Abstract
Let {f(t) : t ∈ T} be a smooth Gaussian random field over a parameter space T, where T may be a subset of Euclidean space or, more generally, a Riemannian manifold. We provide a general formula for the distribution of the height of a local maximum ℙ{f(t 0) > u|t 0 is a local maximum of f(t)} when f is non-stationary. Moreover, we establish asymptotic approximations for the overshoot distribution of a local maximum ℙ{f(t 0) > u+v|t 0 is a local maximum of f(t) and f(t 0) > v} as \(v\to \infty \). Assuming further that f is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when T is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.
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Research partially supported by NIH grant R01-CA157528.
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Cheng, D., Schwartzman, A. Distribution of the height of local maxima of Gaussian random fields. Extremes 18, 213–240 (2015). https://doi.org/10.1007/s10687-014-0211-z
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DOI: https://doi.org/10.1007/s10687-014-0211-z
Keywords
- Height
- Overshoot
- Local maxima
- Riemannian manifold
- Gaussian orthogonal ensemble
- Isotropic field
- Euler characteristic
- Sphere