Research

Preprints

• "Spectral analysis for noisy Hawkes processes inference", A. Bonnet, F. Cheysson, M. Martinez Herrera, M. Sangnier, In Revision, (2024).
  
  Abstract Pdf Github

  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. 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 synthetic 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.
  
  Keywords: Hawkes process, point process, spectral analysis, parametric estimation, superposition, identifiability.

Published papers

• "Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity", A. Bonnet, M. Martinez Herrera, M. Sangnier, Statistics and Computing, (2023).
  
  Abstract Pdf Github

  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.
  
  Keywords: Non-linear Hawkes process, point process, maximum likelihood estimation, identifiability, support recovery, goodness-of-fit.



• "Maximum Likelihood Estimation for Hawkes Processes with self-excitation or inhibition", A. Bonnet, M. Martinez Herrera, M. Sangnier, Statistics and Probability Letters, (2021).
  
  Abstract Pdf Github

  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.

PhD thesis

• "Inference of non-linear or imperfectly observed Hawkes processes", defended on the 12th November 2024.
  
  Abstract Pdf Slides

  The Hawkes point process is a popular statistical tool to analyse temporal patterns. Modern applications propose extensions of this model to account for specificities in each field of study, which in turn complex- ifies the task of inference. In this thesis, we advance different approaches for the parametric estimation of twosubmodelsoftheHawkesprocessinunivariateandmultivariatesettings. Motivatedbythemodelling of complex neuronal interactions observed from spike train data, our first study focuses on accounting for both inhibition and excitation effects between neurons, modelled by the non-linear Hawkes process. We derive a closed-form expression of the log-likelihood in order to implement a maximum likelihood procedure. As a consequence of our approach, we gain access to a goodness-of-fit scheme allowing us to establish ad hoc model selection methods to estimate the interaction network in the multivariate setting. The second part of this thesis focuses on studying Hawkes process data noised by two different alterations: adding or removing points. The absence of knowledge on the noise dynamics makes classical inference procedures intractable or computationally expensive. Our solution is to leverage the spectral analysis of point processes to establish an estimator obtained by maximising the spectral log-likelihood. By deriving the spectral densities of the noised processes and by establishing identifiability conditions on our model, we show that the spectral inference method does not necessitate any information on the structure of the noise, effectively circumventing this issue. An additional result of the study of Hawkes processes with missing points is that it gives access to a subsampling paradigm to enhance the estima- tion methods by introducing a penalisation parameter. We illustrate the efficiency of all of our methods through reproducible numerical implementations.
  
  Keywords: Hawkes processes, parametric inference, identifiability, inhibition, spectral theory, neuronal data.

Activities

• (2023-2024) Representative of PhD students at graduate school (École Doctorale 386 Paris Centre).
• (2022-2023) Co-organiser of the PhD students seminar at LPSM.