Mutual information spectra for comparing categorical maps

Abstract

The continual accumulation of categorical data sets, presented as nominal categories mapped onto regular grids, provides for the increased desire to compare the patterns observed between these maps. We present a measurement scheme for the comparison of categorical maps that decomposes the differences in multidimensional nested coincidence tables according to variables that record occurrence frequencies of categories (Z), at levels of spatial aggregation (Y), on specific maps (X). Sequences of conditional entropies computed according to the specific questions asked (e.g. is there coincidence between colours and locations), characterize the correspondence between the three types of variables in common units (bits) measured by mutual information. The form of these sequences, as a variable runs from coarse to fine detail, referred to as spectra, provide meaningful characterizations of the similarities/differences between categorical maps, including their spatial configuration. We introduce the information theory‐based conceptual framework and illustrate its beneficial application by comparing a series of demonstration maps.

Acknowledgements

We gratefully acknowledge the financial support of the GEOIDE Network of Centres of Excellence (Canada), the Natural Sciences and Engineering Research Council of Canada, and the Ontario Graduate Scholarship programme. Constructive discussions with Sándor Kabos (Eötvös University, Budapest) regarding statistics, entropy‐based measures of similarity, and programming were very useful during project development and prototyping. We also thank Mike Wulder at the Canadian Forest Service for the NFI and EOSD data used to illustrate the application of our approach. Finally, we thank the anonymous reviewers for their comments that led to an improved manuscript.