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 correspond