calCov {magi}

Calculate stationary Gaussian process kernel

Description

Covariance calculations for Gaussian process kernels. Currently supports matern, rbf, compact1, periodicMatern, generalMatern, and rationalQuadratic kernels. Can also return m_phi and other additional quantities useful for ODE inference.

Usage

calCov(
  phi,
  rInput,
  signrInput,
  bandsize = NULL,
  complexity = 3,
  kerneltype = "matern",
  df,
  noiseInjection = 1e-07
)

Arguments

phi

the kernel hyper-parameters. See details for hyper-parameter specification for each kerneltype.

rInput

the distance matrix between all time points s and t, i.e., |s - t|

signrInput

the sign matrix of the time differences, i.e., sign(s - t)

bandsize

size for band matrix approximation. See details.

complexity

integer value for the complexity of the kernel calculations desired:

• 0 includes C only

• 1 additionally includes Cprime, Cdoubleprime, dCdphi

• 2 or above additionally includes Ceigen1over, CeigenVec, Cinv, mphi, Kphi, Keigen1over, KeigenVec, Kinv, mphiLeftHalf, dCdphiCube

See details for their definitions.

kerneltype

must be one of matern, rbf, compact1, periodicMatern, generalMatern, rationalQuadratic. See details for the kernel formulae.

df

degrees of freedom, for generalMatern and rationalQuadratic kernels only. Default is df=2.01 for generalMatern and df=0.01 for rationalQuadratic.

noiseInjection

a small value added to the diagonal elements of C and Kphi for numerical stability

Details

The covariance formulae and the hyper-parameters phi for the supported kernels are as follows. Stationary kernels have C(s,t) = C(r) where r = |s-t| is the distance between the two time points. Generally, the hyper-parameter phi[1] controls the overall variance level while phi[2] controls the bandwidth.

matern

This is the simplified Matern covariance with df = 5/2:

C(r) = phi[1] * (1 + \sqrt 5 r/phi[2] + 5r^2/(3 phi[2]^2)) * \exp(-\sqrt 5 r/phi[2])

rbf

C(r) = phi[1] * \exp(-r^2/(2 phi[2]^2))

compact1

C(r) = phi[1] * \max(1-r/phi[2],0)^4 * (4r/phi[2]+1)

periodicMatern

Define r' = | \sin(r \pi/phi[3])*2 |. Then the covariance is given by C(r') using the Matern formula.

generalMatern

C(r) = phi[1] * 2^(1-df) / \Gamma(df) * ( \sqrt(2.0 * df) * r / phi[2] )^df * besselK( \sqrt(2.0 * df) * r / phi[2] , df)

where besselK is the modified Bessel function of the second kind.

rationalQuadratic

C(r) = phi[1] * (1 + r^2/(2 df phi[2]^2))^(-df)

The kernel calculations available and their definitions are as follows:

C

The covariance matrix corresponding to the distance matrix rInput.

Cprime

The cross-covariance matrix d C(s,t) / ds.

Cdoubleprime

The cross-covariance matrix d^2 C(s,t) / ds dt.

dCdphi

A list with the matrices dC / dphi for each element of phi.

Ceigen1over

The reciprocals of the eigenvalues of C.

CeigenVec

Matrix of eigenvectors of C.

Cinv

The inverse of C.

mphi

The matrix Cprime * Cinv.

Kphi

The matrix Cdoubleprime - Cprime * Kinv * t(Cprime).

Keigen1over

The reciprocals of the eigenvalues of Kphi.

Kinv

The inverse of Kphi.

mphiLeftHalf

The matrix Cprime * CeigenVec.

dCdphiCube

dC / dphi as a 3-D array, with the third dimension corresponding to the elements of phi.

If bandsize is a positive integer, additionally CinvBand, mphiBand, and KinvBand are provided in the return list, which are band matrix approximations to Cinv, mphi, and Kinv with the specified bandsize.

Value

A list containing the kernel calculations included by the value of complexity.

Examples

foo  <- outer(0:40, t(0:40), '-')[, 1, ]
r <- abs(foo)
signr <- -sign(foo)
calCov(c(0.2, 2), r, signr, bandsize = 20, kerneltype = "generalMatern", df = 2.01)