An axiomatic theory of heterogeneity and homogeneity

Summary Indices of heterogeneity and homogeneity are used in biology and ecology as well as in other fields of scientific research, e.g. economics, genetics, antropology, sociology. With respect to these indices, reference is made frequently to information theory. In this paper it is shown that especially recent developments in statistical information theory can contribute to the measurement of heterogeneity and homogeneity. They can lead to a more fundamental and unifying approach and bring together both heterogeneity and homogeneity indices in one axiomatically founded theory. Therefore a generalized heterogeneity index is introduced axiomatically. Algebraic, analytic and geometrical properties are investigated. Special attention is paid to the interpretation of these properties and to the choices of the values of the parameters for practical applications. Based on this generalized heterogeneity index, it is possible to given an axiomatic characterization for a class of homogeneity indices. The properties of this class are the reverse of those of the heterogeneity index. Finally, some potential measures for heterogeneity or homogeneity change are discussed.