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<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Visualising High-dimensional Data with R</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
Dianne Cook <br> Monash University
</div>
</div>
</div>
</section>
<section id="session-1" class="slide level2 transition center center-align">
<h2>Session 1</h2>
</section>
<section id="outline" class="slide level2">
<h2>Outline</h2>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<table>
<colgroup>
<col style="width: 9%">
<col style="width: 90%">
</colgroup>
<thead>
<tr class="header">
<th style="text-align: left;">time</th>
<th style="text-align: left;">topic</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">1:00-1:20</td>
<td style="text-align: left;">Introduction: What is high-dimensional data, why visualise and overview of methods</td>
</tr>
<tr class="even">
<td style="text-align: left;">1:20-1:45</td>
<td style="text-align: left;">Basics of linear projections, and recognising high-d structure</td>
</tr>
<tr class="odd">
<td style="text-align: left;">1:45-2:30</td>
<td style="text-align: left;">Effectively reducing your data dimension, in association with non-linear dimension reduction</td>
</tr>
<tr class="even">
<td style="text-align: left;">2:30-3:00</td>
<td style="text-align: left;">BREAK</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="introduction" class="slide level2 transition center center-align">
<h2>Introduction</h2>
</section>
<section id="what-is-high-dimensional-space" class="slide level2">
<h2>What is high-dimensional space?</h2>
<center>
<img src="https://dicook.github.io/mulgar_book/1-intro_files/figure-html/fig-dimension-cubes-1.png" width="90%">
</center>
<p>Increasing dimension adds an additional orthogonal axis.</p>
<div class="fragment f50">
<p>If you want more high-dimensional shapes there is an R package, <a href="http://schloerke.com/geozoo/all/">geozoo</a>, which will generate cubes, spheres, simplices, mobius strips, torii, boy surface, klein bottles, cones, various polytopes, …</p>
<p>And read or watch <a href="https://en.wikipedia.org/wiki/Flatland">Flatland: A Romance of Many Dimensions (1884) Edwin Abbott</a>.</p>
</div>
</section>
<section id="notation-data" class="slide level2">
<h2>Notation: Data</h2>
<p><span class="math display">\[\begin{eqnarray*}
X_{~n\times p} =
[X_{~1}~X_{~2}~\dots~X_{~p}]_{~n\times p} = \left[ \begin{array}{cccc}
x_{~11} & x_{~12} & \dots & x_{~1p} \\
x_{~21} & x_{~22} & \dots & x_{~2p}\\
\vdots & \vdots & & \vdots \\
x_{~n1} & x_{~n2} & \dots & x_{~np} \end{array} \right]_{~n\times p}
\end{eqnarray*}\]</span></p>
</section>
<section id="notation-projection" class="slide level2">
<h2>Notation: Projection</h2>
<p><span class="math display">\[\begin{eqnarray*}
A_{~p\times d} = \left[ \begin{array}{cccc}
a_{~11} & a_{~12} & \dots & a_{~1d} \\
a_{~21} & a_{~22} & \dots & a_{~2d}\\
\vdots & \vdots & & \vdots \\
a_{~p1} & a_{~p2} & \dots & a_{~pd} \end{array} \right]_{~p\times d}
\end{eqnarray*}\]</span></p>
</section>
<section id="notation-projected-data" class="slide level2">
<h2>Notation: Projected data</h2>
<p><span class="math display">\[\begin{eqnarray*}
Y_{~n\times d} = XA = \left[ \begin{array}{cccc}
y_{~11} & y_{~12} & \dots & y_{~1d} \\
y_{~21} & y_{~22} & \dots & y_{~2d}\\
\vdots & \vdots & & \vdots \\
y_{~n1} & y_{~n2} & \dots & y_{~nd} \end{array} \right]_{~n\times d}
\end{eqnarray*}\]</span></p>
</section>
<section id="why-12" class="slide level2">
<h2>Why? <span class="f70">(1/2)</span></h2>
<div class="columns">
