Expanding the conceptual, mathematical and practical methods for map comparison

Conventional methods of map comparison frequently produce unhelpful results for a variety of reasons. In particular, conventional methods usually analyze pixels at a single default scale and frequently insist that each pixel belongs to exactly one category. The purpose of this paper is to offer improved methods so that scientists can obtain more helpful results by performing multiple resolution analysis on pixels that belong simultaneously to several categories. This paper examines the fundamentals of map comparison beginning from the elementary comparison between two pixels that have partial membership to multiple categories. We examine the conceptual foundation of three methods to create a crosstabulation matrix for a single pair of pixels, and then show how to extend those concepts to compare entire maps at multiple spatial resolutions. This approach is important because the crosstabulation matrix is the basis for numerous popular measurements of spatial accuracy. The three methods show the range of possibilities for constructing a crosstabulation matrix based on possible variations in the spatial arrangement of the categories within a single pixel. A smaller range in the possible spatial distribution of categories within the pixel corresponds to more certainty in the crosstabulation matrix. The quantity of each category within each pixel constrains the range for possible arrangements in subpixel mapping, since there is more certainty for pixels that are dominated by a single category. In this respect, the proposed approach is placed in the context of a philosophy of map comparison that focuses on two separable components of information in a map: 1) information concerning the proportional distribution of the quantity of categories, and 2) information concerning the spatial distribution of the location of categories. The methods apply to cases where a scientist needs to compare two maps that show categories, even when the categories in one map are different from the categories in the other map. We offer a fourth method that is designed for the common case where a scientist needs to compare two maps that show the same set of categories. Results show that the methods can produce extremely different measurements, and that it is possible to interpret the differences at multiple resolutions in a manner that reveals patterns in the maps. The method is designed to present the results graphically in order to facilitate communication. We describe the concepts using simplified examples, and then apply the methods to characterize the change in land cover between 1971 and 1999 in Massachusetts