Nonparametric Markovian Learning of Triggering Kernels for Mutually Exciting and Mutually Inhibiting Multivariate Hawkes Processes

In this paper, we address the problem of fitting multivariate Hawkes processes to potentially large-scale data in a setting where series of events are not only mutually-exciting but can also exhibit inhibitive patterns. We focus on nonparametric learning and propose a novel algorithm called MEMIP (Markovian Estimation of Mutually Interacting Processes) that makes use of polynomial approximation theory and self-concordant analysis in order to learn both triggering kernels and base intensities of events. Moreover, considering that N historical observations are available, the algorithm performs log-likelihood maximization in $O(N)$ operations, while the complexity of non-Markovian methods is in $O(N^{2})$. Numerical experiments on simulated data, as well as real-world data, show that our method enjoys improved prediction performance when compared to state-of-the art methods like MMEL and exponential kernels

Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples

Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data

We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics.Comment: 6 figure

Wavelet penalized likelihood estimation in generalized functional models

The paper deals with generalized functional regression. The aim is to estimate the influence of covariates on observations, drawn from an exponential distribution. The link considered has a semiparametric expression: if we are interested in a functional influence of some covariates, we authorize others to be modeled linearly. We thus consider a generalized partially linear regression model with unknown regression coefficients and an unknown nonparametric function. We present a maximum penalized likelihood procedure to estimate the components of the model introducing penalty based wavelet estimators. Asymptotic rates of the estimates of both the parametric and the nonparametric part of the model are given and quasi-minimax optimality is obtained under usual conditions in literature. We establish in particular that the LASSO penalty leads to an adaptive estimation with respect to the regularity of the estimated function. An algorithm based on backfitting and Fisher-scoring is also proposed for implementation. Simulations are used to illustrate the finite sample behaviour, including a comparison with kernel and splines based methods

Compensator and exponential inequalities for some suprema of counting processes

Talagrand [1996. New concentration inequalities in product spaces. Invent. Math. 126 (3), 505-563], Ledoux [1996. On Talagrand deviation inequalities for product measures. ESAIM: Probab. Statist. 1, 63-87], Massart [2000a. About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 2 (28), 863-884], Rio [2002. Une inégalité de Bennett pour les maxima de processus empiriques. Ann. Inst. H. Poincaré Probab. Statist. 38 (6), 1053-1057. En l'honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov] and Bousquet [2002. A Bennett concentration inequality and its application to suprema of empirical processes. C. R. Math. Acad. Sci. Paris 334 (6), 495-500] have obtained exponential inequalities for suprema of empirical processes. These inequalities are sharp enough to build adaptive estimation procedures Massart [2000b. Some applications of concentration inequalities. Ann. Fac. Sci. Toulouse Math. (6) 9 (2), 245-303]. The aim of this paper is to produce these kinds of inequalities when the empirical measure is replaced by a counting process. To achieve this goal, we first compute the compensator of a suprema of integrals with respect to the counting measure. We can then apply the classical inequalities which are already available for martingales Van de Geer [1995. Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Statist. 23 (5), 1779-1801].Exponential inequalities Counting process Supremum of centered integrals

EXPONENTIAL INEQUALITY FOR CHAOS BASED ON SAMPLING WITHOUT REPLACEMENT

International audienceWe are interested in the behavior of particular functionals, in a framework where the only source of randomness is a sampling without replacement. More precisely the aim of this short note is to prove an exponential concentration inequality for special U-statistics of order 2, that can be seen as chaos