sm.variogram {sm}  R Documentation 
This function constructs an empirical variogram, using the robust form of construction based on squareroot absolute value differences of the data. Flexible regression is used to assess a variety of questions about the structure of the data used to construct the variogram, including independence, isotropy and stationarity. Confidence bands for the underlying variogram, and reference bands for the independence, isotropy and stationarity models, can also be constructed under the assumption that the errors in the data are approximately normally distributed.
sm.variogram(x, y, h, df.se = "automatic", max.dist = NA, original.scale = TRUE, varmat = FALSE, ...)
x 
a vector or twocolumn matrix of spatial location values. 
y 
a vector of responses observed at the spatial locations. 
h 
a smoothing parameter to be used on the distance scale. A normal kernel
function is used and 
df.se 
the degrees of freedom used when smoothing the empirical variogram to estimate standard errors. The default value of "automatic" selects the degrees of smoothing described in the Bowman and Crujeiras (2013) reference below. 
max.dist 
this can be used to constrain the distances used in constructing the variogram. The default is to use all distances. 
original.scale 
a logical value which determines whether the plots are constructed on the original variogram scale (the default) or on the squareroot absolute value scale on which the calculations are performed. 
varmat 
a logical value which determines whether the variance matrix of the estimated variogram is returned. 
... 
other optional parameters are passed to the 
The reference below describes the statistical methods used in the function.
Note that, apart from the simple case of the indpendence model, the calculations required are extensive and so the function can be slow.
The display
argument has a special meaning for
this function. Its default value is "binned"
, which plots the
binned version of the empirical variogram. As usual, the value "none"
will suppress the graphical display. Any other value will lead to a plot
of the individual differences between all observations. This will lead
to a very large number of plotted points, unless the dataset is small.
A list with the following components:

the raw differences and distances 

the binned differences and distances 

the frequencies of the bins 

the values of the estimate at the evaluation points 

the evaluation points 

the value of the smoothing parameter used 

an indicator of the bin in which the distance between each pair of observations was placed 

the indices of the original observations used to construct each pair 
The suitability of a particular model can be assessed by setting the model
argument, in which case the following components may also be returned, determined by
the arguments passed in ... or the settings in sm.options
.

the pvalue of the test 

the standard errors of the binned values
(if the argument 

when an independence model is examined, this gives the standard error of the difference between the smooth estimate and the mean of all the data points, if a reference band has been requested 

the variance matrix of the binned variogram. When 

the standardised difference between the estiamte of the variogram
and the reference model, evaluated at 

the levels of standarised difference at which contours are drawn
in the case of 
a plot on the current graphical device is produced, unless the option
display="none"
is set.
Diblasi, A. and Bowman, A.W. (2001). On the use of the variogram for checking independence in a Gaussian spatial process. Biometrics, 57, 211218.
Bowman, A.W. and Crujeiras, R.M. (2013). Inference for variograms. Computational Statistics and Data Analysis, 66, 1931.
## Not run: with(coalash, { Position < cbind(East, North) sm.options(list(df = 6, se = TRUE)) par(mfrow=c(2,2)) sm.variogram(Position, Percent, original.scale = FALSE, se = FALSE) sm.variogram(Position, Percent, original.scale = FALSE) sm.variogram(Position, Percent, original.scale = FALSE, model = "independent") sm.variogram(East, Percent, original.scale = FALSE, model = "independent") par(mfrow=c(1,1)) }) # Comparison of Co in March and September with(mosses, { nbins < 12 vgm.m < sm.variogram(loc.m, Co.m, nbins = nbins, original.scale = TRUE, ylim = c(0, 1.5)) vgm.s < sm.variogram(loc.s, Co.s, nbins = nbins, original.scale = TRUE, add = TRUE, col.points = "blue") trns < function(x) (x / 0.977741)^4 del < 1000 plot(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean), type = "b", ylim = c(0, 1.5), xlab = "Distance", ylab = "Semivariogram") points(vgm.s$distance.mean  del, trns(vgm.s$sqrtdiff.mean), type = "b", col = "blue", pch = 2, lty = 2) plot(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean), type = "b", ylim = c(0, 1.5), xlab = "Distance", ylab = "Semivariogram") points(vgm.s$distance.mean  del, trns(vgm.s$sqrtdiff.mean), type = "b", col = "blue", pch = 2, lty = 2) segments(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean  2 * vgm.m$se), vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean + 2 * vgm.m$se)) segments(vgm.s$distance.mean  del, trns(vgm.s$sqrtdiff.mean  2 * vgm.s$se), vgm.s$distance.mean  del, trns(vgm.s$sqrtdiff.mean + 2 * vgm.s$se), col = "blue", lty = 2) mn < (vgm.m$sqrtdiff.mean + vgm.s$sqrtdiff.mean) / 2 se < sqrt(vgm.m$se^2 + vgm.s$se^2) plot(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean), type = "n", ylim = c(0, 1.5), xlab = "Distance", ylab = "Semivariogram") polygon(c(vgm.m$distance.mean, rev(vgm.m$distance.mean)), c(trns(mn  se), rev(trns(mn + se))), border = NA, col = "lightblue") points(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean)) points(vgm.s$distance.mean, trns(vgm.s$sqrtdiff.mean), col = "blue", pch = 2) vgm1 < sm.variogram(loc.m, Co.m, nbins = nbins, varmat = TRUE, display = "none") vgm2 < sm.variogram(loc.s, Co.s, nbins = nbins, varmat = TRUE, display = "none") nbin < length(vgm1$distance.mean) vdiff < vgm1$sqrtdiff.mean  vgm2$sqrtdiff.mean tstat < c(vdiff %*% solve(vgm1$V + vgm2$V) %*% vdiff) pval < 1  pchisq(tstat, nbin) print(pval) }) # Assessing isotropy for Hg in March with(mosses, { sm.variogram(loc.m, Hg.m, model = "isotropic") }) # Assessing stationarity for Hg in September with(mosses, { vgm.sty < sm.variogram(loc.s, Hg.s, model = "stationary") i < 1 image(vgm.sty$eval.points[[1]], vgm.sty$eval.points[[2]], vgm.sty$estimate[ , , i], col = topo.colors(20)) contour(vgm.sty$eval.points[[1]], vgm.sty$eval.points[[2]], vgm.sty$sdiff[ , , i], col = "red", add = TRUE) }) ## End(Not run)