Prediction of High-Dimensional Time Series with Exogenous Variables Using Generalized Koopman Operator Framework in Reproducing Kernel Hilbert Space

Abstract

We propose a novel methodology to predict high-dimensional time series with exogenous variables using Koopman operator framework, by assuming that the time series are generated by some underlying unknown dynamical system with input as exogenous variables. In order to do that, we first generalize the definition of the original Koopman operator to allow for input to the underlying dynamical system. We then obtain a formulation of the generalized Koopman operator in reproducing kernel Hilbert space (RKHS) and a new derivation of its numerical approximation methods, namely, Extended Dynamic Mode Decomposition (EDMD) and its kernel-based version. We also obtain a statistical interpretation of kernel-based EDMD developed for deterministic Koopman operator by utilizing the connection between RKHS and Gaussian processes regression, and relate it to the stochastic Koopman and Perron–Frobenius operator. In applications, we found that the prediction performance of this methodology is promising in forecasting real-world high-dimensional time series with exogenous variables, including financial markets data. We believe that this methodology will be of interest to the community of scientists and engineers working on quantitative finance, econometrics, system biology, neurosciences, meteorology, oceanography, system identification and control, data mining, machine learning, computational intelligence, and many other fields involving high-dimensional time series and spatiotemporal data.