matern.specdens: Matern correlation spectral density function
In spectralGP: Approximate Gaussian Processes Using the Fourier Basis

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

1	matern.specdens(omega, param, d = 2)

`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.

The spectral density,

\frac{Γ(ν+d/2)(4ν)^ν}{π^(d/2)Γ(ν)(π ρ)^{2ν}}≤ft(\frac{4ν}{(π ρ)^2}+ω^T ω\right)^{-(ν +d/2)},

corresponds to the following functional form of the Matern correlation function,

\frac{1}{Γ(ν)2^{ν-1}}≤ft(\frac{2√{ν}τ}{ρ}\right)^{ν}\mathcal{K}_{ν}≤ft(\frac{2√{ν}τ}{ρ}\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.

A vector of spectral density values corresponding to the supplied frequencies.

Christopher Paciorek paciorek@alumni.cmu.edu

Type 'citation("spectralGP")' for references.

gp, calc.variances.gp

library(spectralGP)
gp1=gp(128,matern.specdens,c(1,4))
gp2=gp(c(64,64),matern.specdens,c(1,4))
dens1=matern.specdens(gp1$omega,c(1,4),d=1)
dens2=matern.specdens(gp2$omega,c(1,4),d=2)