Local Likelihood Estimation for Nonstationary Random Fields

平稳随机场在空间统计理论和应用中都起着重要的作用。不幸的是，在处理真实数据时，平稳性经常被破坏。这对空间统计学家来说是一个挑战，他们对随机场的依赖结构的估计和建模很感兴趣。研究者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依赖性的参数。应用于利用局部似然来估计宇宙微波背景（CMB）由于引力透镜引起的局部畸变。由于两个原因，检测和估计这种透镜很重要。首先，测量扭曲可以间接测量宇宙中的物质，从而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的更精确的测量。这将为科学家提供对基本宇宙学问题更深入的探索。本提案的具体目标包括：1)发展计算技术，使局部似然估计适用于大型数据集，如CMB；2)发展局部权重构建的估计方程理论，特别关注减轻偏差和自动调整边界和不均匀观测位置；3)在参数函数的局部似然估计中构造不确定性估计；4)利用不规则项原理，建立一般非平稳随机场的柔性通用局部模型。在数据泛滥和高分辨率传感中，非平稳随机场已成为一个普遍存在的特征。不幸的是，非平稳随机场的建模、估计和预测技术尚未得到充分的发展和分析。这一建议将朝着发展一套完整的分析非平稳随机场的方法技术的目标迈出一步。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很有用。这使得这一提议的科学后果产生了双重的广泛影响。一方面，准确估计CMB中的引力透镜效应对理解宇宙结构和宇宙起源具有深远的影响。另一方面，这个项目的工具被期望在现实世界的其他领域有广泛的应用。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，它们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.