<div class="column">
<p><br> Scatterplot matrix</p>
<p><br><br> Here, we see <span class="blue2">linear association</span>, <span class="blue2">clumping</span> and <span class="blue2">clustering</span>, potentially some <span class="blue2">outliers</span>.</p>
</div><div class="column">
<div class="cell" data-layout-align="center">
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</div>
</div>
</div>
</div>
</section>
<section id="why-22" class="slide level2">
<h2>Why? <span class="f70">(2/2)</span></h2>
<div class="columns">
<div class="column" style="width:50%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
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</div>
</div><div class="column" style="width:50%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
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<div style="font-size: 80%;">
<p>There is an outlier in the data on the right, like the one in the left, but it is <span class="orange2">hidden in a combination of variables</span>. It’s not visible in any pair of variables.</p>
</div>
</section>
<section id="and-help-to-see-the-data-as-a-whole" class="slide level2">
<h2>And help to see the data as a whole</h2>
<div class="columns">
<div class="column">
<p>To avoid misinterpretation …</p>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="images/elephant-guided.png" class="quarto-figure quarto-figure-center" width="500"></p>
</figure>
</div>
</div><div class="column">
<p>… see the bigger picture!</p>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="images/elephant-grand.png" class="quarto-figure quarto-figure-center" width="500"></p>
</figure>
</div>
</div>
</div>
<div class="f50">
<p>Image: <a href="https://sketchplanations.com/the-overview-effect">Sketchplanations</a>.</p>
</div>
</section>
<section id="tours-of-linear-projections" class="slide level2">
<h2>Tours of linear projections</h2>
<div class="columns">
<div class="column center" style="font-size: 50%;">
<p><img data-src="gifs/explain_1d.gif" alt="1D tour of 2D data. Data has two clusters, we see bimodal density in some 1D projections." width="500"></p>
<center>
<p>Data is 2D: <span class="math inline">\(~~p=2\)</span></p>
Projection is 1D: <span class="math inline">\(~~d=1\)</span>
</center>
<p><span class="math display">\[\begin{eqnarray*}
A_{~2\times 1} = \left[ \begin{array}{c}
a_{~11} \\
a_{~21}\\
\end{array} \right]_{~2\times 1}
\end{eqnarray*}\]</span></p>
</div><div class="column" style="font-size: 70%;">
<div class="fragment">
<p><br> Notice that the values of <span class="math inline">\(A\)</span> change between (-1, 1). All possible values being shown during the tour.</p>
<p><img data-src="images/explain_1d_axes_1_0.jpg" style="width:30.0%"> <img data-src="images/explain_1d_axes_7_7.jpg" style="width:30.0%"> <img data-src="images/explain_1d_axes_-7_7.jpg" style="width:30.0%"></p>
<p><span style="font-size: 50%;"> <span class="math display">\[\begin{eqnarray*}
A = \left[ \begin{array}{c}
1 \\
0\\
\end{array} \right]
~~~~~~~~~~~~~~~~
A = \left[ \begin{array}{c}
0.7 \\
0.7\\
\end{array} \right]
~~~~~~~~~~~~~~~~
A = \left[ \begin{array}{c}
0.7 \\
-0.7\\
\end{array} \right]
\end{eqnarray*}\]</span></span></p>
</div>
<div class="fragment">
<p><br> watching the 1D shadows we can see:</p>
<ul>
<li>unimodality</li>
<li>bimodality, there are two clusters.</li>
</ul>
</div>
<div class="fragment">
<p><span style="color:#EC5C00"> What does the 2D data look like? Can you sketch it? </span></p>
</div>
</div>
</div>
</section>
<section id="tours-of-linear-projections-1" class="slide level2" data-visibility="uncounted">
