Spatial Stochastic Models Generated by Nested Stochastic Partial Differential Equations

A new class of stochastic field models is constructed using nested stochastic partial differential equations (SPDEs). The model class is computationally efficient, applicable to data on general smooth manifolds, and includes both the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions. Non-stationary covariance models are obtained by spatially varying the parameters in the SPDEs, and the model parameters are estimated using direct numerical optimization, which is more efficient than standard Markov Chain Monte Carlo procedures. As examples of areas of application, the model class is used to approximate popular models in random ocean wave theory, and applied to a large data set of global Total Column Ozone (TCO) data. The TCO data set contains approximately 180 000 measurements, showing that the models allow for efficient inference, even for large environmental data sets.