SuperGauss (version 1.0.2)

Toeplitz-class: Constructor and methods for Toeplitz matrix objects.

Description

The Toeplitz class contains efficient methods for linear algebra with symmetric positive definite (i.e., variance) Toeplitz matrices.

Usage

Toeplitz(n, acf)

Arguments

n

Size of the Toeplitz matrix.

acf

Autocorrelation vector of Toeplitz matrix.

Value

A Toeplitz object.

Methods

If Toep is a Toeplitz object with first row/column given by acf, then:

Toep$setAcf(acf)

Sets the autocorrelation of the matrix.

Toep$getAcf()

Gets the autocorrelation of the matrix.

nrow(Toep), ncol(Toep), dim(Toep)

Selected dimension(s) of the matrix.

Toep %*% X, X %*% Toep

Toeplitz-Matrix and Matrix-Toeplitz multiplication. Also works if X is a vector.

solve(Toep, X), solve(Toep)

Solves Toeplitz systems of equations. When second argument is missing, returns the inverse of the Toeplitz matrix.

determinant(Toep)

Log-determinant of the Toeplitz matrix, i.e., same thing as log(det(toeplitz(acf))).

Toep$traceT2(acf2)

If T1 == toeplitz(acf) and T2 == toeplitz(acf2), computes the trace of solve(T1, T2). This is used in the computation of the gradient of Gaussian likelihoods with Toeplitz variance matrix.

Toep$traceT4(acf2, acf3)

If T1 == toeplitz(acf), T2 == toeplitz(acf2), and T3 == toeplitz(acf3), computes the trace of solve(T1, T2) %*% solve(T1, T3). This is used in the computation of the Hessian of Gaussian likelihoods with Toeplitz variance matrix.

Details

It is assumed that the autocorrelation of the Toeplitz object defines a valid (i.e., positive definite) variance matrix. The multiplication algorithms still work when this is not the case but the other algorithms do not (return values typically contain NaNs).

Examples

# NOT RUN {
# construction
acf <- exp(-(1:5))
Toep <- Toeplitz(acf = acf)
# alternatively, can allocate space first
Toep <- Toeplitz(n = length(acf))
Toep$setAcf(acf = acf)

dim(Toep) # == c(nrow(Toep), ncol(Toep))
Toep # show method
Toep$getAcf() # extract the acf

# linear algebra
X <- matrix(rnorm(10), 5, 2)
Toep %*% X
t(X) %*% Toep
solve(Toep, X)
determinant(Toep) # log-determinant
# }