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Learning Fractional-Order Dynamics from a Single Trajectory

Xiaole Zhang, Ziyi Zhang, Zehao Zhao, Stephen Tu, Guannan Qu, Yorie Nakahira, Paul Bogdan

arXiv:2609.18127v1cs.LGeess.SY

TL;DR

Long-range dependence makes single-trajectory identification of fractional-order systems difficult because the dynamics depend on the full history. The paper introduces FO-GS, a row-wise grid-search and least-squares estimator, and reports high-probability recovery guarantees with both errors scaling as O(t^-1/2).

  • Problem

    The paper asks whether the fractional order and system matrix of a non-Markovian FOLTI system can be identified from one trajectory despite history coupling and temporal dependence.

  • Method

    FO-GS exploits the diagonal Grünwald–Letnikov operator to decouple estimation row-wise, combining coordinate-wise fractional-order grid searches with conditional least squares.

  • Results

    Under stability, FO-GS provides high-probability non-asymptotic recovery guarantees for both parameters, with estimation errors scaling as O(t^-1/2), and outperforms baselines on synthetic and EEG data.

  • Takeaways & Limitations

    The results indicate that direct single-trajectory identification of FOLTI systems is statistically analyzable and practically effective despite non-Markovian dynamics.

  • Takeaways & Limitations

    The analysis assumes stability, and the authors identify relaxing this assumption as future work.

Abstract

from arXiv · show

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.

1 Introduction

Long-range dependence makes Markovian models inadequate because system states retain influence from the full history. The paper formulates single-trajectory identification of fractional-order LTI systems and introduces FO-GS with non-asymptotic recovery guarantees.

  • Long-range dependence appears as power-law temporal correlation decay across natural and technological systems.
  • Fractional-order dynamics replace one-step recursion with a weighted sum over the full history, addressing the limitations of Markovian models.
  • The paper studies estimating the fractional-order vector and system matrix for heterogeneous and homogeneous FOLTI systems from one observed trajectory.
  • Single-trajectory identification must handle history coupling, nonlinear dependence on the fractional order, temporal dependence, and no independent rollouts.
  • FO-GS uses row-wise grid searches over fractional-order candidates and conditional least-squares estimates, selecting each pair by a profiled loss.
  • Under stability, FO-GS recovers both parameters with high probability and estimation errors scaling as O(t^-1/2), with validation on synthetic and real-world data.

2 Related Work

The paper extends ordinary LTI identification ideas to fractional-order systems, where single-trajectory identification remains less developed. It distinguishes its direct treatment of the original problem from prior multi-trajectory and truncated-system approaches.

  • The proposed approach adapts OLS-based identification and grid search to exploit the diagonal fractional operator and estimate each coordinate-row pair.
  • Fractional-order system identification has received substantially less attention than ordinary LTI system identification.
  • Prior work with multiple trajectories estimates fractional parameters before regressing unknown dynamics, including extensions to stochastic settings.
  • A prior single-trajectory approach uses system truncation and provides a sample-complexity result for an augmented system matrix.
  • This paper directly studies the original single-trajectory problem without prescribed data generation or system truncation and gives guarantees for both parameter types.

3 Preliminaries and Problem Formulation

The paper defines discrete-time FOLTI dynamics through the Grünwald–Letnikov operator and poses recovery of the fractional order and system matrix from one noisy trajectory. FO-GS implements this recovery through coordinate-wise grid search and row-wise least squares.

  • 3.1 Grünwald–Letnikov Difference Operator: The Grünwald–Letnikov operator discretizes fractional derivatives as finite differences involving a diagonal coefficient matrix.
  • 3.1 Grünwald–Letnikov Difference Operator: The fractional-order vector contains coordinate-specific values, and the operator coefficients are defined using the gamma function.
  • 3.2 FOLTI System Identification: The FOLTI model relates the fractional difference of the next state to the current state through a constant matrix and independent Gaussian noise.
  • 3.2 FOLTI System Identification: The discrete-time system also admits a solution represented through recursively defined matrices G_s.
  • 3.2 FOLTI System Identification: Given one trajectory, the identification problem is to estimate the fractional order and system matrix while establishing statistical guarantees.
  • 3.2 FOLTI System Identification: FO-GS builds coordinate-specific order grids, forms the data matrix, computes least-squares row estimates for each candidate, and evaluates profiled losses.

4 Main Results

FO-GS identifies fractional orders and system rows through coordinate-wise grid search and row-wise OLS, with guarantees under stability and excitation conditions. Theoretical errors achieve t^-1/2 scaling when grid discretization is chosen comparably, while smaller fractional orders require longer trajectories.

