# 9.3 and 9.4 The Spatial Model And Spatial Prediction and the Kriging Paradigm.

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9.3 and 9.4The Spatial ModelAndSpatial Prediction and the Kriging Paradigm

9.3The Spatial Model

The Spatial Model.What is it?Statistical model decomposing the variability in a random function into a deterministic mean structure and one or more spatial random processes

For Geostatistical data:Z(s) = (s) + W(s) + (s) + e(s)WhereLarge scale variation of Z(s) is expressed through the mean (s)W(s) is the smooth small-scale variation(s) is a spatial process with variogram e(s) represents measurement error

Two Basic Model Types

Signal modelS(s)=(s) + W(s) + (s)

Mean model(s)= W(s) + (s) + e(s)

Signal Model S(s)=(s) + W(s) + (s)Denotes the signal of the processThen:Z(s) = S(s) + e(s)

Mean Model (s)= W(s) + (s) + e(s)Denotes the error of processThen:Z(s) = (s) + (s)Model is the entry point for spatial regression and analysis of varianceFocus is on modeling the mean function (s) as a function of covariates and point location(s) is assumed to have spatial autocorrelation structure

Mean and signal models have different focal pointsSignal model: we are primarily interested in stochastic behavior of the random field and, if it is spatially structured, predict Z(s) or S(s) at observed and unobserved locations(s) is somewhat ancillaryMean model: Interest lies primarily in modeling the large scale trend of the processStochastic structure that arises from (s) is somewhat ancillary

For mean models, the analyst must decide which effects are part of the large-scale structure (s) and which are components of the error structure (s)No solid answer

One modelers fixed effect is someone elses random effect

Cliff and Ords Reaction and Interaction ModelsReaction model: sites react to outside influencesEx: Plants react to the amount of available nutrients in the root zoneInteraction model: sites dont react to outside influences, but rather with each otherEx: Neighboring plants compete with each other for resources

In general:When dominant spatial effects are caused by sites reacting with external forces, include the effects as part of the mean functionInteractive effects call for modeling spatial variability through the spatial autocorrelation structure of the error process

Spatial autocorrelation in a model does not always imply an interactive model over a reactive oneCan be spurious if caused by large-scale trends or real if caused by cumulative small-scale, spatially varying componentsThus, error structure often thought of as the local structure and the mean structure as the glocal structure

9.4Spatial Prediction and the Kriging Paradigm

Prediction vs. EstimationPrediction is the determination of the value of a random variableEstimation is the determination of the value of an unknown constantIf interest lies in the value of a random field at location (s) then we should predict Z(s) or the signal S(s). If the average value at location (s) across all realizations of the random experiment is of interest, we should estimate E[Z(s)].

The goal of geostatistical analysis:Common goal is to map the random function Z(s) in some region of interestSampling produces observations Z(s1),Z(sn) but Z(s) varies continuously throughout the domain DProducing a map requires prediction of Z() at unobserved locations (s0)

The Geostatistical Method: Using exploratory techniques, prior knowledge, and/or anything else, posit a model of possibly nonstationary mean plus second-order or intrinsically stationary error for the Z(s) process that generated the dataEstimate the mean function by ordinary least squares, smoothing, or median polishing to detrend the data. If the mean is stationary this step is not necessary. The methods for detrending employed at this step ususually do not take autocorrelation into account

3. Using the residuals obtained in step 2 (or the original data if the mean is stationary), fit a semivariogram model (h;) by one of the methods in 9.2.44. Statistical estimates of the spatial dependence in hand (from step 3) return to step 2 to re-estimate the parameters of the mean function, now taking into account the spatial autocorrelation5. Obtain new residuals from step 4 and iterate steps 2-4, if necessary6. Predict the attribute Z() at unobserved locations and calculate the corresponding mean square prediction errors

In classical linear model, the predictor (Y) and the estimator (X) are the sameSpatial data exhibit spatial autocorrelations which are a function of the proximity of observationsWe must determine which function of the data best predicts Z(s0) and how to measure the mean square prediction error

KrigingThese methods are solutions to the prediction problem where a predictor is best if itMinimizes the mean square prediction errorIs linear in the observed values Z(s1),,Z(sn)Is unbiased in the sense that the mean of the predicted value at s0 equals the mean of Z(s0)

