Embeddings and Prediction of Dynamical Time Series

Delay coordinate embeddings;Dynamical time series prediction;Mathematics;Science;Applied and Interdisciplinary Mathematics

Navarrete, RaymundoZhou, Shuheng ;

University of Michigan

关键词: Delay coordinate embeddings;    Dynamical time series prediction;    Mathematics;    Science;    Applied and Interdisciplinary Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/145923/raymundo_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

A dynamical time series is a sequence of real-valued observations of a dynamical system. Commonly in applications, the dynamical system of interest is unknown and only a dynamical time series is observed. Dynamical time series arise in ergodic theory in mathematics, nonlinear dynamics in physics, state space modeling in statistics, and control theory in engineering. We consider two common goals in the analysis of dynamical time series. First, it is desirable to construct a faithful representation of the state of the dynamical system using only the time series. Delay-time coordinates are widely used for this purpose. Under certain conditions, the delay map taking the state of the dynamical system to its corresponding delay-time coordinates is generically an embedding of the state space. More precisely, current work shows that for a fixed dynamical system, delay-time coordinates result in embeddings of the state space generically with respect to the observation function. Motivated by common usage of delay-time coordinates, we consider the more difficult situation where the observation function is fixed and genericity is studied with respect to the dynamical system. We prove that delay-time coordinates result in embeddings of the state space for polynomial perturbations of the dynamical system with probability one over the perturbing coefficients. Second, it is desirable to predict time series accurately. Prediction of dynamical time series with additive noise using kernel-based regression is consistent for certain classes of discrete dynamical systems. These methods are effective at computing the expected value of the time series at a future time given the present delay-time coordinates. However, the present coordinates themselves are noisy, so these methods are only optimal when it is not possible to remove noise. We consider the problem of prediction for flows, and show that the use of smoothing splines to reduce noise before using kernel-based regression results in increased prediction accuracy. We prove that our method is consistent, converging to the exact predictor based on the noiseless time series, in the limit as sampling frequency and sampling time go to infinity.