A Bayesian Multivariate Functional Dynamic Linear Model

ABSTRACT

We present a Bayesian approach for modeling multivariate, dependent functional data. To account for the three dominant structural features in the data—functional, time dependent, and multivariate components—we extend hierarchical dynamic linear models for multivariate time series to the functional data setting. We also develop Bayesian spline theory in a more general constrained optimization framework. The proposed methods identify a time-invariant functional basis for the functional observations, which is smooth and interpretable, and can be made common across multivariate observations for additional information sharing. The Bayesian framework permits joint estimation of the model parameters, provides exact inference (up to MCMC error) on specific parameters, and allows generalized dependence structures. Sampling from the posterior distribution is accomplished with an efficient Gibbs sampling algorithm. We illustrate the proposed framework with two applications: (1) multi-economy yield curve data from the recent global recession, and (2) local field potential brain signals in rats, for which we develop a multivariate functional time series approach for multivariate time–frequency analysis. Supplementary materials, including R code and the multi-economy yield curve data, are available online.

KEYWORDS:

• Hierarchical Bayes
• Orthogonality constraint
• Spline
• Time–frequency analysis
• Yield curve

Supplementary Materials

The supplement contains the initialization procedure and MCMC sampling algorithm for the proposed model, provides MCMC diagnostics for the applications, presents additional details and extensions of the common trend model of Section 4.1.1, and shows additional figures relevant to the applications. Additionally, the R code is provided in the supplement.

Acknowledgment

The authors thank the editors and two referees for very helpful comments. We also thank Professor Eve De Rosa and Dr. Vladimir Ljubojevic for providing the LFP data and for their helpful discussions.

Funding

Financial support from NSF grant AST-1312903 (Kowal and Ruppert) and the Cornell University Institute of Biotechnology and the New York State Division of Science, Technology and Innovation (NYSTAR), a Xerox PARC Faculty Research Award, and NSF grant DMS-1455172 (Matteson) is gratefully acknowledged.