Probabilistic Machine Learning Methods for Fractional Brownian Motion Time Series Forecasting

Abstract

This paper explores the capabilities of machine learning for the probabilistic forecasting of fractional Brownian motion (fBm). The focus is on predicting the probability of the value of an fBm time series exceeding a certain threshold after a specific number of time steps, given only the knowledge of its Hurst exponent. The study aims to determine if the self-similarity property is preserved in a forecasting time series and which machine learning algorithms are the most effective. Two types of forecasting methods are investigated: methods with a predefined distribution shape and those without. The results show that the self-similar properties of the fBm time series can be reliably reproduced in the continuations of the time series predicted by machine learning methods. The study also provides an experimental comparison of various probabilistic forecasting methods and their potential applications in the analysis and modeling of fractal time series.

Nonlinear Filter-Based Adaptive Output-Feedback Control for Uncertain Fractional-Order Nonlinear Systems with Unknown External Disturbance

“…As the extension and generalization of integral calculus, fractional calculus, on the one hand, is more applicable for describing systems with memory and history-dependent processes, and can more accurately model and characterize objective phenomena that cannot be described by an integer-order (IO) system [1,2], such as the semi-derivative relationship between heat flow and temperature, and the "trailing" phenomenon of solute transport in porous media, etc. Fractional calculus, on the other hand, has higher design degrees of freedom and can, therefore, exhibit better robustness and transient performance, such as in FO CRONE controllers [3] and PI λ D µ controllers [4].…”