Morten Hjorth-Jensen
396 lines
21 KiB
HTML
396 lines
21 KiB
HTML
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<a class="navbar-brand" href="week37-bs.html">Week 37: Summary of Ridge and Lasso Regression and Resampling Methods</a>
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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="._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#thursday-september-16-summary-of-ridge-and-lasso-regression-and-start-resampling-methods" style="font-size: 80%;">Thursday September 16, Summary of Ridge and Lasso Regression and start Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs003.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-bs004.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-bs005.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-bs006.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-bs007.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.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-bs009.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-bs010.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-bs011.html#invoking-bayes-theorem" style="font-size: 80%;">Invoking Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs012.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs016.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-bs017.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#resampling-methods-jackknife-and-bootstrap" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs024.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-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.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-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs030.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-bs031.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-bs032.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-bs033.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-bs034.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-bs035.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-bs036.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
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<!-- navigation toc: --> <li><a href="#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs038.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-bs039.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-bs040.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs041.html#summing-up" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs042.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="._week37-bs043.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-bs044.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" 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="._week37-bs045.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-bs046.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-bs047.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-bs048.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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</ul>
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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="part0037"></a>
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<!-- !split -->
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<h2 id="the-bias-variance-tradeoff" class="anchor">The bias-variance tradeoff </h2>
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<p>
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We will discuss the bias-variance tradeoff in the context of
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continuous predictions such as regression. However, many of the
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intuitions and ideas discussed here also carry over to classification
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tasks. Consider a dataset \( \mathcal{D} \) consisting of the data
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\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
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<p>
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Let us assume that the true data is generated from a noisy model
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$$
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\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}
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$$
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<p>
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where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \).
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<p>
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In our derivation of the ordinary least squares method we defined then
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an approximation to the function \( f \) in terms of the parameters
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\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model,
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that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \).
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<p>
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Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function
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$$
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C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
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$$
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<p>
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We can rewrite this as
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$$
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
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$$
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<p>
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The three terms represent the square of the bias of the learning
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method, which can be thought of as the error caused by the simplifying
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assumptions built into the method. The second term represents the
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variance of the chosen model and finally the last terms is variance of
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the error \( \boldsymbol{\epsilon} \).
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<p>
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To derive this equation, we need to recall that the variance of \( \boldsymbol{y} \) and \( \boldsymbol{\epsilon} \) are both equal to \( \sigma^2 \). The mean value of \( \boldsymbol{\epsilon} \) is by definition equal to zero. Furthermore, the function \( f \) is not a stochastics variable, idem for \( \boldsymbol{\tilde{y}} \).
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We use a more compact notation in terms of the expectation value
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$$
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right],
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$$
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and adding and subtracting \( \mathbb{E}\left[\boldsymbol{\tilde{y}}\right] \) we get
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$$
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right],
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$$
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which, using the abovementioned expectation values can be rewritten as
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$$
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\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2,
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$$
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that is the rewriting in terms of the so-called bias, the variance of the model \( \boldsymbol{\tilde{y}} \) and the variance of \( \boldsymbol{\epsilon} \).
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<p>
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<p>
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