DeepAI
Log In Sign Up

Estimation of Expected Euler Characteristic Curves of Nonstationary Smooth Gaussian Random Fields

08/07/2019
by   Armin Schwartzman, et al.
0

The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains has been limited because in this case the LKCs are difficult to estimate. In this paper, consistent estimators of the LKCs are proposed as linear projections of "pinned" observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations of 2D fields and illustrated in thresholding of 3D fMRI brain activation maps and cosmological simulations on the 2-sphere.

READ FULL TEXT
11/10/2020

Perturbation of the expected Minkowski functional for weakly non-Gaussian isotropic fields on a bounded domain

The Minkowski functionals (MF) including the Euler characteristic (EC) s...
11/19/2019

On critical points of Gaussian random fields under diffeomorphic transformations

Let {X(t), t∈ M} and {Z(t'), t'∈ M'} be smooth Gaussian random fields pa...
03/16/2020

Clustering of Local Extrema in Planck CMB maps

The clustering of local extrema including peaks and troughs will be expl...
05/15/2020

On approximation theorems for the Euler characteristic with applications to the bootstrap

We study approximation theorems for the Euler characteristic of the Viet...
11/06/2021

Geodesic curves in Gaussian random field manifolds

Random fields are mathematical structures used to model the spatial inte...
01/12/2023

An approximation to peak detection power using Gaussian random field theory

We study power approximation formulas for peak detection using Gaussian ...
01/08/2022

Spatial Confidence Regions for Combinations of Excursion Sets in Image Analysis

The analysis of excursion sets in imaging data is essential to a wide ra...