空间相依数据的统计推断及其应用研究

Spatial dependent data has a wide range of applications, it is frequently used to describe phenomena with spatial dependent structures in many fields such as ecology, geography, image and economics.In this project, we will use a nonparametric technique to capture the spatial dependent structure of the high-dimensional data. In particular, we propose to estimate the spatial covariance via K-L expression of a random field and the penalized smoothing technique, and to test the covariance feactures such as symmetry, stationary, isotropy and banded structure through spatial extreme value theory、empirical likelihood and likelihood ratio. And their corresponding large-sample properties will be considered. Meanwhile, we will consider to apply spatial unit-root, spatial change-point and spatial threshold models to capture the familiar non-stationary phenomena in environment and ecology. And we will devote to studying the estimation and prediction of these models and establishing their limit theories too. Further, we are interested in the goodness of fit test for spatial model by marked empirical process and Cramer-von Mises statistics.

空间相依数据用于模拟和刻画生态、地理信息、图像、经济、医学等领域中具 有空间相依结构的随机现象，具有重要的实际背景和巨大的应用价值。本项目拟通过非参模 型来更合理地刻画高维空间数据复杂的相依结构，借助随机场的 Karhunen-Loeve 表示和惩 罚核光滑等非参手段来探讨空间数据的协方差矩阵的估计及其大样本性质，拟通过空间极值 理论、经验似然、广义似然比等办法来检验空间协方差结构（如对称性、平稳性、各向同态 性和带状结构）；同时，该项目还拟利用空间单位根、空间门限和变点模型来拟合在环境、 生态等实际领域中十分普遍的空间非平稳数据，研究这几种空间模型在不同相依数据下的参 数估计、预测及其相应的大样本性质；此外，我们还拟利用标点经验过程和 Cramer-von Mises 统计量等检验空间数据模型拟合度问题。

空间相依数据在生态、地理信息、图像、经济、医学等领域中具有非常重要的应用。本项目结合项目组在时间序列和随机场渐近理论的工作基础，研究了空间数据分析中若干重要的统计推断问题，包括：（i）通过随机矩阵转化和GLASSO算法解决了结构变化非平稳时空数据估计的计算问题，提出了新的有效检验空间结构变化的方法；（ii）利用潜在低秩模型探讨了高维时空数据的有效降维与最优预测问题；（iii）利用特征分析方法解决了协整空间的估计与短期效应的有效预测问题；（iv）利用得分经验过程探讨了时空模型拟合度的检验；（v）利用经验似然比探讨了厚尾异方差模型尾指数的估计与推断问题。