Heterogeneous Multivariate Functional Time Series Modeling: A State Space Approach

Functional data have been gaining increasing popularity in the field of time series analysis. However, so far modeling heterogeneous multivariate functional time series remains a research gap. To fill it, this paper proposes a time-varying functional state space model (TV-FSSM). It uses functional decomposition to extract features of the functional observations, where the decomposition coefficients are regarded as latent states that evolve according to a tensor autoregressive model. This two-layer structure can on the one hand efficiently extract continuous functional features, and on the other provide a flexible and generalized description of data heterogeneity among different time points. An expectation maximization (EM) framework is developed for parameter estimation, where regularization and constraints are incorporated for better model interoperability. As the sample size grows, an incremental learning version of the EM algorithm is given to efficiently update the model parameters. Some model properties, including model identifiability conditions, convergence issues, time complexities, and bounds of its one-step-ahead prediction errors, are also presented. Extensive experiments on both real and synthetic datasets are performed to evaluate the predictive accuracy and efficiency of the proposed framework.