kernel.matern: Matern Covariance Function

Description

Computes the Matern covariance function with smoothness parameter $\nu$: $$k(s, t) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \left(\sqrt{2\nu}\frac{|s-t|}{\ell}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{|s-t|}{\ell}\right)$$

Usage

kernel.matern(variance = 1, length_scale = 1, nu = 1.5)

Value

A covariance function object of class 'kernel_matern'.

Arguments

variance

Variance parameter $\sigma^2$ (default 1).

length_scale

Length scale parameter $\ell$ (default 1).

nu

Smoothness parameter $\nu$ (default 1.5). Common values:

nu = 0.5: Exponential (continuous, not differentiable)
nu = 1.5: Once differentiable
nu = 2.5: Twice differentiable
nu = Inf: Gaussian/squared exponential (infinitely differentiable)

Details

where $K_\nu$ is the modified Bessel function of the second kind.

The Matern family of covariance functions provides flexible control over the smoothness of sample paths through the $\nu$ parameter. As $\nu$ increases, sample paths become smoother. The Matern family includes the exponential ($\nu = 0.5$) and approaches the Gaussian kernel as $\nu \to \infty$.

For computational efficiency, special cases $\nu \in \{0.5, 1.5, 2.5, \infty\}$ use simplified closed-form expressions.

Examples

Run this code

# Create Matern covariance functions with different smoothness
cov_rough <- kernel.matern(nu = 0.5)    # Equivalent to exponential
cov_smooth <- kernel.matern(nu = 2.5)   # Twice differentiable

t <- seq(0, 1, length.out = 50)

# Compare sample paths
fd_rough <- make.gaussian.process(n = 5, t = t, cov = cov_rough, seed = 42)
fd_smooth <- make.gaussian.process(n = 5, t = t, cov = cov_smooth, seed = 42)