Estimation, Modeling and Prediction of Nonseparable and Nonstationary Space-Time Processes

M. FUENTES：DMS - 0353029 摘要经典地质统计学和傅立叶谱方法是研究平稳和可分离过程时空结构的有力工具。 然而，人们普遍认为，在实际应用中，时空过程很少是平稳和可分离的. 因此，这些谱方法的一个重要的扩展是对非平稳和不可分的过程. 本文提出了一些新的谱估计方法和工具来估计、建模和检验非平稳性和不可分离性，引入了非参数方法和拟合算法来估计定义在连续空间上的非平稳和不可分离空间过程的时空结构，并研究了这些估计的渐近性质。这些方法是基于一个频谱的方法，使用频谱函数是时空相关的。 本研究最重要的科学贡献是：对一般情况下环境过程的复杂时空依赖性进行了参数和非参数估计（非平稳性、各向异性、不可分离性）;采用光谱方法对环境过程进行空间预测的灵活模式;空间过程是许多环境和科学问题的重要建模工具。 然而，从事时空数据工作的环境科学家通常不相信真实的数据满足诸如可分离性和平稳性等目前在实践中使用的简单模型假设。因此，统计学家必须开发不使用这些假设的方法，特别是用于时空（环境）数据集的方法。 通过与科学家的合作，研究人员提出的用于估计和预测时空过程的新统计模型和方法将通过改善天气和空气质量绘图来加强科学。 研究人员将与大气科学家和海洋学家合作开发应用程序，解决数据同化问题，评估天气、海洋和空气质量数值模式的性能。 这里提出的时空过程的方法也适用于其他领域。 PI与环境保护局（EPA），国家海洋和大气管理局（NOAA）和国家大气研究中心（NCAR）的各种科学家过去的互动证明了PI以前的工作对各个领域产生了影响。 在NCSU，与其他统计部门相比，女性、美国人和非洲裔美国学生的比例很高。 目前在PI监督下撰写论文的7名博士生中有5名是女性。PI将继续努力扩大少数民族和妇女的参与。

M. FUENTES: DMS - 0353029 ABSTRACTClassical geostatistics and Fourier spectral methods are powerfultools to study the spatial temporal structure of stationary andseparable processes. However, it is widely recognized that in realapplications spatial temporal processes are rarely stationary andseparable. Thus an important extension of these spectral methods is toprocesses that are nonstationary and nonseparable. In this work, theinvestigator presents some new spectral approaches and tools toestimate, model, and test for nonstationarity and nonseparability.The investigator introduces nonparametric approaches and fittingalgorithms to estimate the spatial temporal structure of anonstationary and nonseparable spatial process defined on a continuousspace, and studies the asymptotic properties of these estimates. Themethods are based on a spectral approach, using spectral functionsthat are space-time dependent. The most important scientificcontributions of the research proposed here are: the parametric andnonparametric estimation of the complex spatial temporal dependence ofenvironmental processes in general situations (nonstationarity,anisotropy, nonseparability); the introduction of flexible models forspatial prediction of environmental processes using spectral methods;and new methodology for spatial prediction and estimation in thepresence of massive data.Spatial processes are an important modeling tool for manyenvironmental and scientific problems. Environmental scientists whowork with spatial temporal data, however, do not typically believethat real data satisfy the simple model assumptions such asseparability and stationarity that are currently used in practice.Therefore, it is is imperative for statisticians to develop methodswithout using those assumptions, especially for use with massivespatial-temporal (environmental) data sets. Through collaborationswith scientists, the new statistical models and methods proposed bythe investigator for estimation and prediction of space-timeprocesses, will enhance science by improving weather and air qualitymapping. The investigator will develop applications in collaborationwith atmospheric scientists and oceanographers on data assimilationproblems and on assessment of the performance of weather, ocean, andair quality numerical models. The methods proposed here for spacetime processes are also applicable to other fields. Past interactionsof the PI with various scientists at the Environmental ProtectionAgency (EPA), the National Oceanic and Atmospheric Administration(NOAA), and the National Center for Atmospheric Research (NCAR) areevidence that previous work of the PI has had an impact on variousfields. At NCSU there is ahigh proportion of women, American and African-American studentscompared to other Statistics departments. Five out of the seven PhDstudents currently working on their dissertations under the PI'ssupervision are women. The PI will continue her efforts to broaden theparticipation of minorities and women.