Preferential sampling provides a formal modeling specification to capture the effect of bias in a set of sampling locations on inference when a geostatistical model is used to explain observed responses at the sampled locations. In particular, it enables modification of spatial prediction adjusted for the bias. Its original presentation in the literature addressed assessment of the presence of such sampling bias while follow on work focused on regression specification to improve spatial interpolation under such bias. All of the work in the literature to date considers the case of a univariate response variable at each location, either continuous or modeled through a latent continuous variable. The contribution here is to extend the notion of preferential sampling to the case of bivariate response at each location. This exposes sampling scenarios where both responses are observed at a given location as well as scenarios where, for some locations, only one of the responses is recorded. That is, there may be different sampling bias for one response than for the other. It leads to assessing the impact of such bias on co-kriging. It also exposes the possibility that preferential sampling can bias inference regarding dependence between responses at a location. We develop the idea of bivariate preferential sampling through various model specifications and illustrate the effect of these specifications on prediction and dependence behavior. We do this both through simulation examples as well as with a forestry dataset that provides mean diameter at breast height (MDBH) and trees per hectare (TPH) as the point-referenced bivariate responses.

当使用地统计模型来解释在采样位置观察到的响应时，优先采样提供了一个正式的建模规范，以捕获一组采样位置中的偏差对推理的影响。特别是，它可以修改针对偏差调整的空间预测。它在文献中的原始介绍涉及评估这种抽样偏差的存在，而后续工作则侧重于回归规范，以改善这种偏差下的空间插值。迄今为止，文献中的所有工作都考虑了每个位置的单变量响应变量的情况，无论是连续的还是通过潜在连续变量建模的。这里的贡献是将优先抽样的概念扩展到每个位置的双变量响应的情况。这揭示了在给定位置观察到两种响应的采样场景，以及对于某些位置仅记录其中一个响应的场景。也就是说，一个响应的采样偏差可能与另一个响应不同。它导致评估这种偏差对联合克里金法的影响。它还揭示了优先抽样可能会使关于某个位置响应之间的依赖关系的推断产生偏差。我们通过各种模型规范发展了双变量优先抽样的想法，并说明了这些规范对预测和依赖行为的影响。我们通过模拟示例以及提供平均胸高 (MDBH) 和每公顷树木 (TPH) 作为点参考双变量响应的林业数据集来做到这一点。