Learning fractional order dynamics from a single observed system trajectory
Learning Fractional-Order Dynamics from a Single Trajectory
Machine Learning
Summary
Some systems remember their whole history instead of just the latest state, making them tricky to understand. The authors study how to figure out how such systems behave by looking at just one long sequence of observations. They created a method called FO-GS that breaks the problem into smaller parts to estimate the system’s memory and rules. They prove their method gets better with more data and performs well in tests compared to older approaches.
What this means in practice
- •For control engineers: Identify controllers for systems with memory effects from limited data, improving control design for processes with long-term dependencies.
- •For financial analysts: Better model asset price dynamics exhibiting long-range dependence by estimating fractional orders from observed price trajectories.
Authors
Xiaole Zhang, Ziyi Zhang, Zehao Zhao, Stephen Tu, Guannan Qu, Yorie Nakahira, Paul Bogdan
Abstract
Many real-world processes exhibit long-range dependence, where the current state depends on a slowly decaying trace of past states rather than on the most recent state alone. This paper studies system identification for discrete-time fractional-order linear time-invariant systems from a single observed trajectory of length $t$, a setting that captures such non-Markovian dynamics through the Grünwald--Letnikov difference operator. Unlike Markovian systems, fractional-order systems couple estimation across the entire history, making both statistical analysis and practical identification more challenging. We propose \emph{Fractional-Order Ordinary-Least-Squares Grid-Search (FO-GS)}, a simple two-stage estimator that exploits the diagonal structure of the fractional-difference operator to decouple the identification problem row-wise. Under the stability assumption, we establish high-probability, non-asymptotic error bounds for estimating both the fractional order and the system matrix in the heterogeneous setting, with both estimation errors scaling as \(\mathcal{O}(t^{-1/2})\). Through experiments, we show that \emph{FO-GS} outperforms existing baselines in recovering both the fractional order and the underlying system dynamics.