Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes

The covariance structure of multivariate functional data can be highly complex, especially if the multivariate dimension is large, making extension of statistical methods for standard multivariate data to the functional data setting quite challenging. For example, Gaussian graphical models have recently been extended to the setting of multivariate functional data by applying multivariate methods to the coefficients of truncated basis expansions. However, a key difficulty compared to multivariate data is that the covariance operator is compact, and thus not invertible. The methodology in this paper addresses the general problem of covariance modeling for multivariate functional data, and functional Gaussian graphical models in particular. As a first step, a new notion of separability for multivariate functional data is proposed, termed partial separability, leading to a novel Karhunen-Lo\`eve-type expansion for such data. Next, the partial separability structure is shown to be particularly useful in order to provide a well-defined Gaussian graphical model that can be identified with a sequence of finite-dimensional graphical models, each of fixed dimension. This motivates a simple and efficient estimation procedure through application of the joint graphical lasso. Empirical performance of the method for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during a motor task.Comment: 39 pages, 5 figure

Inference in Graphical Gaussian Models with Edge and Vertex Symmetries with the gRc Package for R

In this paper we present the R package gRc for statistical inference in graphical Gaussian models in which symmetry restrictions have been imposed on the concentration or partial correlation matrix. The models are represented by coloured graphs where parameters associated with edges or vertices of same colour are restricted to being identical. We describe algorithms for maximum likelihood estimation and discuss model selection issues. The paper illustrates the practical use of the gRc package.

Multivariate approach for brain decomposable connectivity networks

International audienceThis paper deals with the analysis of brain functional network using fMRI data. It recapitulates the concept of decomposable connectivity graph. Graphs are a usual tool to represent complex systems behavior, although edge strength estimation issues have not yet received a universally adopted solution. In the framework of linear Gaussian instantaneous exchanges, the well known partial correlation is usually introduced. However its estimation remains a challenge for highly connected or dense systems. Here, we propose to combine a wavelet decomposition and a graphical Gaussian model approach relying on decomposable graphs. This is shown to improve the estimations of brain function networks in the presence of long range dependence; the results are compared to those obtained with classical partial correlation estimators

Joint Multiple Testing Procedures for Graphical Model Selection with Applications to Biological Networks

Gaussian graphical models have become popular tools for identifying relationships between genes when analyzing microarray expression data. In the classical undirected Gaussian graphical model setting, conditional independence relationships can be inferred from partial correlations obtained from the concentration matrix (= inverse covariance matrix) when the sample size n exceeds the number of parameters p which need to estimated. In situations where n \u3c p, another approach to graphical model estimation may rely on calculating unconditional (zero-order) and first-order partial correlations. In these settings, the goal is to identify a lower-order conditional independence graph, sometimes referred to as a ‘0-1 graphs’. For either choice of graph, model selection may involve a multiple testing problem, in which edges in a graph are drawn only after rejecting hypotheses involving (saturated or lower-order) partial correlation parameters. Most multiple testing procedures applied in previously proposed graphical model selection algorithms rely on standard, marginal testing methods which do not take into account the joint distribution of the test statistics derived from (partial) correlations. We propose and implement a multiple testing framework useful when testing for edge inclusion during graphical model selection. Two features of our methodology include (i) a computationally efficient and asymptotically valid test statistics joint null distribution derived from influence curves for correlation-based parameters, and (ii) the application of empirical Bayes joint multiple testing procedures which can effectively control a variety of popular Type I error rates by incorpo- rating joint null distributions such as those described here (Dudoit and van der Laan, 2008). Using a dataset from Arabidopsis thaliana, we observe that the use of more sophisticated, modular approaches to multiple testing allows one to identify greater numbers of edges when approximating an undirected graphical model using a 0-1 graph. Our framework may also be extended to edge testing algorithms for other types of graphical models (e.g., for classical undirected, bidirected, and directed acyclic graphs)

Inference in Graphical Gaussian Models with Edge and Vertex Symmetries with the gRc Package for R

In this paper we present the R package gRc for statistical inference in graphical Gaussian models in which symmetry restrictions have been imposed on the concentration or partial correlation matrix. The models are represented by coloured graphs where parameters associated with edges or vertices of same colour are restricted to being identical. We describe algorithms for maximum likelihood estimation and discuss model selection issues. The paper illustrates the practical use of the gRc package

Sparse model-based network inference using Gaussian graphical models

We consider the problem of estimating a sparse dynamic Gaussian graphical model with L1 penalized maximum likelihood of structured precision matrix. The structure can consist of specific time dynamics, known presence or absence of links in the graphical model or equality constraints on the parameters. The model is defined on the basis of partial correlations, which results in a specific class precision matrices. A priori L1 penalized maximum likelihood estimation in this class is extremely difficult, because of the above mentioned constraints, the computational complexity of the L1 constraint on the side of the usual positive-definite constraint. The implementation is non-trivial, but we show that the com- putation can be done effectively by taking advantage of an efficient maximum determinant algorithm (SDPT3) developed in convex optimization. For selecting the tuning parameter, we compare several selection criteria and argue that the traditional AIC and BIC should not expect to work. We compare our method with related methods, such as glasso (Friedman et al. 2007)