Adaptive estimation for Hawkes processes; application to genome analysis

TL;DR

This paper introduces a new adaptive estimation method for Hawkes processes applied to genome analysis, providing theoretical guarantees, a novel model selection strategy, and practical calibration for real data, improving biological insights.

Contribution

It develops a nonasymptotic penalized model selection approach with a new theoretical penalty for complex Hawkes models, and introduces the Islands strategy for biological relevance.

Findings

The method achieves adaptive minimax estimation for Hölderian functions.

Simulations help calibrate the penalty constant for practical use.

Results on genomic data align with biological knowledge and refine existing understanding.

Abstract

The aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along DNA sequences. These events are modeled by a Hawkes process. The biological problem is actually complex enough to need a nonasymptotic penalized model selection approach. We provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models. The consecutive theoretical estimator is shown to be adaptive minimax for H\"{o}lderian functions with regularity in $(1/2,1]$: those aspects have not yet been studied for the Hawkes' process. Moreover, we introduce an efficient strategy, named Islands, which is not classically used in model selection, but that happens to be particularly relevant to the biological question we want to answer. Since a multiplicative constant in the theoretical penalty is not computable in practice, we provide extensive simulations to find a data-driven calibration of this constant. The results obtained on real genomic data are coherent with biological knowledge and eventually refine them.