Discrete Manhattan and Chebyshev pair correlation functions in k dimensions.

Area of Science:

• Spatial statistics
• Discrete mathematics
• Geostatistics

Background:

• Pair correlation functions (PCFs) are vital for quantifying spatial correlations in point processes.
• Existing PCF methods primarily focus on continuous spaces, with less attention to discrete, on-lattice scenarios.
• Current on-lattice PCFs rely on ad hoc derivations of distance distributions for specific metrics.

Purpose of the Study:

• To develop a generalized method for deriving pair distance probability distributions in k-dimensional lattices.
• To extend Manhattan and Chebyshev pair correlation functions to k-dimensional discrete spaces.
• To quantify the variability of these discrete PCFs for assessing statistical reliability.

Main Methods:

• Derivation of generalized probability distributions for pair distances using Manhattan and Chebyshev metrics.
• Extension of these distributions to k-dimensional lattices.
• Quantification of the variability in the resulting pair correlation functions.

Main Results:

• A unified framework for calculating pair distance distributions in discrete k-dimensional spaces.
• Generalized Manhattan and Chebyshev pair correlation functions applicable to various lattice dimensions.
• Quantification of PCF variability, offering insights into confidence intervals.

Conclusions:

• The generalized approach provides a standardized method for analyzing spatial correlations in discrete lattice data.
• The extended PCFs offer a more reliable and interpretable measure of spatial patterns in k-dimensions.
• Understanding PCF variability is crucial for accurate interpretation of spatial data analysis.