Estimation of local dependence graphs via Hawkes processes and link with functional connectivity in neurosciences

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. This method is then applied to Hawkes processes as model for spike train analysis. The estimation allows us to recover the functional underlying connectivity as the local dependence graph that has been estimated. Simulations and real data analysis show the excellent performances of our method in practice.

This is a joint work (still in progress) with V. Rivoirard (Dauphine),
C. Tuleau-Malot, F. Grammont (Nice), N.R. Hansen (Copenhagen), T.
Bessaih, R. Lambert, N. Leresche (Paris 6).