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Spectral Density Estimation for Random Fields via Periodic Embeddings

Joseph Guinness

Accepted, Biometrika

We introduce methods for estimating the spectral density of a random field on a d-dimensional lattice from incomplete gridded data. The methods iteratively impute missing data with periodic conditional simulations on an expanded lattice. Periodic conditional simulations are convenient computationally, in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in O(n log n) time and O(n) memory. They also have desirable theoretical properties; we prove that both periodogram bias and correlation in the discrete Fourier transform vector can be completely eliminated in some situations, and in general decays faster than O(1/n) when the spectral density is sufficiently smooth. In addition, we introduce a filtering method that is designed to reduce periodogram smoothing bias. The methods are demonstrated in numerical and simulation studies, and we present an application to gridded satellite data with missing values. We also describe an implementation of the methods in a publicly available R software package.