Simulations: Parallel Zones on the Surface of a Spheroid.
In real world systems this kind of pattern of subdifferentiation of major surface systems is rather common: think, for example, of temperature zones around the earth, or the first order "striping" of surface constituents around the planets Jupiter and Saturn. In such systems, of course, second order deviations from the main "striping" pattern are easy to discern; nevertheless, a primary response is also in effect.
The approach I took, just to have a beginning look, was to set up sets of four "test zones" comprised of multiples of ten percent of the arc distance from the equator to the pole on the spheroid, and summing to the full one hundred percent. I only included combinations that involved zones that were no wider than fifty percent of the total distance. It turns out that a mere 68 different combinations are possible, and I tested all of these.
So, and for example, zones occupying (1) the first 36 degrees of latitude from the equator, (2) the next 9 degrees, (3) the next 18 degrees, and (4) the last 27 degrees (corresponding to 40, 10, 20, and 30 percent of the total distance, respectively) were identified, and those of the 1000 points that fell within these zones assigned to one or another of four classes (with the "southern hemisphere" being treated the same way). Once all the 1000 points were assigned, the spatial autocorrelation properties of the resulting sets could be established, again producing a 4 by 4 matrix of values that could be input to the double standardization algorithm.
Using three different spatial autocorrelation measures, three sets of 68 matrices were double-standardized. The results of the three analyses were not very different; in two of them 24 of the matrices passed the test, and in the third, 21 passed it. Thus, just about one-third of all the resulting z-score configurations could be expressed as an unambiguous three dimensional space.
Recall that I earlier spoke of another statistic that might prove helpful in interpreting the results: the mean correlation value for the correlation matrix associated with the 4 by 4 set of values being input to the double-standardization operation. A very low mean correlation among such vectors suggests low redundancy of function in an operational sense; in the real world this might indicate systems that are approaching internal dynamic equilibrium. In the present test this summary mean correlation statistic ranged considerably (as is usual for any set of these analyses), with the mean r value (Pearson correlation coefficient) being .053, .053, and .035 for the 24, 24, and 21 "passing" matrices. Importantly, several input matrices in each of the three sets of tests had mean r values of under .01, which is very low (by contrast, the mean r value for the 4 by 4 input matrices comprised completely of random numbers, as described earlier, was .175, with a standard deviation of about .113). Thus, one has some basis for thinking that systems of the simple multi-zone nature imagined might be able to express themselves over such surfaces as either equilibrial or disequilibrial entities; further, it would appear that it is easier to generate a three-dimensional space under the assumption of such a starting metric to begin with.
Again, it must be admitted that a variety of other approaches to the same ends here (employing more combinations of zone width, different spatial autocorrelation measures, arc distance as opposed to direct Euclidean, etc.) need to be investigated before anything fully detailed and concrete emerges from this kind of simulation. Yet, the results as just described would seem the best possible encouragement for going ahead and trying out such extensions. In the next simulations, we have a look at another kind of zonal expression: that occurring within spheroidal bodies.
Copyright 2006-2012 by Charles H. Smith.
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