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corrected typo

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Morten Hjorth-Jensen
2024-08-27 05:01:51 +02:00
parent 507c20218a
commit 265ed01ede
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doc/LectureNotes/_build/html/exercisesweek35.html
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@@ -414,11 +414,12 @@ analytical expressions is extremely helpful in case we have simpler
derivatives as well as when we analyze various properties (like second
derivatives) of the chosen cost functions. Vectors are always written
as boldfaced lower case letters and matrices as upper case boldfaced
letters. You will find useful the notes from week 35 on derivatives of vectors and matrices.</p>
letters. You will find useful the notes from week 35 on derivatives of vectors and matrices.
See also the textbook of Faisal at al, chapter 5 and in particular sections 5.3-5.5 at <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/MathMLbook.pdf">https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/MathMLbook.pdf</a></p>
<p>Show that</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b},
\frac{\partial (\boldsymbol{a}^T\boldsymbol{x})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T,
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
@@ -432,8 +433,8 @@ letters. You will find useful the notes from week 35 on derivatives of vectors a
\]</div>
<p>and finally find the second derivative of this function with respect to the vector <span class="math notranslate nohighlight">\(\boldsymbol{s}\)</span>. If we replace the vector <span class="math notranslate nohighlight">\(\boldsymbol{s}\)</span> with the unknown parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> used to define the ordinary least squares method, we end up with the equations that determine these parameters. The matrix <span class="math notranslate nohighlight">\(\boldsymbol{A}\)</span> is then the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> here has to be replaced with the outputs <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>.</p>
<p>The second derivative of the mean squared error is then proportional to the so-called Hessian matrix <span class="math notranslate nohighlight">\(\boldsymbol{H}=\boldsymbol{X}^T\boldsymbol{X}\)</span>.</p>
<p><strong>Hint</strong>: In these exercises it is always useful to write out with summation indices the various quantities.
As an example, consider the function</p>
<p><strong>Hint</strong>: In these exercises it is always useful to write out with summation indices the various quantities. Take also a look at the weekly slides from week 35 and the various examples included there.</p>
<p>As an example, consider the function</p>
<div class="math notranslate nohighlight">
\[
f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x},