### Description

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 this research we propose to study new and challenging directions in extreme value theory, focusing on critical points and excursion probability for Gaussian and related random fields. We then apply these novel theoretical results to important statistical problems involving signal detection in image analysis, multiple hypothesis testing and parameter estimation.

For critical points, we investigate exact computable formulas for their expected number, height distribution and overshoot distribution, and establish approximations to the moments of the number of critical points. For excursion probabilities, we investigate their approximation by the expected Euler characteristic in the case of smooth Gaussian fields with nonconstant variance and for non-Gaussian fields such as F and t fields. We also study asymptotics for excursion probabilities of fractional Brownian motion on manifolds.

In terms of statistical applications, we use testing of local maxima for detecting peaks in nonstationary Gaussian noise, and testing of cluster extent and mass for detecting signal regions, where p-values are computed using the novel theoretical results described above. Statistical applications also include estimation of parameters in the height distribution of local maxima, control of k-FWER for clusters, and detection of change points by performing multiple tests of local extrema of the derivative.

The proposed research will not only lead to novel theoretical results in extreme value theory, but provides efficient signal detection methods for data analysis in many scientific areas.

For critical points, we investigate exact computable formulas for their expected number, height distribution and overshoot distribution, and establish approximations to the moments of the number of critical points. For excursion probabilities, we investigate their approximation by the expected Euler characteristic in the case of smooth Gaussian fields with nonconstant variance and for non-Gaussian fields such as F and t fields. We also study asymptotics for excursion probabilities of fractional Brownian motion on manifolds.

In terms of statistical applications, we use testing of local maxima for detecting peaks in nonstationary Gaussian noise, and testing of cluster extent and mass for detecting signal regions, where p-values are computed using the novel theoretical results described above. Statistical applications also include estimation of parameters in the height distribution of local maxima, control of k-FWER for clusters, and detection of change points by performing multiple tests of local extrema of the derivative.

The proposed research will not only lead to novel theoretical results in extreme value theory, but provides efficient signal detection methods for data analysis in many scientific areas.

Status | Active |
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Effective start/end date | 9/10/18 → 6/30/21 |

### Funding

- National Science Foundation (NSF): $75,463.00

### Fingerprint

critical point

signal detection

oceanography

range (extremes)

random noise

approximation

image analysis

astronomy

statistics

microscopy

moments