Empirical (Matheron) variogram

Matheron method-of-moments semivariogram estimator, \(\hat\gamma(h_k) = \frac{1}{2 |N(h_k)|} \sum_{(i,j) \in N(h_k)} (Z(s_i)-Z(s_j))^2\), where the pair set of lag bin k collects the pairs whose separation distance falls in that bin. With integer bins the bins are equal-width up to half the maximum pairwise distance; a numeric vector gives explicit ascending bin edges.

Usage

Vgrm(coords, values, bins = 15L)

variogram(coords, values, bins = 15L)

Arguments

coords

Site coordinates, n by d matrix.

values

Observations, length n.

bins

Number of lag bins (integer) or explicit bin edges (numeric vector, ascending).

Value

List with lag (mean pair distance per bin), gamma, n_pairs, edges, n.

References

Matheron, G. (1963). Principles of geostatistics. Economic Geology, 58(8), 1246-1266.

Schabenberger, O. and Gotway, C. A. (2005). Statistical Methods for Spatial Data Analysis. Chapman and Hall/CRC, Sec. 4.2, eq. (4.1). Local PDF: WD_BLACK/library/pdf/Statistical_Methods_for_Spatial_Data_Analysis.pdf.

Examples

Vgrm(cbind(0:3, 0), c(0, 1, 3, 6), bins = c(0.5, 1.5, 2.5, 3.5))
#> $lag
#> [1] 1 2 3
#> 
#> $gamma
#> [1]  2.333333  8.500000 18.000000
#> 
#> $n_pairs
#> [1] 3 2 1
#> 
#> $edges
#> [1] 0.5 1.5 2.5 3.5
#> 
#> $n
#> [1] 4
#> 
#> $method
#> [1] "Matheron empirical semivariogram (Schabenberger-Gotway eq. 4.1)"
#>