  • 4.1 FO-GS: FO-GS searches each fractional-order coordinate on a grid and solves a corresponding row-wise OLS problem, then stacks the row estimators to recover A.In the homogeneous case, the coordinate-wise searches reduce to one shared one-dimensional search.
  • 4.2 Theoretical Guarantees: The guarantees rely on a stability assumption stronger than minimal stability; without stability, early errors may be exponentially amplified through stochastic coupling.The assumption also implies invertibility of A and prevents the state from blowing up over time.
  • 4.2 Theoretical Guarantees: Under stability, excitation, and localization conditions, Theorem 1 provides a high-probability error bound for estimating the fractional-order vector over the finite search grid.The trajectory must be sufficiently long for uniform global and local lower-isometry bounds.
  • 4.2 Theoretical Guarantees: Choosing the maximum grid step as ϵmax = O(t^-1/2) balances statistical and discretization errors, yielding t^-1/2 fractional-order accuracy and requiring O(t^1/2) grid points on bounded intervals.Smaller fractional orders require longer trajectories because of stronger long-memory dependence.
  • 4.2 Theoretical Guarantees: The system-matrix error combines an oracle least-squares term with propagation from fractional-order estimation, so A achieves the same t^-1/2 rate when ϵmax = O(t^-1/2).When α is known and equals 1, the result matches the state-of-the-art bound for ordinary LTI system-matrix estimation.

5 Proof Outline

The proof first controls fractional-order estimation through in-sample error and lower isometry, then decomposes system-matrix error into noise and bias terms. Row-wise complexity and grid localization yield the stated t^-1/2 fractional-order rate.

  • Bounding α⋆: The proof bounds fractional-order error by combining an in-sample upper bound with a quadratic lower-isometry bound after localizing each estimator to a neighborhood of the truth.The lower-isometry argument converts prediction error into a quadratic bound on fractional-order error.
  • Bounding α⋆: Row-wise decomposition avoids a supremum over the full Cartesian grid and provides the sharp complexity dependence for the grid-search estimator.This follows from the fractional-order operator’s diagonal structure and separate searches over each αi.
  • Bounding α⋆: Choosing grid spacing ϵmax = O(t^-1/2) yields the t^-1/2 fractional-order estimation rate.The finite-grid approximation term vanishes when the true orders lie on the search grid.
  • Bounding A⋆: For fixed fractional order, the system-matrix identification error is decomposed into separate noise and bias terms.The decomposition uses Wt and Bt(α) to isolate stochastic noise from grid-induced or approximation bias.
  • Proof structure: The proof proceeds in two stages, first bounding α⋆ estimation error and then bounding the system-matrix error under the stability assumption.The detailed bounds for the noise and bias components are supplied through separate lemmas.

6 Experiments

Experiments show that FO-GS consistently outperforms existing baselines on synthetic identification and provides superior predictive performance on EEG data. Its advantages persist across trajectory horizons, noise scales, and moderate grid sizes, while small fractional orders require longer trajectories to approach the predicted convergence rate.

  • Synthetic Experiments: FO-GS outperforms FO-BS and FO-WT in estimating both the fractional order and system matrix across trajectory lengths and noise levels.The advantage is particularly pronounced for shorter trajectories, and all methods’ MSE increases as noise grows.
  • Synthetic Experiments: FO-GS surpasses both baselines with roughly ten grid points for fractional-order estimation and roughly five for system-matrix estimation.Both estimation errors decrease as the number of grid points increases.
  • Synthetic Experiments: In the small-α regime, empirical convergence is slower over short horizons but approaches the predicted t^-1/2 rate as trajectories lengthen.Over long horizons, fitted rates are approximately t^-0.36 and t^-0.38 overall, improving to approximately t^-0.43 and t^-0.42 when t ≥ 3200.
  • Real-World Experiments: On EEG mental-arithmetic data, FO-GS delivers superior predictive performance for subject-level one-step NMSE compared with FO-BS and FO-WT.The evaluation uses subject-level averages over non-overlapping windows.
  • Real-World Experiments: FO-GS learns heterogeneous non-integer memory across EEG channels, with a median window-averaged order of 0.855.A cross-channel order range greater than 0.5 occurs in 74.6% of windows, and 72.8% contain a channel with α < 0.1.

7 Conclusion

The paper introduces FO-GS for identifying fractional-order LTI systems from a single trajectory and establishes its statistical and empirical effectiveness. It concludes that direct identification remains feasible despite non-Markovian dynamics, while identifying grid-search cost and stability as future concerns.