<h2>Tours of linear projections</h2>
<div class="columns">
<div class="column" style="width:60%;">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides1_files/figure-revealjs/unnamed-chunk-9-1.png" class="quarto-figure quarto-figure-center" style="width:70.0%" alt="Scatterplot showing the 2D data having two clusters."></p>
</figure>
</div>
</div>
</div>
</div><div class="column" style="width:30%;">
<p><br><br> <span style="color:#EC5C00"> ⟵ <br> The 2D data </span></p>
<div class="fragment">
<p><img data-src="images/explain_1d_annotated.png" alt="2D two cluster data with lines marking particular 1D projections, with small plots showing the corresponding 1D density."></p>
</div>
</div>
</div>
</section>
<section id="tours-of-linear-projections-2" class="slide level2">
<h2>Tours of linear projections</h2>
<div class="columns">
<div class="column center" style="font-size: 50%;">
<p><img data-src="gifs/explain_2d.gif" alt="Grand tour showing points on the surface of a 3D torus." width="500"></p>
<p>Data is 3D: <span class="math inline">\(p=3\)</span></p>
<p>Projection is 2D: <span class="math inline">\(d=2\)</span></p>
<p><span class="math display">\[\begin{eqnarray*}
A_{~3\times 2} = \left[ \begin{array}{cc}
a_{~11} & a_{~12} \\
a_{~21} & a_{~22}\\
a_{~31} & a_{~32}\\
\end{array} \right]_{~3\times 2}
\end{eqnarray*}\]</span></p>
</div><div class="column" style="font-size: 70%;">
<div class="fragment">
<p><br><br><br><br><br><br> Notice that the values of <span class="math inline">\(A\)</span> change between (-1, 1). All possible values being shown during the tour.</p>
</div>
<div class="fragment">
<p>See:</p>
<ul>
<li>circular shapes</li>
<li>some transparency, reveals middle</li>
<li>hole in in some projections</li>
<li>no clustering</li>
</ul>
</div>
</div>
</div>
</section>
<section id="tours-of-linear-projections-3" class="slide level2">
<h2>Tours of linear projections</h2>
<div class="columns">
<div class="column center" style="font-size: 40%;">
<p><img data-src="gifs/penguins1.gif" alt="Grand tour showing the 4D penguins data. Two clusters are easily seen, and a third is plausible." width="500"></p>
<p>Data is 4D: <span class="math inline">\(p=4\)</span></p>
<p>Projection is 2D: <span class="math inline">\(d=2\)</span></p>
<p><span class="math display">\[\begin{eqnarray*}
A_{~4\times 2} = \left[ \begin{array}{cc}
a_{~11} & a_{~12} \\
a_{~21} & a_{~22}\\
a_{~31} & a_{~32}\\
a_{~41} & a_{~42}\\
\end{array} \right]_{~4\times 2}
\end{eqnarray*}\]</span></p>
</div><div class="column" style="font-size: 70%;">
<p><br> How many clusters do you see?</p>
<div class="fragment">
<ul>
<li>three, right?</li>
<li>one separated, and two very close,</li>
<li>and they each have an elliptical shape.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>do you also see an outlier or two?</li>
</ul>
</div>
</div>
</div>
</section>
<section id="intuitively-tours-are-like" class="slide level2">
<h2>Intuitively, tours are like …</h2>
<center>
<img src="https://dicook.github.io/mulgar_book/images/shadow_puppets.png" width="90%">
</center>
</section>
<section id="anomaly-is-no-longer-hidden" class="slide level2">
<h2>Anomaly is no longer hidden</h2>
<div class="columns">
<div class="column">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides1_files/figure-revealjs/invisible-1.png" class="quarto-figure quarto-figure-center" style="width:70.0%"></p>
</figure>
</div>
</div>
</div>
</div><div class="column">
<center>
<p><img data-src="gifs/anomaly2.gif" width="500"></p>
Wait for it!
