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Morten Hjorth-Jensen
2023-09-05 05:57:31 +02:00
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@@ -445,7 +445,7 @@ const thebe_selector_output = ".output, .cell_output"
<div>
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<!-- dom:TITLE: Exercises week 36 --><div class="tex2jax_ignore mathjax_ignore section" id="exercises-week-36">
<h1>Exercises week 36<a class="headerlink" href="#exercises-week-36" title="Permalink to this headline"></a></h1>
<p><strong>September 4-8, 2023</strong></p>
@@ -516,12 +516,13 @@ a finite number larger than zero. We will not implement that here.</p>
</div>
<div class="section" id="b-the-singular-value-decomposition">
<h3>b) The singular value decomposition<a class="headerlink" href="#b-the-singular-value-decomposition" title="Permalink to this headline"></a></h3>
<p>Use the singular value decomposition of an <span class="math notranslate nohighlight">\(n\times p\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> (our design matrix)</p>
<p>Here we will use the singular value decomposition of an <span class="math notranslate nohighlight">\(n\times p\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> (our design matrix)</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
\]</div>
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{V}\)</span> are orthogonal matrices of dimensions
<p>to study properties of Ridge regression and ordinary least squares regression.
Here <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{V}\)</span> are orthogonal matrices of dimensions
<span class="math notranslate nohighlight">\(n\times n\)</span> and <span class="math notranslate nohighlight">\(p\times p\)</span>, respectively, and <span class="math notranslate nohighlight">\(\boldsymbol{\Sigma}\)</span> is an
<span class="math notranslate nohighlight">\(n\times p\)</span> matrix which contains the singular values only. This material was discussed during the lectures of week 35.</p>
<p>Show that you can write the