Multi-resolution lattice models and theory for spatial process estimators

空间过程估计器的多分辨率点阵模型和理论

• 批准号: 0707069
• 负责人: Stephan Sain
• 金额: $ 22.78万
• 依托单位: University Corporation For Atmospheric Res
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707069&HistoricalAwards=false
• 关键词: Multi; resolution; lattice; models; theory

Estimating a smooth function from noisy observations is a core problem in mathematical statistics and supports the areas of nonparametric regression and spatial data analysis. However, there are still gaps in our knowledge of the statistical properties of methods such as smoothing splines and geostatistical estimators (Kriging), and there is also limited understanding of estimators that adapt to heterogeneous structure in the function. In this proposal, the investigators address some of the issues in nonparametric function estimation through a spatial statistics framework. The approach is based on a new model for nonstationary covariance functions that combines a multiresolution (or wavelet) basis with a multivariate lattice model. This multiresolution lattice (MRL) model builds off previous work on lattice models for spatial fields and the use of multiresolution bases for representing nonstationary covariance functions. The key innovation is that the lattice model describes dependence on the coefficients of the basis, not the spatial field. The multivariate extension allows for connections of basis coefficients between different scales, and the localization of the basis functions in space facilitates modeling nonstationary covariance. An important component is the extension of large sample statistical theory to analyze spatial estimators applied to irregular locations and with nonstationary covariance. Ultimately, in addition to methodological and practical advances, this research seeks to break new ground in the theoretical understanding of how nonparametric and spatial smoothers behave, and, in effect, unifying a broad area of mathematical statistics.The interpretation of spatial observations or fields is a fundamental data analysis problem that is ubiquitous in the geosciences. A specific example is the study of regional climate change where complex numerical models are coupled to simulate climate at local scales. These numerical models are a primary tool to quantify specific impacts of climate change at a scale that can be understood by the general public. The fields produced by these simulations have a great deal of large-scale structure, are noisy, and often exhibit heteroscedastic and nonstationary behavior. Drawing inferences about these fields to provide, for example, a probabilistic assessment of the projected climate change, requires a deliberate statistical approach. The research outlined in this proposal seeks to expand the tools available to analyze geophysical data, in particular the complex outputs of regional climate models.

从噪声观测中估计平滑函数是数理统计中的核心问题，并支持非参数回归和空间数据分析领域。然而，我们对平滑样条和地统计估计器（克里金法）等方法的统计特性的了解仍然存在差距，并且对适应函数中异构结构的估计器的理解也很有限。在该提案中，研究人员通过空间统计框架解决了非参数函数估计中的一些问题。 该方法基于非平稳协方差函数的新模型，该模型将多分辨率（或小波）基础与多元晶格模型相结合。这种多分辨率点阵 (MRL) 模型建立在先前关于空间场点阵模型的工作以及使用多分辨率基础来表示非平稳协方差函数的基础上。 关键的创新在于，晶格模型描述了对基系数的依赖，而不是空间场。多元扩展允许不同尺度之间的基系数连接，并且基函数在空间中的定位有利于对非平稳协方差进行建模。一个重要组成部分是大样本统计理论的扩展，以分析应用于不规则位置和非平稳协方差的空间估计器。最终，除了方法论和实践方面的进步之外，这项研究还试图在非参数和空间平滑器行为的理论理解上开辟新天地，并实际上统一数学统计的广泛领域。空间观测或场的解释是地球科学中普遍存在的基本数据分析问题。 一个具体的例子是区域气候变化的研究，其中耦合复杂的数值模型来模拟局部尺度的气候。 这些数值模型是在公众可​​以理解的范围内量化气候变化具体影响的主要工具。这些模拟产生的场具有大量的大规模结构，充满噪声，并且经常表现出异方差和非平稳行为。例如，对这些领域进行推论以提供对预计气候变化的概率评估，需要采用深思熟虑的统计方法。该提案中概述的研究旨在扩展可用于分析地球物理数据的工具，特别是区域气候模型的复杂输出。