Connecting Markov Random Fields with Geostatistical Models

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。最近推导出的内在自回归和deWijs过程之间的连接开辟了进一步探索和利用高斯马尔可夫随机场和连续模型之间的连接，以促进空间数据的统计分析的可能性。在区域数据的分析中，马尔可夫随机场经常受到批评，因为它假设了基于两个地区（县，区或州）之间共享边界的依赖结构，而忽略了实际的区域信息。相比之下，地质统计模型可以通过考虑区域积分来纳入实际区域信息，但面临着大数据的计算挑战。该项目开发了一系列新的方法，通过使用马尔可夫随机场在大型数据集上拟合地质统计模型。研究者的工作为定常和内禀过程的标度极限提供了新的理论结果，并为时空过程的构建提供了新的见解。研究者应用这些方法研究了大面积农田的Fairfield-Smith生育率定律，获得了西雅图人口密度图，调查了伊利诺斯州的各种癌症风险，并插值了风矢量。这些研究出现了两种方法，一种是基于马尔可夫随机场，其他地质统计学。在这个项目中，研究人员结合了马尔可夫随机场和地质统计学的思想，获得了一系列新的方法来分析空间数据。研究者的理论工作将20世纪50年代南非金矿研究中的数学与量子物理学和最近的空间统计理论联系起来。调查员方法的实际效果是巨大的。例如，在研究癌症风险的地理差异时，这一新方法克服了过去的计算挑战，将详细的地区和人口信息纳入统计模型，从而为揭示不同地理区域的风险差异提供了新的方向，指出了可能需要筛查和干预的相对风险较高的地区，并检测了社会经济条件和环境危害的影响。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Recently derived connections between the intrinsic autoregressions and the de Wijs process open up the possibility to further explore and exploit connections between Gaussian Markov random fields and the continuum models in geostatistics to facilitate the statistical analysis of spatial data. In the analysis of areal data, Markov random fields often draw criticism by assuming a dependence structure based on sharing boundaries between two regions (counties, districts or states) and ignoring actual area information. In contrast, geostatistical models can incorporate actual area information by considering integrals over regions, but are faced with computational challenges for large data. This project develops a wide range of new methodologies to fit geostatistical models on large data sets by using Markov random fields. The investigator's work provides new theoretical results for scaling limits of stationary and intrinsic processes and new insight into construction of space-time processes. The investigator applies these methodologies to study the Fairfield-Smith law of fertility on large agricultural fields, obtain the pycnophylactic map of Seattle population, investigate the risk of various cancers in the state of Illinois and interpolate wind vectors.Recent decades have seen much progress and interest in studying spatial variability with applications in image analysis, agriculture, epidemiology, geology and other areas of environmental science. Two approaches to these studies have emerged, one based on Markov random fields, the other on geostatistics. In this project, the investigator combines the ideas of both Markov random fields and geostatistics to obtain a range of new methodologies to analyze spatial data. The investigator's theoretical work connects the mathematics that originated in the studies of gold mines in South Africa in the nineteen fifties to quantum physics and to recent theory of spatial statistics. The practical effect of investigator's approach is enormous. For example, in studies of geographic variations of risk of cancers, this new methodology overcomes past computational challenges to include detailed area and population information in statistical models and thus provides new directions in revealing differences in risk across geographic regions, indicating areas of high relative risk that may require screening and intervention and detecting effects of socio-economic conditions and exposure to environmental hazards.