Estimation of expected Euler characteristic curves of nonstationary smooth random fields

Abstract

The expected Euler characteristic (EEC) of excursion sets of a smooth Gaussian-related random field over a compact manifold approximates the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC of a Gaussian-related field is expressed by the Gaussian kinematic formula (GKF) as a finite sum of known functions multiplied by the Lipschitz–Killing curvatures (LKCs) of the generating Gaussian field. This paper proposes consistent estimators of the LKCs as linear projections of “pinned” Euler characteristic (EC) curves obtained from realizations of zero-mean, unit variance Gaussian processes. As observed, data seldom is Gaussian and the exact mean and variance is unknown, yet the statistic of interest often satisfies a CLT with a Gaussian limit process; we adapt our LKC estimators to this scenario using a Gaussian multiplier bootstrap approach. This yields consistent estimates of the LKCs of the possibly nonstationary Gaussian limiting field that have low variance and are computationally efficient for complex underlying manifolds. For the EEC of the limiting field, a parametric plug-in estimator is presented, which is more efficient than the nonparametric average of EC curves. The proposed methods are evaluated using simulations of 2D fields, and illustrated on cosmological observations and simulations on the 2-sphere and 3D fMRI volumes.