TY - JOUR
T1 - ESTIMATION OF EXPECTED EULER CHARACTERISTIC CURVES OF NONSTATIONARY SMOOTH RANDOM FIELDS
AU - Telschow, Fabian J.E.
AU - Cheng, Dan
AU - Pranav, Pratyush
AU - Schwartzman, Armin
N1 - Publisher Copyright:
© Institute of Mathematical Statistics, 2023.
PY - 2023/10
Y1 - 2023/10
N2 - 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.
AB - 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.
KW - Euler characteristic
KW - Gaussian related fields
KW - Lipschitz-Killing curvatures
KW - Random field theory
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U2 - 10.1214/23-AOS2337
DO - 10.1214/23-AOS2337
M3 - Article
AN - SCOPUS:85180734492
SN - 0090-5364
VL - 51
SP - 2272
EP - 2297
JO - Annals of Statistics
JF - Annals of Statistics
IS - 5
ER -