An empirical study of tests for uniformity in multidimensional data

An important problem in high-dimensional data analysis is determining whether sample points are uniformly distributed (i.e., exhibit complete spatial randomness) over some compact support, or rather possess some underlying structure (e.g., clusters or other nonhomogeneities). We propose two new graph-theoretic tests of uniformity which utilize the minimum spanning tree and a snake (a short non-branching acyclic path connecting each data point). We compare the powers of statistics based on these graphs with other statistics from the literature on an array of non-uniform alternatives in a variety of supports. For data in a hypercube, we find that test statistics based on the minimum spanning tree have superior power when the data displays regularity (e.g., results from an inhibition process). For arbitrarily shaped or unknown supports, we use run length statistics of the sequence of segment lengths along the snake’s path to test uniformity. The snake is particularly useful because no knowledge or estimation of the support is required to compute the test statistic, it can be computed quickly for any dimension, and it shows what kinds of non-uniformities are present. These properties make the snake unique among multivariate tests of uniformity since others only function on specific and known supports, have computational difficulties in high dimension, or have inconsistent type I error rates.