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Abstract:
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An important yet not often studied problem in high-dimensional data analysis is determining whether data points are uniformly distributed over some compact domain, or rather possess some underlying structure instead (e.g., clumps, clusters, or other nonhomogeneities). We use the total length of the minimum spanning tree to confirm or reject uniformity in the unit hypercube. We also consider the number of runs and maximum run length of the sequence of segment lengths of a short Hamiltonian path (or "snake") constructed on the data to confirm or reject uniformity on an arbitrarily shaped compact domain in R^d. We find that both methods can detect clumps or clusters in otherwise uniformly distributed data with higher power than recent alternative methods proposed in the literature.
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