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(t0)>u∣t0 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(t0)>u+v∣t0 is a local maximum of f(t) and f(t0) > v} as v → ∞. 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.

Let $X=\{X(x): x\in\mathbb{S}^N\}$ be a real-valued, centered Gaussian random field indexed on the $N$-dimensional unit sphere $\mathbb{S}^N$. Approximations to the excursion probability ${\mathbb{P}}\{\sup_{x\in\mathbb{S}^N}X(x)\ge u\}$, as $u\to\infty$, are obtained for two cases: (i) $X$ is locally isotropic and its sample functions are non-smooth and; (ii) $X$ is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367--379) and Chan and Lai (Ann. Probab. 34 (2006) 80--121). It is shown that the asymptotics of ${\mathbb{P}}\{\sup_{x\in\mathbb {S}^N}X(x)\ge u\}$ is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.Comment: Published at http://dx.doi.org/10.3150/14-BEJ688 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic Gaussian noise, where tests are performed at local maxima of the smoothed observed signals. The procedure generalizes the one-dimensional scheme of [20] to Euclidean domains of arbitrary dimension. Two methods are developed according to two different ways of computing p-values: (i) using the exact distribution of the height of local maxima [6], available explicitly when the noise field is isotropic; (ii) using an approximation to the overshoot distribution of local maxima above a pre-threshold [6], applicable when the exact distribution is unknown, such as when the stationary noise field is non-isotropic. The algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency, with specific rates, as the search space and signal strength get large. The optimal smoothing bandwidth and optimal pre-threshold are obtained to achieve maximum power. Simulations show that FDR levels are maintained in non-asymptotic conditions. The methods are illustrated in a nanoscopy image analysis problem of detecting fluorescent molecules against the image background.

We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (spheres), and include the special boundary case of random Laplacian eigenfunctions.

Let X = {X(t), t ∈ R N } be a centered Gaussian random field with stationary increments and X(0) = 0. For any compact rectangle T ⊂ R N and u ∈ R, denote by Au = {t ∈ T : X(t) ≥ u} the excursion set. Under X(·) ∈ C 2 (R N ) and certain regularity conditions, the mean Euler characteristic of Au, denoted by E{ϕ(Au)}, is derived. By applying the Rice method, it is shown that, as u → ∞, the excursion probability P{sup t∈T X(t) ≥ u} can be approximated by E{ϕ(Au)} such that the error is exponentially smaller than E{ϕ(Au)}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.1. Introduction. Let X = {X(t), t ∈ T } be a real-valued Gaussian random field on probability space (Ω, F, P), where T is the parameter set. The study of the excursion probability P{sup t∈T X(t) ≥ u} is a classical but very important problem in probability theory and has many applications in statistics and related areas. Many authors have developed various methods for precise approximations of P{sup t∈T X(t) ≥ u}. These include the double sum method [Piterbarg (1996a)], the tube method [Sun (1993)], the Euler characteristic method [Adler