At
the time I was primarily interested in seeing whether the patterns of
growth in the area could suggest some refinements of what geographers
refer to as "Central Place Theory," a conceptualization of the
way hierarchical economic forces shape the development of regional structures
over time. My particular approach was to try to move away from the analysis
of interactions between individual places within the system (as was commonly
being done at the time using the entropy maximization methods developed
by mathematician/geographer Alan Wilson in England, and others), and instead
to concentrate on the interactions among Despite the fact that this is one of the most ideal regional areas on earth for such studies (having had a fairly short history, and lying within a physical expanse uninterrupted by any major water or topographical barriers), a fairly daunting initial problem presented itself: although U. S. Census data existed for the whole history of the area, the basic township units that covered it were not quite satisfactorily constant in area over the whole expanse of the region (for example, too many larger-than-average townships existed in the southwestern quarter of it). It was therefore necessary to do some lumping and splitting of the initial units until roughly the same average size of fundamental units prevailed across all quarters of the region (less than ten per cent of the results were affected, largely in uniformly rural areas). I
might have stopped there, before performing various analyses, but reasoned
that it would be preferable to make use of a further assumption: that
not only were the township units of the same average size from one portion
to another of the region, but that they were I next took the resulting thirteen sets of 387 adjusted populations and subjected each set to the same nonhierarchical clustering algorithm discussed in the stream drainage basins analysis. Specifically, for each set the solutions consisted of splitting the population sizes up into two through six classes of population size. Figure 1 shows two of these systemizations, the four- and five-class classifications for the 1970 data.
In any case, the populations at particular locations within the region act, in calculation terms, very much like the sampled elevation locations do within the stream basins: in each case, variations in value/level mean variations in potential for interaction. Indeed, there is a term called "population potential" in geography which describes mathematically, over space, how a given center's population "mass" creates a potential for interaction with other such masses, and how that function of potential weakens the further removed one is from the center. Once the "1,1" problem is overcome, matrices describing group- (class-) level potentials could be constructed, and then double-standardized. I did try some such analyses using various estimates for the 1,1 value, and initial results suggest that the four-class partitions from the earlier years of the region do in fact double-standardize to the symmetric pattern looked for here (for the same reason as in the other analyses), but that starting around 1950 they no longer do. If this bears out under future examinations, I offer the following possible explanation. For
the earlier years, it could be safely assumed that the pattern of resident
populations was essentially equivalent to the pattern of population potential
Nevertheless,
I have some other evidence that the adjustments to the raw data have resulted
in a clearer picture of overall regional system organization--and that
this is, in fact, dominated by an If one standardizes all of the adjusted populations such that their new "populations" now represent a proportion of that year's total regional population, and then uses these figures to calculate gravity/population potentials across all town to town interactions in the region, one comes up with figures that when totalled can be compared from census year to census year for the system as a whole. In the present instance there is in fact a continuous increase from census to census in the sum gravity/population potential for the system, all the way through 1980. This means that there has been a directional movement throughout the life of the system to locate individual people in such a fashion as to minimize the mean distance from all persons to all persons. To accomplish this, the system has described a complex clustering process that seems to cut across all other trends (urbanization, suburbanization, etc.), suggesting some overriding principle of organization in effect. That such a principle has been operating, and that it is related to the evolution of a four size-class set of internal relationships, is more directly suggested by the degree to which the four-class solutions for each set of data 1860-1980 dominate, at least with respect to percentage of variation explained. The overall trends of explanation in the two- through six-class classifications of the data are indicated in Figure 2, which precisely graphs census year against proportion of total variation explained.
The way the four-class classification stands out is most evident by calculating the means of improvement in explanation across all thirteen of the periods. The average two-class explanation accounts for 67.14 percent of the variation; an average of 60.18 percent of the remaining 32.86 percent is then accounted for by adding the third class; an average of 55.54 percent of the remaining 13.08 percent is then accounted for by adding the fourth class. Note now that adding the fifth class only accounts on the average for 39.46 percent of the remaining 5.90 percent, and that the sixth class only adds an average of 32.20 percent of the then remaining 3.60 percent. There is thus a considerable gap in this series after the fourth class level (i.e., 67.14 -- 60.18 -- 55.54 -(here)- 39.46 -- 32.20), and this gap is even greater when one examines the 1960, 1970, and 1980 data alone. This strongly suggests that there is something special about the four class solutions here; i.e., that there is something resembling four evenly spread clusterings of values among the 387 elements of the set, taken in order. This
can be examined further and debated at some later point. For the moment
these results can be taken as (modestly) supporting the general thesis
here regarding the forces underlying organization of complex systems.
I don't claim to have _________________________
Copyright 2006-2014 by Charles H. Smith.
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