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

## S3 method for class 'Krig':
predict.se(object, x = NULL, cov = FALSE, verbose = FALSE,...)

Arguments

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 it 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 estimate 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 is interpreted as the true covariance and the prediction variances are evaluated accordingly.

Examples

Run this code

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

  fit<- Krig(ozone$x,ozone$y,cov.function="Exp.cov", theta=10)    # Krig fit 
  predict.se.Krig(fit)      # std errors of predictions at obs.

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

#at the grid points out is a vector of length 400 
#reshape the grid points into a 20X20 matrix etc.  

   out.p<-as.surface( xg, out) 
   surface( out.p, type="C") 

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

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