Multi-resolution lattice models and theory for spatial process estimators

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，仍然有差距，我们的知识的统计特性的方法，如平滑样条和地质统计估计（克里格），也有有限的理解估计，适应异质结构的功能。在这个建议中，调查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。 该方法是基于一个新的模型，结合了多分辨率（或小波）的基础上的非平稳协方差函数的多元格子模型。这种多分辨率格（MRL）模型建立了以前的工作格模型的空间领域和使用多分辨率基础代表非平稳协方差函数。 关键的创新在于格点模型描述了对基系数的依赖，而不是空间场。多变量扩展允许不同尺度之间的基系数的连接，并且基函数在空间中的本地化有助于对非平稳协方差进行建模。一个重要的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。最终，除了方法和实践的进步，这项研究旨在打破新的地面在理论上的理解如何非参数和空间平滑的行为，并在效果上，统一了广泛的领域的数理statistics.The空间观测或字段的解释是一个基本的数据分析问题，是无处不在的地球科学。 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。 这些数值模型是以公众能够理解的尺度量化气候变化具体影响的主要工具。由这些模拟产生的场具有大量的大尺度结构，是有噪声的，并且经常表现出异方差和非平稳行为。例如，对这些领域进行推断以提供对预测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大分析地球物理数据的可用工具，特别是区域气候模型的复杂输出。

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.