Simulations: Concentric Zones Within a Spheroid.
We have already seen that some patterns distributed over the surface of a sphere or spheroid do in fact generate an entropy-maximized, three-dimensional space. Now the question becomes whether the most common internal pattern seen in such systems--concentric zoning--will do the same thing. To look at this I tested both spherical- and spheroidal-based zonation patterns, along the way employing three measures of spatial autocorrelation, and several different sizes and shifts of the sampling grid.
The sample grid shifts were included to try to make the best out of a sub-ideal situation. For the various simulations run, the number of coordinate locations "within" the spheres and spheroids totalled between around 16,000 to 20,000. This may seem like a good-sized sample, but there is an extenuating sampling problem: for any zone identified that is on the narrow side--say, less than ten percent of the overall diameter--not that much of the grid coincides with it: perhaps as little as the equivalent of only one row of coordinates. Thus, moving the grid only slightly in such instances can significantly shift the sample, and change the results. Ideally, a test of this type should probably employ at least 200,000 sample points, but as all of this was being run on a pc, this was out of the question. In view of this, a total of eighteen sets of studies were performed employing slight variations on the size and shift of the sampling grid, across three (actually two, as one of the three was used for reasons of perspective only) spatial autocorrelation measures. An algorithm that generated random combinations of zone widths was constructed, and 150 double-standardizations performed according to the eighteen starting point conditions.
There ended up being only minor differences between the simulations run on spherical and spheroidal settings, or among the various shifts and coarseness changes in the sampling grid. Much larger differences turned up according to which spatial autocorrelation measure was used. For the tests run on spherical volume zonations, an average of about eight of the 150 of the input matrices passed the three-dimensionality test using the first spatial autocorrelation measure, 47 of 150 for the second measure, and 122 of 150 for the third. For the tests on spheroidal volume zonations, the corresponding averages were 18, 70, and 128 (of 150). So again we have a situation in which, largely irregardless of the spatial autocorrelation measure applied, results are produced all of which support the general idea that three-dimensional spatial structure can be generated in this fashion.
One last set of simulations is now described, an extension of the "internal zones" investigation just described here.
Copyright 2006-2012 by Charles H. Smith.
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