Lognormal Estimates of Macroregional City-Size Distributions, 1950–1970

Cola, Lee De

doi:10.1068/a171637

Cited by 4 publications

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“…Define the size of any region E , to be si = Ex ,&x), the number of cells in the region, and let the rank ri = Nr( E : s > s,) + 1, the number of regions larger than E i . The size distribution { s} of the n regions may be summarized by the order statistic max( s}, the size of the region of rank ri = 1, which statistic is likely to be quite descriptive of underlying process but obviously will not be very robust (De Cola 1985). One approach (Goodchild and Mark 1987) is the Korcak formulation of the Pareto distribution describing the size of the regions from their unique ranks: s j = A ( r j ) -b , corresponding to the linear model…”

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Pareto and Fractal Description of Regions from a Binomial Lattice

Cola¹

1989

Geographical Analysis

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74 / Geographical Analysis rectangular scaling of latitude and longitude to 18 MAADS for the most general projective transformation. If we assume M, equal to 4, spherical generalization will be less costly whenever more than 25 percent of the points are removed (Figure 3, upper curve). For M equal to 18, spherical generalization is advantageous if as few as 5 percent of d e original points are excluded. For r less than 0.5, spherical generalization before projection is anywhere from slightly to nearly five times less expensive than planar generalization after projection. The analysis is only approximate, but it does show that the cost of spherical generalization is certainly not high, and may in fact be considerably lower than the traditional approach.In summary, it is argued that spherical line generalization is useful, even necessary, for both analytical and display applications at global scales. Fortunately, the precepts developed by cartographers and embodied in planar generalization are applicable to line data on the sphere, thus planar methods can be adapted to the sphere without great difficulty. For the Douglas method, the cost of spherical generalization is low relative to the cost of projection, and there is therefore a net computational gain associated with spherical generalization. Given the ease with which spherical generalization can be performed, it should be used whenever analytic and/or cartographic concerns so dictate. LITERATURE CITED Buttenfield, B. (1985). "Treatment of the Cartographic Line." Carfographica 22, 1-26.Digital spatial analysis is often concerned with the description of rectangular images representing integral-valued measurements taken at regularly spaced sites: indeed, such data underlie raster-based geographic information systems (Burrough.The author gratefully acknowledges the support of John Parr, the assistance of Larry Haugh, and the helpful comments of several referees of previous versions of this paper. Lee DeCola is assistant professor of geography, University of Vernwnt.

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“…De Cola 1985).Let 4, = Nr(E: s < s,) be cumulative counts corresponding to the distinct quantiles of { s} and consider the following model In(sj) = p + uz(qj/n) I ' I ' I ' I ' I ' I ' 1 ' 1 ' I 0 00 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 1 00 PROPORTN FIG. 2.…”

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