Analysis of Qualitative Similarity between Surfaces

This paper develops a method for analyzing surfaces with a focus on their qualitative similarity. The method describes the qualitative similarity between surfaces defined in the same region in both quantitative and qualitative ways. Given a location and a direction in the region, mathematical functions evaluate the similarity between surfaces. Integrals of the functions with respect to location and direction give quantitative measures of the total similarity between surfaces. A qualitative method uses spatial characteristics shared by surfaces: a-peak regions, a-pit regions, and p monotonic lines. a-peak and a-pit regions indicate approximate locations where many surfaces have peaks and pits, respectively. p-monotonic lines are line segments on which most surfaces change monotonically in the same direction. Those spatial objects reveal the spatial structure shared by surfaces. The method is applied to the analysis of the daily market structure of a supermarket in Japan as an empirical study.

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