ウィーバーの組合せ分析法の再検討と修正

抄録

In his ingenious papers (footnote 1 to 4), John C. Weaver has not only contributed much to the advancement of the agricultural geography of American Middle West, but also presented to geographers in general an important method of the analysis of percentage distributions, namely the method of combination analysis. However, it should have been noted that, although his method was almost perfect for his study area where, thanks to the crop rotation, near-equal share of cropland percentages among major crops was the rule, his method is not always so effective if applied to other types of percentage sequences. Therefore, while many geographers of other countries adopted his method in the research of their particular subjects (footnote 5-8), some of them modified it to correct its most serious deficiency, i.e., its tendency to give “fragmented combinations” that include quite many elments with large as well as very small percentages.
The writer is one of those modifiers and has demonstrated in an earlier paper (footnote 12, 13) that his modified method is superior to the original one in that (1) it prevents the fragmentation and produces combinations with the same or fewer number of elements, and (2) it saves much calculating work by the use of a table of critical percentage values instead of squares and divisions of percentages.
In this paper the writer re-examines the original method to find the cause of the fragmented combinations resulting from the successive decrease, with the increase of the number of elements, of the variances used by Weaver (Weaver's variance).
Weaver's variance can be divided into two parts, both being non-negative: (a) true variance or squared standard deviation, and (b) the correction term, i.e., the square of the difference between mean and the theoretical norm. Because the correction term always decreases with the increase of the number of the elements in the combination, either successive increase of the standard deviation (as in the case of Ionia County in Table 1) or the existence of a distinctive maximum of the standard deviation is the necessary condition for the occurrence of a minimum of Weaver's variance which determines the combination. Analysis of the percentage sequences incorporating various mathematical progressions has shown that only arithmetic and cosine progrssions show succesive increase of standard deviation. While these two progressions are represented in a semilogarithmic graph by convex curves, other types of progressions have very weak or no maximum of standard deviation and some of them are shown as straight lines and others, as concave curves (Fig. 1).
Actual sequences of percentages have, of coutse, no such mathematical regularity, but plotting them on a semi-logarithmic graph may suggest the existence or non-existence of a minimum of standard deviation. However, convexity of the semi-logarithmic curve is only one condition for the existence of a minimum of Weaver's variance, and another condition is the sizes of the percentages of the first to, say, third ranking elements. If they, or the first of them, are large enough, Weaver's variances will show a minimum at the first, second, or the third element. Formula (II-1), another expression of Weaver's variance, leads to the condition of the occurrence of one-element combination, Formula (II-2). Solution of this formula is shown as the area I in Fig. 2 A, which also shows the ranges of percentages of the first and the second ranking elements of the two- or more-element combinations (II, II-III-, III-).
Some geographers criticised Weaver's standard norm, an exactly equal division of 100% by the number of elements. Of course, other norms such as 50, 25, 25%, or 40, 20, 20, 10, 10% or even a mathematical progression are feasible, but the multiplicity of these norms and the subjectivity in selecting one from these render the merit of these alternative norms doubtful.