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Morten Hjorth-Jensen
2023-09-04 16:27:23 +02:00
parent 3aec5cbb80
commit 14e3712190
8 changed files with 113 additions and 101 deletions
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doc/LectureNotes/_build/.doctrees/environment.pickle
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"### b) The singular value decomposition\n",
"\n",
"Use the singular value decomposition of an n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)"
"Use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)"
]
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"where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n",
"$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n",
"$n\\times p$ matrix which contains the ingular values only. This material was discussed during the lectures of week 35.\n",
"$n\\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35.\n",
"\n",
"Show that you can write the \n",
"OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix $\\boldsymbol{U}$ as"
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doc/LectureNotes/_build/html/exercisesweek36.html
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@@ -279,6 +279,18 @@ const thebe_selector_output = ".output, .cell_output"
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Projects
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="project1.html">
Project 1 on Machine Learning, deadline October 9 (midnight), 2023
</a>
</li>
</ul>
</div>
</nav> <!-- To handle the deprecated key -->
@@ -504,14 +516,14 @@ 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 n\times p<span class="math notranslate nohighlight">\( matrix \)</span>\boldsymbol{X}$ (our design matrix)</p>
<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>
<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
<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 ingular values only. This material was discussed during the lectures of week 35.</p>
<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
OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span> as</p>
<div class="math notranslate nohighlight">
doc/LectureNotes/_build/html/searchindex.js
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doc/LectureNotes/_build/jupyter_execute/exercisesweek36.ipynb
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"source": [
"### b) The singular value decomposition\n",
"\n",
"Use the singular value decomposition of an n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)"
"Use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)"
]
},
{
"cell_type": "markdown",
"id": "8761ed23",
"id": "40651a93",
"metadata": {
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@@ -253,14 +253,14 @@
},
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"where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n",
"$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n",
"$n\\times p$ matrix which contains the ingular values only. This material was discussed during the lectures of week 35.\n",
"$n\\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35.\n",
"\n",
"Show that you can write the \n",
"OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix $\\boldsymbol{U}$ as"
@@ -268,7 +268,7 @@
},
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doc/LectureNotes/exercisesweek36.ipynb
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"source": [
"### b) The singular value decomposition\n",
"\n",
"Use the singular value decomposition of an n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)"
"Use the singular value decomposition of an $n\\times p$ matrix $\\boldsymbol{X}$ (our design matrix)"
]
},
{
"cell_type": "markdown",
"id": "8761ed23",
"id": "40651a93",
"metadata": {
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@@ -253,14 +253,14 @@
},
{
"cell_type": "markdown",
"id": "92f8479e",
"id": "9c32138e",
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"where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal matrices of dimensions\n",
"$n\\times n$ and $p\\times p$, respectively, and $\\boldsymbol{\\Sigma}$ is an\n",
"$n\\times p$ matrix which contains the ingular values only. This material was discussed during the lectures of week 35.\n",
"$n\\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35.\n",
"\n",
"Show that you can write the \n",
"OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix $\\boldsymbol{U}$ as"
@@ -268,7 +268,7 @@
},
{
"cell_type": "markdown",
"id": "9df91bda",
"id": "b9051ac7",
"metadata": {
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@@ -280,7 +280,7 @@
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@@ -290,7 +290,7 @@
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doc/src/week36/exercisesweek36.do.txt
+2 -2
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@@ -85,7 +85,7 @@ The ordinary least squares result is
=== b) The singular value decomposition ===
Use the singular value decomposition of an n\times p$ matrix $\bm{X}$ (our design matrix)
Use the singular value decomposition of an $n\times p$ matrix $\bm{X}$ (our design matrix)
!bt
\[
\bm{X}=\bm{U}\bm{\Sigma}\bm{V}^T,
@@ -94,7 +94,7 @@ Use the singular value decomposition of an n\times p$ matrix $\bm{X}$ (our desig
where $\bm{U}$ and $\bm{V}$ are orthogonal matrices of dimensions
$n\times n$ and $p\times p$, respectively, and $\bm{\Sigma}$ is an
$n\times p$ matrix which contains the ingular values only. This material was discussed during the lectures of week 35.
$n\times p$ matrix which contains the singular values only. This material was discussed during the lectures of week 35.
Show that you can write the
OLS solutions in terms of the eigenvectors (the columns) of the orthogonal matrix $\bm{U}$ as