Collaborative Research: Critical Points and Excursion Probability of Random Fields: Theory and Statistical Applications

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Random fields are playing increasingly important roles in statistics, due to their use as spatial models in many scientific areas such as neuroimaging, astronomy, geosciences, oceanography and microscopy, where many problems involve dependent data at spatial locations. In these applications, researchers are interested in detecting signals hidden in a noisy background which can usually be modeled as a random field. This project aims to derive theoretical properties of random fields and use these results to create efficient statistical tools to extract important and useful information from spatial data. This is done by focusing on specific features of random fields such as peaks, which serve as local representatives of the signal in their immediate vicinity. Thus, local signals can be discovered by detecting peaks whose height is above what would be expected by chance. This project develops such signal detection procedures with rigorous statistical inference theory. By identifying regions of brain activity in brain images to finding stellar objects against the cosmic background radiation, the aforementioned scientific disciplines will benefit from the proposed methods, which provide new efficient tools to analyze spatially dependent data and detect signals in the presence of noise.This project will study critical points and excursion probabilities of Gaussian and related random fields, and then apply the obtained theoretical results to important statistical problems involving signal detection in image analysis, multiple hypothesis testing and parameter estimation. For critical points, the project will investigate exact computable formulas for the expected number, height distribution, and overshoot distribution, and establish approximations to the moments of the number of critical points. For excursion probabilities, the project will investigate the expected Euler characteristic approximation for smooth Gaussian fields with non-constant variance and for non-Gaussian fields, and fractional Brownian motion on manifolds. As statistical applications, the project will devise tests for local maxima in non-stationary Gaussian noise and testing of cluster extent and mass for detecting signal regions. In these applications, p-values are computed using the developed theory. Statistical applications also include estimation of parameters in the height distribution of local maxima and detection of change-points. This project uses interdisciplinary tools from probability, statistics, and geometry to develop the desired theoretical results and statistical methods. It will create interesting connections between several mathematical areas involving topology and random matrices theory, and to other disciplines, including neuroimaging, cosmology, and beyond.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.