esv: Compute the empirical semivariogram

In spmodel: Spatial Statistical Modeling and Prediction

Compute the empirical semivariogram

Description

Compute the empirical semivariogram for varying bin sizes and cutoff values.

Usage

esv(
  formula,
  data,
  xcoord,
  ycoord,
  cloud = FALSE,
  robust = FALSE,
  bins = 15,
  cutoff,
  dist_matrix,
  partition_factor
)

## S3 method for class 'esv'
plot(x, ...)

Arguments

formula	A formula describing the fixed effect structure.
data	A data frame or sf object containing the variables in formula and geographic information.
xcoord	Name of the variable in data representing the x-coordinate. Can be quoted or unquoted. Not required if data is an sf object.
ycoord	Name of the variable in data representing the y-coordinate. Can be quoted or unquoted. Not required if data is an sf object.
cloud	A logical indicating whether the empirical semivariogram should be summarized by distance class or not. When cloud = FALSE (the default), pairwise semivariances are binned and averaged within distance classes. When cloud = TRUE, all pairwise semivariances and distances are returned (this is known as the "cloud" semivariogram).
robust	A logical indicating whether the robust semivariogram (Cressie and Hawkins, 1980) is used. The default is FALSE.
bins	The number of equally spaced bins. The default is 15. Ignored if cloud = TRUE.
cutoff	The maximum distance considered. The default is half the diagonal of the bounding box from the coordinates.
dist_matrix	A distance matrix to be used instead of providing coordinate names.
partition_factor	An optional formula specifying the partition factor. If specified, semivariances are only computed for observations sharing the same level of the partition factor.
x	An object from esv().
...	Other arguments passed to other methods.

Details

The empirical semivariogram is a tool used to visualize and model spatial dependence by estimating the semivariance of a process at varying distances. For a constant-mean process, the semivariance at distance h is denoted \gamma(h) and defined as 0.5 * Var(z1 - z2). Under second-order stationarity, \gamma(h) = Cov(0) - Cov(h), where Cov(h) is the covariance function at distance h. Typically the residuals from an ordinary least squares fit defined by formula are second-order stationary with mean zero. These residuals are used to compute the empirical semivariogram. At a distance h, the empirical semivariance is 1/N(h) \sum (r1 - r2)^2, where N(h) is the number of (unique) pairs in the set of observations whose distance separation is h and r1 and r2 are residuals corresponding to observations whose distance separation is h. The robust version is described by Cressie and Hawkins (1980). In spmodel, these distance bins actually contain observations whose distance separation is h +- c, where c is a constant determined implicitly by bins. Typically, only observations whose distance separation is below some cutoff are used to compute the empirical semivariogram (this cutoff is determined by cutoff).

When using splm() with estmethod as "sv-wls", the empirical semivariogram is calculated internally and used to estimate spatial covariance parameters.

Value

If cloud = FALSE, a tibble (data.frame) with distance bins (bins), the average distance (dist), the average semivariance (gamma), and the number of (unique) pairs (np). If cloud = TRUE, a tibble (data.frame) with distance (dist) and semivariance (gamma) for each unique pair.

References

Cressie, N & Hawkins, D.M. 1980. Robust estimation of the variogram. Journal of the International Association for Mathematical Geology, 12, 115-125.

Examples

esv(sulfate ~ 1, sulfate)
plot(esv(sulfate ~ 1, sulfate))

spmodel documentation built on July 3, 2025, 9:08 a.m.