Semivariogram Analysis and Estimation

1. Semivariogram Analysis and Estimation Tanya, Nick Caroline

2. Semivariogram • Gives information about the nature and structure of spatial dependency in a random field → must be estimated from the data • Estimating a semivariogram: • Derive empirical estimate from data • Fit theoretical semivariogram model to empirical estimate

3. Properties • Evenness • Passes through the origin • Conditionally negative-definite

4. Second order stationary Random Field • Sill – upper asymptote • Range – distance at which semivariogram meets asymptote → covariance = zero • Observations spatially separated by more than the range are uncorrelated • Spatial autocorrelation exists only for pairs of points separated by less than the range • The more quickly semivariogram rises from the origin to the sill, the more quickly autocorrelation declines

5. Intrinsically but not second order stationary random field • Semivariogram never reaches upper asymptote • No range defined increase of semivariogram cannot be arbitrary

6. Nugget Only Model White noise process Void of spatial structure Relative structure variability is zero Second order stationary in any dimension Appropriate if smallest sample distance in the data is greater than the range of the spatial process Linear Model Stationary → parameters θ0 and β1 → must be positive Could be initial increase of second order stationary model that is linear near the origin → not enough samples far enough apart to capture range and sill Parametric Isotropic Semivariogram Models

7. Spherical Model second order stationary semivariogram model that behaves linearly near the origin At distance α semivariogram meets sill and remains flat → range α, not practical range Exponential Model Approaches sill asymptotically as h goes to infinity For same range and sill as spherical, rises more quickly from the origin and yields autocorrelations at short lag distances smaller than those of spherical Models cont.

8. Gaussian Exhibits quadratic behavior near origin and produces short range correlations that are higher than for any second order stationary models with same range Only difference between this model and exponential model is the square in the exponent Most continuous near origin → very smooth Power Intrinsically stationary model for 0 ≤ λ < 2 β must be positive Wave (Cardinal Sine) Model Permits positive and negative autocorrelation Fluctuates about sill and fluctuations decrease with increasing lag All parameters must be positive Models cont.

9. Structure • Related to degree of smoothness or continuity • The slower the increase of semivariogram near origin → smoother and more spatially structured process • Nugget effect – discontinuity at origin → greatest absence of structure • Practical range is often interpreted as zone of influence

10. Estimation and Fitting • Empirical Semivariogram estimators • Methods of Fitting: • Least squares • Maximum Likelihood • Composite Likelihood

11. Fitting by Least Squares • OLS requires data points to be uncorrelated and homoscedastic • Values of robust estimator are less correlated than those of Matheron estimator • Problem with generalized least squares approach →determination of variance – covariance matrix V

12. Fitting by Maximum Likelihood • Requires knowledge of distribution of the data • Data assumed to be Gaussian • Negatively biased • Has asymptotic efficiency

13. Composite Likelihood • Fairly new idea, only 10 years old • We have n Random variables • Uses Log Likelihood to Maximize • Calculated using computing power

14. Nested Models and Nonparametric Fitting • Non-Parametric models have one solid advantage • There could be a model inside one of the already known models, this is known as nesting • We start with a general model and see if we can find a model nested inside there

15. Homework • How many parametric semivariogram models are there and what are they? • What is the nugget effect? • List the different fitting methods. • Good luck on finals and have a great summer!!!