Multifractal of random permutation set

Random permutation sets (RPS) was recently proposed, considering all possible permutations of elements in the power set of the Dempster-Shafer evidence theory. In order to explore the fractal characteristics of RPS, the information fractal dimension of RPS is proposed to reveal the fractal characteristics of RPS entropy under scale invariance. However, exploring the information fractal dimension of RPS at different scales and being fully compatible with the Rényi information dimension is still an open problem. To solve this problem, the multifractal of RPS is proposed. In the proposed method, the multifractal spectrum of RPS is first proposed to provide a new method for revealing the structure of RPS by representing permutation mass function (PMF) through coordinate points. Then, the multifractal of RPS is proposed and its strong compatibility in integrating information dimension and multifractal in RPS, evidence theory and probability theory is demonstrated. Meanwhile, its numerator is defined as the Generalized RPS entropy, which is compatible with RPS entropy corresponding to the information fractal dimension of RPS. After that, an interesting property is to be found: the multifractal dimension of the maximum RPS entropy distribution is 2 no matter how α varies, which is equal to the fractal dimensions of Julia sets, Mandelbrot set, Peano curve, and Brownian motion. In particular, under the exclusive subset distributions, the multifractal of RPS can acquire arbitrary dimension as α varies. At last, we illustrate the validity of the model with properties and numerical examples.