Causality in Time Series - ClopiNet

Causal analysis of fMRIthat estimation of mathematical models from time-series data generally has two importantaspects: model selection and model identification (Ljung, 1999). In the modelselection stage a class of models is chosen by the researcher that is deemed suitablefor the problem at hand. In the model identification stage the parameters in the chosenmodel class are estimated from the observed data record. In practice, model selectionand identification often occur in a somewhat interactive fashion where, for instance,model selection can be informed by the fit of different models to the data achieved inan identification step. The important point is that model selection involves a mixtureof choices and assumptions on the part of the researcher and the information gainedfrom the data-record itself. These considerations indicate that an important distinctionmust be made between exploratory and confirmatory approaches, especially in structuralmodel selection procedures for brain connectivity. Exploratory techniques useinformation in the data to investigate the relative applicability of many models. Assuch, they have the potential to detect ‘missing’ regions in structural models. Confirmatoryapproaches, such as DCM, test hypotheses about connectivity within a set ofmodels assumed to be applicable. Sources of common input or intervening causes aretaken into account in a multivariate confirmatory model, but only if the employed structuralmodel allows it (i.e. if the common input or intervening node is incorporated inthe model).The technique of Granger Causality Mapping (GCM) was developed to exploreall regions in the brain that interact with a single selected reference region using autoregressivemodeling of fMRI time-series (Roebroeck et al., 2005). By employing asimple bivariate model containing the reference region and, in turn, every other voxel inthe brain, the sources and targets of influence for the reference region can be mapped.It was shown that such an ‘exploratory’ mapping approach can form an important toolin structural model selection. Although a bivariate model does not discern direct fromindirect influences, the mapping approach locates potential sources of common inputand areas that could act as intervening network nodes. In addition, by settling fora bivariate model one trivially avoids the conflation of direct and indirect influencesthat can arise in discrete AR model due to temporal aggregation, as discussed above.Other applications of autoregressive modeling to fMRI data have considered full multivariatemodels on large sets of selected brain regions, illustrating the possibility toestimate high-dimensional dynamical models. For instance, Valdes-Sosa (2004) andValdes-Sosa et al. (2005b) applied these models to parcellations of the entire cortex inconjunction with sparse regression approaches that enforce an implicit structural modelselection within the set of parcels. In another more recent example (Deshpande et al.,2008) a full multivariate model was estimated over 25 ROIs (that were found to be activatedin the investigated task) together with an explicit reduction procedure to pruneregions from the full model as a structural model selection procedure. Additional variantsof VAR model based causal inference that has been applied to fMRI include timevarying influence (Havlicek et al., 2010), blockwise (or ‘cluster-wise’) influence fromone group of variables to another (Barrett et al., 2010; Sato et al., 2010) and frequencydecomposedinfluence (Sato et al., 2009).91