Brain covariance selection: better individual functional connectivity models using population prior

Spontaneous brain activity, as observed in functional neuroimaging, has been shown to display reproducible structure that expresses brain architecture and carries markers of brain pathologies. An important view of modern neuroscience is that such large-scale structure of coherent activity reflects modularity properties of brain connectivity graphs. However, to date, there has been no demonstration that the limited and noisy data available in spontaneous activity observations could be used to learn full-brain probabilistic models that generalize to new data. Learning such models entails two main challenges: i) modeling full brain connectivity is a difficult estimation problem that faces the curse of dimensionality and ii) variability between subjects, coupled with the variability of functional signals between experimental runs, makes the use of multiple datasets challenging. We describe subject-level brain functional connectivity structure as a multivariate Gaussian process and introduce a new strategy to estimate it from group data, by imposing a common structure on the graphical model in the population. We show that individual models learned from functional Magnetic Resonance Imaging (fMRI) data using this population prior generalize better to unseen data than models based on alternative regularization schemes. To our knowledge, this is the first report of a cross-validated model of spontaneous brain activity. Finally, we use the estimated graphical model to explore the large-scale characteristics of functional architecture and show for the first time that known cognitive networks appear as the integrated communities of functional connectivity graph.Comment: in Advances in Neural Information Processing Systems, Vancouver : Canada (2010

Sufficient Dimension Reduction and Modeling Responses Conditioned on Covariates: An Integrated Approach via Convex Optimization

Given observations of a collection of covariates and responses $(Y, X) \in \mathbb{R}^p \times \mathbb{R}^q$, sufficient dimension reduction (SDR) techniques aim to identify a mapping $f: \mathbb{R}^q \rightarrow \mathbb{R}^k$ with $k \ll q$ such that $Y|f(X)$ is independent of $X$. The image $f(X)$ summarizes the relevant information in a potentially large number of covariates $X$ that influence the responses $Y$. In many contemporary settings, the number of responses $p$ is also quite large, in addition to a large number $q$ of covariates. This leads to the challenge of fitting a succinctly parameterized statistical model to $Y|f(X)$, which is a problem that is usually not addressed in a traditional SDR framework. In this paper, we present a computationally tractable convex relaxation based estimator for simultaneously (a) identifying a linear dimension reduction $f(X)$ of the covariates that is sufficient with respect to the responses, and (b) fitting several types of structured low-dimensional models -- factor models, graphical models, latent-variable graphical models -- to the conditional distribution of $Y|f(X)$. We analyze the consistency properties of our estimator in a high-dimensional scaling regime. We also illustrate the performance of our approach on a newsgroup dataset and on a dataset consisting of financial asset prices.Comment: 34 pages, 1 figur

Estimation and Inference for High-Dimensional Gaussian Graphical Models with Structural Constraints.

This work discusses several aspects of estimation and inference for high-dimensional Gaussian graphical models and consists of two main parts. The first part considers network-based pathway enrichment analysis based on incomplete network information. Pathway enrichment analysis has become a key tool for biomedical researchers to gain insight into the underlying biology of differentially expressed genes, proteins and metabolites. We propose a constrained network estimation framework that combines network estimation based on cell- and condition-specific high-dimensional Omics data with interaction information from existing data bases. The resulting pathway topology information is subsequently used to provide a framework for simultaneous testing of differences in expression levels of pathway members, as well as their interactions. We study the asymptotic properties of the proposed network estimator and the test for pathway enrichment, and investigate its small sample performance in simulated experiments and illustrate it on two cancer data sets. The second part of the thesis is devoted to reconstructing multiple graphical models simultaneously from high-dimensional data. We develop methodology that jointly estimates multiple Gaussian graphical models, assuming that there exists prior information on how they are structurally related. The proposed method consists of two steps: in the first one, we employ neighborhood selection to obtain estimated edge sets of the graphs using a group lasso penalty. In the second step, we estimate the nonzero entries in the inverse covariance matrices by maximizing the corresponding Gaussian likelihood. We establish the consistency of the proposed method for sparse high-dimensional Gaussian graphical models and illustrate its performance using simulation experiments. An application to a climate data set is also discussed.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113495/1/mjing_1.pd