Deep learning the Hurst parameter of linear fractional processes and assessing its reliability

TL;DR

This paper evaluates the effectiveness of LSTM deep learning models in estimating the Hurst parameter of various fractional stochastic processes, demonstrating superior performance over traditional methods for some processes but limited accuracy for others.

Contribution

It introduces a deep learning approach for Hurst parameter estimation across multiple fractional processes and assesses its reliability and limitations.

Findings

LSTM outperforms traditional methods for fBm and fOU processes.

Limited accuracy of LSTM on linear fractional stable motions.

Training data quality and process type significantly affect estimation accuracy.

Abstract

This research explores the reliability of deep learning, specifically Long Short-Term Memory (LSTM) networks, for estimating the Hurst parameter in fractional stochastic processes. The study focuses on three types of processes: fractional Brownian motion (fBm), fractional Ornstein-Uhlenbeck (fOU) process, and linear fractional stable motions (lfsm). The work involves a fast generation of extensive datasets for fBm and fOU to train the LSTM network on a large volume of data in a feasible time. The study analyses the accuracy of the LSTM network's Hurst parameter estimation regarding various performance measures like RMSE, MAE, MRE, and quantiles of the absolute and relative errors. It finds that LSTM outperforms the traditional statistical methods in the case of fBm and fOU processes; however, it has limited accuracy on lfsm processes. The research also delves into the implications of training length and valuation sequence length on the LSTM's performance. The methodology is applied by estimating the Hurst parameter in Li-ion battery degradation data and obtaining confidence bounds for the estimation. The study concludes that while deep learning methods show promise in parameter estimation of fractional processes, their effectiveness is contingent on the process type and the quality of training data.