matern.specdens: Matern correlation spectral density function

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

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

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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.

Details

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.

Value

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

Author(s)

Christopher Paciorek paciorek@alumni.cmu.edu

References

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

See Also

gp, calc.variances.gp

Examples

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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)

spectralGP documentation built on May 2, 2019, 2:40 a.m.