Spatial Stochastic Models Generated by Nested SPDEs

A new class of stochastic field models is constructed using nested stochastic partial differential equations (SPDEs). The model class is applicable to data on general smooth manifolds and includes both the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions. Computationally efficient approximations are obtained using Hilbert space approximations, which gives more accurate approximations than both convolution fields approximations and covariance tapering for a given computational cost. 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. Results are illustrated with a large data set of spatially irregular global Total Column Ozone (TCO) data. The TCO data set contains approximately 180000 measurements, showing that the models allow for efficient inference, even for large environmental data sets.