Abstract
In this article, we propose a review of studies evaluating the distribution of distances between elements of a random set independently and uniformly distributed over a region of space in a normed R -vector space (for example, point events generated by a homogeneous Poisson process in a compact set). The distribution of distances between individuals is present in many situations when interaction depends on distance and concerns many disciplines, such as statistical physics, biology, ecology, geography, networking, etc. After reviewing the solutions proposed in the literature, we present a modern, general and unified resolution method using convolution of random vectors. We apply this method to typical compact sets: segments, rectangles, disks, spheres and hyperspheres. We show, for example, that in a hypersphere the distribution of distances has a typical shape and is polynomial for odd dimensions. We also present various applications of these results and we show, for example, that variance of distances in a hypersphere tends to zero when space dimension increases.
Highlights
The distribution of distances between elements in a set of points is present in many problems, in spatial analysis, and in various fields of application: ecology, epidemiology, forestry, biology, astronomy, economics, particle physics, network applications, etc
Most methods used to measure spatial autocorrelation or to model spatial interactions are based on a weighted average of a variable between pairs of elements in a disk [2,3,4,5]
Nobelmethod prize physicist solvedmathematical a slightly simpler problem than propose a unified of resolution which uses only standard objects, and which the one studied in this article: he modelled the distribution
Summary
The distribution of distances between elements in a set of points is present in many problems, in spatial analysis, and in various fields of application: ecology, epidemiology, forestry, biology, astronomy, economics, particle physics, network applications, etc. [1]. Given two points randomly selected in a set of points independently and uniformly distributed in space, we aim to know the probability of the distance between these two points inside the set of distances between all the pairs of points (Figure 1) This question is important when trying to evaluate or model spatial interactions between elements, such as clustering of objects, spatial autocorrelation of a variable across a set of locations, or neighbor relationships and connectivity [2]. Most methods used to measure spatial autocorrelation or to model spatial interactions are based on a weighted average of a variable between pairs of elements in a disk [2,3,4,5]. Distances between points, how likely would this distance be? Would it be below or above average?
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