Estimation of the average number of continuous crossings for non-stationary non-diffusion processes

Romain Azaïs (1, 2) , Alexandre Genadot (3, 4)

1 Probabilités et statistiques
2 BIGS - Biology, genetics and statistics
3 CQFD - Quality control and dynamic reliability
4 IMB - Institut de Mathématiques de Bordeaux

Assume that you observe trajectories of a non-diffusive non-stationary process and that you are interested in the average number of times where the process crosses some threshold (in dimension d = 1) or hypersurface (in dimension d ≥ 2). Of course, you can actually estimate this quantity by its empirical version counting the number of observed crossings. But is there a better way? In this paper, for a wide class of piecewise smooth processes, we propose estimators of the average number of continuous crossings of an hypersurface based on Kac-Rice formulae. We revisit these formulae in the uni-and multivariate framework in order to be able to handle non-stationary processes. Our statistical method is tested on both simulated and real data.

Domaines: Méthodologie [stat.ME]; Probabilités [math.PR]

Romain Azaïs, Alexandre Genadot. Estimation of the average number of continuous crossings for non-stationary non-diffusion processes. 2017. ⟨hal-01499914v1⟩