Lasso and probabilistic inequalities for multivariate point processes

Abstract

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. Nonasymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activity inference, we finally carry out a simulation study for multivariate Hawkes processes and compare our methodology with the adaptive Lasso procedure proposed by Zou in (J. Amer. Statist. Assoc. 101 (2006) 1418-1429). We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of (J. Amer. Statist. Assoc. 101 (2006) 1418-1429), our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.Comment: Published at http://dx.doi.org/10.3150/13-BEJ562 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Multiple Tests Based on a Gaussian Approximation of the Unitary Events Method with Delayed Coincidence Count

“…It is already known that this model can easily deal with more than two neurons (Daley & Vere-Jones, 2003;Pernice et al, 2011Pernice et al, , 2012Chornoboy et al, 1988). However, it is only in a recent work that we have proposed theoretical statistical methods to deal with several neurons and large delays of interaction through Lasso methods (Hansen et al, 2012). We aim at generalizing those results to non stationary data in time.…”

Cox process functional learning

“…It is stressed that the originality of our work is that it takes advantage of the theory of stochastic processes to handle a functional data analysis problem-in that sense, it differs from other studies devoted to nonparametric estimation of Cox process intensity (see for instance Hansen et al, 2013, and the references therein).…”

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

“…(24) shows that for all ε > 0 and for all starting points x ∈ R n+1 , the diffusion Y N , starting from Y N (0) = x, visits tubes B ε (Γ) = {y ∈ R n+1 : dist(y, Γ) < ε} infinitely often, almost surely. Moreover, for large N, the diffusion stays within such tubes around the limit cycle during long periods, before eventually leaving such a tube after a time which is of order e NV , whereV is related to the control problem of our diffusion and to the cost of steering the process from the limit cycle to the boundary of the tube around the limit cycle (see [30] for details).…”

“…In particular, (24) shows how the invariant measure µ N of the diffusion approximation Y N concentrates around the periodic orbit Γ. To obtain (24) we rely on the approach of Freidlin and Wentzell [21] to sample path large deviations of diffusions, developed further in Dembo and Zeitouni [14] and extended to the case of degenerate diffusions in Rey-Bellet and Thomas [32]; we refer the interested reader to [30] for more details.…”

“…Hawkes processes provide good models of this synaptic integration phenomenon by the structure of their intensity processes, see (1) below. We refer to Chevallier et al (2015) [7], Chornoboy et al (1988) [8], Hansen et al (2015) [24] and to Reynaud-Bouret et al (2014) [35] for the use of Hawkes processes in neuronal modeling. For an overview of point processes used as stochastic models for interacting neurons both in discrete and in continuous time and related issues, see also Galves and Löcherbach (2016) [22].…”