kergp (version 0.5.7)

prinKrige: Principal Kriging Functions

Description

Principal Kriging Functions.

Usage

prinKrige(object)

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.

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 \(e_1\), \(\dots\), \(e_p\).

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)