A test for multidimensional clustering tendency

摘要

This paper examines a d-dimensional extension of the Cox-Lewis statistic and investigates its power as a function of dimensionality in discriminating among random, aggregated and regular arrangements of points in d-dimensions. It was motivated by the Clustering Tendency problem which attempts to prevent the inappropriate application of clustering algorithms and other exploratory procedures. After reviewing the literature, the d-dimensional Cox-Lewis statistic is defined and its distribution under a randomness hypothesis of a Poisson spatial point process is given. Analytical expressions for the densities of the Cox-Lewis statistic under lattice regularity and under extreme clustering are also provided. The powers of Neyman-Pearson tests of hypotheses based on the Cox-Lewis statistic are derived and compared. Power is a unimodal function of dimensionality in the test of lattice regularity, with the minimum occurring at 12 dimensions.The power of the Cox-Lewis statistic is also examined under hard-core regularity and under Neyman-Scott clustering with Monte Carlo simulations. The Cox-Lewis statistic leads to one-sided tests for regularity having reasonable power and provides a sharper discrimination between random and clustered data than other statistics. The choice of sampling window is a critical factor. The Cox-Lewis statistic shows great promise for assessing the gross structure of data.