rkriging (version 1.0.2)

RQ.Kernel: Rational Quadratic (RQ) Kernel

Description

This function specifies the Rational Quadratic (RQ) kernel.

Usage

RQ.Kernel(lengthscale, alpha = 1)

Value

A Rational Quadratic (RQ) Kernel Class Object.

Arguments

lengthscale

a vector for the positive length scale parameters

alpha

a positive scalar for the scale mixture parameter that controls the relative weighting of large-scale and small-scale variations

Author

Chaofan Huang and V. Roshan Joseph

Details

The Rational Quadratic (RQ) kernel is given by $$k(r;\alpha)=\left(1+\frac{r^2}{2\alpha}\right)^{-\alpha},$$ where \(\alpha\) is the scale mixture parameter 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 \(\alpha\to\infty\), it converges to the Gaussian.Kernel.

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

MultiplicativeRQ.Kernel, Get.Kernel, Evaluate.Kernel.

Examples

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

# approach 1
kernel <- RQ.Kernel(lengthscale, alpha=1)
Evaluate.Kernel(kernel, X)

# approach 2
kernel <- Get.Kernel(lengthscale, type="RQ", parameters=list(alpha=1))
Evaluate.Kernel(kernel, X)