</center>
</div>
</div>
</section>
<section id="how-to-use-a-tour-in-r" class="slide level2">
<h2>How to use a tour in R</h2>
<div class="columns">
<div class="column" style="font-size: 80%;">
<p>This is a <span class="orange2">basic tour</span>, which will run in your RStudio plot window.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tourr)</span>
<span id="cb1-2"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">animate_xy</span>(flea[, <span class="dv">1</span><span class="sc">:</span><span class="dv">6</span>], <span class="at">rescale=</span><span class="cn">TRUE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="fragment">
<p>This data has a class variable, <code>species</code>.</p>
<div class="f70">
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="" aria-hidden="true" tabindex="-1"></a>flea <span class="sc">|></span> <span class="fu">slice_head</span>(<span class="at">n=</span><span class="dv">3</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> species tars1 tars2 head aede1 aede2 aede3
1 Concinna 191 131 53 150 15 104
2 Concinna 185 134 50 147 13 105
3 Concinna 200 137 52 144 14 102</code></pre>
</div>
</div>
</div>
<p>Use this to <span class="orange2">colour points</span> with:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">animate_xy</span>(flea[, <span class="dv">1</span><span class="sc">:</span><span class="dv">6</span>], </span>
<span id="cb4-2"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">col =</span> flea<span class="sc">$</span>species, </span>
<span id="cb4-3"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">rescale=</span><span class="cn">TRUE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div><div class="column" style="font-size: 80%;">
<div class="fragment">
<p>You can specifically <span class="orange2">guide</span> the tour choice of projections using</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">animate_xy</span>(flea[, <span class="dv">1</span><span class="sc">:</span><span class="dv">6</span>], </span>
<span id="cb5-2"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">tour_path =</span> <span class="fu">guided_tour</span>(<span class="fu">holes</span>()), </span>
<span id="cb5-3"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">col =</span> flea<span class="sc">$</span>species, </span>
<span id="cb5-4"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">rescale =</span> <span class="cn">TRUE</span>, </span>
<span id="cb5-5"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">sphere =</span> <span class="cn">TRUE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
<div class="fragment">
<p>and you can <span class="orange2">manually</span> choose a variable to control with:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">915</span>)</span>
<span id="cb6-2"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">animate_xy</span>(flea[, <span class="dv">1</span><span class="sc">:</span><span class="dv">6</span>], </span>
<span id="cb6-3"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="fu">radial_tour</span>(<span class="fu">basis_random</span>(<span class="dv">6</span>, <span class="dv">2</span>), </span>
<span id="cb6-4"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">mvar =</span> <span class="dv">6</span>), </span>
<span id="cb6-5"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">rescale =</span> <span class="cn">TRUE</span>,</span>
<span id="cb6-6"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">col =</span> flea<span class="sc">$</span>species)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div>
</div>
</section>
<section id="how-to-save-a-tour" class="slide level2">
<h2>How to save a tour</h2>
<div class="columns">
<div class="column">
<center>
<img data-src="gifs/penguins1.gif" alt="Grand tour showing the 4D penguins data. Two clusters are easily seen, and a third is plausible." width="500">
</center>
</div><div class="column">
<p><span class="f80">To save as an animated gif:</span></p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">645</span>)</span>
<span id="cb7-2"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">render_gif</span>(penguins_sub[,<span class="dv">1</span><span class="sc">:</span><span class="dv">4</span>],</span>
<span id="cb7-3"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="fu">grand_tour</span>(),</span>
<span id="cb7-4"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="fu">display_xy</span>(<span class="at">col=</span><span class="st">"#EC5C00"</span>,</span>
<span id="cb7-5"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">half_range=</span><span class="fl">3.8</span>, </span>
<span id="cb7-6"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">axes=</span><span class="st">"bottomleft"</span>, <span class="at">cex=</span><span class="fl">2.5</span>),</span>
