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<h1>Multi-Dimensional Scaling</h1>
<blockquote>
<p>If you have multiple features for each observation (row) in a dataset and would like to reduce the number of features in the data so as to visualize which observations are similar, <em>Multi Dimensional Scaling (MDS)</em> will help.</p>
</blockquote>
<h2>The Advantage and Disadvantage of MDS</h2>
<p>The advantage with MDS is that you can specify the number of dimensions you want in the output data. The disadvantage however is that it is not possible to deal with un-ordered categorical features.</p>
<h2>How to implement MDS?</h2>
<p>It can be easily implemented using the <code>cmdscale()</code> in <code>{stats}</code> and the <code>isoMDS()</code> and <code>sammon()</code> from <code>{MASS} package</code>. All these functions take the dissimilarity object of class <code>dist</code> as the main argument and <code>k</code> is the desired number of dimensions in the scaled output.</p>
<p>Below is the code that demonstrates these functions on <code>swiss</code> data that contains fertility and socio-economic data on 47 French speaking provinces in Switzerland.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">head</span>(swiss) <span class="co"># first 6 rows of swiss</span>
<span class="co"># Fertility Agriculture Examination Education Catholic Infant.Mortality</span>
<span class="co"># Courtelary 80.2 17.0 15 12 9.96 22.2</span>
<span class="co"># Delemont 83.1 45.1 6 9 84.84 22.2</span>
<span class="co"># Franches-Mnt 92.5 39.7 5 5 93.40 20.2</span>
<span class="co"># Moutier 85.8 36.5 12 7 33.77 20.3</span>
<span class="co"># Neuveville 76.9 43.5 17 15 5.16 20.6</span>
<span class="co"># Porrentruy 76.1 35.3 9 7 90.57 26.6</span></code></pre></div>
<h3>1. <code>cmdscale()</code>: Classical MDS</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">d <-<span class="st"> </span><span class="kw">dist</span>(swiss) <span class="co"># compute distance matrix</span>
scaled_2 <-<span class="st"> </span><span class="kw">cmdscale</span>(d) <span class="co"># perform MDS. k defaults to 2</span>
<span class="kw">head</span>(scaled_2) <span class="co"># first 6 features</span>
<span class="co"># [,1] [,2]</span>
<span class="co"># Courtelary 37.032433 -17.434879</span>
<span class="co"># Delemont -42.797334 -14.687668</span>
<span class="co"># Franches-Mnt -51.081639 -19.274036</span>
<span class="co"># Moutier 7.716707 -5.458722</span>
<span class="co"># Neuveville 35.032658 5.126097</span>
<span class="co"># Porrentruy -44.161953 -25.922412</span>
scaled_3 <-<span class="st"> </span><span class="kw">cmdscale</span>(d, <span class="dt">k=</span><span class="dv">3</span>) <span class="co"># setting k=3 to get 3 features.</span>
<span class="kw">head</span>(scaled_3)
<span class="co">#> [,1] [,2] [,3]</span>
<span class="co">#> Courtelary 37.032433 -17.434879 -22.609928</span>
<span class="co">#> Delemont -42.797334 -14.687668 -12.063389</span>
<span class="co">#> Franches-Mnt -51.081639 -19.274036 -22.541458</span>
<span class="co">#> Moutier 7.716707 -5.458722 -20.799893</span>
<span class="co">#> Neuveville 35.032658 5.126097 -9.218281</span>
<span class="co">#> Porrentruy -44.161953 -25.922412 -10.045238</span></code></pre></div>
<h3>2. <code>MASS::isoMDS()</code>: Non-metric Multi-dimensional scaling</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(MASS)
swiss.dist <-<span class="st"> </span><span class="kw">dist</span>(swiss)
swiss.mds <-<span class="st"> </span><span class="kw">isoMDS</span>(swiss.dist)
<span class="kw">head</span>(swiss.mds$points)
<span class="co">#> [,1] [,2]</span>
<span class="co">#> Courtelary 38.850496 -16.154674</span>
<span class="co">#> Delemont -42.676573 -13.720989</span>
<span class="co">#> Franches-Mnt -53.587659 -21.335763</span>
<span class="co">#> Moutier 6.735536 -4.604116</span>
<span class="co">#> Neuveville 35.622307 4.633972</span>
<span class="co">#> Porrentruy -44.739479 -25.495702</span>
<span class="kw">plot</span>(swiss.mds$points, <span class="dt">type =</span> <span class="st">"n"</span>)
<span class="kw">text</span>(swiss.mds$points, <span class="dt">labels =</span> <span class="kw">as.character</span>(<span class="dv">1</span>:<span class="kw">nrow</span>(swiss)))</code></pre></div>
<h3>3. <code>MASS::sammon()</code>: Another form of non-metric MDS</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(MASS)
swiss.dist <-<span class="st"> </span><span class="kw">dist</span>(swiss)
swiss.sam <-<span class="st"> </span><span class="kw">sammon</span>(swiss.dist)
<span class="kw">head</span>(swiss.sam$points)
<span class="co">#> [,1] [,2]</span>
<span class="co">#> Courtelary 37.032433 -17.434879</span>
<span class="co">#> Delemont -42.797334 -14.687668</span>
<span class="co">#> Franches-Mnt -51.081639 -19.274036</span>
<span class="co">#> Moutier 7.716707 -5.458722</span>
<span class="co">#> Neuveville 35.032658 5.126097</span>
<span class="co">#> Porrentruy -44.161953 -25.922412</span></code></pre></div>
<h2>Cluster with k-Means and plot</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">kmeans_clust <-<span class="st"> </span><span class="kw">kmeans</span>(swiss.sam$points, <span class="dv">3</span>) <span class="co"># k-means wihth 3 clusters.</span>
<span class="kw">plot</span>(swiss.sam$points, <span class="dt">type =</span> <span class="st">"n"</span>, <span class="dt">main=</span><span class="st">"MDS with sammon() and clustered"</span>, <span class="dt">xlab =</span> <span class="st">"X-Dim"</span>, <span class="dt">ylab=</span><span class="st">"Y-Dim"</span>)
<span class="kw">text</span>(swiss.sam$points, <span class="dt">labels =</span> <span class="kw">rownames</span>(swiss), <span class="dt">col =</span> kmeans_clust$cluster) <span class="co"># set color using k-means output</span></code></pre></div>
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