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 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.
There has been remarkable progress over the past decade in establishing finite-sample, non-asymptotic bounds on recovering unknown system parameters from observed system behavior. Surprisingly, however, we show that the current state-of-the-art bounds do not accurately capture the statistical complexity of system identification, even in the most fundamental setting of estimating a discrete-time linear dynamical system (LDS) via ordinary least-squares regression (OLS). Specifically, we utilize asymptotic normality to identify classes of problem instances for which current bounds overstate the squared parameter error, in both spectral and Frobenius norm, by a factor of the state-dimension of the system. Informed by this discrepancy, we then sharpen the OLS parameter error bounds via a novel second-order decomposition of the parameter error, where crucially the lower-order term is a matrix-valued martingale that we show correctly captures the CLT scaling. From our analysis we obtain finite-sample bounds for both (i) stable systems and (ii) the many-trajectories setting that match the instance-specific optimal rates up to constant factors in Frobenius norm, and polylogarithmic state-dimension factors in spectral norm.
We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.
Does a neuromorphic system need a true power-law memory kernel, and if so, can anyone build one? Neuromorphic systems process signals spanning many timescales at once, from milliseconds to tens of seconds. Integer-order circuits buy each additional timescale with an additional state variable. Fractional-order dynamics offer a different bargain: one operator whose power-law kernel carries a continuum of timescales, tuned by one parameter, the order alpha. A fractional derivative is non-local, so evaluating it costs storage and arithmetic that grow with the retained history, where an integer-order derivative costs a constant. This review organizes the hardware literature around that cost. We derive the retained history needed to hold the truncation error below a tolerance epsilon, show that it scales as epsilon^(-1/alpha), and set beside it a second and independent limit on the direct form: in fixed point the weights themselves underflow, so word length caps the usable history however long the buffer is. The two limits move at very different rates with the order, and where they cross decides whether a word length can serve an order at all. We use both to sort published hardware into three strategies, note a fourth the numerical literature has developed and this hardware has not, and survey digital, analog and device work. Along the way we ask whether the field is worried about the right obstacle. It is not. Fabricated constant-phase devices already span the orders two groups identify as task-optimal, so the order gap has largely closed, leaving a residual gap near 0.1 and at the lower order describing cortical adaptation. What remains is a frequency-band gap of about three decades at the low end. That corner is not empty, since double-layer electrodes work there, but every device in it is discrete, and no integrable thin-film element has been characterized there.