Information fractal dimension of mass function

TL;DR

This paper introduces an information fractal dimension for mass functions, extending the concept of fractal dimension from probability distributions to a more general framework, with numerical validation and a notable property related to Deng entropy.

Contribution

It proposes a novel information fractal dimension for mass functions, addressing an open problem and linking it to known fractal dimensions like that of the Sierpiński triangle.

Findings

The proposed dimension effectively characterizes mass functions.

Maximum Deng entropy corresponds to a fractal dimension of approximately 1.585.

The dimension aligns with the fractal dimension of the Sierpiński triangle.

Abstract

Fractal plays an important role in nonlinear science. The most important parameter to model fractal is fractal dimension. Existing information dimension can calculate the dimension of probability distribution. However, given a mass function which is the generalization of probability distribution, how to determine its fractal dimension is still an open problem of immense interest. The main contribution of this work is to propose an information fractal dimension of mass function. Numerical examples are illustrated to show the effectiveness of our proposed dimension. We discover an important property in that the dimension of mass function with the maximum Deng entropy is $\frac{ln3}{ln2}\approx 1.585$, which is the well-known fractal dimension of Sierpi\'nski triangle.