In this work, we present an additive model for space-time data that splits the data into a temporally correlated component and a spatially correlated component. We model the spatially correlated portion using a time-varying Gaussian graphical model. Under assumptions on the smoothness of changes in covariance matrices, we derive strong single sample convergence results, confirming our ability to estimate meaningful graphical structures as they evolve over time. We apply our methodology to the discovery of time-varying spatial structures in human brain fMRI signals.
@cite_3 proposed a Bayesian additive time-varying graphical model, where the spatially-correlated noise term is a parameter of the driving noise covariance in a temporal dynamical model. Unlike our method, they did not estimate the temporal correlation, instead requiring the dynamical model to be pre-set. Our proposed method has wholly independent spatial and temporal models, directly estimating an inverse covariance graphical model for the temporal relationships of the data. This allows for a much richer temporal model and increases its applicability.
@cite_3
2074686342 2100241850 2112517951 2096018935
We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior. Computational challenges in this construction arise from the need for repeated evaluations of the forward model (e.g., in the context of Markov chain Monte Carlo) and are compounded by high dimensionality of the posterior. We address these challenges by introducing truncated Karhunen-Loeve expansions, based on the prior distribution, to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior. We seek a solution of this problem using Galerkin projection on a polynomial chaos basis, and use the solution to construct a reduced-dimensionality surrogate posterior density that is inexpensive to evaluate. We demonstrate the formulation on a transient diffusion equation with prescribed source terms, inferring the spatially-varying diffusivity of the medium from limited and noisy data. We consider the problem of Bayesian inference of graphical structure describing the interactions among multiple vector time-series. A directed temporal interaction model is presented which assumes a fixed dependence structure among time-series. Using a conjugate prior over this model’s structure and parameters, we focus our attention on characterizing the exact posterior uncertainty in the structure given data. The model is extended via the introduction of a dynamically evolving latent variable which indexes dependence structures over time. Performing inference using this model yields promising results when analyzing the interaction of multiple tracked moving objects. We develop a Bayesian modeling approach for tracking people in 3D from monocular video with unknown cameras. Modeling in 3D provides natural explanations for occlusions and smoothness discontinuities that result from projection, and allows priors on velocity and smoothness to be grounded in physical quantities: meters and seconds vs. pixels and frames. We pose the problem in the context of data association, in which observations are assigned to tracks. A correct application of Bayesian inference to multi-target tracking must address the fact that the model's dimension changes as tracks are added or removed, and thus, posterior densities of different hypotheses are not comparable. We address this by marginalizing out the trajectory parameters so the resulting posterior over data associations has constant dimension. This is made tractable by using (a) Gaussian process priors for smooth trajectories and (b) approximately Gaussian likelihood functions. Our approach provides a principled method for incorporating multiple sources of evidence, we present results using both optical flow and object detector outputs. Results are comparable to recent work on 3D tracking and, unlike others, our method requires no pre-calibrated cameras. This paper introduces a novel class of Bayesian models for multivariate time series analysis based on a synthesis of dynamic linear models and graphical models. The synthesis uses sparse graphical modelling ideas to introduce struc- tured, conditional independence relationships in the time-varying, cross-sectional covariance matrices of multiple time series. We dene this new class of models and their theoretical structure involving novel matrix-normal hyper-inverse Wishart distributions. We then describe the resulting Bayesian methodology and compu- tational strategies for model tting and prediction. This includes novel stochastic evolution theory for time-varying, structured variance matrices, and the full se- quential and conjugate updating, ltering and forecasting analysis. The models are then applied in the context of nancial time series for predictive portfolio analysis. The improvements dened in optimal Bayesian decision analysis in this example context vividly illustrate the practical benets of the parsimony induced via appro- priate graphical model structuring in multivariate dynamic modelling. We discuss theoretical and empirical aspects of the conditional independence structures in such models, issues of model uncertainty and search, and the relevance of this new framework as a key step towards scaling multivariate dynamic Bayesian modelling methodology to time series of increasing dimension and complexity.