Spatial data are often derived from multiple sources (e.g. satellites, in-situ sensors, survey samples) with different supports, but associated with the same properties of a spatial phenomenon of interest. It is common for predictors to also be measured on different spatial supports than the response variables. Although there is no standard way to work with spatial data with different supports, a prevalent approach used by practitioners has been to use downscaling or interpolation to project all the variables of analysis towards a common support, and then using standard spatial models. The main disadvantage with this approach is that simple interpolation can introduce biases and, more importantly, the uncertainty associated with the change of support is not taken into account in parameter estimation. In this article, we propose a Bayesian spatial latent Gaussian model that can handle data with different rectilinear supports in both the response variable and predictors. Our approach allows to handle changes of support more naturally according to the properties of the spatial stochastic process being used, and to take into account the uncertainty from the change of support in parameter estimation and prediction. We use spatial stochastic processes as linear combinations of basis functions where Gaussian Markov random fields define the weights. Our hierarchical modelling approach can be described by the following steps: (i) define a latent model where response variables and predictors are considered as latent stochastic processes with continuous support, (ii) link the continuous-index set stochastic processes with its projection to the support of the observed data, (iii) link the projected process with the observed data. We show the applicability of our approach by simulation studies and modelling land suitability for improved grassland in Rhondda Cynon Taf, a county borough in Wales.

翻译：空间数据通常来源于多种不同支撑的渠道（如卫星、原位传感器、调查样本），但均与所关注空间现象的相同属性相关。预测变量与响应变量常在不同空间支撑上测量。尽管目前缺乏处理不同支撑空间数据的标准化方法，实践者普遍采用降尺度或插值技术将所有分析变量投影至统一支撑后使用标准空间模型。该方法的主要缺陷在于：简单插值可能引入偏差，更关键的是参数估计过程中未考虑支撑变化带来的不确定性。本文提出一种贝叶斯空间潜在高斯模型，可同时处理响应变量与预测变量中不同矩形支撑的数据。该方法能根据所用空间随机过程的特性更自然地应对支撑变化，并在参数估计与预测中纳入支撑变化的不确定性。我们采用基函数线性组合表示空间随机过程，并用高斯马尔可夫随机场定义权重。层级建模方法可通过以下步骤描述：(i) 定义潜在模型，将响应变量与预测变量视为连续支撑的潜在随机过程；(ii) 建立连续指标集随机过程与其向观测数据支撑投影之间的关联；(iii) 将投影过程与观测数据关联。通过模拟研究及对威尔士朗达卡农塔夫县改良草地的土地适宜性建模，验证了该方法的适用性。