Semivariogram Analysis and Semivariogram Analysis and EstimationEstimation

TanyaTanya, Nick , Nick CarolineCaroline

SemivariogramSemivariogram

• Gives information about the nature Gives information about the nature and structure of spatial dependency and structure of spatial dependency in a random field in a random field → must be → must be estimated from the data estimated from the data

• Estimating a semivariogram:Estimating a semivariogram:1.1. Derive empirical estimate from dataDerive empirical estimate from data

2.2. Fit theoretical semivariogram model to Fit theoretical semivariogram model to empirical estimateempirical estimate

PropertiesProperties

• EvennessEvenness

• Passes through the originPasses through the origin

• Conditionally negative-definite Conditionally negative-definite

Second order stationary Second order stationary Random FieldRandom Field

• Sill – upper asymptoteSill – upper asymptote• Range – distance at which semivariogram Range – distance at which semivariogram

meets asymptote meets asymptote → covariance = zero → covariance = zero • Observations spatially separated by more Observations spatially separated by more

than the range are uncorrelatedthan the range are uncorrelated• Spatial autocorrelation exists only for pairs Spatial autocorrelation exists only for pairs

of points separated by less than the range of points separated by less than the range • The more quickly semivariogram rises from The more quickly semivariogram rises from

the origin to the sill, the more quickly the origin to the sill, the more quickly autocorrelation declines autocorrelation declines

Intrinsically but not second order Intrinsically but not second order stationary random field stationary random field

• Semivariogram never reaches upper Semivariogram never reaches upper asymptote asymptote

• No range defined increase of No range defined increase of semivariogram cannot be arbitrary semivariogram cannot be arbitrary

Parametric Isotropic Parametric Isotropic Semivariogram ModelsSemivariogram Models

• Nugget Only ModelNugget Only Model– White noise processWhite noise process– Void of spatial structureVoid of spatial structure– Relative structure Relative structure

variability is zero variability is zero – Second order stationary Second order stationary

in any dimension in any dimension – Appropriate if smallest Appropriate if smallest

sample distance in the sample distance in the data is greater than the data is greater than the range of the spatial range of the spatial process process

• Linear ModelLinear Model– Stationary Stationary → parameters → parameters

θθ0 0 and and ββ1 1 → must be → must be positivepositive

– Could be initial increase Could be initial increase of second order of second order stationary model that is stationary model that is linear near the origin → linear near the origin → not enough samples far not enough samples far enough apart to capture enough apart to capture range and sill range and sill

Models cont.Models cont.

• Spherical ModelSpherical Model– second order stationary second order stationary

semivariogram model semivariogram model that behaves linearly that behaves linearly near the origin near the origin

– At distance At distance αα semivariogram meets sill semivariogram meets sill and remains flat → range and remains flat → range αα, not practical range , not practical range

• Exponential ModelExponential Model– Approaches sill Approaches sill

asymptotically as h goes asymptotically as h goes to infinity to infinity

– For same range and sill For same range and sill as spherical, rises more as spherical, rises more quickly from the origin quickly from the origin and yields and yields autocorrelations at short autocorrelations at short lag distances smaller lag distances smaller than those of spherical than those of spherical

Models cont. Models cont.

• GaussianGaussian– Exhibits quadratic Exhibits quadratic

behavior near origin and behavior near origin and produces short range produces short range correlations that are correlations that are higher than for any higher than for any second order stationary second order stationary models with same range models with same range

– Only difference between Only difference between this model and this model and exponential model is the exponential model is the square in the exponent square in the exponent

– Most continuous near Most continuous near origin origin → very smooth→ very smooth

• PowerPower– Intrinsically stationary Intrinsically stationary

model for 0 model for 0 ≤ ≤ λλ < 2 < 2– ββ must be positive must be positive

• Wave (Cardinal Sine) Wave (Cardinal Sine) ModelModel– Permits positive and Permits positive and

negative autocorrelationnegative autocorrelation– Fluctuates about sill and Fluctuates about sill and

fluctuations decrease fluctuations decrease with increasing lag with increasing lag

– All parameters must be All parameters must be positive positive

Structure Structure

• Related to degree of smoothness or Related to degree of smoothness or continuitycontinuity

• The slower the increase of The slower the increase of semivariogram near origin → smoother semivariogram near origin → smoother and more spatially structured process and more spatially structured process

• Nugget effect – discontinuity at origin Nugget effect – discontinuity at origin → greatest absence of structure→ greatest absence of structure

• Practical range is often interpreted as Practical range is often interpreted as zone of influence zone of influence

Estimation and Fitting Estimation and Fitting

• Empirical Semivariogram estimators Empirical Semivariogram estimators

• Methods of Fitting:Methods of Fitting:– Least squares Least squares – Maximum Likelihood Maximum Likelihood – Composite LikelihoodComposite Likelihood

Fitting by Least SquaresFitting by Least Squares

• OLS requires data points to be OLS requires data points to be uncorrelated and homoscedastic uncorrelated and homoscedastic

• Values of robust estimator are less Values of robust estimator are less correlated than those of Matheron correlated than those of Matheron estimator estimator

• Problem with generalized least squares Problem with generalized least squares approach approach →→ determination of variance determination of variance – covariance matrix V– covariance matrix V

Fitting by Maximum LikelihoodFitting by Maximum Likelihood

• Requires knowledge of distribution of Requires knowledge of distribution of the data the data

• Data assumed to be Gaussian Data assumed to be Gaussian

• Negatively biased Negatively biased

• Has asymptotic efficiency Has asymptotic efficiency

Composite LikelihoodComposite Likelihood

• Fairly new idea, only 10 years oldFairly new idea, only 10 years old

• We have n Random variables We have n Random variables

• Uses Log Likelihood to MaximizeUses Log Likelihood to Maximize

• Calculated using computing powerCalculated using computing power

Nested Models and Nested Models and Nonparametric Fitting Nonparametric Fitting

• Non-Parametric models have one solid Non-Parametric models have one solid advantageadvantage

• There could be a model inside one of There could be a model inside one of the already known models, this is the already known models, this is known as nestingknown as nesting

• We start with a general model and see We start with a general model and see if we can find a model nested inside if we can find a model nested inside therethere

Homework Homework

• How many parametric semivariogram How many parametric semivariogram models are there and what are they?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 Good luck on finals and have a great summer!!!summer!!!