`Kriging`

Description

Create a Kriging Object representing a Trend \(+\) GP Model

Usage

Just build the model:

Kriging(kernel)
# later, call fit(y,X,...)

or, build and fit at the same time:

Kriging(
  y,
  X,
  kernel,
  regmodel = "constant",
  normalize = FALSE,
  optim = "BFGS",
  objective = "LL",
  parameters = NULL
)

Arguments

Argument	Description
`y`	Numeric vector of response values.
`X`	Numeric matrix of input design.
`kernel`	Character defining the covariance model: `"gauss"` , `"exp"` , `"matern3_2"` , `"matern5_2"`.
`regmodel`	Universal Kriging linear trend: `"constant"`, `"linear"`, `"interactive"`, `"quadratic"`.
`normalize`	Logical. If `TRUE` both the input matrix `X` and the response `y` in normalized to take values in the interval \([0, 1]\) .
`optim`	Character giving the Optimization method used to fit hyper-parameters. Possible values are: `"BFGS"` , `"Newton"` and `"none"` , the later simply keeping the values given in `parameters` . The method `"BFGS"` uses the gradient of the objective (note that `"BGFS10"` means 10 multi-start of BFGS). The method `"Newton"` uses both the gradient and the Hessian of the objective.
`objective`	Character giving the objective function to optimize. Possible values are: `"LL"` for the Log-Likelihood, `"LOO"` for the Leave-One-Out sum of squares and `"LMP"` for the Log-Marginal Posterior.
`parameters`	Initial values for the hyper-parameters. When provided this must be named list with elements `"sigma2"` and `"theta"` containing the initial value(s) for the variance and for the range parameters. If `theta` is a matrix with more than one row, each row is used as a starting point for optimization.

Details

The hyper-parameters (variance and vector of correlation ranges) are estimated thanks to the optimization of a criterion given by objective , using the method given in optim .

Value

An object "Kriging" . Should be used with its predict , simulate , update methods.

Examples

f <- function(x) 1 - 1 / 2 * (sin(12 * x) / (1 + x) + 2 * cos(7 * x) * x^5 + 0.7)
set.seed(123)
X <- as.matrix(runif(10))
y <- f(X)
## fit and print
k <- Kriging(y, X, kernel = "matern3_2")
k

x <- as.matrix(seq(from = 0, to = 1, length.out = 101))
p <- k$predict(x = x, return_stdev = TRUE, return_cov = FALSE)

plot(f)
points(X, y)
lines(x, p$mean, col = "blue")
polygon(c(x, rev(x)), c(p$mean - 2 * p$stdev, rev(p$mean + 2 * p$stdev)),
border = NA, col = rgb(0, 0, 1, 0.2))

s <- k$simulate(nsim = 10, seed = 123, x = x)

matlines(x, s, col = rgb(0, 0, 1, 0.2), type = "l", lty = 1)

Results

* data: 10x[0.0455565,0.940467] -> 10x[0.194057,1.00912]
* trend constant (est.): 0.433954
* variance (est.): 0.0873685
* covariance:
  * kernel: matern3_2
  * range (est.): 0.240585
  * fit:
    * objective: LL
    * optim: BFGS