Functional Data Graphical Models Hongxiao Zhu Virginia Tech July 2, 2015 BIRS Workshop 1 (Joint work with Nate Strawn and David B. Dunson)

2 Graphical models. Graphical models for functional data -- a theoretical framework for Bayesian inference. Gaussian process graphical models. Simulation and EEG application. Outline

Graphical models Used to characterize complex systems in a structured, compact way. Model the dependence structures: 3 GenomicsSocial Networks Brain NetworksEconomics Networks

Graphical models 4

Graphical model theory A marriage between probability theory and graph theory (Jordan, 1999). Key idea is to factorize the joint distribution according to the structure of an underlying graph. In particular, there is a one-to-one map between “separation” and conditional independence: 5 P is a Markov distributi on.

Graphical models – some concepts A graph/subgraph is complete if all possible vertices are connected. Maximal complete subgraphs are called cliques. If C is complete and separate A and B, then C is a separator. The pair (A, B ) forms a decomposition of G. 6

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Graphical models – some concepts 8

Graphical models – the Gaussian case 9 A special case of Hyper- Markov Law defined in Dawid and Lauritzen (93)

Graphical models for functional data Potential applications: 10 Neuroimaging Data ERP Senor Nodes EEG EEG Signals MRI/fMRI Brain Regions MRI 2D Slice

The Construction: Graphical models for multivariate functional data 11

Conditional independence between random functional object 12

Markov distribution of functional objects 13

Construct a Markov distribution 14 This is called a Markov combination of P1 and P2.

Construct a probability distribution with Markov property – Cont’d 15

A Bayesian Framework 16

Hyper Markov Laws 17

Hyper Markov Laws – a Gaussian process example 18

Hyper Markov Laws – a Gaussian process example (cont’d ) 19

Simulation See video.

An application to EEG data (at alpha-frequency band) 21 The posterior modes of alcoholic group (a) and control group (b), the edges with >0.5 difference i n marginal probabilities (c), the boxplots of the number of edges per node (d) and the total numbe r of edges (e), the boxplots of the number of asymmetric pairs per node (f) and the total number o f asymmetric pairs (g).

Reference Zhu, H., Strawn, N. and Dunson, D. B. Bayesian graphical models for multivariate functional data. (arXiv: ) M. I. Jordan, editor. Learning in Graphical Models. MIT Press, Dawid, A. P. and Lauritzen, S. L. (1993). Hyper Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21, 3, 1272– Contact: Hongxiao Zhu Department of Statistics, Virginia Tech 406-A Hutcheson Hall Blacksburg, VA United States