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)

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

Graphical models • Used ed to to ch characte terize co complex syste tems in in a stru tructured, co compact way . . • Mod odel el th the e dep epen endence stru tructu tures es: Brain Networks Economics Networks Genomics Social 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 oint dis istr trib ibution acc ccordin ing to to th the e stru tructu ture of of an under erlying gra graph. • In In part rtic icula lar, th there is is a on one-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 ossib 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 of 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 MRI/f MRI /fMRI EEG EEG MRI RI 2D 2D Slice Brain in Regio ions Senor or No Nodes EEG EEG Sign gnals

Graphical models for multivariate functional data The Co Construction:

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