Fractal Dimension of Fractional Brownian Motion Based on Random Sets

Abstract

The fractal dimension of fractional Brownian motion can effectively describe random sets, reflecting the regularity implicit in complex random sets. Data mining algorithms based on fractal theory usually follow the calculation of the fractal dimension of fractional Brownian motion. However, the existing fractal dimension calculation methods of fractal Brownian motion have high time complexity and space complexity, which greatly reduces the efficiency of the algorithm and makes it difficult for the algorithm to adapt to high-speed and massive data flow environments. Therefore, several existing fractal dimension calculation methods of fractional Brownian motion are summarized and analyzed, and a random method is proposed, which uses a fixed memory space to quickly estimate the associated dimension of the data stream. Finally, a comparison experiment with existing algorithms proves the effectiveness of this random algorithm. Second, in the sense of two different measures, based on the principle of stochastic comparison, the stability of the stochastic fuzzy differential equations is derived using the stability of the comparison equations, and the practical stability criterion of two measures according to probability is obtained. Then, the stochastic fuzzy differential equations are discussed. The definition of stochastic exponential stability is given and the stochastic exponential stability criterion is proved.