Abstract
This article presents some mathematical properties of the AGsi-curve (AGglomerative-curve using single-linkage, Takai et al., 2017), which is used for graphical classification of spatial point patterns. Although, the AGsi-curves are constructed from the lengths of the Graph Edges of the Minimum Spanning Tree (GEMST), the exact distributions of the GEMST have not been established at this time. In this article, two kinds of approximate distributions of the AGsi-curve are established from deriving the approximate distribution of the GEMST in the multidimensional space; the exact distribution is also evaluated using the Monte Carlo method. Comparing the approximate and exact distributions reveals that two approximate distributions are useful in a limited range of intensity (the expectations of the Monte Carlo distributions exist between these two distributions) and that the edge effect of the minimum spanning tree affects the AGsi-curve.
