rkriging (version 1.0.2)

MultiplicativeMatern.Kernel: Multiplicative Generalized Matern Kernel

Description

This function specifies the Multiplicative Generalized Matern kernel.

Usage

MultiplicativeMatern.Kernel(lengthscale, nu = 2.01)

Value

A Multiplicative Generalized Matern Kernel Class Object.

Arguments

lengthscale

a vector for the positive length scale parameters

nu

a positive scalar parameter that controls the smoothness

Author

Chaofan Huang and V. Roshan Joseph

Details

The Multiplicative Generalized Matern kernel is given by $$k(r;\nu)=\prod_{i=1}^{p}\frac{2^{1-\nu}}{\Gamma(\nu)}(\sqrt{2\nu}r_{i})^{\nu}K_{\nu}(\sqrt{2\nu}r_{i}),$$ where \(\nu\) is the smoothness parameter, \(K_{\nu}\) is the modified Bessel function, \(\Gamma\) is the gamma function, and $$r_{i}(x,x^{\prime})=\sqrt{\left(\frac{x_{i}-x_{i}^{\prime}}{l_{i}}\right)^2}$$ is the dimension-wise euclidean distances between \(x\) and \(x^{\prime}\) weighted by the length scale parameters \(l_{i}\)'s.

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

Matern.Kernel, Get.Kernel, Evaluate.Kernel.

Examples

n <- 5
p <- 3
X <- matrix(rnorm(n*p), ncol=p)
lengthscale <- c(1:p)

# approach 1
kernel <- MultiplicativeMatern.Kernel(lengthscale, nu=2.01)
Evaluate.Kernel(kernel, X)

# approach 2
kernel <- Get.Kernel(lengthscale, type="MultiplicativeMatern", parameters=list(nu=2.01))
Evaluate.Kernel(kernel, X)