<span id="cb7-7"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">gif_file =</span> <span class="st">"gifs/penguins1.gif"</span>,</span>
<span id="cb7-8"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">apf =</span> <span class="dv">1</span><span class="sc">/</span><span class="dv">60</span>,</span>
<span id="cb7-9"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">frames =</span> <span class="dv">1500</span>,</span>
<span id="cb7-10"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">width =</span> <span class="dv">500</span>, </span>
<span id="cb7-11"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">height =</span> <span class="dv">400</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div>
</section>
<section id="your-turn" class="slide level2">
<h2><span class="orange2">Your turn</span></h2>
<p>Use a grand tour on the data set <code>c1</code> in the <code>mulgar</code> package. What shapes do you see?</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tourr)</span>
<span id="cb8-2"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(mulgar)</span>
<span id="cb8-3"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">animate_xy</span>(c1)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p><br><br> Have a look at <code>c3</code> or <code>c7</code> also. How are the structures different.</p>
<p></p><div class="countdown" id="timer_f14549a2" data-update-every="1" tabindex="0" style="right:0;bottom:0;"> <div class="countdown-controls"><button class="countdown-bump-down">−</button><button class="countdown-bump-up">+</button></div> <code class="countdown-time"><span class="countdown-digits minutes">05</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code> </div><p></p>
</section>
<section id="dimension-reduction" class="slide level2 transition center center-align">
<h2>Dimension reduction</h2>
</section>
<section id="what-is-dimensionality" class="slide level2">
<h2>What is dimensionality?</h2>
<img data-src="slides1_files/figure-revealjs/unnamed-chunk-24-1.png" class="quarto-figure quarto-figure-center r-stretch" style="width:100.0%"><p>When an axis extends out of a direction where the points are collapsed, it means that this variable is partially responsible for the reduced dimension.</p>
</section>
<section id="in-high-dimensions" class="slide level2">
<h2>In high-dimensions</h2>
<div class="columns">
<div class="column" style="width:33%;">
<center>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="https://dicook.github.io/mulgar_book/gifs/plane.gif" width="300"></p>
<figcaption>2D plane in 5D</figcaption>
</figure>
</div>
</center>
</div><div class="column" style="width:33%;">
<div class="fragment">
<center>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="https://dicook.github.io/mulgar_book/gifs/box.gif" width="300"></p>
<figcaption>3D plane in 5D</figcaption>
</figure>
</div>
</center>
</div>
</div><div class="column" style="width:33%;">
<div class="fragment">
<center>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="https://dicook.github.io/mulgar_book/gifs/cube5d.gif" width="300"></p>
<figcaption>5D plane in 5D</figcaption>
</figure>
</div>
</center>
</div>
</div>
</div>
<div class="fragment">
<p>Principal component analysis (PCA) will detect these dimensionalities.</p>
</div>
</section>
<section id="example-womens-track-records-13" class="slide level2">
<h2>Example: womens’ track records <span class="f70">(1/3)</span></h2>
<div class="columns">
<div class="column">
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides1_files/figure-revealjs/unnamed-chunk-26-1.png" class="quarto-figure quarto-figure-center" style="width:100.0%"></p>
</figure>
</div>
</div>
</div>
<p><span class="f50"><em>Source</em>: Johnson and Wichern, Applied multivariate analysis</span></p>
</div><div class="column">
<div class="fragment">
<center>
<img data-src="gifs/track.gif" width="600">
</center>
</div>
</div>
</div>
</section>
<section id="example-pca-summary-23" class="slide level2">
<h2>Example: PCA summary <span class="f70">(2/3)</span></h2>
<div class="columns">
<div class="column">
<p>Variances/eigenvalues</p>
<div class="cell" data-layout-align="center">
<div class="cell-output cell-output-stdout">
<pre><code>[1] 5.806 0.654 0.300 0.125 0.054 0.039 0.022</code></pre>
</div>
</div>
<p>Component coefficients</p>
<div class="cell" data-layout-align="center">
<div class="cell-output cell-output-stdout">
<pre><code> PC1 PC2 PC3 PC4
m100 0.37 0.49 -0.286 0.319
m200 0.37 0.54 -0.230 -0.083
m400 0.38 0.25 0.515 -0.347
m800 0.38 -0.16 0.585 -0.042
m1500 0.39 -0.36 0.013 0.430
m3000 0.39 -0.35 -0.153 0.363
marathon 0.37 -0.37 -0.484 -0.672</code></pre>
</div>
</div>
</div><div class="column">
<p>How many PCs?</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides1_files/figure-revealjs/unnamed-chunk-31-1.png" class="quarto-figure quarto-figure-center" style="width:100.0%"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="example-visualise-33" class="slide level2">
