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<h1 id="mds-top" class="title">Multi-Dimensional Scaling</h1>
<p>Multidimensional scaling is a set of related statistical techniques
often used in information visualization for exploring similarities or
dissimilarities in data. An MDS algorithm starts with a matrix of item-item
similarities, then assigns a location to each item in low-dimensional space.
For sufficiently low dimension, the resulting locations may be displayed in a
graph or 3D visualization.</p>
<p>The major types of MDS algorithms include:</p>
<dl>
<dt>Classical multi-dimensional scaling</dt>
<dd><p>It takes an input matrix giving dissimilarities between pairs of items and
outputs a coordinate matrix whose configuration minimizes a loss function
called strain.</p></dd>
<dt>Metric multi-dimensional scaling</dt>
<dd><p>A superset of classical MDS that generalizes the optimization procedure
to a variety of loss functions and input matrices of known distances with
weights and so on. A useful loss function in this context is called stress
which is often minimized using a procedure called stress majorization.</p></dd>
<dt>Non-metric multi-dimensional scaling</dt>
<dd><p>In contrast to metric MDS, non-metric MDS finds both a non-parametric
monotonic relationship between the dissimilarities in the item-item matrix
and the Euclidean distances between items, and the location of each item in
the low-dimensional space. The relationship is typically found using isotonic
regression.</p></dd>
<dt>Generalized multi-dimensional scaling</dt>
<dd><p>An extension of metric multidimensional scaling, in which the target
space is an arbitrary smooth non-Euclidean space. In case when the
dissimilarities are distances on a surface and the target space is another
surface, GMDS allows finding the minimum-distortion embedding of one surface
into another.</p></dd>
</dl>
<h2 id="classical">Classical Multi-dimensional Scaling</h2>
<p>Classical multidimensional scaling is also known as principal coordinates
analysis. Given a matrix <code>A</code> of dissimilarities (e.g. pairwise distances), MDS
finds a set of points in low dimensional space that well-approximates the
dissimilarities in <code>A</code>. We are not restricted to using a Euclidean
distance metric. However, when Euclidean distances are used, classical MDS is
equivalent to PCA.</p>
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<li class="active"><a href="#java_1" data-toggle="tab">Java</a></li>
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</ul>
<div class="tab-content">
<div class="tab-pane" id="scala_1">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
def mds(proximity: Array[Array[Double]], k: Int, add: Boolean = false): MDS
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_1">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
public class MDS {
public static MDS fit(double[][] proximity, Options options);
}
</code></pre>
</div>
</div>
<div class="tab-pane" id="kotlin_1">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-kotlin"><code>
fun mds(proximity: Array<DoubleArray>, k: Int, add: Boolean = false): MDS
</code></pre>
</div>
</div>
</div>
<p>where <code>proximity</code> is the non-negative proximity matrix of dissimilarities.
The diagonal should be zero and all other elements should be positive and
symmetric. For pairwise distances matrix, it should be just the plain
distance, not squared.
The parameter <code>k</code> is the dimension of output space.
If the parameter <code>add</code> is true, the method estimates an appropriate constant to be added
to all the dissimilarities, apart from the self-dissimilarities, that
makes the learning matrix positive semi-definite. The other formulation of
the additive constant problem is as follows. If the proximity is
measured in an interval scale, where there is no natural origin, then there
is not a sympathy of the dissimilarities to the distances in the Euclidean
space used to represent the objects. In this case, we can estimate a constant c
such that proximity + c may be taken as ratio data, and also possibly
to minimize the dimensionality of the Euclidean space required for
representing the objects.</p>
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</ul>
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<div class="tab-pane" id="scala_2">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
val eurodist = read.csv("data/mds/eurodist.txt", delimiter = "\t", header = false)
val dist = eurodist.drop(0).toArray()
val citys = eurodist.stringVector(0).toArray()
val x = mds(dist, 2).coordinates
show(text(citys, x))
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_2">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
var eurodist = Read.csv("data/mds/eurodist.txt", CSVFormat.DEFAULT.withDelimiter('\t'));
var dist = eurodist.drop(0).toArray();
var citys = eurodist.stringVector(0).toArray();
var x = MDS.fit(dist, 2).coordinates();
TextPlot.of(citys, x).canvas().window();
</code></pre>
</div>
</div>
<div class="tab-pane" id="kotlin_2">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-kotlin"><code>
import smile.*;
import smile.manifold.*;
import smile.plot.swing.*;
val eurodist = read.csv("data/mds/eurodist.txt", delimiter = '\t', header = false)
val dist = eurodist.drop(0).toArray()
val citys = eurodist.stringVector(0).toArray()
val x = mds(dist, 2).coordinates
TextPlot.of(citys, x).canvas().window()
</code></pre>
</div>
</div>
</div>
<p>In the above example, we apply MDS to a distance matrix of some European cities.</p>
<div style="width: 100%; display: inline-block; text-align: center;">
<img src="images/mds.png" class="enlarge" style="width: 480px;" />
<div class="caption" style="min-width: 480px;">MDS</div>
