Robust Estimation of Point Process Intensity Features using k - minimal Spanning Trees 1

Minimal spanning trees (MST) have been applied to multi-dimensional random processes for pattern recognition and randomness testing (See [l] for references). In this paper we present a robust version of the MST to estimate complexity features of a point process intensity function under an epsilon contaminated model for the intensity. The principal feature considered is the Renyi entropy of the mixture and a strongly consistent entropy estimator is given which depends on the data only through the total length of the MST passing through the data points. Robustification of the MST estimator is achieved by applying the theory of k-minimum MST’s [Z]. I. RENYI FEATURES AND FRACTIONAL MOMENTS Let d N ( z ) , 2: E X, be a multi-dimensional Poisson point process having normalized multivariate intensity A(z) which is the mixture: X(z) = (1 e)X,(z) + cA,(z), 0 5 t < 1. We assume that the “noise component” A, is a uniform intensity and e is unknown. A spanning tree is a connected acyclic graph which passes through all coordinates associated with the point cloud generated by the process. It consists of an ordered list of normalized edge lengths along with a list of edge adjacency relations. The total length of the tree is defined as the sum of all edge lengths. The minimal spanning tree (MST) is the spanning tree which posesses minimal total length. The kminimum spanning tree is the minimum length MST among those that pass through any k of the n points thus the standard MST is equivalent to the n-minimum MST. The features are defined as the Renyi entropy H,(dN) of fractional orders a, 0 < a 5 1, associated with A, The Renyi entropy equals zero for a = 0 and converges to the Shannon entropy s, A, In A, as a approaches 1. For any a the Renyi entropy is maximized for a uniform intensity and minimized for an intensity concentrated a t a single point. Define the fractional moment of the edge lengths {Z,}y=-: of an MST passing through n points of an observed p dimensional point cloud