## Project Details

### Description

Extremes of random fields: critical points, excursion components and their statistical applications Extremes of random fields: critical points, excursion components and their statistical applications 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 components 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 detection of change points and structural breaks. 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 components, the project will investigate the distributions of extent and volume of excursion components of Gaussian random fields. 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. Interestingly, by performing kernel smoothing, the detection of change points and structural breaks can be transformed into the problem of peak detection, which provides a novel and efficient tool for detecting changing patterns. Statistical applications also include estimation of parameters in the height distribution of local maxima. 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.

Status | Active |
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Effective start/end date | 9/1/21 → 8/31/26 |

### Funding

- Simons Foundation: $42,000.00

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