Optimal PredictionTo find the optimal predictor p(Z;s0) for Z(s0) requires a measure for the loss incurred by using p(Z;s0) for prediction at s0. Different loss functions result in different BEST predictorsThe loss function of greatest importance in statistics is squared-error loss{Z(s0)-p(Z;s0)}2Mean Square prediction error is the expected value\Conditional mean minimizes the MSPEE[Z(s0)|Z(s)]

Basic Kriging MethodsOrdinary and Universal KrigingKriging predictors are the best linear unbiased predictors under squared error lossSimple, ordinary, and universal kriging differ in their assumption about the mean structure u(s) of the spatial model

Classical Kriging techniquesMethods for predicting Z(s0) based on combining assumptions about the spatial model with requirements about the predictor p(Z;s0)p(Z;s0) is a linear combination of the observed values Z(s1),,Z(sn)p(Z;s0) is unbiased in the sense that E[p(Z;s0)]=E[Z(s0)]p(Z;s0) minimizes the mean square prediction error

MethodAssumption about u(s)Delta (s)Simple KrigingU(s) is knownSecond order or intrinsically stationaryOrdinary KrigingU(s) = u, u unknownSecond order or intrinsically stationaryUniversal KrigingU(s)= x(s)B, B unknownSecond order or intrinsically stationary

Simple KrigingUnbiasedThe optimal method of spatial prediction (under squared error loss) in a Gaussian random field

The minimized mean square prediction error of an unbiased kriging predictor is often called the kriging variance or the kriging error.

For simple kriging, the kriging variance is:sigma2SK(s0)= sigma2-cepsilon-1cUseful in that it determines the benchmark for other kriging methods

Universal and Ordinary KrigingMean of the random field is not known and can be expressed by a linear modelOrdinary kriging predictor minimizes the mean square prediction error subject to an unbiasedness constraint

Notes on KrigingKriging is not a perfect interpolator when predicting the signal at observed locations and the nugget effect contains a measurement error componentKriging shouldnt be done on a process with linear drift and exponential semivariogramUse trend removal, use a method that does not require

Local Kriging and the Kriging NeighborhoodKriging predictors must be calculated for each location at which predictions are desired.Matrix can become formidableSolution: Consider for prediction of Z(s0) only observed data points within a neighborhood of s0, called the Kriging Neighborhood (aka local kriging)

Advantages

Computational efficiency

Reasonable to assume that the mean is at least locally stationaryDisadvantages

User needs to decide on the size and shape of the neighborhood

Local kriging predictors are no longer best

Kriging VarianceVariance-covariance matrix is usually unknownAn estimate of is substituted in expressionsHowever, the uncertainty associated with the estimation of the semivariances or covariances is typically not accounted for in the determination of the mean square prediction errorTherefore, the kriging variance obtained is an underestimate of the mean square prediction error

Cokriging and Spatial RegressionIf a spatial data set consists of more than one attribute and stochastic relationships exist among them, these relationships can be exploited to improve predictive abilityOne attribute, Z1(s), is designated the primary attribute and Z2(s),,Zk(s) are the secondary attributes

Cokriging is a multivariate spatial prediction method that relies on the spatial autocorrelation of the primary and secondary attributes as well as the cross-covariances among the primary and the secondary attributesSpatial regression is a multiple spatial prediction method where the mean of the primary attribute is modeled as a function of secondary attributesAn advantage of spatial regression over cokriging is that it only requires the spatial covariance function of the primary attributeA disadvantage is that only colocated samples of Z1(s) and all secondary attributes can be used in estimationIf only one of the secondary attributes has not been observed at a particular location the information collected on any attribute at that location will be lost

HomeworkRead the applications section: Spatial prediction-Kriging of Lead Concentrates (9.8.3):Why are the estimates of total lead conservative?Why would an estimate of total lead based on the sample average be positively biased?What does the top panel of Figure 9.41 tell us? Why do we perform universal kriging?What does figure 9.43 represent? Does it show the total lead concentration?What is the estimate of the total amount of lead obtained by universal block-kriging?Please email the answers

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