<h2>Example: Visualise <span class="f70">(3/3)</span></h2>
<div class="columns">
<div class="column">
<p><span class="f70">Biplot: data in the model space</span></p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="slides1_files/figure-revealjs/unnamed-chunk-32-1.png" class="quarto-figure quarto-figure-center" style="width:100.0%"></p>
</figure>
</div>
</div>
</div>
</div><div class="column">
<p><span class="f70">2D model in data space</span></p>
<center>
<img data-src="gifs/track_model.gif" width="350">
</center>
<div class="f60">
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="" aria-hidden="true" tabindex="-1"></a>track_model <span class="ot"><-</span> mulgar<span class="sc">::</span><span class="fu">pca_model</span>(track_std_pca, <span class="at">d=</span><span class="dv">2</span>, <span class="at">s=</span><span class="dv">2</span>)</span>
<span id="cb11-2"><a href="" aria-hidden="true" tabindex="-1"></a>track_all <span class="ot"><-</span> <span class="fu">rbind</span>(track_model<span class="sc">$</span>points, track_std[,<span class="dv">1</span><span class="sc">:</span><span class="dv">7</span>])</span>
<span id="cb11-3"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">animate_xy</span>(track_all, <span class="at">edges=</span>track_model<span class="sc">$</span>edges,</span>
<span id="cb11-4"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">edges.col=</span><span class="st">"#E7950F"</span>, </span>
<span id="cb11-5"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">edges.width=</span><span class="dv">3</span>, </span>
<span id="cb11-6"><a href="" aria-hidden="true" tabindex="-1"></a> <span class="at">axes=</span><span class="st">"off"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div>
</div>
</section>
<section id="non-linear-dimension-reduction-12" class="slide level2">
<h2>Non-linear dimension reduction <span class="f70">(1/2)</span></h2>
<div class="columns">
<div class="column f60">
<p>Find some low-dimensional layout of points which approximates the distance between points in high-dimensions, with the purpose being to have a <span class="orange2">useful representation that reveals high-dimensional patterns</span>, like clusters.</p>
<p><span class="blue2">Multidimensional scaling (MDS)</span> is the original approach:</p>
<p><span class="math display">\[
\mbox{Stress}_D(x_1, ..., x_n) = \left(\sum_{i, j=1; i\neq j}^n (d_{ij} - d_k(i,j))^2\right)^{1/2}
\]</span> where <span class="math inline">\(D\)</span> is an <span class="math inline">\(n\times n\)</span> matrix of distances <span class="math inline">\((d_{ij})\)</span> between all pairs of points, and <span class="math inline">\(d_k(i,j)\)</span> is the distance between the points in the low-dimensional space.</p>
</div><div class="column f60">
<p>PCA is a special case of MDS. The result from PCA is a linear projection, but generally MDS can provide some non-linear transformation.</p>
<p>Many variations being developed:</p>
<ul>
<li><span class="blue2">t-stochastic neighbourhood embedding (t-SNE)</span>: compares interpoint distances with a standard probability distribution (eg <span class="math inline">\(t\)</span>-distribution) to exaggerate local neighbourhood differences.</li>
<li><span class="blue2">uniform manifold approximation and projection (UMAP)</span>: compares the interpoint distances with what might be expected if the data was uniformly distributed in the high-dimensions.</li>
</ul>
<p>NLDR can be useful but it can also make some misleading representations.</p>
</div>
</div>
</section>
<section id="non-linear-dimension-reduction-22" class="slide level2">
<h2>Non-linear dimension reduction <span class="f70">(2/2)</span></h2>
<div class="columns">
<div class="column">
<center>
<span class="f70">UMAP 2D representation</span>
</center>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
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<p><img data-src="slides1_files/figure-revealjs/penguins-umap-1.png" class="quarto-figure quarto-figure-center" style="width:70.0%"></p>
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<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(uwot)</span>
<span id="cb12-2"><a href="" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">253</span>)</span>
<span id="cb12-3"><a href="" aria-hidden="true" tabindex="-1"></a>p_tidy_umap <span class="ot"><-</span> <span class="fu">umap</span>(p_tidy_std[,<span class="dv">2</span><span class="sc">:</span><span class="dv">5</span>], <span class="at">init =</span> <span class="st">"spca"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><span class="f70">Tour animation of the same data</span></p>
<img data-src="gifs/penguins1.gif" alt="Grand tour showing the 4D penguins data. Two clusters are easily seen, and a third is plausible." width="500">