Morten Hjorth-Jensen
426 lines
29 KiB
HTML
426 lines
29 KiB
HTML
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week37-bs001.html#plans-for-week-37" style="font-size: 80%;">Plans for week 37</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs002.html#material-from-last-week-and-relevant-for-the-weekly-exercises" style="font-size: 80%;">Material from last week and relevant for the weekly exercises</a></li>
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<!-- navigation toc: --> <li><a href="._week37-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="._week37-bs004.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week37-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="._week37-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="._week37-bs007.html#material-for-lecture-thursday-september-14" style="font-size: 80%;">Material for lecture Thursday September 14</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.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="._week37-bs009.html#independent-and-identically-distrubuted-iid" style="font-size: 80%;">Independent and Identically Distrubuted (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs010.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs011.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs012.html#more-basic-statistics-and-bayes-theorem" style="font-size: 80%;">More basic Statistics and Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs014.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs015.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs016.html#interpretations-of-bayes-theorem" style="font-size: 80%;">Interpretations of Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#example-of-usage-of-bayes-theorem" style="font-size: 80%;">Example of Usage of Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#bayes-theorem-and-ridge-and-lasso-regression" style="font-size: 80%;">Bayes' Theorem and Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs022.html#test-function-for-what-happens-with-ols-ridge-and-lasso" style="font-size: 80%;">Test Function for what happens with OLS, Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs023.html#rerunning-the-above-code" style="font-size: 80%;">Rerunning the above code</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs027.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs029.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.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="._week37-bs027.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs028.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs029.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs031.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs032.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs033.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs034.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs035.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs036.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs037.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="._week37-bs038.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs039.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="._week37-bs040.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs041.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs042.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="._week37-bs043.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs044.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs045.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="._week37-bs046.html#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="._week37-bs047.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs048.html#summing-up" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="#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="._week37-bs050.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="._week37-bs051.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs052.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="._week37-bs053.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="._week37-bs054.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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</ul>
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</li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="another-example-from-scikit-learn-s-repository" class="anchor">Another Example from Scikit-Learn's Repository </h2>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">"""</span>
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<span style="color: #BA2121; font-style: italic">============================</span>
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<span style="color: #BA2121; font-style: italic">Underfitting vs. Overfitting</span>
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<span style="color: #BA2121; font-style: italic">============================</span>
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<span style="color: #BA2121; font-style: italic">This example demonstrates the problems of underfitting and overfitting and</span>
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<span style="color: #BA2121; font-style: italic">how we can use linear regression with polynomial features to approximate</span>
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<span style="color: #BA2121; font-style: italic">nonlinear functions. The plot shows the function that we want to approximate,</span>
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<span style="color: #BA2121; font-style: italic">which is a part of the cosine function. In addition, the samples from the</span>
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<span style="color: #BA2121; font-style: italic">real function and the approximations of different models are displayed. The</span>
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<span style="color: #BA2121; font-style: italic">models have polynomial features of different degrees. We can see that a</span>
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<span style="color: #BA2121; font-style: italic">linear function (polynomial with degree 1) is not sufficient to fit the</span>
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<span style="color: #BA2121; font-style: italic">training samples. This is called **underfitting**. A polynomial of degree 4</span>
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<span style="color: #BA2121; font-style: italic">approximates the true function almost perfectly. However, for higher degrees</span>
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<span style="color: #BA2121; font-style: italic">the model will **overfit** the training data, i.e. it learns the noise of the</span>
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<span style="color: #BA2121; font-style: italic">training data.</span>
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<span style="color: #BA2121; font-style: italic">We evaluate quantitatively **overfitting** / **underfitting** by using</span>
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<span style="color: #BA2121; font-style: italic">cross-validation. We calculate the mean squared error (MSE) on the validation</span>
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<span style="color: #BA2121; font-style: italic">set, the higher, the less likely the model generalizes correctly from the</span>
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<span style="color: #BA2121; font-style: italic">training data.</span>
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<span style="color: #BA2121; font-style: italic">"""</span>
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<span style="color: #008000">print</span>(<span style="color: #19177C">__doc__</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">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">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
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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.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
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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> cross_val_score
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">true_fun</span>(X):
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>pi <span style="color: #666666">*</span> X)
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
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n_samples <span style="color: #666666">=</span> <span style="color: #666666">30</span>
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degrees <span style="color: #666666">=</span> [<span style="color: #666666">1</span>, <span style="color: #666666">4</span>, <span style="color: #666666">15</span>]
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n_samples))
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y <span style="color: #666666">=</span> true_fun(X) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n_samples) <span style="color: #666666">*</span> <span style="color: #666666">0.1</span>
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plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">14</span>, <span style="color: #666666">5</span>))
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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>(<span style="color: #008000">len</span>(degrees)):
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ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(degrees), i <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
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plt<span style="color: #666666">.</span>setp(ax, xticks<span style="color: #666666">=</span>(), yticks<span style="color: #666666">=</span>())
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polynomial_features <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=</span>degrees[i],
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include_bias<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
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linear_regression <span style="color: #666666">=</span> LinearRegression()
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pipeline <span style="color: #666666">=</span> Pipeline([(<span style="color: #BA2121">"polynomial_features"</span>, polynomial_features),
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(<span style="color: #BA2121">"linear_regression"</span>, linear_regression)])
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pipeline<span style="color: #666666">.</span>fit(X[:, np<span style="color: #666666">.</span>newaxis], y)
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<span style="color: #408080; font-style: italic"># Evaluate the models using crossvalidation</span>
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scores <span style="color: #666666">=</span> cross_val_score(pipeline, X[:, np<span style="color: #666666">.</span>newaxis], y,
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scoring<span style="color: #666666">=</span><span style="color: #BA2121">"neg_mean_squared_error"</span>, cv<span style="color: #666666">=10</span>)
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X_test <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">100</span>)
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plt<span style="color: #666666">.</span>plot(X_test, pipeline<span style="color: #666666">.</span>predict(X_test[:, np<span style="color: #666666">.</span>newaxis]), label<span style="color: #666666">=</span><span style="color: #BA2121">"Model"</span>)
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plt<span style="color: #666666">.</span>plot(X_test, true_fun(X_test), label<span style="color: #666666">=</span><span style="color: #BA2121">"True function"</span>)
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plt<span style="color: #666666">.</span>scatter(X, y, edgecolor<span style="color: #666666">=</span><span style="color: #BA2121">'b'</span>, s<span style="color: #666666">=20</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Samples"</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"x"</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"y"</span>)
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plt<span style="color: #666666">.</span>xlim((<span style="color: #666666">0</span>, <span style="color: #666666">1</span>))
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plt<span style="color: #666666">.</span>ylim((<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>))
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plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"best"</span>)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"Degree </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">MSE = </span><span style="color: #BB6688; font-weight: bold">{:.2e}</span><span style="color: #BA2121">(+/- </span><span style="color: #BB6688; font-weight: bold">{:.2e}</span><span style="color: #BA2121">)"</span><span style="color: #666666">.</span>format(
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degrees[i], <span style="color: #666666">-</span>scores<span style="color: #666666">.</span>mean(), scores<span style="color: #666666">.</span>std()))
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plt<span style="color: #666666">.</span>show()
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</pre>
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