Local Likelihood Estimation for Nonstationary Random Fields

平稳随机场在空间统计学的理论和应用中都扮演着重要的角色。不幸的是，在处理真实数据时，平稳性经常被破坏。这对空间统计学家提出了一个挑战，他们对随机场中的相关性结构的估计和建模感兴趣。研究人员和他的合作者将研究并随后应用最近开发的局部似然估计版本，用于估计控制非平稳随机场中局部相关性的参数。其应用是使用局部似然来估计由于引力透镜引起的宇宙微波背景(CMB)的局部失真。检测和估计这种透镜很重要，原因有两个。首先，测量扭曲给出了宇宙中物质的间接测量，这可以提供暗物质分布的详细地图。其次，人们可以利用引力透镜的估计来获得对未透镜的CMB的更准确的测量。这将为科学家提供对基本宇宙学问题的更深层次的探索。这一建议的具体目标包括：1)发展计算技术，使局部似然估计适用于像CMB这样的大数据集；2)发展局部权重构造的估计方程的理论，特别关注减轻偏差并自动调整边界和不均匀的观测位置；3)构造参数函数的局部似然估计的不确定性估计；4)使用基本的不规则项为一般非平稳随机场建立灵活且通用的局部模型。在最近的数据洪流和高分辨率传感中，非平稳随机场正成为一个普遍存在的特征。遗憾的是，非平稳随机场的建模、估计和预测技术还没有得到充分的发展和分析。这项提议将朝着开发一套完整的分析非平稳随机场的方法学技术的目标迈进一步。此外，这样构建的工具不仅对天文学和宇宙学有用，而且对其他科学和技术分支也是有用的。这使得这一提议的科学后果产生的更广泛的影响是双重的。一方面，准确估计CMB中的引力透镜效应对于理解宇宙结构和宇宙的起源具有深远的影响。另一方面，来自该项目的工具有望在现实世界的其他应用领域中得到广泛使用。最近开发的技术，如功能磁共振成像和扩散张量成像，是其他科学领域的两个例子，这些领域将受益于为非平稳随机场开发的方法。

Stationary random fields play a fundamental role in both theoretical and applied spatial statistics. Unfortunately, stationarity is often violated when working with real data. This presents a challenge for the spatial statistician who is interested in estimating and modeling dependence structure in random fields. The investigator and his collaborator will study, and subsequently apply, a recently developed version of local likelihood estimation for estimating parameters that govern the local dependency in nonstationary random fields. The application is to use local likelihoods for the estimation of the local distortion of the cosmic microwave background (CMB) due to gravitational lensing. Detecting and estimating this lensing is important for two reasons. First, measuring the distortion gives an indirect measurement of matter in the Universe which can provide detailed maps of the distribution of Dark Matter. Secondly, one can use the estimates of gravitational lensing to obtain a more accurate measurement of the un-lensed CMB. This will provide scientists with a deeper probe of fundamental cosmological questions. Specific aims of this proposal include: 1) develop computational techniques which will make local likelihood estimation applicable to large data sets such as the CMB; 2) develop the theory of estimating equations for local weight construction with a particular focus on mitigating bias and automatically adjusting for boundaries and uneven observation locations; 3) construct estimates of uncertainty in the local likelihood estimates of the parameter function; 4) use the principle irregular term to develop flexible and universal local models for general nonstationary random fields. Nonstationary random fields are becoming a ubiquitous feature in the recent data deluge and high resolution sensing. Unfortunately, the techniques for modeling, estimating and predicting nonstationary random fields have yet to be fully developed and analyzed. This proposal will make steps towards the goal of developing a complete set of methodological techniques for analyzing nonstationary random fields. Moreover, the tools thus constructed will be useful, not only for Astronomy and Cosmology, but other branches of science and technology as well. This makes the broader impact of the scientific consequences of this proposal two fold. On the one hand, accurately estimating gravitational lensing in the CMB has far reaching consequences for the understanding of cosmic structure and the beginnings of the Universe. On the other hand, the tools from this project are expected to have broad use in other areas of real world application. Recently developed technologies such as fMRI and diffusion tensor imaging are two examples of other scientific areas that will benefit from the methodologies developed for nonstationary random fields.