Spectral estimation of a structural thin-plate smoothing model

A nonstationary structural spatial model that explicitly sets the data to evolve across a rectangular lattice constrained by second-order smoothing restrictions is presented. The model exemplifies the concept of model-based spatial smoothing and, in particular, it provides a rationale for the popular discrete thin-plate smoothing method. It is further shown how to use a frequency-domain approach to estimate the spatial model via maximum likelihood. In essence, the approach allows both dimensions to be treated separately from each other so that the computational burden for the estimation of two-dimensional models is dramatically reduced both in terms of the computing time and the memory required. Besides, this spectral approach allows straightforward construction of analytic derivatives and an expression for the asymptotic variance of the estimated smoothing parameter is derived with which to construct confidence intervals. Some numerical Monte–Carlo evidence and one example illustrate the results given.