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Morten Hjorth-Jensen
2023-09-19 11:52:10 +02:00
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@@ -451,8 +451,8 @@ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_
\]</div>
<p>Here the expected value <span class="math notranslate nohighlight">\(\mathbb{E}\)</span> is the sample value.</p>
<p>Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a
term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.</p>
<p>That is, show that</p>
term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.
That is, show that</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2,
@@ -465,10 +465,8 @@ term which measures the deviation from the true data and the mean value of the m
<p>and</p>
<div class="math notranslate nohighlight">
\[
\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
\]</div>
<p>The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.
Explain what the terms mean and discuss their interpretations.</p>
<p>Explain what the terms mean and discuss their interpretations.</p>
<p>Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by
studying the MSE value as function of the complexity of your model. Use ordinary least squares only.</p>