Brain imaging data such as EEG and fMRI represent a challenge for statisticians as well as clinicians due to their pervasive yet still unknown dependency structure. The emerging field of topological data analysis (TDA) can provide new insights on provides new insights on brain activity; as it characterize data through connected structures of any dimension (i.e. connected components in dimension 0, loops in dimension 1 and so on), in fact, TDA provides information not only what on areas of the brain are connected (i.e. work together) but also what kind of connection relates them.
Topological features can be considered as an alternative representation of the association between brain areas, hence understanding the determinants of the topology of brain imaging data can inform us on the determinants of brain activity itself. This allows to capture more complex forms of dependence, such as cyclical dependence, which is typically neglected by existing network-based methods for brain imaging data.
Our goal is to provide a toolbox to investigate and assess the impact of external covariates on topological invariants, with a special emphasis on their persistence, i.e. how prominent they are in the characterisation of the brain activity. In addition, we plan to release opensource software for the implementation of our proposed methodology.
From a methodological stand point our main contribution is to use covariates in the characterization of topological signatures.
Topological methods have been exploited in the context of supervised learning by means of feature extraction, either explicitely as in, either implicitely by means of kernel methods. The literature however focuses only on the case where the topological feature is the covariate; our goal is to analyse the case where the topological feature is the response, which, as of now, is still uncharted territory.
As it is a very recent branch of study, in fact, the literature on TDA has been focused on showing that these tools were useful, neglecting the reason why. Our goal is to extend the use of covariates to the study of topological invariants commonly used in the TDA framework to allow for a deeper understanding of these tools.
Moreover, understanding and assessing the dependence of topological feature with respect to a given set of covariates is important not only per se but also for its possible use in multilevel modelling, when the topological feature itself can be modelled and used as a covariate.
From an applied perspective our main contributions regards the gain of new insights on brain activity.
As connected components are clusters, loops are periodic structures and so on, assessing the dependency of those topological features with respect to covariates, consists in a distribution free way of characterising complex relationships between brain areas. The quantification of the covariate effect on topological summaries is instrumental in understanding the effect of those covariates on brain activity itself. This framework allows to capture complex form of dependency, such as cyclical dependence without restrictive assumption on the generative model.