Bayesian Inference for High-dimensional Time Series by Latent Process Modeling
arxiv(2024)
摘要
Time series data arising in many applications nowadays are high-dimensional.
A large number of parameters describe features of these time series. We propose
a novel approach to modeling a high-dimensional time series through several
independent univariate time series, which are then orthogonally rotated and
sparsely linearly transformed. With this approach, any specified intrinsic
relations among component time series given by a graphical structure can be
maintained at all time snapshots. We call the resulting process an
Orthogonally-rotated Univariate Time series (OUT). Key structural properties of
time series such as stationarity and causality can be easily accommodated in
the OUT model. For Bayesian inference, we put suitable prior distributions on
the spectral densities of the independent latent times series, the orthogonal
rotation matrix, and the common precision matrix of the component times series
at every time point. A likelihood is constructed using the Whittle
approximation for univariate latent time series. An efficient Markov Chain
Monte Carlo (MCMC) algorithm is developed for posterior computation. We study
the convergence of the pseudo-posterior distribution based on the Whittle
likelihood for the model's parameters upon developing a new general posterior
convergence theorem for pseudo-posteriors. We find that the posterior
contraction rate for independent observations essentially prevails in the OUT
model under very mild conditions on the temporal dependence described in terms
of the smoothness of the corresponding spectral densities. Through a simulation
study, we compare the accuracy of estimating the parameters and identifying the
graphical structure with other approaches. We apply the proposed methodology to
analyze a dataset on different industrial components of the US gross domestic
product between 2010 and 2019 and predict future observations.
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