Schabenberger & Gotway (2005) eqs (6.94)-(6.98), after Kitanidis (1986) and
Le & Zidek (1992): with Z ~ G(X beta, sigma2 V), V known, and (beta,
sigma2) ~ NIG(a, d, m, Q), Z(s0) | Z is multivariate t with mean X0 m* +
V0z V^-1 (Z - X m*) and variance a* / (nu - 2) times (V00 - V0z V^-1 Vz0 + (X0 -
V0z V^-1 X) Q* (X0 - V0z V^-1 X)'), where Q* = (Q^-1 + X' V^-1 X)^-1, m* =
Q* (Q^-1 m + X' V^-1 Z), a* = a + m' Q^-1 m + Z' V^-1 Z - m*' Q*^-1 m*, and
nu = n + d (n + d - p for the flat prior, Qinv = NULL, whose mean is the
universal kriging predictor). The printed nu* of (6.98) weights beta_gls - m
by X' V^-1 X; the exact a* weights it by (Q + (X' V^-1 X)^-1)^-1.
Usage
bkrnig(X, z, X0, V, V0z, V00, a = 0, d = 0, m = NULL, Qinv = NULL)
Arguments
- X, X0
Design matrices for the data and the prediction sites.
- z
Numeric responses.
- V, V0z, V00
Correlation blocks: data, prediction-data, prediction.
- a, d
Inverse-gamma hyperparameters.
- m
Prior mean of beta (proper prior).
- Qinv
Prior precision of beta up to sigma2; NULL is flat.
Value
Named list: mean, variance, scale, df, a_star, m_star, Q_star.
References
Kitanidis, P. K. (1986). WRR 22, 499-507. Le, N. D. & Zidek,
J. V. (1992). JMVA 43, 351-374. Schabenberger & Gotway (2005), eqs
(6.94)-(6.98), pp. 391-394.
Examples
xy <- cbind(1:6, c(2, 4, 1, 5, 3, 6))
v <- exp(-as.matrix(dist(xy)))
v0 <- exp(-sqrt(colSums((t(xy) - c(3.5, 3.5))^2)))
bkrnig(cbind(1, xy[, 1]), sin(1:6), cbind(1, 3.5), v, matrix(v0, 1), matrix(1))$mean
#> [1] -0.1082095
