Academic journal article Cartography and Geographic Information Science

# Supporting the Comparison of Choropleth Maps Using an Evolutionary Algorithm

Academic journal article Cartography and Geographic Information Science

# Supporting the Comparison of Choropleth Maps Using an Evolutionary Algorithm

## Article excerpt

Introduction

The spatial distribution of different variables can be compared by placing multiple small maps side by side (see Tufte 1990). This approach is particularly effective if the displayed variables are correlated. When this idea is applied to multiple classed choropleth maps, however, the original variables are not displayed. Instead, the patterns formed by the classification of the variables are conveyed to a reader. Since changing the classification of each variable will result in the formation of different mapped patterns, classification decisions are critical when comparisons are made.

The purpose of this paper is to develop an optimization approach to the search for classifications that can be used to support the visual comparison of multiple choropleth maps. Though the search for such classifications can be formulated as a problem that can be solved using traditional optimization methods such as linear programming (Cromley 1996), these methods are computationally intensive and their use may be limited to small datasets. In addition, exact methods are typically designed to yield only a single optimal solution (i.e., a set of class intervals) in terms of the objective function formulation. To overcome these limitations, we adopt principles of evolutionary algorithms to generate a diverse set of solutions that vary in visual similarity and areal equality.

In the remainder of this paper, we discuss how the problem of searching for choropleth map classes is formulated in a multi-objective optimization framework. The following section introduces the concept of evolutionary algorithms and describes the design of the evolutionary algorithm used in our research. Next we provide results obtained from a set of computational experiments. We conclude by placing this research in the context of recent developments in geographical visualization.

Background and Problem Formulation

Monmonier (1975; 1976) has discussed the issue of maximizing the visual correlation of choropleth maps. In his approach, the goal is to search for a classification that produces a choropleth map that is visually similar to a referent map. He emphasized 1) the development of a metric that can be used to measure visual correlation between two maps; 2) a heuristic approach that can be used to find a good classification for a map so that it is visually similar to a referent map; and 3) the modification of commonly used classification objectives so that visual correlation can be incorporated. These methods are effective if a referent map can be determined a priori.

When a referent map is not specified, however, the search for classifications that yield similar maps for two or more variables becomes more difficult. It has been suggested in the literature that quantile, equal area, mean-standard deviation, and nested means classification methods are appropriate (especially the first two) to support comparison of multiple choropleth maps (Brewer and Pickle 2002; Slocum et al. 2005, 342-345). Though these methods have some theoretical potential, they were not designed explicitly to maximize the visual con-elation among choropleth maps. Moreover, for the purpose of visual exploration (MacEachren and Kraak 1997; Kraak and MacEachren 1999; Dykes et al. 2005), especially during the early stages of research (DiBiase 1990), one may wish to examine different levels of visual correlation in a collection of maps.

It is also reasonable to assume that more than one combination of choropleth maps may share the same level of visual correlation. For example, two pairs of choropleth maps may have the same level of visual correlation, but the spatial units of these pairs may cover different amounts of area (Figure 1). In such cases, visually equivalent combinations of choropleth maps will exhibit different areal configurations that may be interesting to map readers. Simply exploring the statistical similarity among choropleth maps, however, will not reveal such information. …

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