Geographical Differentiation and its Measures in a Discrete System

The preceding preliminary paper is expanded into a full-fledged theory of differentiation. Two types are considered: feature-based differentiation and areal differentiation, both in their geographical and nongeographical contexts. Each type is examined both in a broad sense, without weighting, and in a narrow sense, with the values of a selected feature or other weights attached to the parts of a whole. Measures of differentiation are then borrowed from information theory. Differentiation in the broad sense (without weights) can be measured by the simpler Hartley measure of information transmission, which applies when any one of distinct possible outcomes of an event is equally likely to occur. Differentiation in the narrow sense (weighted) can be measured by the more complicated Shannon measure of entropy, which takes into consideration the probability that a given event will have a particular outcome.