fields (version 1.2)

predict.se.Krig: Standard errors of predictions for Krig spatial process estimate

Description

Finds the standard error ( or covariance) of prediction based on a linear combination of the observed data. The linear combination is usually the "Best Linear Unbiased Estimate" (BLUE) found from the Kriging equations. There are also provisions to use a different covariance for evaluation than the one used to define the BLUE.

Usage

predict.se.Krig(out, x, cov.function, rho, sigma2, weights=NULL, cov=FALSE,  
stationary=TRUE, fixed.mean=TRUE, ...)

Arguments

out

A Krig object.

x

Points to compute the SE.

cov.function

Name of the true covariance function for the random surface if different from the one used to generate predictions. If omitted, then the function from the Krig object

rho

Parameter that multiplies the covariance function. If omitted the value out$rho, estimated from the data from the data is used.

sigma2

Variance of the measurement error. If omitted the value out$sigma2, estimated from the data is used.

weights

stationary

If true the covariance function is assumed to be stationary and more efficient computations can be used.

cov

If true the full covariance matrix for the predicted values is returned. Make sure this will not be big is this option is used. ( e.g. 50X50 grid will return a matrix that is 2500X2500!) If false just the marginal standard deviations of the predicted val

fixed.mean

If true the mean function is assumed fixed and the variance of this component is taken to be zero. If False the variance for this component is added to the variance of prediction.

...

These additional arguments are given to cov.function the "true" covariance function, not the one associated with the Kriging weights.

Value

• A vector of standard errors for the predicted values of the Kriging fit.

References

See Case Studies in Environmental Statistics

Details

The predictions are represented as a linear combination of the dependent variable, Y. Call this LY. Based on this representation the conditional variance is the same as the expected value of (P(x) + Z(X) - LY)**2. where P(x)+Z(x) is the value of the surface at x and LY is the linear combination that estimates this point. Finding this expected value is straight forward given the unbiasedness of LY for P(x) and the covariance for Z and Y.

In these calculations is assumed that the covariance parameters are fixed. This is an approximation since in most cases they have been estimated from the data. It should also be noted that if one assumes a Gaussian field and known parameters in the covariance, the usual Kriging estimating is the conditional mean of the field given the data. This function finds the conditional standard deviations (or full covariance matrix) of the fields given the data.

There are two useful extensions supported by this function. Adding the variance to the estimate of the spatial mean if this is a correlation model. (See help file for Krig) and calculating the variances under covariance misspecification. Note that the linear combination is based on the covariance function from the Krig object. One can view this first step as simply defining a spatial estimator. If the covariance used is correct it is BLUE, otherwise the MSE for the spatial estimate will be larger than optimal. The cov.function argument in this function defaults to the same covariance used to determine the spatial prediction but it also can be specified separately, in this case it interpreted as the true covariance and the prediction variances are evaluated accordingly.

See Also

Krig, predict.Krig, predict.surface.se

Examples

# 
# Note: in these examples predict.se will default to predict.se.Krig using 
# a Krig object  

fit<- Krig(ozone$x,ozone$y,exp.cov, theta=10)    # krig fit 
predict.se.Krig(fit)                        # std errors of predictions 

# make a  grid of X's  
xg<-make.surface.grid( list(seq(-27,34,,40),seq(-20,35,,40)))     
out<- predict.se.Krig(fit,xg)     # std errors of predictions 

#at the grid points out is a vector of length 1600 
# reshape the grid points into a 40X40 matrix etc.  
out.p<-as.surface( xg, out) 
image.plot( out.p) 

# this is equivalent to  the single step function  
# (but default is not to extrapolation beyond data
out<- predict.surface.se( fit) 
image.plot( out) 


# Investigate misspecification 
# 
# first call Krig to create the Krig object.  
# 
Krig( ozone$x, ozone$y, cov.function=exp.cov, theta=100)-> fit 
# note how the new cov. parameters are specified just like in Krig  
predict.se(fit,xg)-> look 
predict.se( fit, xg, cov.function=exp.cov, theta=2.0, sigma2=1)-> look2 

set.panel( 2,1)
image.plot( as.surface( xg, look))
points( fit$x)
image.plot( as.surface( xg, look2))
set.panel( 1,1)