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

协作研究：随机场的临界点和偏移概率：理论和统计应用

• 批准号: 1811659
• 负责人: Armin Schwartzman
• 金额: $ 7.5万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811659&HistoricalAwards=false
• 关键词: Collaborative; Research; Critical; Points; Excursion

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.

随机场在统计学中发挥着越来越重要的作用，因为它们被用作许多科学领域的空间模型，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上的相关数据。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，噪声背景通常可以建模为随机场。该项目旨在推导随机场的理论属性，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。这是通过关注诸如峰值的随机场的特定特征来实现的，该随机场充当其附近信号的局部代表。因此，可以通过检测其高度高于预期高度的峰来发现局部信号。这个项目用严格的统计推理理论开发了这样的信号检测程序。通过识别大脑图像中的大脑活动区域来寻找宇宙背景辐射中的恒星对象，上述科学学科将受益于所提出的方法，这些方法为分析空间相关数据和在存在噪声的情况下检测信号提供了新的有效工具。本项目将研究高斯场和相关随机场的临界点和漂移概率，然后将所获得的理论结果应用于图像分析中的信号检测、多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准确可计算公式，并建立临界点数量的矩的近似。对于漂移概率，该项目将研究具有非常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，该项目将为非平稳高斯噪声中的局部极大值设计测试，并为检测信号区域设计集群范围和质量测试。在这些应用中，p值是使用发展的理论来计算的。统计应用还包括估计局部极大值高度分布中的参数和检测变化点。该项目使用概率、统计学和几何学中的跨学科工具来开发所需的理论结果和统计方法。它将在几个涉及拓扑学和随机矩阵理论的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、宇宙学和其他学科建立有趣的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

On critical points of Gaussian random fields under diffeomorphic transformations

微分同胚变换下高斯随机场的临界点

• DOI: 10.1016/j.spl.2019.108672
• 发表时间: 2020
• 期刊: Statistics & Probability Letters
• 影响因子: 0.8
• 作者: Cheng, Dan;Schwartzman, Armin
• 通讯作者: Schwartzman, Armin

Multiple testing of local maxima for detection of peaks on the (celestial) sphere

多次测试局部最大值以检测（天体）球上的峰值

• DOI: 10.3150/18-bej1068
• 发表时间: 2020
• 期刊: Bernoulli
• 影响因子: 1.5
• 作者: Cheng, Dan;Cammarota, Valentina;Fantaye, Yabebal;Marinucci, Domenico;Schwartzman, Armin
• 通讯作者: Schwartzman, Armin

Multiple testing of local extrema for detection of change points

局部极值的多重测试以检测变化点

• DOI: 10.1214/20-ejs1751
• 发表时间: 2020
• 期刊: Electronic Journal of Statistics
• 影响因子: 1.1
• 作者: Cheng, Dan;He, Zhibing;Schwartzman, Armin
• 通讯作者: Schwartzman, Armin

Point source detection and false discovery rate control on CMB maps

CMB 地图上的点源检测和错误发现率控制

• DOI: 10.1016/j.ascom.2019.100310
• 发表时间: 2019
• 期刊: Astronomy and Computing
• 影响因子: 2.5
• 作者: Carrón Duque, J.;Buzzelli, A.;Fantaye, Y.;Marinucci, D.;Schwartzman, A.;Vittorio, N.
• 通讯作者: Vittorio, N.