matern.specdens: Matern correlation spectral density function
Description
Calculates the Matern spectral density for supplied frequencies and
Matern correlation parameters. Spectral density is evaluated for each
supplied frequency or pair of frequencies. The output is generally
used as the prior variances for spectral GP basis coefficients.
Usage
matern.specdens(omega, param, d = 2)
Arguments
omega
Vector or two-column matrix-like object of frequencies, with the
first column the frequencies in the first dimension and the second
column in the second dimension.
param
Vector of two Matern parameter values, first the spatial
range and second the differentiability parameter.
d
Dimension of the domain.
Value
A vector of spectral density values corresponding to the supplied frequencies.
Details
The spectral
density,$$\frac{\Gamma(\nu+d/2)(4\nu)^\nu}{\pi^(d/2)\Gamma(\nu)(\pi \rho)^{2\nu}}\left(\frac{4\nu}{(\pi \rho)^2}+\omega^T \omega\right)^{-(\nu +d/2)},$$
corresponds to the following functional form of
the Matern correlation function,
$$\frac{1}{\Gamma(\nu)2^{\nu-1}}\left(\frac{2\sqrt{\nu}\tau}{\rho}\right)^{\nu}\mathcal{K}_{\nu}\left(\frac{2\sqrt{\nu}\tau}{\rho}\right),$$ where rho is the range and nu the differentiability. Rho is interpreted on the scale $(0,1)^d$. Nu of 0.5 is the exponential correlation, and as nu goes to infinity the correlation approaches the squared exponential (Gaussian). Nu of 0.5 gives Gaussian processes with continuous but not differentiable sample paths, while nu of infinity gives infinitely-differentiable (and analytic) sample paths. In the spectral GP approximation, the frequencies are a sequence of integers from 0 to half the gridsize in each dimension.