Spatial and Spatio-temporal Processes: Asymptotics, Misspecification and Multivariate Extension

研究者开发适当的填充或固定域渐近结果，以评估空间数据的近似方法。对于空间数据，通常首选填充渐近框架。然而，估计的填充渐近结果通常难以导出，并且只有少数特定模型的显式填充渐近结果。在这一建议中，概述了建立填充渐近性质的一般方法，并遵循建立最大化一些近似似然函数（如Vecchia近似和协方差逐渐变细）的估计量的渐近分布。这些填充渐近结果保证了逼近方法可以得到渐近有效的估计量。此外，对于时空数据，研究者考虑了一个新的渐近框架，其中时域增加，而空间采样域是固定的。在这个渐近框架下，研究者表明一个不正确的协方差函数（如协方差逐渐变细）通常会导致有偏估计。研究者建立了允许偏差校正的渐近结果。期望调整后的估计量是渐近正态和无偏的。这些渐近结果允许人们使用不正确但更简单的时空协方差函数，然后调整偏差。最后，研究者将结果从单变量情况扩展到跨空间和/或随时间观察多个空间变量的多变量情况。在全球变暖、环境监测、精准农业、流行病学、国土安全热点探测等许多对社会非常重要的科学研究中，经常观察到跨空间和跨时间的数据。巨大的数据量和跨时空的相关性对这些时空数据的建模和分析提出了新的挑战。这个项目的主要目标是通过研究计算上可行和有效的方法来分析大量的时空数据来解决这些挑战，这些方法也不会或很少损失统计效率。预计该项目将使更广泛学科的科学家更容易获得和可行地分析巨大的空间和时空数据，从而使科学家能够从大量数据中检索有意义和可靠的信息。

The investigator develops appropriate infill or fixed domain asymptotic results for the evaluation of approximation methods for spatial data. The infill asymptotic framework is generally preferred for spatial data. However, infill asymptotic results for estimation are in general difficult to derive, and there exist only a few explicit infill asymptotic results pertaining to specific models. In this proposal, a general approach to establishing the infill asymptotic properties is outlined and followed to establish asymptotic distributions of estimators that maximize some approximated likelihood functions such as Vecchia's approximation and covariance tapering. These infill asymptotic results assure that the approximation methods may yield asymptotically efficient estimators. In addition, for spatio-temporal data, the investigator considers a new asymptotic framework in which the temporal domain is increasing while the spatial sampling domain is fixed. Under this asymptotic framework, the investigator shows that an incorrect covariance function (such as covariance tapering) generally results in biased estimators. The investigator establishes asymptotic results that allow for the correction of biases. The adjusted estimators are expected to be asymptotically normal and unbiased. These asymptotic results allow one to employ incorrect but simpler spatio-temporal covariance functions and then adjust for the bias. Finally, the investigator extends the results from the univariate case to the multivariate case when multiple spatial variables are observed across space and/or over time.Data across space and time are routinely observed in many scientific studies that are very important to the society such as those on global warming, environmental monitoring, precision agriculture, epidemiology, hot spot detection in homeland security, etc. The immense amount of data and the correlation across space and time have raised new challenges to the modeling and analysis of such space-time data. The primary goal of this project is tackle these challenges by studying computationally feasible and efficient approaches to the analysis of vast space-time data, which also bear no or little loss of statistical efficiency. It is expected that this project will make more accessible and feasible the analysis of huge spatial and spatial-temporal data to scientists in broader disciplines, and thus enables scientists to retrieve significant and reliable information from the vast amount of data.