Generalized.Rational.Kriging {rkriging}

Generalized Rational Kriging

Description

This functions fits the generalized rational kriging model to the data.

Usage

Generalized.Rational.Kriging(
  X,
  y,
  fit = TRUE,
  kernel = NULL,
  kernel.parameters = list(),
  nlopt.parameters = list()
)

Arguments

X

a matrix for input (feature)

y

a vector for output (target), only one-dimensional output is supported

fit

whether to fit the length scale parameters from data

kernel

a kernel class object

kernel.parameters

a list of parameters required for the kernel, if no kernel class object is provided

nlopt.parameters

a list of parameters required for NLopt, including choice of optimization algorithm and maximum number of evaluation

Details

Ordinary kriging and rational kriging can be obtained as special cases of generalized rational kriging. Please see Joseph (2024) for details. The Spectra library is used for fast computation of the first eigenvalues/vectors. Only interpolation is available. Noisy output is not supported for generalized rational kriging.

The kernel choices are required and can be specified by (i) providing the kernel class object to kernel or (ii) specifying the kernel type and other parameters in kernel.parameters. Please see examples section of Fit.Kriging for detail usages.

When the lengthscale / correlation parameters are unknown, all parameters including the constant mean can be estimated via Maximum Likelihood method by setting fit=TRUE. The initial / lower bound / upper bound of the lengthscale parameters can be provided in kernel.parameters, otherwise a good initial and range would be estimated from the data. The optimization is performed via NLopt library, a open-source library for nonlinear optimization. All gradient-free optimization methods in NLopt are supported and can be specified in nlopt.parameters. See nloptr::nloptr.print.options() for the list of available derivative-free algorithms (prefix with NLOPT_GN or NLOPT_LN). The maximum number of optimization steps can also be defined in nlopt.parameters. Please see examples section of Fit.Kriging for detail usages.

Value

A Generalized Rational Kriging Class Object.

Author(s)

Chaofan Huang and V. Roshan Joseph

References

Joseph, V. R. (2024). Rational Kriging. Journal of the American Statistical Association.

Qiu, Y., Guennebaud, G., & Niesen, J. (2015). Spectra: C++ library for large scale eigenvalue problems.

See Also

Fit.Kriging, Predict.Kriging, Get.Kriging.Parameters.

Examples

# one dimensional example 
f <- function(x) {
  x <- 0.5 + 2*x
  y <- sin(10*pi*x)/(2*x) + (x-1)^4
  return (y)
}

set.seed(1234)
# train set
n <- 30
p <- 1
X <- matrix(runif(n),ncol=p)
y <- apply(X, 1, f)
newX <- matrix(seq(0,1,length=1001), ncol=p)

# approach 1
kriging <- Generalized.Rational.Kriging(X, y, fit=TRUE, 
                                        kernel.parameters=list(type="RQ",alpha=1))
pred <- Predict.Kriging(kriging, newX)
plot(newX, f(newX), "l")
points(X, y, pch=16, col="blue")
lines(newX, pred$mean, col="red", lty=2)
lines(newX, pred$mean-2*pred$sd, col="red", lty=3)
lines(newX, pred$mean+2*pred$sd, col="red", lty=3)
Get.Kriging.Parameters(kriging)

# approach 2
kriging <- Fit.Kriging(X, y, interpolation=TRUE, fit=TRUE, model="GRK",
                       kernel.parameters=list(type="RQ",alpha=1))
pred <- Predict.Kriging(kriging, newX)
plot(newX, f(newX), "l")
points(X, y, pch=16, col="blue")
lines(newX, pred$mean, col="red", lty=2)
lines(newX, pred$mean-2*pred$sd, col="red", lty=3)
lines(newX, pred$mean+2*pred$sd, col="red", lty=3)
Get.Kriging.Parameters(kriging)