TY - JOUR

T1 - Lasso and probabilistic inequalities for multivariate point processes

AU - Hansen, Niels Richard

AU - Reynaud-Bouret, Patricia

AU - Rivoirard, Vincent

PY - 2015/2

Y1 - 2015/2

N2 - 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 ℓ1-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.

AB - 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 ℓ1-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.

KW - adaptive estimation

KW - Bernstein-type inequalities

KW - Hawkes processes

KW - Lasso procedure

KW - multivariate counting process

U2 - 10.3150/13-BEJ562

DO - 10.3150/13-BEJ562

M3 - Journal article

VL - 21

SP - 83

EP - 143

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 1

ER -