Bayesian kriging over the Matern class (Handcock and Stein 1993)

Z ~ G(X beta, sigma2 V(theta, nu)) with V the Matern correlation (4.9), prior pi(theta, nu) / sigma2 with bounded uniform priors on the scale theta and smoothness nu (Schabenberger & Gotway 2005, p. 394). Integrating out beta and sigma2 leaves p(theta, nu | Z) proportional to pi(theta, nu) |V|^-1/2 |X' V^-1 X|^-1/2 (S2)^-(n - p)/2; at each grid point the predictive is the flat-prior t of (6.97)-(6.98). There is no closed form over (theta, nu), so the posterior is evaluated on the grids and the mixture's mean and variance are returned.

Usage

hsbkrg(coords, z, X, coords0, X0, theta_grid, nu_grid, prior_weights = NULL)

Arguments

coords, coords0

Coordinates of the data and the prediction sites.

z

Numeric responses.

X, X0

Design matrices.

theta_grid, nu_grid

Grid points of the uniform priors.

prior_weights

Optional matrix of prior quadrature weights.

Value

Named list: mean, variance, posterior, theta_grid, nu_grid.

References

Handcock, M. S. & Stein, M. L. (1993). Technometrics 35, 403-410. Berger, J. O., De Oliveira, V. & Sanso, B. (2001). JASA 96, 1361-1374. Schabenberger & Gotway (2005), p. 394.

Examples

xy <- cbind(1:8, c(2, 4, 1, 5, 3, 6, 8, 7))
hsbkrg(xy, sin(1:8), cbind(1, xy[, 1]), rbind(c(4, 4)), cbind(1, 4), c(0.5, 1, 2), 1.5)$mean
#> [1] -0.6448198