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

随机场在统计学中发挥着越来越重要的作用，因为它们在许多科学领域中用作空间模型，如神经成像，天文学，地球科学，海洋学和显微镜，其中许多问题涉及空间位置的相关数据。在这些应用中，研究人员感兴趣的是检测隐藏在噪声背景中的信号，通常可以建模为一个随机场。该项目旨在推导随机场的理论性质，并利用这些结果创建有效的统计工具，从空间数据中提取重要和有用的信息。这是通过关注随机场的特定特征来完成的，例如峰值，其作为其紧邻区域的信号的局部代表。因此，可以通过检测其高度高于偶然预期的高度的峰值来发现局部信号。本计画以严谨的统计推论理论，发展此等讯号侦测程序。通过识别大脑图像中的大脑活动区域，以在宇宙背景辐射中找到恒星物体，上述科学学科将受益于所提出的方法，这些方法提供了分析空间相关数据和检测存在噪声的信号的新的有效工具。然后将所得理论结果应用于图像分析中的信号检测、多重假设检验和参数估计等重要统计问题。对于临界点，该项目将研究预期数量，高度分布和超调分布的精确计算公式，并建立临界点数量的近似值。对于偏移概率，该项目将研究具有非常数方差的光滑高斯场和非高斯场的预期欧拉特征近似，以及流形上的分数布朗运动。 作为统计应用，该项目将设计非平稳高斯噪声中局部最大值的测试以及用于检测信号区域的集群范围和质量的测试。 在这些应用中，使用开发的理论计算p值。 统计应用还包括局部最大值高度分布中的参数估计和变点检测。该项目使用概率，统计和几何的跨学科工具来开发所需的理论结果和统计方法。它将在涉及拓扑学和随机矩阵理论的几个数学领域之间建立有趣的联系，并与其他学科，包括神经成像，宇宙学等建立联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.