Statistical Inference for Spatial Trends

Classical statistical methods for spatial data primarily focus on second-order stationary spatial statistical models with a constant mean and a stationary covariance structure. However, in many scenarios of scientific research, such as biological sciences, environmental sciences and geographical sciences, data could be affected by topographical structures, abrupt policy changes, sudden events, or other local issues. In these cases, the data in different regions may follow different spatial models, and thus the second order stationarity assumption becomes questionable. To model these types of data, it is important to test whether abrupt changes exists, and estimate the change boundaries that partition the data into stationary sub-regions. Testing and estimation of change-boundaries has profound applications not only in spatial data analysis of economic or environmental data, but also in medical imaging, such as the detection of eye diseases in optical coherence tomography (OCT) fundus images, where identifying change boundaries plays a crucial role in determining the onset and monitoring the progression of glaucoma diseases. Many existing spatial boundary detection procedures are only applicable to lattice data which allows simple theoretical analysis. However, in practice, spatial data are often collected at irregularly positioned locations. For example, pollution monitoring sites are irregularly positioned on the earth surface, and buildings are located at irregular sites in housing prices analysis. To fill this important gap in the literature, in this proposal we address the basic issue of inference for abrupt change boundaries in irregularly spaced spatial data. The proposed work, upon successful implementations, has the following three important contributions. First, a novel methodology based on local measures will be developed for measuring discrepancies in the spatial trend function in different regions. Consequently, two test statistics, respectively based on integrated and extreme-valued discrepancies, will be developed to test for abrupt changes in spatial situations. Theoretical results including consistency of the tests will be studied in details under increasing domain and infill asymptotic frameworks. Second, an estimation procedure for the change-boundaries will be developed. Some combinations of increasing domain and infill asymptotic are needed to ensure sufficient nearby data for each of the concerned locations is available so that estimation consistency can be established. Finally, extensions of the aforementioned methodologies to a regression setting with possibly functional coefficients will be pursued. The proposed methods will be applied to infer optical coherence tomography (OCT) fundus images susceptible with glaucoma and related eye diseases.