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 dataGives information about the nature and structure of spatial dependency in a random field → must be estimated from the data Estimating a semivariogram:Estimating a semivariogram: 1.Derive empirical estimate from data 2.Fit theoretical semivariogram model to empirical estimate

3. Properties EvennessEvenness Passes through the originPasses through the origin Conditionally negative-definiteConditionally negative-definite

4. Second order stationary Random Field Second order stationary Random Field Sill – upper asymptoteSill – upper asymptote Range – distance at which semivariogram meets asymptote → covariance = zeroRange – distance at which semivariogram meets asymptote → covariance = zero Observations spatially separated by more than the range are uncorrelatedObservations spatially separated by more than the range are uncorrelated Spatial autocorrelation exists only for pairs of points separated by less than the rangeSpatial 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 declinesThe 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 asymptoteSemivariogram never reaches upper asymptote No range defined increase of semivariogram cannot be arbitraryNo range defined increase of semivariogram cannot be arbitrary

6. Parametric Isotropic Semivariogram Models Nugget Only ModelNugget 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 ModelLinear Model –Stationary → parameters θ 0 and β 1 → 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

7. Models cont. Spherical ModelSpherical 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 ModelExponential 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

8. Models cont. GaussianGaussian –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 PowerPower –Intrinsically stationary model for 0 ≤ λ < 2 –β must be positive Wave (Cardinal Sine) ModelWave (Cardinal Sine) Model –Permits positive and negative autocorrelation –Fluctuates about sill and fluctuations decrease with increasing lag –All parameters must be positive

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

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

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

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

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

14. Nested Models and Nonparametric Fitting Non-Parametric models have one solid advantageNon-Parametric models have one solid advantage There could be a model inside one of the already known models, this is known as nestingThere 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 thereWe 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?How many parametric semivariogram models are there and what are they? What is the nugget effect?What is the nugget effect? List the different fitting methods.List the different fitting methods. Good luck on finals and have a great summer!!!Good luck on finals and have a great summer!!!