Functional Data Graphical Models Hongxiao Zhu Virginia Tech July 2, 2015 BIRS Workshop 1 (Joint work with Nate Strawn and David B. Dunson)
Jan 19, 2016
Functional Data Graphical Models
Hongxiao Zhu
Virginia Tech
July 2, 2015 BIRS Workshop
1(Joint work with Nate Strawn and David B. Dunson)
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• 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:
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Genomics Social Networks Brain Networks Economics Networks
Graphical models
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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:
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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.
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Graphical models – some concepts
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Graphical models – the Gaussian case
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A special case of Hyper-Markov Law defined in Dawid and Lauritzen (93)
Graphical models for functional data
Potential applications:
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Neuroimaging Data
ERP
Senor Nodes
EEG
EEG Signals
MRI/fMRI
Brain Regions MRI 2D Slice
The Construction:
Graphical models for multivariate functional data
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Conditional independence between random functional object
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Markov distribution of functional objects
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Construct a Markov distribution
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This is called a Markov combination of P1 and P2.
Construct a probability distribution with Markov property – Cont’d
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A Bayesian Framework
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Hyper Markov Laws
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Hyper Markov Laws – a Gaussian process example
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Hyper Markov Laws – a Gaussian process example (cont’d )
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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.
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Contact:Hongxiao [email protected]
Department of Statistics, Virginia Tech406-A Hutcheson HallBlacksburg, VA 24061-0439 United States