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Morten Hjorth-Jensen
2025-09-01 14:00:15 +02:00
parent eb31c10d1c
commit 55beb4d9b8
8 changed files with 143 additions and 101 deletions
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doc/LectureNotes/_build/html/exercisesweek37.html
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@@ -427,7 +427,7 @@ doconce format html exercisesweek37.do.txt -->
<p>We start with a very simple function</p>
<div class="math notranslate nohighlight">
\[
\f(x)= 2-x+5x^2,
f(x)= 2-x+5x^2,
\]</div>
<p>defined for <span class="math notranslate nohighlight">\(x\in [-2,2]\)</span>. You can add noise if you wish.</p>
<p>We are going to fit this function with a polynomial ansatz. The easiest thing is to set up a second-order polynomial and see if you can fit the above function.
@@ -496,10 +496,10 @@ print(&quot;Closed-form OLS coefficients:&quot;, theta_closed_form)
</div>
</div>
<p>This computes the Ridge and OLS regression coefficients directly. The identity
matrix <span class="math notranslate nohighlight">\(I\)</span> has the same size as <span class="math notranslate nohighlight">\(X^T X\)</span>. It adds <span class="math notranslate nohighlight">\(\lambda\)</span> to the diagonal of <span class="math notranslate nohighlight">\(X^T X for Ridge regression. We
then invert this matrix and multiply by \)</span>X^T y<span class="math notranslate nohighlight">\(. The result
for \)</span>\boldsymbol{\theta}<span class="math notranslate nohighlight">\( is a NumPy array of shape (n\)</span>_<span class="math notranslate nohighlight">\(features,) containing the
fitted parameters \)</span>\boldsymbol{\theta}$..</p>
matrix <span class="math notranslate nohighlight">\(I\)</span> has the same size as <span class="math notranslate nohighlight">\(X^T X\)</span>. It adds <span class="math notranslate nohighlight">\(\lambda\)</span> to the diagonal of <span class="math notranslate nohighlight">\(X^T X\)</span> for Ridge regression. We
then invert this matrix and multiply by <span class="math notranslate nohighlight">\(X^T y\)</span>. The result
for <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> is a NumPy array of shape (n<span class="math notranslate nohighlight">\(\_\)</span>features,) containing the
fitted parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span>.</p>
<section id="id1">
<h3>3a)<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
<p>Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span>.</p>