arXiv subjects

Miguel Martinez Herrera

Publications and source records attributed to Miguel Martinez Herrera.

4 recordsLinked to original sources

arXiv2026

Ridge-penalised spectral least-squares estimation for point processes

Penalised estimation methods for point processes usually rely on a large amount of independent repetitions for cross-validation purposes. However, in the case of a single realisation of the process, existing cross-validation methods may be impractical depending on the chosen model. To overcome this issue, this paper presents a Ridge-penalised spectral least-squares estimation method for second-order stationary point processes. This is achieved through two novel approaches: a p-thinning-based cross-validation method to tune the penalisation parameter, relying on the spectral representation of the process; and the introduction of a spectral least-squares contrast based around the asymptotic properties of the periodogram of the sample. The proposed method is then illustrated by a simulation study on linear Hawkes processes in the context of parametric estimation, highlighting its performances against more traditional approaches, specifically when working with short observation windows.

stat.ME↗

arXiv2025

Spectral analysis for the inference of noisy Hawkes processes

Classic estimation methods for Hawkes processes rely on the assumption that observed event times are indeed a realisation of a Hawkes process, without considering any potential perturbation of the model. However, in practice, observations are often altered by some noise, the form of which depends on the context. It is then required to model the alteration mechanism in order to infer accurately such a noisy Hawkes process. While several models exist, we consider, in this work, the observations to be the indistinguishable union of event times coming from a Hawkes process and from an independent Poisson process. Since standard inference methods (such as maximum likelihood or Expectation-Maximisation) are either unworkable or numerically prohibitive in this context, we propose an estimation procedure based on the spectral analysis of second order properties of the noisy Hawkes process. Novel results include sufficient conditions for identifiability of the ensuing statistical model with exponential interaction functions for both univariate and bivariate processes, along with consistency and asymptotic normality guarantees of our estimator in the univariate case. Although we mainly focus on the exponential scenario, other types of kernels are investigated and discussed. A new estimator based on maximising the spectral log-likelihood is then described, and its behaviour is numerically illustrated on both synthetic data and neuronal data. Besides being free from knowing the source of each observed time (Hawkes or Poisson process), the proposed estimator is shown to perform accurately in estimating both processes.

stat.ME↗

arXiv2023

Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity

The multivariate Hawkes process is a past-dependent point process used to model the relationship of event occurrences between different phenomena.Although the Hawkes process was originally introduced to describe excitation effects, which means that one event increases the chances of another occurring, there has been a growing interest in modelling the opposite effect, known as inhibition.In this paper, we focus on how to infer the parameters of a multidimensional exponential Hawkes process with both excitation and inhibition effects. Our first result is to prove the identifiability of this model under a few sufficient assumptions. Then we propose a maximum likelihood approach to estimate the interaction functions, which is, to the best of our knowledge, the first exact inference procedure in the frequentist framework.Our method includes a variable selection step in order to recover the support of interactions and therefore to infer the connectivity graph.A benefit of our method is to provide an explicit computation of the log-likelihood, which enables in addition to perform a goodness-of-fit test for assessing the quality of estimations.We compare our method to standard approaches, which were developed in the linear framework and are not specifically designed for handling inhibiting effects.We show that the proposed estimator performs better on synthetic data than alternative approaches. We also illustrate the application of our procedure to a neuronal activity dataset, which highlights the presence of both exciting and inhibiting effects between neurons.

stat.ME↗

arXiv2021

Maximum Likelihood Estimation for Hawkes Processes with self-excitation or inhibition

In this paper, we present a maximum likelihood method for estimating the parameters of a univariate Hawkes process with self-excitation or inhibition. Our work generalizes techniques and results that were restricted to the self-exciting scenario. The proposed estimator is implemented for the classical exponential kernel and we show that, in the inhibition context, our procedure provides more accurate estimations than current alternative approaches.

math.ST↗