Minimax and adaptive tests for detecting abrupt and possibly transitory changes in a Poisson process

TL;DR

This paper develops minimax and adaptive statistical tests for detecting abrupt or transitory changes in the intensity of a Poisson process, with applications in cybersecurity and epidemiology, analyzing various regimes and proposing practical testing procedures.

Contribution

It provides a comprehensive theoretical framework for minimax and adaptive change detection in Poisson processes, including new tests and analysis of adaptation costs.

Findings

Identified three regimes of minimax separation rates.

Proposed near-CUSUM linear and quadratic tests for different scenarios.

Demonstrated the effectiveness of adaptive tests through simulations.

Abstract

Motivated by applications in cybersecurity and epidemiology, we consider the problem of detecting an abrupt change in the intensity of a Poisson process, characterised by a jump (non transitory change) or a bump (transitory change) from constant. We propose a complete study from the nonasymptotic minimax testing point of view, when the constant baseline intensity is known or unknown. The question of minimax adaptation with respect to each parameter (height, location, length) of the change is tackled, leading to a comprehensive overview of the various minimax separation rate regimes. We exhibit three such regimes and identify the factors of the two phase transitions, by giving the cost of adaptation to each parameter. For each alternative hypothesis, depending on the knowledge or not of each change parameter, we propose minimax or minimax adaptive tests based on linear statistics, close to CUSUM statistics, or quadratic statistics more adapted to the L 2-distance considered in our minimax criteria and typically more powerful in practice, as our simulation study shows. When the change location or length is unknown, our adaptive tests are constructed from a scan aggregation principle combined with Bonferroni or min-p level correction, and a conditioning trick when the baseline intensity is unknown.