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<html>
<head>
<title>Histogram Distance</title>
</head>
<body>
<h1>Histogram Distance</h1>
<p>
In the <a href="VisualizationTables.htm">previous section</a> we
studied the transient behavior of the Schlogl model by recording the
populations in a series of histograms. We saw that the populations
settled into a bi-modal distribution. The plot showing the probability
distributions for the species population is reproduced below.
</p>
<p align="center">
<!--Saved as PNG. Reduced from 8 to 6 inches and saved as JPG.-->
<img src="VisualizationPlottingHistogramsSchloglNonzero.jpg">
</p>
<p>
The total variation metric may be used to measure the distance between
two probability distributions. For two histograms (with the same lower
bound and bin width) having bin values of <i>x<sub>i</sub></i>
and <i>y<sub>i</sub></i>, this metric is half the sum of the absolute
values of the bin differences.
</p>
<p>
<i>D</i> = Σ|<i>x<sub>i</sub></i> - <i>y<sub>i</sub></i>| / 2
</p>
<p>
Because the histogram represents a probability distribution, the sum
of the bins is unity, Σ <i>x<sub>i</sub></i> = 1. The factor of
1/2 in the distance metric normalizes the distance to be between 0
and 1.
</p>
<p>
Hitting the histogram distance button
<img src="HistogramDistance.png"> in the simulation output panel
will bring up a window that allows us to select a pair of histograms.
(One may also select a set of histograms, in which case the average
distance will be computed. In this way one may, for instance,
determine the average distance for a set of species.)
Below we see that as time advances the distance beween successive
frames decreases.
</p>
<p align="center">
<img src="VisualizationHistogramSchloglDistance_0_2.jpg"><br>
<img src="VisualizationHistogramSchloglDistance_2_4.jpg"><br>
<img src="VisualizationHistogramSchloglDistance_4_6.jpg"><br>
<img src="VisualizationHistogramSchloglDistance_6_8.jpg"><br>
<img src="VisualizationHistogramSchloglDistance_8_10.jpg">
</p>
</body>
</html>
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