Conditional formulation for the calibration of multi-level random fields with incomplete data

Random fields are widely used in the modeling of spatially varying quantities. As an extension, multi-level random fields capture not only the spatial variability, but also the relationship between inter-dependent quantities. To calibrate such a model, we need data of different variables measured at the same locations in each level. However, measurements of different quantities are usually made independently at inconsistent locations. Recent work addressed this issue by predicting the missing data first, and calibrating the model using the completed dataset. However, existing formulations cannot capture the spatial correlation and inter-level dependency. This paper proposes a conditional random field formulation that overcomes the limitations. Beginning from the lowest level, we calibrate the model using observed data and make predictions conditioned on the observations. The observed data and predictions are then used in the calibration and prediction of the higher levels. To propagate the uncertainties, we develop predictive distributions based on total probability rule for the predictions. We discuss how the formulation works for two missing data scenarios: 1) data partially missing, and 2) data completely missing at some levels. An example of the modeling of multi-level earthquake intensities is provided to illustrate how to implement the proposed formulation in practice.