
Data obtained from Gomez and Hazen (1970, Tables 19 and 20) on coal ash for the Robena Mine Property in Greene County Pennsylvania. Overall plot of the data. #  Create an image plot of the data. For fun, add the contour lines. look < as.image( coalash$y, x=coalash$x) image.plot( look) contour( look, add=T, labex=0) #  Perform a Krig fit using exponential covariance function with range #  paramter, theta=2.5 (choice of theta will be explained below). fit < Krig( coalash$x, coalash$y, cov.function = exp.cov.S, theta=2.5) summary( fit) Call: plot( fit) #  We will use the SPlus generic function, predict, to make predictions, #  using the Krig object fit. #  We also added the contours of the standard errors of the estimates. fit.pred < predict( fit) look < as.image( fit.pred, x=coalash$x) image.plot( look) fit.se < predict.se( fit) look2 < as.image (fit.se, x=coalash$x) contour( look2, add=T, labex=0)
coalash.vgram < vgram( coalash$x, coalash$y) #  coalash.vgram is a list object. Where "d" are the distances, #  coalash.vgram$vgram is the variogram, and call tells how the vgram function was called. names( coalash.vgram) [1] "d" "vgram" "call" bplot.xy( coalash.vgram$d, coalash.vgram$vgram)
Now we use the SPlus function, nls (Non Linear Least Squares Regression) to search for a suitable range parameter. Simultaneously, we search for possible values for sigma2 and rho which we can subsequently fix, if desired, for the Krig fit. vgram.fit < nls( vgram ~ sigma2*(1rho*exp(d/theta)), coalash.vgram, start=list( sigma2= 9, rho=0.95, theta=1)) summary( vgram.fit) Formula: vgram ~ sigma2 * (1  rho * exp(  d/theta)) Parameters: Value Std. Error t value sigma2 1.696190 0.0334569 50.69770 rho 0.750553 0.2807990 2.67292 theta 1.635050 0.5580010 2.93019 Residual standard error: 3.60828 on 21523 degrees of freedom Correlation of Parameter Estimates: sigma2 rho rho 0.346 theta 0.566 0.877 fit < Krig( coalash$x, coalash$y, theta=1.635050, sigma2=1.696190, rho=0.750553) summary( fit) CALL: Krig(x = coalash$x, Y = coalash$y, rho = 0.750553, sigma2 = 1.69619, theta = 1.63505) Number of Observations: 208 Number of unique points: 208 Degree of polynomial null space ( base model): 1 Number of parameters in the null space 3 Effective degrees of freedom: 48.8 Residual degrees of freedom: 159.2 MLE sigma 0.9604 GCV est. sigma 0.9647 MLE rho 0.4082 Scale used for covariance (rho) 0.7506 Scale used for nugget (sigma^2) 1.696 lambda (sigma2/rho) 2.26 RESIDUAL SUMMARY: min 1st Q median 3rd Q max 1.899 0.5541 0.1009 0.3808 5.486 COVARIANCE MODEL: exp.cov DETAILS ON SMOOTHING PARAMETER: Method used: user Cost: 1 lambda trA GCV GCV.one GCV.model shat 2.26 48.8 1.216 1.216 NA 0.9647 Summary of estimates for lambda lambda trA GCV shat GCV 3.422 37.27 1.213 0.9979 GCV.one 3.422 37.27 1.213 0.9979 