Semiparametric Estimation of Spectral Density With Irregular Observations

Abstract

We propose a semiparametric method for estimating spectral densities of isotropic Gaussian processes with scattered data. The spectral density function (Fourier transform of the covariance function) is modeled as a linear combination of B-splines up to a cutoff frequency and, from this point, a truncated algebraic tail. We calculate an analytic expression for the covariance function and tackle several numerical issues that arise when calculating the likelihood. The parameters are estimated by maximizing the likelihood using the simulated annealing method. Our method directly estimates the tail behavior of the spectral density, which has the greatest impact on interpolation properties. The use of the likelihood in parameter estimation takes the correlations between observations fully into account. We compare our method with a kernel method proposed by Hall et al. and a parametric method using the Matérn model. Simulation results show that our method outperforms the other two by several criteria. Application to rainfall data shows that our method outperforms the kernel method.