videos" style="font-size: 80%;">Readings and Videos</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs003.html#linking-the-regression-analysis-with-a-statistical-interpretation" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs004.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs005.html#expectation-value-and-variance" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs006.html#expectation-value-and-variance-for-boldsymbol-beta" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs007.html#deriving-ols-from-a-probability-distribution" style="font-size: 80%;">Deriving OLS from a probability distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs008.html#independent-and-identically-distributed-iid" style="font-size: 80%;">Independent and Identically Distributed (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs009.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs010.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs014.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs016.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs013.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs014.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs015.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs016.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs017.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs018.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs019.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs020.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs021.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs023.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs024.html#standard-approach-based-on-the-normal-distribution" style="font-size: 80%;">Standard Approach based on the Normal Distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs025.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs026.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs028.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs030.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs031.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs032.html#a-way-to-read-the-bias-variance-tradeoff" style="font-size: 80%;">A way to Read the Bias-Variance Tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs034.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#summing-up" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#code-example-for-cross-validation-and-k-fold-cross-validation" 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="._week38-bs040.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs041.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs042.html#material-for-the-lab-sessions" style="font-size: 80%;">Material for the lab sessions</a></li>
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<!-- !split -->
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<h2 id="example-code-for-bias-variance-tradeoff" class="anchor">Example code for Bias-Variance tradeoff </h2>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<pre style="line-height: 125%;"><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">500</span>
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n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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degree <span style="color: #666666">=</span> <span style="color: #666666">18</span> <span style="color: #408080; font-style: italic"># A quite high value, just to show.</span>
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noise <span style="color: #666666">=</span> <span style="color: #666666">0.1</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">-1</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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<span style="color: #408080; font-style: italic"># Hold out some test data that is never used in training.</span>
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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: #408080; font-style: italic"># Combine x transformation and model into one operation.</span>
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<span style="color: #408080; font-style: italic"># Not neccesary, but convenient.</span>
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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; font-weight: bold">False</span>))
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<span style="color: #408080; font-style: italic"># The following (m x n_bootstraps) matrix holds the column vectors y_pred</span>
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<span style="color: #408080; font-style: italic"># for each bootstrap iteration.</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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<span style="color: #408080; font-style: italic"># Evaluate the new model on the same test data each time.</span>
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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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<span style="color: #408080; font-style: italic"># Note: Expectations and variances taken w.r.t. different training</span>
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<span style="color: #408080; font-style: italic"># data sets, hence the axis=1. Subsequent means are taken across the test data</span>
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<span style="color: #408080; font-style: italic"># set in order to obtain a total value, but before this we have error/bias/variance</span>
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<span style="color: #408080; font-style: italic"># calculated per data point in the test set.</span>
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<span style="color: #408080; font-style: italic"># Note 2: The use of keepdims=True is important in the calculation of bias as this </span>
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<span style="color: #408080; font-style: italic"># maintains the column vector form. Dropping this yields very unexpected results.</span>
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error <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; font-weight: bold">True</span>) )
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bias <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; font-weight: bold">True</span>))<span style="color: #666666">**2</span> )
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variance <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; font-weight: bold">True</span>) )
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Error:'</span>, error)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Var:'</span>, variance)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> >= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(error, bias, variance, bias<span style="color: #666666">+</span>variance))
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plt<span style="color: #666666">.</span>plot(x[::<span style="color: #666666">5</span>, :], y[::<span style="color: #666666">5</span>, :], label<span style="color: #666666">=</span><span style="color: #BA2121">'f(x)'</span>)
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plt<span style="color: #666666">.</span>scatter(x_test, y_test, label<span style="color: #666666">=</span><span style="color: #BA2121">'Data points'</span>)
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plt<span style="color: #666666">.</span>scatter(x_test, np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>), label<span style="color: #666666">=</span><span style="color: #BA2121">'Pred'</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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