Finite-Time Model Inference From A Single Noisy Trajectory

Abstract

This paper proposes a novel model inference procedure to identify system matrix from a single noisy trajectory over a finite-time interval. The proposed inference procedure comprises an observation data processor, a redundant data processor and an ordinary least-square estimator, wherein the data processors mitigate the influence of observation noise on inference error. We first systematically investigate the comparisons with naive least-square-regression based model inference and uncover that 1) the same observation data has identical influence on the feasibility of the proposed and the naive model inferences, 2) the naive model inference uses all of the redundant data, while the proposed model inference optimally uses the basis and the redundant data. We then study the sample complexity of the proposed model inference in the presence of observation noise, which leads to the dependence of the processed bias in the observed system trajectory on time and coordinates. Particularly, we derive the sample-complexity upper bound (on the number of observations sufficient to infer a model with prescribed levels of accuracy and confidence) and the sample-complexity lower bound (high-probability lower bound on model error). Finally, the proposed model inference is numerically validated and analyzed.

Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems

“…where S t 1 is defined in (25). Now taking expectation, and using lemma 6 and Caucy-Schwarz inequality for the first and second terms, respectively, in (33) we obtain the claim.…”

“…[23] considers the non linear dynamical systems of the form x t+1 = Aφ(x t , u t ) + η t which φ is a known non-linearity and matrix A is to be estimated. [24,25] consider essentially linear dynamics but allow for certain non-linearities that can be modeled as process noise. All these again differ from the model we consider.…”