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

9.3 The 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) = μ(s) + W(s) + η(s) + e(s) Where Large scale variation of Z(s) is expressed through the mean μ(s) W(s) is the smooth small-scale variation η(s) is a spatial process with variogram γ η e(s) represents measurement error

Two Basic Model Types Signal model S(s)=μ(s) + W(s) + η(s) Mean model δ(s)= W(s) + η(s) + e(s)

Signal Model S(s)=μ(s) + W(s) + η(s) Denotes the signal of the process Then: –Z(s) = S(s) + e(s)

Mean Model δ(s)= W(s) + η(s) + e(s) Denotes the error of process Then: –Z(s) = μ(s) + δ(s) Model is the entry point for spatial regression and analysis of variance Focus is on modeling the mean function μ(s) as a function of covariates and point location –δ(s) is assumed to have spatial autocorrelation structure

Mean and signal models have different focal points Signal 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 –μ(s) is somewhat ancillary Mean model: Interest lies primarily in modeling the large scale trend of the process –Stochastic structure that arises from δ(s) is somewhat ancillary

For mean models, the analyst must decide which effects are part of the large-scale structure μ(s) and which are components of the error structure δ(s) No solid answer One modeler’s fixed effect is someone else’s random effect

Cliff and Ord’s Reaction and Interaction Models Reaction model: sites react to outside influences –Ex: Plants react to the amount of available nutrients in the root zone Interaction model: sites don’t react to outside influences, but rather with each other –Ex: 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 function Interactive 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 one –Can be spurious if caused by large-scale trends or real if caused by cumulative small-scale, spatially varying components –Thus, error structure often thought of as the local structure and the mean structure as the glocal structure

9.4 Spatial Prediction and the Kriging Paradigm

Prediction vs. Estimation Prediction is the determination of the value of a random variable Estimation is the determination of the value of an unknown constant If 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 interest Sampling produces observations Z(s 1 ),…Z(s n ) but Z(s) varies continuously throughout the domain D Producing a map requires prediction of Z() at unobserved locations (s 0 )

The Geostatistical Method: 1. 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 data 2.Estimate 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 γ(h;θ) by one of the methods in 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 autocorrelation 5. Obtain new residuals from step 4 and iterate steps 2-4, if necessary 6. 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 same Spatial data exhibit spatial autocorrelations which are a function of the proximity of observations –We must determine which function of the data best predicts Z(s 0 ) and how to measure the mean square prediction error

Kriging These methods are solutions to the prediction problem where a predictor is best if it –Minimizes the mean square prediction error –Is linear in the observed values Z(s 1 ),…,Z(s n ) –Is unbiased in the sense that the mean of the predicted value at s 0 equals the mean of Z(s 0 )

Optimal Prediction To find the optimal predictor p(Z;s 0 ) for Z(s 0 ) requires a measure for the loss incurred by using p(Z;s 0 ) for prediction at s 0. Different loss functions result in different BEST predictors The loss function of greatest importance in statistics is squared-error loss{Z(s 0 )-p(Z;s 0 )} 2 –Mean Square prediction error is the expected value\ Conditional mean minimizes the MSPE –E[Z(s0)|Z(s)]

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

Classical Kriging techniques Methods for predicting Z(s 0 ) based on combining assumptions about the spatial model with requirements about the predictor p(Z;s 0 ) –p(Z;s 0 ) is a linear combination of the observed values Z(s 1 ),…,Z(s n ) –p(Z;s 0 ) is unbiased in the sense that E[p(Z;s 0 )]=E[Z(s 0 )] –p(Z;s 0 ) minimizes the mean square prediction error

MethodAssumption about u(s) Delta (s) Simple KrigingU(s) is knownSecond order or intrinsically stationary Ordinary KrigingU(s) = u, u unknown Second order or intrinsically stationary Universal Kriging U(s)= x’(s)B, B unknown Second order or intrinsically stationary

Simple Kriging Unbiased The 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: sigma 2 SK (s 0 )= sigma 2 -c’epsilon -1 c Useful in that it determines the benchmark for other kriging methods

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

Notes on Kriging Kriging is not a perfect interpolator when predicting the signal at observed locations and the nugget effect contains a measurement error component Kriging shouldn’t be done on a process with linear drift and exponential semivariogram –Use trend removal, use a method that does not require Σ

Local Kriging and the Kriging Neighborhood Kriging predictors must be calculated for each location at which predictions are desired. –Matrix can become formidable Solution: 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 stationary Disadvantages User needs to decide on the size and shape of the neighborhood Local kriging predictors are no longer best

Kriging Variance Variance-covariance matrix Σ is usually unknown An estimate Σˆ of Σ is substituted in expressions However, the uncertainty associated with the estimation of the semivariances or covariances is typically not accounted for in the determination of the mean square prediction error Therefore, the kriging variance obtained is an underestimate of the mean square prediction error

Cokriging and Spatial Regression If a spatial data set consists of more than one attribute and stochastic relationships exist among them, these relationships can be exploited to improve predictive ability One attribute, Z 1 (s), is designated the primary attribute and Z 2 (s),…,Z k (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 attributes Spatial regression is a multiple spatial prediction method where the mean of the primary attribute is modeled as a function of secondary attributes An advantage of spatial regression over cokriging is that it only requires the spatial covariance function of the primary attribute A disadvantage is that only colocated samples of Z1(s) and all secondary attributes can be used in estimation –If 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

Homework Read 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 the answers to Thanks!