Level sets estimation and Vorob'ev expectation of random compact sets

Philippe Heinrich (1) , Radu Stoica (1) , Viet Chi Tran (1, 2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

The issue of a ''mean shape'' of a random set $X$ often arises, in particular in image analysis and pattern detection. There is no canonical definition but one possible approach is the so-called Vorob'ev expectation $\E_V(X)$, which is closely linked to quantile sets. In this paper, we propose a consistent and ready to use estimator of $\E_V(X)$ built from independent copies of $X$ with spatial discretization. The control of discretization errors is handled with a mild regularity assumption on the boundary of $X$: a not too large 'box counting' dimension. Some examples are developed and an application to cosmological data is presented.

Mots clés: Stochastic geometry; Random closed sets; Level sets; Vorob'ev expectation

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Philippe Heinrich, Radu Stoica, Viet Chi Tran. Level sets estimation and Vorob'ev expectation of random compact sets. Spatial Statistics, 2012, 2, pp.47-61. ⟨10.1016/j.spasta.2012.10.001⟩. ⟨hal-00495449v2⟩