A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.

中文翻译：

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在地统计插值中结合面数据和点数据：在土壤科学和医学地理学中的应用。

空间插值中的一个常见问题是在不同空间支持上测量的数据的组合。例如，可用于绘制疾病风险图的信息通常包括点数据（例如患者和对照的住所）和汇总数据（例如在人口普查轨道级别记录的社会人口统计和经济属性）。类似地，田间离散位置的土壤测量通常用等值线图（例如土壤或地质图）作为补充，这些图将土壤属性的空间分布建模为具有恒定值的多边形（区域）并列。本文介绍了克里金法的一般公式，它允许通过在克里金法系统中使用面到面、面到点和点到点协方差来组合点和面数据。该过程使用两个数据集进行说明：(1) 瑞士侏罗地区表土中记录的地质图和重金属浓度，以及 (2) 每个人口普查区的晚期乳腺癌诊断发生率和三个患者住所的位置密歇根州的县。在第二种情况下，克里金系统包括一个根据二项式分布得出的误差方差项，以说明发生率的不同程度的可靠性取决于这些区域中记录的病例总数。除了在二项式克里金法框架下，面积和点克里金法确保预测的连贯性，使得每个映射单元内插值的平均值等于原始面积基准。二项式克里金法、泊松克里金法、在人口规模和空间支持的不同情景下讨论了指标克里金法和指标克里金法。基于局部均值在每个映射单元内是常数的假设，敏感性分析表明新程序比普通和传统残差克里金法具有更小的平滑度和更高的预测精度。