Calculates the sample variogram from data, or in case of a linear model is given, for the residuals, with options for directional, robust, and pooled variogram, and for irregular distance intervals.

In case spatio-temporal data is provided, the function `variogramST`

is called with a different set of parameters.

```
# S3 method for gstat
variogram(object, ...)
# S3 method for formula
variogram(object, locations = coordinates(data), data, ...)
# S3 method for default
variogram(object, locations, X, cutoff, width = cutoff/15,
alpha = 0, beta = 0, tol.hor = 90/length(alpha), tol.ver =
90/length(beta), cressie = FALSE, dX = numeric(0), boundaries =
numeric(0), cloud = FALSE, trend.beta = NULL, debug.level = 1,
cross = TRUE, grid, map = FALSE, g = NULL, ..., projected = TRUE,
lambda = 1.0, verbose = FALSE, covariogram = FALSE, PR = FALSE,
pseudo = -1)
# S3 method for gstatVariogram
print(x, ...)
# S3 method for variogramCloud
print(x, ...)
```

object

object of class `gstat`

; in this form, direct
and cross (residual) variograms are calculated for all variables and
variable pairs defined in `object`

; in case of `variogram.formula`

,
formula defining the response vector and (possible)
regressors, in case of absence of regressors, use e.g. `z~1`

;
in case of `variogram.default`

: list with for each variable
the vector with responses (should not be called directly)

data

data frame where the names in formula are to be found

locations

spatial data locations. For variogram.formula: a
formula with only the coordinate variables in the right hand (explanatory
variable) side e.g. `~x+y`

; see examples.

For variogram.default: list with coordinate matrices, each with the number of rows matching that of corresponding vectors in y; the number of columns should match the number of spatial dimensions spanned by the data (1 (x), 2 (x,y) or 3 (x,y,z)).

...

any other arguments that will be passed to variogram.default (ignored)

X

(optional) list with for each variable the matrix with regressors/covariates; the number of rows should match that of the correspoding element in y, the number of columns equals the number of regressors (including intercept)

cutoff

spatial separation distance up to which point pairs are included in semivariance estimates; as a default, the length of the diagonal of the box spanning the data is divided by three.

width

the width of subsequent distance intervals into which data point pairs are grouped for semivariance estimates

alpha

direction in plane (x,y), in positive degrees clockwise from positive y (North): alpha=0 for direction North (increasing y), alpha=90 for direction East (increasing x); optional a vector of directions in (x,y)

beta

direction in z, in positive degrees up from the (x,y) plane;

tol.hor

horizontal tolerance angle in degrees

tol.ver

vertical tolerance angle in degrees

cressie

logical; if TRUE, use Cressie''s robust variogram estimate; if FALSE use the classical method of moments variogram estimate

dX

include a pair of data points $y(s_1),y(s_2)$ taken at locations $s_1$ and $s_2$ for sample variogram calculation only when $||x(s_1)-x(s_2)|| < dX$ with and $x(s_i)$ the vector with regressors at location $s_i$, and $||.||$ the 2-norm. This allows pooled estimation of within-strata variograms (use a factor variable as regressor, and dX=0.5), or variograms of (near-)replicates in a linear model (addressing point pairs having similar values for regressors variables)

boundaries

numerical vector with distance interval upper boundaries; values should be strictly increasing

cloud

logical; if TRUE, calculate the semivariogram cloud

trend.beta

vector with trend coefficients, in case they are known. By default, trend coefficients are estimated from the data.

debug.level

integer; set gstat internal debug level

cross

logical or character; if FALSE, no cross variograms are computed
when object is of class `gstat`

and has more than one variable; if
TRUE, all direct and cross variograms are computed; if
equal to "ST", direct and cross variograms are computed for all pairs
involving the first (non-time lagged) variable; if equal to "ONLY",
only cross variograms are computed (no direct variograms).

formula

formula, specifying the dependent variable and possible covariates

x

object of class `variogram`

or `variogramCloud`

to be printed

grid

grid parameters, if data are gridded (not to be called directly; this is filled automatically)

map

logical; if TRUE, and `cutoff`

and `width`

are given, a variogram map is returned. This requires package
sp. Alternatively, a map can be passed, of class SpatialDataFrameGrid
(see sp docs)

g

NULL or object of class gstat; may be used to pass settable parameters and/or variograms; see example

projected

logical; if FALSE, data are assumed to be unprojected,
meaning decimal longitude/latitude. For projected data, Euclidian
distances are computed, for unprojected great circle distances
(km). In `variogram.formula`

or `variogram.gstat`

, for data
deriving from class Spatial, projection is detected automatically using
`is.projected`

lambda

test feature; not working (yet)

verbose

logical; print some progress indication

pseudo

integer; use pseudo cross variogram for computing time-lagged spatial variograms? -1: find out from coordinates -- if they are equal then yes, else no; 0: no; 1: yes.

covariogram

logical; compute covariogram instead of variogram?

PR

logical; compute pairwise relative variogram (does NOT check whether variable is strictly positive)

If map is TRUE (or a map is passed), a grid map is returned containing the (cross) variogram map(s). See package sp.

In other cases, an object of class "gstatVariogram" with the following fields:

the number of point pairs for this estimate;
in case of a `variogramCloud`

see below

the average distance of all point pairs considered for this estimate

the actual sample variogram estimate

the horizontal direction

the vertical direction

the combined id pair

If cloud is TRUE: an object of class variogramCloud, with the field np encoding the numbers of the point pair that contributed to a variogram cloud estimate, as follows. The first point is found by 1 + the integer division of np by the .BigInt attribute of the returned object, the second point by 1 + the remainder of that division. as.data.frame.variogramCloud returns no np field, but does the decoding into:

for variogramCloud: data id (row number) of one of the data pair

for variogramCloud: data id (row number) of the other data in the pair

In case of a spatio-temporal variogram is sought see variogramST for details.

Cressie, N.A.C., 1993, Statistics for Spatial Data, Wiley.

Cressie, N., C. Wikle, 2011, Statistics for Spatio-temporal Data, Wiley.

Pebesma, E.J., 2004. Multivariable geostatistics in S: the gstat package. Computers \& Geosciences, 30: 683-691.

print.gstatVariogram,
plot.gstatVariogram,
plot.variogramCloud;
for variogram models: vgm,
to fit a variogram model to a sample variogram:
fit.variogram
`variogramST`

for details on the spatio-temporal sample variogram.

```
# NOT RUN {
library(sp)
data(meuse)
# no trend:
coordinates(meuse) = ~x+y
variogram(log(zinc)~1, meuse)
# residual variogram w.r.t. a linear trend:
variogram(log(zinc)~x+y, meuse)
# directional variogram:
variogram(log(zinc)~x+y, meuse, alpha=c(0,45,90,135))
variogram(log(zinc)~1, meuse, width=90, cutoff=1300)
# GLS residual variogram:
v = variogram(log(zinc)~x+y, meuse)
v.fit = fit.variogram(v, vgm(1, "Sph", 700, 1))
v.fit
set = list(gls=1)
v
g = gstat(NULL, "log-zinc", log(zinc)~x+y, meuse, model=v.fit, set = set)
variogram(g)
if (require(rgdal)) {
proj4string(meuse) = CRS("+init=epsg:28992")
meuse.ll = spTransform(meuse, CRS("+proj=longlat +datum=WGS84 +ellps=WGS84"))
# variogram of unprojected data, using great-circle distances, returning km as units
variogram(log(zinc) ~ 1, meuse.ll)
}
# }
```

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