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.

Keywords

• fractional Brownian motion
• hurst exponent
• time series
• probabilistic forecast
• quantile regression
• distributional modelling