geoR (version 1.1-5)

variog: Compute Empirical Variograms

Description

Computes sample (empirical) variograms with options for the classical or robust estimator. Output can be returned as a binned variogram, a variogram cloud or a smoothed variogram. Data transformation (Box-Cox) is allowed. Trends fitted by ordinary least squares can be removed. In this case variograms are computed using the residuals.

Usage

variog(geodata, coords=geodata$coords, data=geodata$data, 
       uvec = "default", trend = "cte", lambda = 1, 
       option = c("bin", "cloud", "smooth"), 
       estimator.type = c("classical", "modulus"), 
       nugget.tolerance = 0, max.dist = NULL, pairs.min = 2, 
       bin.cloud = FALSE, direction = "omnidirectional",
       tolerance = pi/8, unit.angle = c("radians", "degrees"),
       messages.screen = TRUE, ...)

Arguments

geodata

a list containing element coords as described next. Typically an object of the class "geodata" - a geoR data-set. If not provided the arguments coords must be provided instead.

coords

an $n \times 2$ matrix containing coordinates of the $n$ data locations in each row. Defaults to geodata$coords, if provided.

data

a vector or matrix with data values. If a matrix is provided, each column is regarded as one variable or realization. Defaults to geodata$data, if provided.

uvec

a vector with values defining the variogram binning. Only used when option = "bin". The values of uvec defines the mid-points of the bins. If $uvec[1] > 0$ the first bin is: $0 < u

trend

specifies the mean part of the model. The options are: "cte" (constant mean), "1st" (a first degree polynomial on the coordinates), "2nd" (a second degree polynomial on the coordinates), or a form

lambda

values of the Box-Cox transformation parameter. Defaults to $1$ (no transformation). If another value is provided the variogram is computed after transforming the data. A case of particular interest is $\lambda = 0$ which corresponds

option

defines the output type: the options "bin" returns values of binned variogram, "cloud" returns the variogram cloud and "smooth" returns the kernel smoothed variogram. Defaults to "bin".

estimator.type

"classical" computes the classical method of moments estimator. "modulus" returns the variogram estimator suggested by Hawkins and Cressie (see Cressie, 1993, pg 75). Defaults to "classical".

nugget.tolerance

a numeric value. Points which are separated by a distance less than this value are considered co-located. Defaults to zero.

max.dist

a numerical value defining the maximum distance for the variogram. Pairs of locations separated for distance larger than this value are ignored for the variogram calculation. Defaults to the maximum distance among the pairs of data locatio

pairs.min

a integer number defining the minimum numbers of pairs for the bins. For option = "bin", bins with number of pairs smaller than this value are ignored. Defaults to NULL.

bin.cloud

logical. If TRUE and option = "bin" the cloud values for each class are included in the output. Defaults to FALSE.

direction

a numerical value for the directional (azimuth) angle. This used to specify directional variograms. Default defines the omnidirectional variogram. The value must be in the interval $[0, 180]$ degrees.

tolerance

numerical value for the tolerance angle, when computing directional variograms. The value must be in the interval $[0, 90]$ degrees. Defaults to $\pi/8$.

unit.angle

defines the unit for the specification of angles in the two previous arguments. Options are "degrees" and "radians".

messages.screen

logical. Indicates whether status messages should be printed on the screen (or output device) while the function is running.

...

arguments to be passed to the function ksmooth, if option = "smooth".

Value

• An object of the class variogram which is a list with the following components:
• u
  a vector with distances.
• v
  a vector with estimated variogram values at distances given in u.
• n
  number of pairs in each bin, if option = "bin".
• var.mark
  variance of the data.
• output.type
  echoes the option argument.
• estimator.type
  echoes the type of estimator used.
• direction
  direction for which the variogram was computed.
• tolerance
  tolerance angle for directional variogram.
• uvec
  lags provided in the function call.
• call
  the function call.

Details

Variograms are widely used in geostatistical analysis for exploratory purposes, to estimate covariance parameters and/or to compare theoretical and fitted models against sample variograms.

The two estimators currently implemented are:

• classicalestimator:$$\gamma(h) = \frac{1}{N_h} \sum_{i=1}^{N_h}[Y(x_{(i+h)}) - Y(x_i)]^2$$
• Hawkins and Cressie'smodulusestimator$$\gamma(h) = [\frac{1}{N_h} \sum_{i=1}^{N_h}|Y(x_{i+h}) - Y(x_i)|^{\frac{1}{2}}]^4$$

References

Cressie, N.A.C (1993) Statistics for Spatial Data. New York: Wiley. Further information about geoR can be found at: http://www.maths.lancs.ac.uk/~ribeiro/geoR.html.

See Also

variog4 for more on computation of directional variograms, variog.model.env and variog.mc.env for variogram envelopes, variofit for variogram based parameter estimation and plot.variogram for graphical output.

Examples

# Loading data:
if(is.R()) data(s100)
#
# computing variograms:
#
# binned variogram
vario.b <- variog(s100, max.dist=1)
# variogram cloud
vario.c <- variog(s100, max.dist=1, op="cloud")
#binned variogram and stores the cloud
vario.bc <- variog(s100, max.dist=1, bin.cloud=TRUE)
# smoothed variogram
vario.s <- variog(s100, max.dist=1, op="sm", band=0.2)
#
#
# plotting the variograms:
par(mfrow=c(2,2))
plot(vario.b, main="binned variogram") 
plot(vario.c, main="variogram cloud")
plot(vario.bc, bin.cloud=TRUE, main="clouds for binned variogram")  
plot(vario.s, main="smoothed variogram") 

# computing a directional variogram
vario.0 <- variog(s100, max.dist=1, dir=0, tol=pi/8)
plot(vario.b, type="l", lty=2)
lines(vario.0)
legend(0, 1.2, legend=c("omnidirectional", expression(0 * degree)), lty=c(2,1))
#