  • 7 Conclusion: FO-GS decouples row-wise estimation of the fractional order and system matrix by exploiting the Grünwald–Letnikov operator’s diagonal structure.The estimator is a two-stage procedure for single-trajectory stochastic FOLTI identification.
  • 7 Conclusion: Under a stability assumption, FO-GS has high-probability non-asymptotic guarantees for recovering both parameters and outperforms existing baselines on synthetic and EEG data.These results support direct single-trajectory identification of FOLTI systems.
  • 7 Conclusion: Future work should reduce grid-search cost and relax the stability assumption.

A.1 Synthetic Experiments

Synthetic experiments generate controlled fractional-order LTI trajectories while varying trajectory length, noise, and FO-GS grid resolution. The evaluation compares FO-GS with FO-BS and FO-WT under specified simulation settings.

  • A.1 Synthetic Experiments: Synthetic systems are two-dimensional, with independently sampled fractional orders in [0.1, 0.5] and system eigenvalues in [−0.5, 0.5].
  • A.1 Synthetic Experiments: FO-GS is compared with FO-BS using truncation and binary search and FO-WT using Haar-wavelet order estimation.
  • A.1 Synthetic Experiments: Trajectory-horizon experiments vary t from 50 to 500 while fixing σ = 0.1.Initial states use zero-mean Gaussian sampling with standard deviation 4.0.
  • A.1 Synthetic Experiments: Noise-robustness experiments fix t = 200 and vary σ from 0.02 to 0.40.Initial states again use a zero-mean Gaussian distribution with standard deviation 4.0.
  • A.1 Synthetic Experiments: Grid-resolution experiments fix t = 100 and σ = 0.01 while varying the number of grid points from 3 to 25.Initial states use standard deviation 2.0.

A.2 Real-World Experiments

Real-world evaluation uses multichannel EEG recordings segmented into fixed non-overlapping windows. FO-GS, FO-BS, and FO-WT are assessed through one-step training and test NMSE under fixed search and approximation settings.

  • A.2 Real-World Experiments: The EEG dataset contains 36 subjects and 19 channels, with each trajectory split into non-overlapping windows of length 150.Each window contributes 105 training samples and 45 test samples.
  • A.2 Real-World Experiments: The first 70% of each window is used for training and the remaining 30% for testing, without additional normalization.
  • A.2 Real-World Experiments: FO-GS searches [0.05, 0.95] with 50 equally spaced points per row, while FO-BS uses p = 40 truncation and FO-WT uses Haar-wavelet regression.
  • A.2 Real-World Experiments: Methods are evaluated by one-step prediction using training and test NMSE.

B.1 Controlling In-Sample Error via Martingale Offset Complexity

The analysis adapts offset martingale complexity to FO-GS’s row-wise grid search, accounting for finite-grid misspecification and exploiting coordinate-wise decomposition. High-probability in-sample bounds are then combined with lower-isometry arguments to control fractional-order estimation.

  • Grid discretization: A finite search grid adds a discretization error term because the true fractional parameter need not lie on the grid.When every true parameter lies on its corresponding grid, this term vanishes and the continuous ERM offset inequality is recovered.
  • Row-wise complexity: FO-GS’s row-wise structure decomposes the offset complexity across coordinates instead of requiring a supremum over the full Cartesian grid.This decomposition yields the sharper complexity dependence of the grid-search estimator.
  • Error control: The resulting high-probability row-wise in-sample bounds are combined with global and local lower-isometry arguments to absorb quadratic estimation-error terms.The global bound excludes grid points outside a separation neighborhood, while local curvature transfers population separation to empirical profiled error.

C.1 Proof of Lemma 3

The proof establishes concentration and dependence controls for the fractional-order system by combining block martingale small-ball arguments with decay bounds for fractional coefficients and state covariances. These ingredients support the theorem’s high-probability error analysis under the stability assumption.

  • Small-ball control: The state process satisfies a block martingale small-ball condition with Γ_sb = σ^2H_⌊k/2⌋ and p = 3/20.The condition follows from conditional Gaussianity, monotonicity of the covariance matrices, and a Paley–Zygmund bound.
  • Dependence control: Fractional coefficients decay as max_i |ψ(α_i,⋆, m)| ≤ m^−(1+α_min), yielding polynomial covariance decay.The resulting covariance bound is ∥Cov(x_p, x_{p+k})∥op ≤ e^C_x(k + 1)^−(1+α_min).
  • Uniform concentration: Gaussian quadratic-form concentration and net arguments provide simultaneous high-probability bounds across rows, while stationary spectral analysis supplies population curvature controls.The proof uses union bounds over coordinates and derives curvature-related lower bounds under invertibility on the unit circle.
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