Unfortunately, it has been a lot of years since I performed these analyses, and some of the documentation is now missing (or perhaps filed somewhere inappropriately?). I can say, however, that three variations on the initial input data were submitted to double-standardization, and that two of three attained symmetric results, with the third, just missing. For the sake of record, the three sets of results were as follows: #1. means of the four vectors of the correlation matrix for the input (spatial autocorrelation measures) data: .147 / .325 / .188 / -.197 (overall mean = .116; mean of absolute values of the means = .214) double-standardized scores:
double-standardized scores:
double-standardized scores:
My records, such that they are, do show that I began an effort to confirm these findings with another more extensive sampling of elevations, but ran out of gas before this could be completed--I probably decided at the time that my guess-timates as to elevation were a bigger problem in the analysis than was small sample size. It is also not clear whether distances between sample points should be based on Euclidean straight lines, or portions of surface arcs (in the completed analysis, I think I used Euclidean distances). These results of themselves are inconclusive, but reasonably suggestive. Under better circumstances (that is, with better data sources and a finer sampling grid) it seems fairly likely that the system will indeed again measure out in a fashion consistent with the model under exploration here. As to whether more precision will result in further lowerings of the mean correlation values (as happened in the internal zones of the earth analysis) is difficult to predict, however: in the stream basins analyses the mean correlations varied from around .01 to over .14, and it is unlikely that like many of these systems the earth as a whole is greatly in disequilibrium with respect to the sum details of its surface topography. It is also interesting here that each vector of z scores contains a large, positive, diagonal value, and then three negative values that are fairly closely equal (especially in the last two models, in combination ranging from about -.482 to -.648). I suspect that with better data, the off-diagonals in this instance will converge to nearly the same figure, reflecting the lack of boundary conditions connected with the system's basic topology (the surface of a near sphere). By contrast, the typical z scores for the internal-zones-of-the-earth results were about 1.50 / .25 / -1.05 / -.70 , and for the streams, about 1.55 / .10 / -1.10 / -.55 . _________________________
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