Probability of constructing prediction model for observable of a dynamical process via time series

TL;DR

This paper demonstrates that, for linear dynamical systems, it is almost surely possible to construct prediction models for observables solely from observed time series data, even without prior knowledge of the system.

Contribution

It shows that in linear systems, one can almost surely determine the prediction model and characteristic polynomial from time series data alone, using Lebesgue measure theory.

Findings

Prediction models can be constructed with probability 1.

Characteristic polynomials of system matrices can be obtained.

Applicable to both discrete and continuous linear systems.

Abstract

One fundamental problem in studying dynamical process is whether it is possible and how to construct prediction model for an unknown system via sampled time series, especially in the modern big data era. The research in this area is beneficial to experimentalists in physics, chemistry, especially, in biological science, where it is hard to construct a prediction models by first principles. Therefore constructing prediction model for the observable of a complex system is of great practical significance in various areas of science and engineering. In the present paper, we show in terms of Lebesgue measure that, at least in the linear case, one can almost surely construct by linear algebra approach the prediction model about observable of unknown systems only via observed time series in the settings of discrete time and continuous time linear systems, even in the situation that one has no information about the underlying dynamical process and output function that generates the observed time series. Interestingly, we can obtain the characteristic polynomial of the unknown matrices for both of discrete time and continuous time linear systems with probability 1.