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This paper considers a problem of estimating brain effective connectivity from EEG signals using a Granger causality (GC) concept characterized on state-space models. We propose a state-space model for explaining coupled dynamics of the source and EEG signals where EEG is a linear combination of sources according to the characteristics of volume conduction. Our formulation has a sparsity prior on the source output matrix that can further classify active and inactive sources. The scheme is comprised of two main steps: model estimation and model inference to estimate brain connectivity. The model estimation consists of performing a subspace identification and the active source selection based on a group-norm regularized least-squares. The model inference relies on the concept of state-space GC that requires solving a discrete-time Riccati equation for the covariance of estimation error. We verify the performance on simulated data sets that represent realistic human brain activities under several conditions including percentages of active sources, a number of EEG electrodes and the location of active sources. The performance of estimating brain networks is compared with a two-stage approach using source reconstruction algorithms and VAR-based Granger analysis. Our method achieved better performances than the two-stage approach under the assumptions that the true source dynamics are sparse and generated from state-space models. The method is applied to a real EEG SSVEP data set and we found that the temporal lobe played a role of a mediator of connections between temporal and occipital areas, which agreed with findings in previous studies.

This paper proposes an estimation for learning a common Granger network of panel data. When vector time series are collected from several subjects belonging to a homogeneous group, a relationship structure of variables can be assumed to share the same topology, while the model parameters of individual subjects could be varied. The formulation is a regularized least-squares estimation of multiple vector autoregressive (VAR) models with a non-convex l 2,1/2 -norm penalty. The common sparsity pattern of VAR coefficients from all models can reveal a consistent network in a group level. Simulation results show that the proposed formulation achieves better accuracy of learning Granger networks than the convex group lasso formulation. Preliminary results of discovering brain connectivities from ABIDE fMRI data sets showed that sparse common networks among subjects from normal and autism groups are distinctive in some brain regions.

This paper aims to explain relationships between time series by using the Granger causality (GC) concept through autoregressive (AR) models and to assure the model stability. Examining such GC relationship is performed on the model parameters using the Wald test and the model stability is guaranteed by the infinity-norm constraint on the dynamic matrix of the AR process. The proposed formulation is a least-squares estimation with Granger causality and stability constraints which is a convex program with a quadratic objective subject to linear equality and inequality norm constraints. We show by simulations that various typical factors could lead to unstable estimated models when using an unconstrained method. Estimated models from our approach are guaranteed to be stable but the model fitting error could be conservatively increased due to the selected stability condition.

Granger graphical models explain Granger causality between variables in time series through an estimation of zero pattern of coefficients in multivariate autoregressive (AR) models. In this paper, we consider a problem of estimating multiple Granger graphical models simultaneously that share similar topology structures from a set of time series data belonging to distinct classes. This is achieved by estimating a group of AR models and employing group fused lasso penalties to promote sparsity in AR coefficients of each model and sparsity in the difference between AR coefficients from two adjacent models. The resulting problem is in a class of group fused lasso formulation which fits nicely in a convex framework and then can be solved by a fast alternating directions method of multipliers (ADMM) algorithm. Advantages of the proposed method and the performance of the algorithm are illustrated through randomly generated data in a high-dimensional setting.

This paper considers a problem of estimating multivariate autoregressive (AR) models with sparse coefficient matrices. A joint zero pattern of AR coefficients reveals a Granger causality structure of the variables, which is typically depicted as a graphical model. The problem of estimating the graph topology is then formulated as a least-squares problem with an l1-type regularization to promote a sparsity in the AR coefficients. We obtain a convex framework of the estimation problem which can become challenging to solve in a large scale setting due to the nondifferentiability of the cost function. We apply a recent powerful algorithm, namely, the alternating direction method of multipliers (ADMM) for solving topology selection problems in Granger graphical models of AR processes. We illustrate the idea and verify the performance of the ADMM algorithm on randomly generated data sets. This approach is finally applied on Google Flu Trends data learn a causal structure of flu activities in the 51 states of the USA.

We propose a convex optimization method for maximum likelihood estimation of autoregressive models, subject to conditional independence constraints. This problem is an extension to times series of the classical covariance selection problem in graphical modeling. The conditional independence constraints impose quadratic equalities on the autoregressive model parameters, which makes the maximum likelihood estimation problem nonconvex and difficult to solve. We formulate a convex relaxation and prove that it is exact when the sample covariance matrix is block-Toeplitz. We also observe experimentally that in practice the relaxation is exact under much weaker conditions. We discuss applications to topology selection in graphical models of time series, by enumerating all possible topologies, and ranking them using information-theoretic model selection criteria. The method is illustrated by an example of air pollution data.