Equivalent Relation between Normalized Spatial Entropy and Fractal Dimension

TL;DR

This paper establishes a theoretical and empirical equivalence between normalized spatial entropy and fractal dimension, enabling conversion and unified analysis of complex spatial systems using the functional box-counting method.

Contribution

It reveals a new equivalence relation between spatial entropy and fractal dimension, providing a basis for unified spatial measurement and analysis of complex systems.

Findings

Normalized spatial entropy equals normalized fractal dimension.

The ratio of actual to maximum entropy equals the ratio of actual to maximum dimension.

Empirical data of urban form supports the theoretical equivalence.

Abstract

Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy to fractal dimension is not yet clear in both theory and practice. This paper is devoted to revealing the new equivalence relation between spatial entropy and fractal dimension using functional box-counting method. Based on varied regular fractals, the numerical relationship between spatial entropy and fractal dimension is examined. The results show that the ratio of actual entropy (Mq) to the maximum entropy (Mmax) equals the ratio of actual dimension (Dq) to the maximum dimension (Dmax). The spatial entropy and fractal dimension of complex spatial systems can be converted into one another by means of functional box-counting method. The theoretical inference is verified by observational data of urban form. A conclusion is that normalized spatial entropy is equal to normalized fractal dimension. Fractal dimensions proved to be the characteristic values of entropies. In empirical studies, if the linear size of spatial measurement is small enough, a normalized entropy value is infinitely approximate to the corresponding normalized fractal dimension value. Based on the theoretical result, new spatial measurements of urban space filling can be defined, and multifractal parameters can be generalized to describe both simple systems and complex systems.