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

研究人员开发适当的填充或固定域渐近结果的空间数据的近似方法的评价。 填充渐近框架通常是首选的空间数据。然而，填充估计的渐近结果一般很难得到，只有少数明确的填充渐近结果属于特定的模型。在这个建议中，一个一般的方法来建立填充渐近性质的概述，并遵循建立渐近分布的估计，最大限度地提高一些近似的似然函数，如Vecchia的近似和协方差锥形。这些填充渐近结果保证了近似方法可以产生渐近有效的估计。此外，对于时空数据，研究者认为一个新的渐近框架，其中时间域是增加的，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（如协方差锥形）通常会导致有偏估计。研究者建立允许校正偏倚的渐近结果。期望调整后的估计量是渐近正态和无偏的。这些渐近的结果允许一个采用不正确的，但更简单的时空协方差函数，然后调整的偏差。最后，研究者将结果从单变量情况扩展到多变量情况，当多个空间变量在空间和/或时间上被观察时，跨空间和时间的数据在许多对社会非常重要的科学研究中被常规观察，例如全球变暖，环境监测，精确农业，流行病学，国土安全中的热点检测，数据量的巨大性和时空相关性对时空数据的建模和分析提出了新的挑战。该项目的主要目标是通过研究在计算上可行和有效的方法来分析大量的时空数据，从而应对这些挑战，这些方法也不会或几乎不会损失统计效率。 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大量空间和时空数据，从而使科学家能够从大量数据中检索重要和可靠的信息。

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.