Spectral analysis for noisy Hawkes processes inference

2024-05-21

AI Summary

This work addresses the realistic scenario where observed event streams are contaminated by both Hawkes processes and independent Poisson noise—a setting violating the conventional “clean event stream” assumption. We establish the first identifiability theory for Hawkes processes under Poisson noise, covering univariate and bivariate exponential kernels. Furthermore, we propose a source-label-free spectral-domain maximum likelihood estimator: leveraging second-order moment-based spectral analysis, it jointly optimizes the spectral log-likelihood to estimate both Hawkes kernel functions (including exponential and generalized kernels) and the Poisson noise intensity. On synthetic data, our method achieves efficient and robust joint estimation without requiring event-source annotations, significantly improving inference accuracy and practicality in noisy settings.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

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.

Problem

• Inferring noisy Hawkes processes with Poisson noise contamination
• Overcoming computational limitations of traditional estimation methods
• Developing spectral-based estimator without requiring event source identification

Innovation

• Spectral analysis for noisy Hawkes process estimation
• Identifiability conditions with exponential interaction functions
• Maximizing spectral log-likelihood for accurate estimation