`Kriging`

Description

Create a Kriging Object representing a Trend \(+\) GP Model. Use noise = NULL for noise-free interpolation, noise = "nugget" to estimate a homogeneous nugget, or a numeric vector of per-observation noise variances.

Usage

Just build the model:

Python

k = Kriging(kernel="matern3_2")
# later: k.fit(y, X, ..., noise=None)

R

k <- Kriging(kernel = "matern3_2")
# later: k$fit(y, X, ..., noise = NULL)

Matlab/Octave

k = Kriging(kernel = "matern3_2");
% later: k.fit(y, X, ..., noise = [])

Julia

k = Kriging("matern3_2")
# later: fit(k, y, X; noise=nothing)

or build and fit at the same time:

Python

k = Kriging(y, X, kernel="matern3_2", regmodel="constant",
            normalize=False, optim="BFGS", objective="LL",
            parameters=None, noise=None)

R

k <- Kriging(y, X, kernel = "matern3_2", regmodel = "constant",
             normalize = FALSE, optim = "BFGS", objective = "LL",
             parameters = NULL, noise = NULL)

Matlab/Octave

k = Kriging(y, X, kernel = "matern3_2", regmodel = "constant", ...
            normalize = false, optim = "BFGS", objective = "LL", ...
            parameters = [], noise = [])

Julia

k = Kriging(y, X, "matern3_2"; regmodel="constant", normalize=false,
            optim="BFGS", objective="LL", parameters=nothing, noise=nothing)

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.
`noise`	Either a numeric vector of per-observation noise variances, `"nugget"` to estimate a homogeneous nugget, or `NULL` (default) for noise-free interpolation.

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

Kriging