Nonparametric modeling of dependencies for spatial interpolation

Abstract

Crucial in spatial interpolation of stochastic processes is the determination of the underlying dependency of the data. The dependency can be represented by an underlying covariogram, variogram, or generalized covariogram. Estimating this function in a nonparametric way is the theme of this thesis. If the function can be found accurately, then kriging is the optimal linear interpolation technique. A nev,· technique for variogram model selection using the derivative of the empirical variogram and non-negative least squares is discussed. The eigenstructure of the spatial design matrix, the key matrix in Matheron's variogram estimator is determined. Then a nonparametric estimator of the variogram and covariogram of a spatial stochastic process is found. The optimal node selection is determined as well as conditions when the spectral coefficients can be found without a non-linear algorithm. A method of extending isotropic positive definite functions in ]Rd is determined in order to avoid a Gibbs effect on the Fourier-Bessel expansion. Finally, a nonparametric estimator of the generalized covariance is discussed.