Local Properties of Irregularly Observed Gaussian Fields
September 20, 2012
2:30 p.m.
Myoungji Lee
Abstract
The behavior of a variogram at the origin is important in spatial interpolation as it determines the smoothness of a random process. The power-law variogram at the origin is capable of capturing the local behaviors of many known parametric covariance functions and self-similar processes. In addition, the fractal dimension, a scale invariant measure of surface roughness, is a function of the exponent in the power-law model. The estimation of the fractal dimension using the empirical variogram and least squares method has been well studied for a stationary Gaussian process, observed at evenly spaced sampling locations.
This talk presents the estimation of the fractal dimension for observations at unevenly spaced sites by approximating restricted likelihood, conditioning on observations at neighboring sites only. Finite sample performances of the estimates from the approximated and full restricted likelihoods are compared through mean squared errors and Godambe’s information measure. The numerical and asymptotic study shows that the locally approximated likelihood is computationally inexpensive yet capturing the local behavior of the variogram well, without requiring the specification of the full variogram. This is a joint work with Michael Stein in the University of Chicago.