Asymptotic moments of near–neighbour distance distributions

Abstract

Let C be a compact convex body in R m and consider a set of points selected at random from C according to some well-behaved sampling distribution. We obtain an asymptotic expression for the positive moments of the kth near-neighbour distance distribution as the number of points increases to in nity.

Asymptotic Moments of Near Neighbor Distances for the Gaussian Distribution

“…These issues have been addressed by the concrete approach in [2], where it was shown that if inf x∈X p(x) > 0 and p(x) has a bounded gradient on X , then under rather weak conditions on the space X , we have…”

“…ℜ n p(x) 1−α/n dx in the limit M → ∞ if 0 < α < n [12,2]. However, the case α > n is quite different and usually a boundedness condition must be imposed on the support of p(x).…”

A law of large numbers for nearest neighbour statistics

“…. It is well known, see for example Percus & Martin (1998) or Evans et al (2002), that provided the sampling distribution is continuous, the expected probability measure of any k -nearest neighbour ball (over all sample realizations) is equal to k/n.…”

“…Suppose also that F satisfies a smooth density condition in the sense that its second partial derivatives are bounded at every point of [0,1] d . Under these conditions, it is known (Evans et al 2002) that for all eO0, the moments of the k -nearest neighbour distance distribution satisfies For any set of vertices {x 1 , ., x n } in R d , the (undirected) Euclidean k -nearest neighbours graph is constructed by including an edge between every point and its k -nearest neighbours in the set, where the nearest neighbour relations are defined with respect to the Euclidean metric ( pZ2). Let L n (X ) denote the total length of the Euclidean k -nearest neighbours graph of the random sample X n ,…”

Nearest neighbors method for detecting transient disturbances in process and electromechanical systems

“…For this reason, it is reasonable to assume that this distribution can model the anomaly index vector, given that anomaly indices are distances, hence sums of squared differences. Several authors also refer to the use of the gamma function to model the frequency distribution of nearest neighbor distances [34,35,36]. Furthermore, the gamma distribution presents a number of properties that agree with anomaly indices, such as being defined only for positive real numbers, being positively skewed and converging to a Gaussian distribution when its skewness tends to zero.…”