</div>
<h2 id="kruskal">Kruskal's Nonmetric MDS</h2>
<p>Kruskal's nonmetric MDS. In non-metric MDS, only the rank order of entries
in the proximity matrix (not the actual dissimilarities) is assumed to
contain the significant information. Hence, the distances of the final
configuration should as far as possible be in the same rank order as the
original data. Note that a perfect ordinal re-scaling of the data into
distances is usually not possible. The relationship is typically found
using isotonic regression.</p>
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<div class="tab-pane" id="scala_3">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
def isomds(proximity: Array[Array[Double]], k: Int, tol: Double = 0.0001, maxIter: Int = 200): IsotonicMDS
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_3">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
public class IsotonicMDS {
public static IsotonicMDS fit(double[][] proximity, Options options);
}
</code></pre>
</div>
</div>
<div class="tab-pane" id="kotlin_3">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-kotlin"><code>
fun isomds(proximity: Array<DoubleArray>, k: Int, tol: Double = 0.0001, maxIter: Int = 200): IsotonicMDS
</code></pre>
</div>
</div>
</div>
<p>where <code>tol</code> is the tolerance for stopping iterations, and
<code>maxIter</code> is the maximum number of iterations.</p>
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<div class="tab-pane" id="scala_4">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
val x = isomds(dist, 2).coordinates
show(text(citys, x))
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_4">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
var x = IsotonicMDS.fit(dist, 2).coordinates();
TextPlot.of(citys, x).canvas().window();
</code></pre>
</div>
</div>
<div class="tab-pane" id="kotlin_4">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-kotlin"><code>
val x = isomds(dist, 2).coordinates
TextPlot.of(citys, x).canvas().window()
</code></pre>
</div>
</div>
</div>
<div style="width: 100%; display: inline-block; text-align: center;">
<img src="images/isomds.png" class="enlarge" style="width: 480px;" />
<div class="caption" style="min-width: 480px;">Kruskal's Nonmetric MDS</div>
</div>
<h2 id="sammon">Sammon's Mapping</h2>
<p>Sammon's mapping is an iterative technique for making interpoint
distances in the low-dimensional projection as close as possible to the
interpoint distances in the high-dimensional object. Two points close
together in the high-dimensional space should appear close together in the
projection, while two points far apart in the high dimensional space should
appear far apart in the projection. The Sammon's mapping is a special case of
metric least-square multidimensional scaling.</p>
<p>Ideally when we project from a high dimensional space to a low dimensional
space the image would be geometrically congruent to the original figure.
This is called an isometric projection. Unfortunately it is rarely possible
to isometrically project objects down into lower dimensional spaces. Instead of
trying to achieve equality between corresponding inter-point distances we
can minimize the difference between corresponding inter-point distances.
This is one goal of the Sammon's mapping algorithm. A second goal of the Sammon's
mapping algorithm is to preserve the topology as good as possible by giving
greater emphasize to smaller interpoint distances. The Sammon's mapping
algorithm has the advantage that whenever it is possible to isometrically
project an object into a lower dimensional space it will be isometrically
projected into the lower dimensional space. But whenever an object cannot
be projected down isometrically the Sammon's mapping projects it down to reduce
the distortion in interpoint distances and to limit the change in the
topology of the object.</p>
<p>The projection cannot be solved in a closed form and may be found by an
iterative algorithm such as gradient descent suggested by Sammon. Kohonen
also provides a heuristic that is simple and works reasonably well.</p>
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</ul>
<div class="tab-content">
<div class="tab-pane" id="scala_5">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
def sammon(proximity: Array[Array[Double]], k: Int, lambda: Double = 0.2, tol: Double = 0.0001, maxIter: Int = 100): SammonMapping
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_5">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
public class SammonMapping {
public static SammonMapping fit(double[][] proximity, Options options);
}
</code></pre>
</div>
</div>
<div class="tab-pane" id="kotlin_5">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
fun sammon(proximity: Array<DoubleArray>, k: Int, lambda: Double = 0.2, tol: Double = 0.0001, maxIter: Int = 100): SammonMapping
</code></pre>
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<p>where <code>lambda</code> is the initial value of the step size constant in diagonal Newton method.</p>
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<pre class="prettyprint lang-scala"><code>
val x = sammon(dist, 2).coordinates
show(text(citys, x))
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<pre class="prettyprint lang-java"><code>
var x = SammonMapping.fit(dist, 2).coordinates();
TextPlot.of(citys, x).canvas().window();
</code></pre>
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<pre class="prettyprint lang-kotlin"><code>
val x = sammon(dist, 2).coordinates
TextPlot.of(citys, x).canvas().window()
</code></pre>
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<div class="caption" style="min-width: 480px;">Sammon's Mapping</div>
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