Learning the dependency structure between spatially distributed observations of a spatio-temporal process is an important problem in many fields such as geology, geophysics, atmospheric sciences, oceanography, etc. . However, estimation of such systems is complicated by the fact that they exhibit dynamics at multiple scales of space and time arising due to a combination of diffusion and convection/advection. As we show, time-series graphical models based on vector auto-regressive processes are inefficient in capturing such multi-scale structure. In this paper, we present a hierarchical graphical model with physically derived priors that better represents the multi-scale character of these dynamical systems. We also propose algorithms to efficiently estimate the interaction structure from data. We demonstrate results on a general class of problems arising in exploration geophysics by discovering graphical structure that is physically meaningful and provide evidence of its advantages over alternative approaches.

Identifying patterns from the neuroimaging recordings of brain activity related to the unobservable psychological or mental state of an individual can be treated as a unsupervised pattern recognition problem. The main challenges, however, for such an analysis of fMRI data are: a) defining a physiologically meaningful feature-space for representing the spatial patterns across time; b) dealing with the high-dimensionality of the data; and c) robustness to the various artifacts and confounds in the fMRI time-series. In this paper, we present a network-aware feature-space to represent the states of a general network, that enables comparing and clustering such states in a manner that is a) meaningful in terms of the network connectivity structure; b)computationally efficient; c) low-dimensional; and d) relatively robust to structured and random noise artifacts. This feature-space is obtained from a spherical relaxation of the transportation distance metric which measures the cost of transporting ``mass'' over the network to transform one function into another. Through theoretical and empirical assessments, we demonstrate the accuracy and efficiency of the approximation, especially for large problems. While the application presented here is for identifying distinct brain activity patterns from fMRI, this feature-space can be applied to the problem of identifying recurring patterns and detecting outliers in measurements on many different types of networks, including sensor, control and social networks.