Showing papers on "Spatial analysis published in 1972"

Entropy and spatial geometry

Abstract: Summary. The concept of entropy as used in explaining locational phenomena is briefly reviewed and it is suggested that the design of a zoning system for measuring such pheno mena is a non-trivial matter. An aggregation procedure based on entropy-maximizing is suggested and applied to the Reading sub-region, and the resulting geometries are con trasted with certain idealized schemes. In the last decade, several researchers have suggested that the concept of entropy is a relevant statistic for measuring the spatial distribution of various geographic phenomena. For example, Leopold and Langbein (1962) use a measure of entropy in deriving the fact that the most probable longitudinal profile of rivers has a negative-exponential form. Curry (1964) has shown that the rank-size distribution of cities can be explained by considerations involving the definition of entropy, and more recently, Wilson (1970) has developed a procedure for maximizing a function of entropy which can be used to describe a host of locational phenomena ranging from distributions of trip-making behaviour to distributions of population. Furthermore, Mogridge (1972), in an excellent review of the concept, demonstrates that entropy ' is of great, indeed essential, use in understanding economic and spatial systems'. Yet in all of this work, the spatial dimension in which the various phenomena are recorded and statistics computed is implicit rather than explicit. There is little concern for the way in which space is partitioned, for most of these entropy models appear to be based on the assumption that distributions are measured on spaces partitioned into equal areas or intervals. However, this assumption is not necessarily the most appropriate; geographers have long recognized that different and often conflicting statistical patterns can be interpreted from similar distributions measured on different areal systems. Although this problem has been quite widely studied in recent years under various guises such as that of spatial auto-correlation, there has been very'little research into the design of optimal spatial systems for geographical analysis or for locational planning. For example, Neft (1966) in his classic book on spatial analysis, hardly broaches the problem which is surprising in view of its importance. It is the purpose of this paper to investigate briefly the geometry of spatial systems in relation to the measurement of locational phenomena using the statistic of entropy. It is likely that there are ideal geometries associated with different spatial statistics, and in this paper, a geometry consistent with the definition of entropy will be outlined from both empirical and theoretical standpoints. In this quest, it is useful to begin by briefly reviewing the use of entropy in locational analysis. The discrete entropy of spatial distribution The definition of entropy most widely used in geographic research is due to Shannon (Shannon and Weaver, 1949) and it is also referred to as a measure of information or uncertainty. This measure can be written as