rkriging (version 1.0.2)

Gaussian.Kernel: Gaussian Kernel

Description

This function specifies the Gaussian / Squared Exponential (SE) / Radial Basis Function (RBF) kernel.

Usage

Gaussian.Kernel(lengthscale)

Value

A Gaussian Kernel Class Object.

Arguments

lengthscale

a vector for the positive length scale parameters

Author

Chaofan Huang and V. Roshan Joseph

Details

The Gaussian kernel is given by $$k(r)=\exp(-r^2/2),$$ where $$r(x,x^{\prime})=\sqrt{\sum_{i=1}^{p}\left(\frac{x_{i}-x_{i}^{\prime}}{l_{i}}\right)^2}$$ is the euclidean distance between \(x\) and \(x^{\prime}\) weighted by the length scale parameters \(l_{i}\)'s.

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

Get.Kernel, Evaluate.Kernel.

Examples

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

# approach 1
kernel <- Gaussian.Kernel(lengthscale)
Evaluate.Kernel(kernel, X)

# approach 2
kernel <- Get.Kernel(lengthscale, type="Gaussian")
Evaluate.Kernel(kernel, X)