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

Genomics Social Networks Brain Networks Economics 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 dis-tribution.

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

7

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)

21The posterior modes of alcoholic group (a) and control group (b), the edges with >0.5 difference in marginal probabilities (c), the boxplots of the number of edges per node (d) and the total number of edges (e), the boxplots of the number of asymmetric pairs per node (f) and the total number of asymmetric pairs (g).

Reference

• Zhu, H., Strawn, N. and Dunson, D. B. Bayesian graphical models for multivariate functional data. (arXiv: 1411.4158)

• M. I. Jordan, editor. Learning in Graphical Models. MIT Press, 1999.

• Dawid, A. P. and Lauritzen, S. L. (1993). Hyper Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21, 3, 1272–1317.

22

Contact:Hongxiao Zhuhongxiao@vt.edu1-540-231-0400

Department of Statistics, Virginia Tech406-A Hutcheson HallBlacksburg, VA 24061-0439 United States