A generalized cross-tabulation matrix to compare soft-classified maps at multiple resolutions

This paper addresses two grand challenges in the development of methods for Geographic Information Science (GIS). First, this paper presents techniques to compute a cross-tabulation matrix for soft-classified pixels. Second, it shows how to compute the cross-tabulation matrix at multiple scales. The traditional approach to construct the cross-tabulation matrix uses a Boolean operator to analyse pixels that are hard-classified. For soft-classified pixels, the contemporary approach uses a Multiplication operator; the fuzzy approach uses a Minimum operator; whereas this paper proposes a multiple-resolution approach that uses a Composite operator. There are difficulties with the traditional, contemporary, and fuzzy methods of computing the cross-tabulation matrix. The proposed multiple-resolution method resolves those difficulties. Furthermore, the proposed method facilitates multiple-resolution analysis, so it can examine how results change as a function of scale. The paper derives the equations to compute cross-tabulation matrices at multiple resolutions and connects those equations to ontological foundations of GIS.