Random Fields with a Time-Varying Graphical Structure

• Activity Number: 257
• Type: Contributed
• Date/Time: Tuesday, August 13, 2002 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing*
• Author(s): Makram Talih*+ and Nicolas Hengartner
• Affiliation(s): Yale University and Yale University
• Keywords: Covariance matrix estimation ; Markov random field ; Decomposable graphical model ; Multivariate stochastic volatility ; Markov chain Monte Carlo ; Spatio-temporal modeling

Abstract

In the multivariate Normal case, a graph constrains the covariance matrix of (large-dimensional) data by imposing a pattern of zeroes in its inverse, and its decomposability ensures comparison between two nested models can be localized. Heteroscedascity can then be thought of as resulting from a slowly varying sequence of underlying graphs. For $B-1$ changepoints, the state space consists of: (i) inter-arrival times $N = (N_1, \cdots, N_B); $ (ii) a Markov chain of decomposable graphs $G = (G_1, \cdots, G_B), $ with one-step transition probabilities $P(h,g) $ given by $1/d(h), $ where $d(h)$ is the number of decomposable graphs $g$ such that at most one change (dropping or adding one edge) separates $g$ from $h;$ (iii) and an unconstrained covariance matrix $\Sigma^*, $ which, for each $b,$ is the completion of the constrained covariance matrix $\Sigma_b$ relative to $G_b.$ A carefully designed proposal strategy takes advantage of the form of the transition matrix $\{P(h,g)\}, $ and allows for a relatively expeditious exploration of the state space via a Metropolis-Hastings chain, facilitating posterior estimation of $\Sigma^*$ and prediction of $G.$