# variogram: Calculate Sample or Residual Variogram or Variogram Cloud In gstat: Spatial and Spatio-Temporal Geostatistical Modelling, Prediction and Simulation

## Description

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.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## S3 method for class 'gstat' variogram(object, ...) ## S3 method for class 'formula' variogram(object, locations = coordinates(data), data, ...) ## Default S3 method: 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 class 'gstatVariogram' print(x, ...) ## S3 method for class 'variogramCloud' print(x, ...) ```

## Arguments

 `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;

optional a vector of directions

 `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)

## Value

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:

 `np` the number of point pairs for this estimate; in case of a `variogramCloud` see below `dist` the average distance of all point pairs considered for this estimate `gamma` the actual sample variogram estimate `dir.hor` the horizontal direction `dir.ver` the vertical direction `id` 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:

 `left` for variogramCloud: data id (row number) of one of the data pair `right` 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.

## Note

`variogram.default` should not be called by users directly, as it makes many assumptions about the organization of the data, that are not fully documented (but of course, can be understood from reading the source code of the other `variogram` methods)

Successfully setting `gridded() <- TRUE` may trigger a branch that will fail unless dx and dy are identical, and not merely similar to within machine epsilon.

## Note

`variogram.line` is DEPRECATED; it is and was never meant as a variogram method, but works automatically as such by the R dispatch system. Use variogramLine instead.

Edzer Pebesma

## References

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.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```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")) # variogram of unprojected data, using great-circle distances, returning km as units variogram(log(zinc) ~ 1, meuse.ll) } ```