prinKrige {kergp}

Principal Kriging Functions

Description

Principal Kriging Functions.

Usage

prinKrige(object)

Arguments

object

An object with class "gp".

Details

The Principal Kriging Functions (PKF) are the eigenvectors of a symmetric positive matrix \mathbf{B} named the Bending Energy Matrix which is met when combining a linear trend and a covariance kernel as done in gp. This matrix has dimension n \times n and rank n - p. The PKF are given in the ascending order of the eigenvalues e_i

e_1 = e_2 = \dots = e_p = 0 < e_{p+1} \leq e_{p+2} \leq \dots \leq e_n.

The p first PKF generate the same space as do the p columns of the trend matrix \mathbf{F}, say \textrm{colspan}(\mathbf{F}). The following n-p PKFs generate a supplementary of the subspace \textrm{colspan}(\mathbf{F}), and they have a decreasing influence on the response. So the p +1-th PKF can give a hint on a possible deterministic trend functions that could be added to the p existing ones.

The matrix \mathbf{B} is such that \mathbf{B} \mathbf{F} = \mathbf{0}, so the columns of \mathbf{F} can be thought of as the eigenvectors that are associated with the zero eigenvalues e_1, \dots, e_p.

Value

A list

values

The eigenvalues of the energy bending matrix in ascending order. The first p values must be very close to zero, but will not be zero since they are provided by numerical linear algebra.

vectors

A matrix \mathbf{U} with its columns \mathbf{u}_i equal to the eigenvectors of the energy bending matrix, in correspondence with the eigenvalues e_i.

B

The Energy Bending Matrix \mathbf{B}. Remind that the eigenvectors are used here in the ascending order of the eigenvalues, which is the reverse of the usual order.

Note

When an eigenvalue e_i is such that e_{i-1} < e_i < e_{i+1} (which can happen only for i > p), the corresponding PKF is unique up to a change of sign. However a run of r > 1 identical eigenvalues is associated with a r-dimensional eigenspace and the corresponding PKFs have no meaning when they are considered individually.

References

Sahu S.K. and Mardia K.V. (2003). A Bayesian kriged Kalman model for short-term forecasting of air pollution levels. Appl. Statist. 54 (1), pp. 223-244.

Examples

library(kergp)
set.seed(314159)
n <- 100
x <- sort(runif(n))
y <- 2 + 4 * x  + 2 * x^2 + 3 * sin(6 * pi * x ) + 1.0 * rnorm(n)
nNew <- 60; xNew <- sort(runif(nNew))
df <- data.frame(x = x, y = y)

##-------------------------------------------------------------------------
## use a Matern 3/2 covariance and a mispecified trend. We should guess
## that it lacks a mainily linear and slightly quadratic part.
##-------------------------------------------------------------------------

myKern <- k1Matern3_2
inputNames(myKern) <- "x"
mygp <- gp(formula = y ~ sin(6 * pi * x),
           data = df, 
           parCovLower = c(0.01, 0.01), parCovUpper = c(10, 100),
           cov = myKern, estim = TRUE, noise = TRUE)
PK <- prinKrige(mygp)

## the third PKF suggests a possible linear trend term, and the
## fourth may suggest a possible quadratic linear trend

matplot(x, PK$vectors[ , 1:4], type = "l", lwd = 2)