SpatialStatisticsIV.pptVIP

Spatial Statistics IV Stat 518 Sp08 Estimation of variograms Recall Method of moments: square of all pairwise differences, smoothed over lag bins Problems: Not necessarily a valid variogram Not very robust A robust empirical variogram estimator (Z(x)-Z(y))2 is chi-squared for Gaussian data Fourth root is variance stabilizing Cressie and Hawkins: Least squares Minimize Alternatives: fourth root transformation weighting by 1/?2 generalized least squares Maximum likelihood Z~Nn(?,?) ?= ?[?(si-sj;?)] = ? V(?) Maximize and q maximizes the profile likelihood Parana data A peculiar ml fit Some more fits All together now... Geometric anisotropy If we have an isotropic covariance (circular isocorrelation curves). If for a linear transformation A, we have geometric anisotropy (elliptical isocorrelation curves). General nonstationary correlation structures are typically locally geometrically anisotropic. The deformation idea In the geometric anisotropic case, write where f(x) = Ax. This suggests using a general nonlinear transformation . Usually d=2 or 3. G-plane D-space We do not want f to fold. Do a Bayesian implementation using thin plate splines California ozone Posterior samples ml ls