Discrete Manhattan and Chebyshev pair correlation functions in $k$ dimensions

Pair correlation functions provide a summary statistic which quantifies the amount of spatial correlation between objects in a spatial domain. While pair correlation functions are commonly used to quantify continuous-space point processes, the on-lattice discrete case is less studied. Recent work has brought attention to the discrete case, wherein on-lattice pair correlation functions are formed by normalizing empirical pair distances against the probability distribution of random pair distances in a lattice with Manhattan and Chebyshev metrics. These distance distributions are typically derived on an ad hoc basis as required for specific applications. Here we present a generalized approach to deriving the probability distributions of pair distances in a lattice with discrete Manhattan and Chebyshev metrics, extending the Manhattan and Chebyshev pair correlation functions to lattices in $k$ dimensions. We also quantify the variability of the Manhattan and Chebyshev pair correlation functions, which is important to understanding the reliability and confidence of the statistic.

Figure 1

Schematic of the Manhattan and Chebyshev distance metric in two dimensions: (a) Manhattan; (b) Chebyshev.

Figure 2

Frequency and pair correlation function versus pair distance, $s$, for a two-dimensional $60\times 30$ spatial domain. The gray shading indicates the region between the 2.5th and 97.5th percentiles. Realizations $R=1000$, density $\rho =0.5$ ($\approx 900$ points). (a), (c), (e), (g) Left column: Pair distance frequency. Sample mean (solid curves) and distance distribution scaled by ${\rho }_2$ (dots). (b), (d), (f), (h) Right column: Pair correlation function. (a), (b) Top row: Nonperiodic Manhattan. (c), (d) Top-middle row: Periodic Manhattan. (e), (f) Bottom-middle row: Nonperiodic Chebyshev. (g), (h) Bottom row: Periodic Chebyshev.

Figure 3

Frequency and pair correlation function versus pair distance, $s$, for a three-dimensional $60\times 30\times 40$ spatial domain. The gray shading indicates the region between the 2.5th and 97.5th percentiles. Realizations $R=1000$, density $\rho =0.01$ ($\approx 720$ points). (a), (c), (e), (g) Left column: Pair distance frequency. Sample mean (solid curves) and distance distribution scaled by ${\rho }_2$ (dots). (b), (d), (f), (h) Right column: Pair correlation function. (a), (b) Top row: Nonperiodic Manhattan. (c), (d) Top-middle row: Periodic Manhattan. (e), (f) Bottom-middle row: Nonperiodic Chebyshev. (g), (h) Bottom row: Periodic Chebyshev.

Figure 4

Analysis of two-dimensional $50\times 50$ diagonal bar patterns with Manhattan pair correlation function (solid curves). The gray shading indicates the region between the 2.5th and 97.5th percentiles of the domains populated uniformly at random with the same density as in the diagonal patterns, realizations $R=1000$. (a), (c), (e) Left column: Thickened central bar pattern analysis. (b), (d), (f) Right column: Thickened off-center bar pattern analysis. (a), (b) Top row: Diagonal bar patterns. (c), (d) Middle row: Nonperiodic Manhattan pair correlation function. (e), (f) Bottom row: Periodic Manhattan pair correlation function.