Uniform Deconvolution for Poisson Point Processes

Anna Bonnet (1) , Claire Lacour (2) , Franck Picard (3, 4, 5) , Vincent Rivoirard (6)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
3 PEGASE - Département PEGASE [LBBE]
4 Statistique en grande dimension pour la génomique
5 LBBE - Laboratoire de Biométrie et Biologie Evolutive - UMR 5558
6 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We focus on the estimation of the intensity of a Poisson process in the presence of a uniform noise. We propose a kernel-based procedure fully calibrated in theory and practice. We show that our adaptive estimator is optimal from the oracle and minimax points of view, and provide new lower bounds when the intensity belongs to a Sobolev ball. By developing the Goldenshluger-Lepski methodology in the case of deconvolution for Pois-son processes, we propose an optimal data-driven selection of the kernel's bandwidth, and we provide a heuristic framework to calibrate the estimator in practice. Our method is illustrated on the spatial repartition of replication origins along the human genome.

Mots clés: Convolution; Poisson process; Adaptive estimation; Poisson Point Process

Domaines: Bio-informatique [q-bio.QM]; Apprentissage [cs.LG]; Modélisation et simulation; Logiciel mathématique [cs.MS]; Statistiques [math.ST]; Génomique, Transcriptomique et Protéomique [q-bio.GN]; Applications [stat.AP]; Méthodologie [stat.ME]; Machine Learning [stat.ML]

Anna Bonnet, Claire Lacour, Franck Picard, Vincent Rivoirard. Uniform Deconvolution for Poisson Point Processes. Journal of Machine Learning Research, 2022, 23 (194), pp.1--36. ⟨hal-02964117v2⟩