Some Methodological Problems in the Use of the Neyman Type A and the Negative Binomial Probability Distributions for the Analysis of Spatial Point Patterns

Abstract

PROBABILITY THEORY provides us with a powerful abstract logical system which we may use as a model for describing and analysing geographical distributions. The abstract formulation given to the mathematical theory of probability by A. Kolmogorov (I933) and subsequent axiomatic treatments (such asJ. M. McCord and R. M. Moroney, I964) have served to increase the power and rigour of the mathematical theory. Nevertheless, the initial development of the theory relied very heavily upon the logical problems inherent in the analysis of games of chance. Mathematicians still rely upon simple 'games' or 'urn models', with rules and conditions fully specified, to give an intuitive meaning to the probability distributions which they derive. W. Feller's (I950) treatment of probability theory is an excellent example of the use of such simple games and urn models to facilitate understanding. But the range of problems which has been effectively tackled using the mathematical theory of probability is very much more extensive than would be implied by the relatively simple coin-tossing or card-dealing experiments used to help identify probability distributions. The mathematical theory of probability is, therefore, a neat demonstration of the point made by C. H. Coombs, H. Raiffa and R. M. Thrall (1954) that the mathematician, primarily concerned with the properties of an abstract theoretical system, requires some kind of simple concrete model to help interpret that abstract system, whereas the social scientist, primarily concerned with empirical problems, looks to the abstract mathematical system to provide a model into which he can 'map' difficult empirical problems so as to facilitate analysis. The basic methodological difficulty which faces the geographer is to establish criteria for judging the validity of this 'mapping' process. Only when such criteria are firmly established may we judge whether the use of a particular mathematical model (be it a system of differential equations, a set of probability distributions, a system of simultaneous equations, a recursive cause-and-effect model, and so on) is appropriate to the particular empirical problem under investigation and, further, judge whether the inferences established by way of the mathematical model are truly representative of the phenomena being observed. The general methodological problem of applying a priori analytical knowledge such as that contained in pure mathematical systems, to a posteriori synthetic empirical problems has exercised both philosophers and empirically minded scientists (S. K6rner, I960;J. S. Coleman, 1964). It is the intention of this paper to examine a particular case of this general methodological problem, namely, the analysis of spatial point patterns by quadrat sampling procedures.