Pair correlation functions for identifying spatial correlation in discrete domains

Abstract

Identifying and quantifying spatial correlation are important aspects of studying the collective behavior of multiagent systems. Pair correlation functions (PCFs) are powerful statistical tools that can provide qualitative and quantitative information about correlation between pairs of agents. Despite the numerous PCFs defined for off-lattice domains, only a few recent studies have considered a PCF for discrete domains. Our work extends the study of spatial correlation in discrete domains by defining a new set of PCFs using two natural and intuitive definitions of distance for a square lattice: the taxicab and uniform metric. We show how these PCFs improve upon previous attempts and compare between the quantitative data acquired. We also extend our definitions of the PCF to other types of regular tessellation that have not been studied before, including hexagonal, triangular, and cuboidal. Finally, we provide a comprehensive PCF for any tessellation and metric, allowing investigation of spatial correlation in irregular lattices for which recognizing correlation is less intuitive.

Persistent exclusion processes: Inertia, drift, mixing, and correlation

“…We first consider the case of a swarm of walkers uniformly distributed on a two-dimensional lattice, and then extend discussion to the non-uniform case of a swarm of walkers diffusing from an initial condition. Gavagnin, Owen and Yates [34] recently provided relevant formulae for computing a on-lattice pair correlation function based on the 1 (Manhattan) norm. Following this, we define the pair correlation function for a given distance metric d as…”

Corrected pair correlation functions for environments with obstacles

“…The alternative choice is periodic boundary conditions; however, we do not expect an individual passing through one boundary to arrive at the opposing boundary. Recently, Gavagnin et al [24] derived the counts of pair distances for a domain without obstacles, D NO (m), for Introducing inaccessible sites into the domain increases the distances between pairs of sites ( Figure 3), and hence we require an expression for the counts of pair distances for m ≥ min(L x , L y ). For an arbitrary L x by L y domain with no obstacles, the counts of pair distances are (see Appendix A for the derivation)…”

“…Various methods for quantifying spatial structure have been proposed previously (for example, see the review by Perry et al [16], and references therein). Here we focus on the use of pair correlation functions (PCFs), which are a powerful and versatile tool for analysing spatial structure and spatial correlation [17][18][19][20][21][22][23][24][25][26][27]. Pair correlation functions have been successfully employed in astrophysics [18], particle physics [23], ecology [27][28][29] and cell biology [19,25], amongst others.…”

“…As it is unlikely that any two pairs of individuals are separated by exactly the same distance, m is typically divided into bins [20]. This can take the form of considering the environment as continuous space and binning the measured distance between pairs of individuals, or by mapping individuals onto a discrete domain such that there is a finite number of possible distances between pairs of individuals [19,20,24,25]. There has been significant recent focus on PCFs for discrete, or lattice-based, domains [17,19,20,[24][25][26], and deriving analytic expressions for the normalization term under various distance metrics [19,24].…”

“…This can take the form of considering the environment as continuous space and binning the measured distance between pairs of individuals, or by mapping individuals onto a discrete domain such that there is a finite number of possible distances between pairs of individuals [19,20,24,25]. There has been significant recent focus on PCFs for discrete, or lattice-based, domains [17,19,20,[24][25][26], and deriving analytic expressions for the normalization term under various distance metrics [19,24]. In particular, Binder and Simpson [19] present a normalization term for rectilinear distance in x and y, Under the taxicab distance metric and the square uniform distance metric, the distance between two lattice sites can be thought of as the number of "jumps" between the two sites under movement occurring in a von Neumann neighborhood (four nearest-neighbors) and a Moore neighborhood (eight nearest-neighbors), respectively [24].…”