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。最近导出的内在自回归与de Wijs过程之间的联系，为进一步探索和利用高斯马尔可夫随机场与地质统计学中的连续体模型之间的联系提供了可能性，以促进空间数据的统计分析。在对区域数据的分析中，马尔可夫随机场经常因假设基于两个区域（县、区或州）之间共享边界的依赖结构而忽略实际的区域信息而受到批评。相比之下，地质统计模型可以通过考虑区域积分来整合实际面积信息，但对于大数据来说，计算方面存在挑战。该项目开发了广泛的新方法，通过使用马尔可夫随机场来拟合大型数据集上的地质统计模型。研究者的工作为固定过程和固有过程的尺度极限提供了新的理论结果，并对时空过程的构造提供了新的见解。研究者将这些方法应用于研究大型农田的费尔菲尔德-史密斯生育力定律，获得西雅图人口的生育地图，调查伊利诺伊州各种癌症的风险，并插值风载体。近几十年来，空间变异性研究在图像分析、农业、流行病学、地质学和其他环境科学领域的应用取得了很大进展和兴趣。这些研究的两种方法已经出现，一种基于马尔可夫随机场，另一种基于地质统计学。在这个项目中，研究者结合了马尔可夫随机场和地质统计学的思想，以获得一系列分析空间数据的新方法。这位研究者的理论工作将20世纪50年代起源于南非金矿研究的数学与量子物理学和最近的空间统计学理论联系起来。研究者的方法的实际效果是巨大的。例如，在癌症风险的地理差异研究中，这种新方法克服了过去在统计模型中纳入详细地区和人口信息的计算挑战，从而为揭示地理区域之间的风险差异提供了新的方向，指出可能需要筛选和干预的相对风险较高的地区，并检测社会经济条件和暴露于环境危害的影响。

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.