Multivariable-based correlation dimension analysis for generalized space

TL;DR

This paper extends fractal geometry and correlation dimension analysis to abstract geographical spaces defined by multivariable distance metrics, supported by mathematical derivation and empirical data.

Contribution

It develops a new correlation dimension analysis method for generalized geographical space using multivariable metrics, broadening fractal theory applications.

Findings

Mathematically proved the correlation between correlation function and correlation length in generalized space.

Demonstrated the applicability of fractal dimension analysis to abstract geographical spaces.

Validated the analytical method with observational data.

Abstract

Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been reported. An abstract space can be defined by multivariable distance metrics, which is frequently met in scientific research. Based on the ideas from fractals, this paper is devoted to developing correlation dimension analysis method for generalized geographical space by means of mathematical derivation and empirical analysis. Defining a mathematical distance or statistical distance, we can construct a generalized correlation function. If the relationship between correlation function and correlation lengths follows a power law, the power exponent can be demonstrated to associate with correlation dimension. Thus fractal dimension can be employed to analyze the structure and nature of generalized geographical space. This suggests that fractal geometry can be generalized to explore scale-free abstract geographical space. The theoretical model was proved mathematically, and the analytical method was illustrated by using observational data. This research is helpful to expand the application of fractal theory in geographical analysis, and the results and conclusions can be extended to other scientific fields.