Area-to-point kriging (ATPK) is a geostatistical method for creating high-resolution raster maps using data of the variable of interest with a much lower resolution. The data set of areal means is often considerably smaller (\(<\,50 \) observations) than data sets conventionally dealt with in geostatistical analyses. In contemporary ATPK methods, uncertainty in the variogram parameters is not accounted for in the prediction; this issue can be overcome by applying ATPK in a Bayesian framework. Commonly in Bayesian statistics, posterior distributions of model parameters and posterior predictive distributions are approximated by Markov chain Monte Carlo sampling from the posterior, which can be computationally expensive. Therefore, a partly analytical solution is implemented in this paper, in order to (i) explore the impact of the prior distribution on predictions and prediction variances, (ii) investigate whether certain aspects of uncertainty can be disregarded, simplifying the necessary computations, and (iii) test the impact of various model misspecifications. Several approaches using simulated data, aggregated real-world point data, and a case study on aggregated crop yields in Burkina Faso are compared. The prior distribution is found to have minimal impact on the disaggregated predictions. In most cases with known short-range behaviour, an approach that disregards uncertainty in the variogram distance parameter gives a reasonable assessment of prediction uncertainty. However, some severe effects of model misspecification in terms of overly conservative or optimistic prediction uncertainties are found, highlighting the importance of model choice or integration into ATPK.

中文翻译：

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从贝叶斯角度出发的基于模型的地统计：使用小数据集研究点对点克里金法

区域对点克里金法（ATPK）是一种地理统计方法，用于使用分辨率低得多的目标变量的数据来创建高分辨率栅格地图。面积均值的数据集通常要小得多（\（<\，50 \）观测数据），而不是地统计学分析中通常处理的数据集。在当代的ATPK方法中，预测中没有考虑到变异函数参数的不确定性。通过在贝叶斯框架中应用ATPK可以解决此问题。通常在贝叶斯统计中，模型参数的后验分布和后验预测分布通过后验的马尔可夫链蒙特卡洛采样进行近似，这在计算上可能是昂贵的。因此，本文采用了部分分析的解决方案，以（i）探索先验分布对预测和预测方差的影响，（ii）研究是否可以忽略不确定性的某些方面，简化必要的计算，以及（iii）测试各种模型规格不正确的影响。比较了几种使用模拟数据，汇总的现实世界点数据以及布基纳法索农作物总产量的案例研究的方法。发现先验分布对分类的预测影响最小。在大多数已知短程行为的情况下，忽略方差图距离参数不确定性的方法可以对预测不确定性进行合理评估。但是，发现过分保守或乐观的预测不确定性会导致模型错误指定的严重后果，这突出表明了模型选择或集成到ATPK中的重要性。在大多数情况下，如果具有已知的短程行为，则可以忽略方差图距离参数中的不确定性，从而对预测不确定性进行合理评估。但是，发现过分保守或乐观的预测不确定性会导致模型错误指定的严重影响，从而突出了模型选择或集成到ATPK中的重要性。在大多数情况下，如果具有已知的短程行为，则可以忽略方差图距离参数中的不确定性，从而对预测不确定性进行合理评估。但是，发现过分保守或乐观的预测不确定性会导致模型错误指定的严重影响，从而突出了模型选择或集成到ATPK中的重要性。