Complexity and Redundancy in Binary Maps

Abstract

The complexity of binary maps that is provided by the areal arrangement of colors is considered, and measured using information theory. In addition, information theory provides other measures that have an interpretation in a map context. One of these, redundancy, is examined and found to bear a striking empirical relationship to a spatial autocorrelation statistic. It is argued that spatial autocorrelation is, conceptual1 y as well as empirically, the two-dimensional equivalent of redundancy. It too measures the extent to which the occurrence of an event (color) in an areal unit constrains, or makes more probable, the occurrence of an event in a neighboring areal unit.The observation that maps are devices for the communication of information has gained wide currency in the cartographic literature [ 19, 231. As natural languages fulfill the same purpose, it is reasonable to argue that maps may be conceived, in a general sense, as output from languages. Given such a conceptualization it seems worthwhile to explore the possibilities of using the ideas and methods of linguistics in map analysis. In this paper, information theory is used to examine two syntactic or grammatical aspects of maps, complexity and redundancy.Harvey asserts that map syntax embraces "the internal structure of the map statements and its basic form as an abstract calculus" [15, pp. 373-741. In natural language it is concerned, broadly speaking, with the way in which elements of the language are put together, that is, with the rules of sentence construction. Extending this into two dimensions, it can be argued that map syntax involves notions of spatial association, periodicity [23, pp. 5-61, and the search for the rules that generate a map pattern. INFORMATION CONTENT AND REDUNDANCY IN ONE AND Two DIMENSIONS Information theory [27] is becoming well known in the geographic literature [3, 171. It can be used to measure the uncertainty or "informa-*This paper is based on my Master's thesis, "Complexity and Redundancy in Choropleth Maps," The Pennsylvania State University, 1974. I would like to thank Professor Peter Gould and Dr. Conrad Strack for their helpful advice and encouragement.Anthony C. Gatrell is lecturer in geography, Uiiicersity of Salford.

Exploring the effects of background image transparency on the usability of image maps

“…Entropy for spatial information Bjørke (1996), Li and Huang (2002) Landscape shape index Fairbairn (2006) Raster Statistical Compositional entropy Sukhov (1970), Gatrell (1977), Rosenholtz, Li, and Nakano (2007) Uniformity (energy) Gatrell (1977), Gonzalez and Woods (2002), Fairbairn (2006) Variance (contrast) MacEachren (1982), Gonzalez and Woods (2002), Harrie et al (2015) Correlation Olson (1975), Gonzalez and Woods (2002) Structural Boltzmann entropy Gao et al (2017), Gao and Li (2019) Relational descriptors Gonzalez and Woods (2002) M × N, the number of microstates is computed as the product of all possible decompositions of every aggregated cell:…”

Simulating Two‐dimensional Autocorrelated Surfaces

“…Provided that the measure distinguishes between patterns where similar, as opposed to dissimilar, values cluster, values of R, in the tails of this distribution relate to patterns that display distinctive structure. Earlier studies already have established links between spatial autocorrelation and redundancy (Gattrell 1977;Haining 1979). In this case, spatial autocorrelation is taken as a measure of the degree of structure or predictability in a given map pattern.…”

Generalized Procedures for Evaluating Spatial Autocorrelation

“…Given a set S containing n geographical units, spatial autocorrelation (SA) refers to the relationship between some variable observed in each of the n localities and a measure of geographical proximity defined for all n ( n -1) pairs chosen from S. Stsatistical methods developed for indexing this correspondence have traditionally been associated with the field of geography and more specifically with the subfield concerned with the assessment of spatial pattern (Brandsma and Ketellapper 1979;Cliff andOrd 1973, 1981;Curry 1966;Gatrell 1977). The techniques for analyzing the effects of geographical proximity, however, are really very general, and when interpreted appropriately they offer a valuable set of inference strategies in many other disciplines and for problems that are far removed from a concern with spatial phenomena.…”