Cressie-Hawkins robust semivariogram estimator

Implements equation (4.26) of Schabenberger & Gotway (2005), p. 160: $$\hat\gamma(h) = \frac{1}{2}\left\{\frac{1}{|N(h)|}\sum |Z(s_i)-Z(s_j)|^{1/2}\right\}^4 / \left(0.457 + \frac{0.494}{|N(h)|}\right).$$

Usage

morie_cressie_hawkins(
  coords,
  z,
  bins = NULL,
  cutoff = NULL,
  exact = FALSE,
  full_correction = FALSE
)

Arguments

coords

Matrix of coordinates, one row per location.

z

Observed values.

bins

Lag-class count, or explicit bin edges.

cutoff

Largest separation used; half the maximum by default.

exact

Treat every distinct separation as its own lag class.

full_correction

Include the 0.045/|N(h)|^2 term.

Value

A list with lag, gamma, n_pairs, matheron, ratio.

Details

Averaging the square-root differences BEFORE raising to the fourth power is what limits an outlier's leverage; the denominator restores approximate unbiasedness. Robust here means resistant to slight contamination of a Gaussian field only: the influence function is unbounded and the breakdown point is zero, and on clean data this estimator is the MORE variable of the two.

The bias correction is derived on p. 160 as 0.457 + 0.494/|N| + 0.045/|N|^2 and then written without the last term in equation (4.26). The printed equation is the default because it is what the book's own worked Example 4.3 evaluates: its factor 0.704 at |N(h)| = 2 is 0.457 + 0.494/2 exactly.

References

Schabenberger O, Gotway CA (2005), Sec 4.4.2, eq (4.26), pp. 159-161. Cressie N, Hawkins DM (1980).

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
morie_cressie_hawkins(V, V)
#> $lag
#> [1] 1 2 3
#> 
#> $gamma
#> [1] 0.947739 3.708282 8.096438
#> 
#> $n_pairs
#> [1] 7 6 5
#> 
#> $matheron
#> [1] 0.5 2.0 4.5
#> 
#> $ratio
#> [1] 1.895478 1.854141 1.799208
#> 
#> $full_correction
#> [1] FALSE
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Cressie-Hawkins robust semivariogram estimator"
#>