Spatio-Temporal Graphical Model Selection

Submitted to the Annals of Applied Statistics SPATIO-TEMPORAL GRAPHICAL MODEL SELECTION∗ By Patrick Harrington†, Alfred Hero† University of Michigan† We consider the problem of estimating the topology of spatial in- teractions in a discrete state, discrete time spatio-temporal graphical model where the interactions affect the temporal evolution of each agent in a network. Among other models, the susceptible, infected, recovered (SIR) model for interaction events fall into this framework. We pose the problem as a structure learning problem and solve it us- ing an ℓ1-penalized likelihood convex program. We evaluate the solu- tion on a simulated spread of infectious over a complex network. Our topology estimates outperform those of a standard spatial Markov random field graphical model selection using ℓ1-regularized logistic regression. 1. Introduction. This paper treats the problem of learning the inter- action structure of a spatio-temporal graphical model for a discrete state and discrete time stochastic process known as the susceptible, infected, recovered (SIR) model. The presence of spatial interactions cause adjacent nodes in the graph to affect each others states over time. Learning the topology of this graph is known as model selection. We cast this graphical model selection problem as a penalized likelihood problem, resulting in a convex program for which convex optimization solvers can be applied. SIR spatio-temporal graphical models are commonly used in modeling the random propagation of information between nodes in large networks in bioinformatics, signal processing, public health, and national security (4; 9; 21). Knowing the net- work link structure allows accurate prediction of individual node states and can aid the development of control and intervention strategies for such net- works. This paper develops a tractable method to estimate the topology of the network for the SIR spatio-temporal graphical model from empirical data. Exact solutions of the graphical model selection problem is NP hard due to the combinatorial nature of enumeration through the discrete space of possible graph topologies. Researchers studying Bayesian networks, both ∗This research was partially supported by the AFRL ATR Center through a Signal Innovations Group subcontract, grant number SIG FA8650-07-D-1221-TO1 AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35 Keywords and phrases: SIR Models, Dynamic Bayesian Networks, Model Selection, ℓ1- Regularization, Structure Learning, Convex Risk Minimization 1 2 HARRINGTON AND HERO static and dynamic, have developed exact and approximate methods for se- lecting a good candidate topology (5; 8; 20). Such methods are appropriate for networks of small size and of unknown generative models for the obser- vations. However, they are difficult to scale to larger graphs. SIR processes are used to model transmission events on complex networks which tend to be sparse in their interactions (22; 3; 21; 7; 4), so that there are relatively few edges in the graph. Over the past decade sparse regularization methods have been developed for graphical model selection using ℓ1-regularization and other approaches. Examples include Gaussian graphical models (GMMs) (18; 10; 33; 26; 24) and Markov random fields (MRFs) (16; 31; 26). The SIR model used throughout this paper is both discrete state and dis- crete time and thus any ℓ1-penalized GMM method that is designed for real valued Gaussian random vectors would not be appropriate for this model. The structure learning algorithms for MRFs discussed in (16; 31; 26) are designed for discrete samples drawn from a MRF and most are limited to binary states (the SIR model has three states and is a different genera- tive model than the MRF). Research in MRFs and GMMs have successfully used the ℓ1-penalty to control the sparseness of the estimated graphical model topologies and we will adopt this approach for the SIR model. The method presented in this paper also scales to large networks more easily than traditional Bayesian network structure learning algorithms (5; 8; 20). The proposed sparse structure learning method is designed for graphs that incorporate random causal transmission events affecting the evolution of the node states, such occurs in the propagation of infectious disease. Identifying the structure of social networks in tracking epidemics has received increased attention due to the global response to pandemic influenza A (H1N1) 2009. We illustrate the accuracy of the proposed network structure learning on two moderate sized complex networks using real-world epidemiological pa- rameters that approximate an H1N1 flu inspired outbreak (32). We compare performance of the proposed estimation method against a MRF graphical model selection using ℓ1-regularized logistic regression (31). The proposed method is more accurate than generic approaches such as (31) for detection of anomalous network structure given sampled data from a spatio-tempo

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