Functional DataGraphical Models Hongxiao Zhu Virginia Tech July 2, - - PowerPoint PPT Presentation

Functional DataGraphical Models

Hongxiao Zhu Virginia Tech July 2, 2015 BIRS Workshop

(Jo (Join int work wit ith Na Nate Str Strawn an and Da David id B. . Du Dunson)

• 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

ed to to ch characte terize co complex syste tems in in a stru tructured, co compact way . .

• Mod
• del

el th the e dep epen endence stru tructu tures es:

Genomics Social Networks Brain Networks Economics Networks

Graphical models

Graphical model theory

• A marriage

ge betw etween pro robabil ilit ity th theo eory and gra graph th theo eory (Jo (Jordan, 1999).

• Ke

Key id idea ea is is to to fa fact ctorize th the e joi

• int dis

istr trib ibution acc ccordin ing to to th the e stru tructu ture of

• f an under

erlying gra graph.

• In

In part rtic icula lar, th there is is a on

• ne-to

to-one map between “separation” and con conditional in independence:

P P is s a a Mark arkov dis distrib ibution.

Graphical models –some concepts

• A gra

graph/s /subgraph is is com comple lete if if all ll pos

• ssib

ible le ve vert rtices are re con connec ected.

• Maximal

l co complete subgr graphs are re ca calle lled cliq cliques.

• If

If C C is is co comple lete and sep eparate A and B, B, th then en C C is is a sep

• eparator. The

e pair ir (A (A , , B B ) ) form forms s a a dec ecomposi siti tion of

• f

G.

Graphical models –some concepts

Graphical models – the Gaussian case

A special case of Hyper- Markov Law defined in Dawid and Lauritzen (93)

Graphical models for functional data

Pot Potential ap appli licatio ions:

Neu euroimaging Da Data

ERP ERP

Senor

• r No

Nodes

EEG EEG

EEG EEG Sign gnals

MRI MRI/f /fMRI

Brain in Regio ions MRI RI 2D 2D Slice

The Co Construction:

Graphical models for multivariate functional data

Conditional independence between random functional object

Markov distribution of functional objects

Construct a Markov distribution

This is called a Markov combination of P1 and P2.

Construct a probability distribution with Markov property –Cont’d

A Bayesian Framework

Hyper Markov Laws

Hyper Markov Laws –a Gaussian process example

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

Simulation

See video.

An application to EEG data (at alpha-frequency band)

The posterior modes of alcoholic group (a) and control group (b), the edges with >0.5 difference in margina l 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.

Contact: Hongxiao Zhu hongxiao@vt.edu 1-540-231-0400 Department of Statistics, Virginia Tech 406-A Hutcheson Hall Blacksburg, VA 24061-0439 United States