Repairing covariance functions on metric trees through a Hausdorff-type distance

Abstract

A compact metric tree is the natural domain of spatial data recorded along a river network, a root system, or the branched processes of a cell, and a suitable distance between two points is the length of the path joining them. That distance admits almost none of the covariance models of spatial statistics, since only the roughest members of each family stay nonnegative definite, and the resulting fields are nowhere mean square differentiable. The repair proposed here is a change of metric. Comparing two points through their distances to the leaves gives a Hausdorff-type metric built from the shortest-path one. The new metric has a closed form, agrees with the shortest-path one inside every edge, and is strictly smaller across branching points. Any correlation function valid in a Euclidean space of dimension equal to the number of leaves may then be composed with it, returning the Matérn, powered exponential, and generalized Cauchy families at every degree of smoothness, and, on trees with few leaves, also correlations that change sign or have compact support. The same construction yields the multivariate Matérn and space-time models of the Gneiting class. In simulations covering a range of smoothness and of spatial scale, maximum likelihood provides estimates without appreciable bias, and on a reconstructed microglial cell the smooth models reduce the cross-validated prediction error by half.