| Matern.Kernel {rkriging} | R Documentation |
Generalized Matern Kernel
Description
This function specifies the (Generalized) Matern kernel with any smoothness parameter \nu.
Usage
Matern.Kernel(lengthscale, nu = 2.01)
Arguments
lengthscale |
a vector for the positive length scale parameters |
nu |
a positive scalar parameter that controls the smoothness |
Details
The Generalized Matern kernel is given by
k(r;\nu)=\frac{2^{1-\nu}}{\Gamma(\nu)}(\sqrt{2\nu}r)^{\nu}K_{\nu}(\sqrt{2\nu}r),
where \nu is the smoothness parameter,
K_{\nu} is the modified Bessel function,
\Gamma is the gamma function,
and
r(x,x^{\prime})=\sqrt{\sum_{i=1}^{p}\left(\frac{x_{i}-x_{i}^{\prime}}{l_{i}}\right)^2}
is the euclidean distance between x and x^{\prime} weighted by
the length scale parameters l_{i}'s.
As \nu\to\infty, it converges to the Gaussian.Kernel.
Value
A Generalized Matern Kernel Class Object.
Author(s)
Chaofan Huang and V. Roshan Joseph
References
Duvenaud, D. (2014). The kernel cookbook: Advice on covariance functions.
Rasmussen, C. E. & Williams, C. K. (2006). Gaussian Processes for Machine Learning. The MIT Press.
See Also
Matern12.Kernel, Matern32.Kernel, Matern52.Kernel, MultiplicativeMatern.Kernel, Get.Kernel, Evaluate.Kernel.
Examples
n <- 5
p <- 3
X <- matrix(rnorm(n*p), ncol=p)
lengthscale <- c(1:p)
# approach 1
kernel <- Matern.Kernel(lengthscale, nu=2.01)
Evaluate.Kernel(kernel, X)
# approach 2
kernel <- Get.Kernel(lengthscale, type="Matern", parameters=list(nu=2.01))
Evaluate.Kernel(kernel, X)