Collaborative Research: Novel modeling and Bayesian analysis of high-dimensional time series

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Every aspect of modern life including economy and finance, communication, and medical records, is associated with large amounts of data on several measurements, often evolving over time. Understanding the progress over time, finding an intrinsic relationship among different variables, and predicting future observations are essential components of decision and policy-making. However, apparent relations between two variables can appear in data caused by their shared association with other components. The principal investigators (PIs) will develop a model to re-express the multi-dimensional time series in independent, one-dimensional, latent time series. The representation will explain the evolution of the data over time and the intrinsic relations present in the component variables. It can also help find a more accurate, efficiently computable prediction formula for future observations by pulling information across different components and time. The approach's simplicity and generality will make it widely applicable and adaptable to diverse fields in economics, finance, social sciences, communications, networks, neuroimaging, and others. The PIs plan to develop free software packages to disseminate the results. They are committed to supporting young researchers and promoting diversity through graduate student training and involvement in the REU program.The developed framework is based on representing an observed multi-dimensional time series as a linear combination of several independent stationary latent processes. The individual latent time series are modeled flexibly with unspecified spectral densities. The PIs will study the conditional independence structure among component time series and the causality of the time series over the temporal domain using a Bayesian approach. They will put independent priors on individual spectral densities through a finite random series prior, and on the matrix of the linear transformation decomposed as a product of a sparse matrix and an orthogonal matrix, the former of which induces a graphical structure for conditional independence among component series. Through this representation, desirable stationarity and causality structures can be imposed. Decoupling through the Whittle likelihood approximation and Hamiltonian Monte-Carlo methods will allow efficient posterior sampling. The causality over nodal time series will be addressed by a Direct Acyclic Graph modeling of the residual process. The formulation seamlessly addresses a mixed frequency sampling situation, difficult to incorporate into competing methods. The developed framework efficiently addresses both temporal and nodal causality respectively by characterization in terms of the Schur-complementation and using a directed acyclic graph, allowing a natural interpretation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.