Robust methods for inferring sparse network structures

This is the post-print version of the final paper published in Computational Statistics & Data Analysis. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2013 Elsevier B.V.Networks appear in many fields, from finance to medicine, engineering, biology and social science. They often comprise of a very large number of entities, the nodes, and the interest lies in inferring the interactions between these entities, the edges, from relatively limited data. If the underlying network of interactions is sparse, two main statistical approaches are used to retrieve such a structure: covariance modeling approaches with a penalty constraint that encourages sparsity of the network, and nodewise regression approaches with sparse regression methods applied at each node. In the presence of outliers or departures from normality, robust approaches have been developed which relax the assumption of normality. Robust covariance modeling approaches are reviewed and compared with novel nodewise approaches where robust methods are used at each node. For low-dimensional problems, classical deviance tests are also included and compared with penalized likelihood approaches. Overall, copula approaches are found to perform best: they are comparable to the other methods under an assumption of normality or mild departures from this, but they are superior to the other methods when the assumption of normality is strongly violated

Learning and comparing functional connectomes across subjects

Functional connectomes capture brain interactions via synchronized fluctuations in the functional magnetic resonance imaging signal. If measured during rest, they map the intrinsic functional architecture of the brain. With task-driven experiments they represent integration mechanisms between specialized brain areas. Analyzing their variability across subjects and conditions can reveal markers of brain pathologies and mechanisms underlying cognition. Methods of estimating functional connectomes from the imaging signal have undergone rapid developments and the literature is full of diverse strategies for comparing them. This review aims to clarify links across functional-connectivity methods as well as to expose different steps to perform a group study of functional connectomes

Persistent Homology in Sparse Regression and its Application to Brain Morphometry

Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dimension, persistent homological structures over the parameter space is introduced and explored. The structures are then further exploited in speeding up the computation using the proposed soft-thresholding technique. The topological structures are further used as multivariate features in the tensor-based morphometry (TBM) in characterizing white matter alterations in children who have experienced severe early life stress and maltreatment. These analyses reveal that stress-exposed children exhibit more diffuse anatomical organization across the whole white matter region.Comment: submitted to IEEE Transactions on Medical Imagin

Simulating interventions in graphical chain models for longitudinal data

Simulating the outcome of an intervention is a central problem in many fields as this allows decision-makers to quantify the effect of any given strategy and, hence, to evaluate different schemes of actions. Simulation is particularly relevant in very large systems where the statistical model involves many variables that, possibly, interact with each other. In this case one usually has a large number of parameters whose interpretation becomes extremely difficult. Furthermore, in a real system, although one may have a unique target variable, there may be a number of variables which might, and often should, be logically considered predictors of the target outcome and, at the same time, responses of other variables of the system. An intervention taking place on a given variable, therefore, may affect the outcome either directly and indirectly though the way in which it affects other variables within the system. Graphical chain models are particularly helpful in depicting all of the paths through which an intervention may affect the final outcome. Furthermore, they identify all of the relevant conditional distributions and therefore they are particularly useful in driving the simulation process. Focussing on binary variables, we propose a method to simulate the effect of an intervention. Our approach, however, can be easily extended to continuous and mixed responses variables. We apply the proposed methodology to assess the effect that a policy intervention may have on poorer health in early adulthood using prospective data provided by the 1970 British Birth Cohort Study (BCS70).chain graph, conditional approach, Gibbs Sampling, Simulation of interventions, age at motherhood, mental health

A Graph Algorithmic Approach to Separate Direct from Indirect Neural Interactions

Network graphs have become a popular tool to represent complex systems composed of many interacting subunits; especially in neuroscience, network graphs are increasingly used to represent and analyze functional interactions between neural sources. Interactions are often reconstructed using pairwise bivariate analyses, overlooking their multivariate nature: it is neglected that investigating the effect of one source on a target necessitates to take all other sources as potential nuisance variables into account; also combinations of sources may act jointly on a given target. Bivariate analyses produce networks that may contain spurious interactions, which reduce the interpretability of the network and its graph metrics. A truly multivariate reconstruction, however, is computationally intractable due to combinatorial explosion in the number of potential interactions. Thus, we have to resort to approximative methods to handle the intractability of multivariate interaction reconstruction, and thereby enable the use of networks in neuroscience. Here, we suggest such an approximative approach in the form of an algorithm that extends fast bivariate interaction reconstruction by identifying potentially spurious interactions post-hoc: the algorithm flags potentially spurious edges, which may then be pruned from the network. This produces a statistically conservative network approximation that is guaranteed to contain non-spurious interactions only. We describe the algorithm and present a reference implementation to test its performance. We discuss the algorithm in relation to other approximative multivariate methods and highlight suitable application scenarios. Our approach is a tractable and data-efficient way of reconstructing approximative networks of multivariate interactions. It is preferable if available data are limited or if fully multivariate approaches are computationally infeasible.Comment: 24 pages, 8 figures, published in PLOS On

Graphical Models for Structural Vector Autoregressions

In this paper a method to identify the causal structure associated with a VAR model is proposed. The structure is described by means of a graph, which provides a rigorous language to analyze the statistical and logical properties of causal relations. Under some general assumptions, causal relations are associated with a set of vanishing partial correlations among the variables that constitute them. In order to infer the causal structure among the contemporaneous variable, tests on vanishing partial correlations among the estimated residuals of a VAR are used, jointly with background knowledge. This method is applied to an updated version of the King et al. (1991) dataset and it allows to obtain an orthogonalization of the residuals coherent with the causal structure among the contemporaneous variables and alternative to the standard one, which is based on the Choleski factorization of the covariance matrix of the residuals. The impulse response functions calculated, with the method proposed here, for the King et al. (1991) model confirm their results about the fact that US macroeconomic data do not support the hypothesis that real permanent shocks are the dominant source of business-cycle fluctuations.Causality, Directed Acyclic Graphs, Identification Problem, Residuals Orthogonalization, Impulse Response Functions.