The Toeplitz class contains efficient methods for linear algebra with symmetric positive definite (i.e., variance) Toeplitz matrices.
is.Toeplitz(x)as.Toeplitz(x)
# S3 method for Toeplitz
dim(x)
An R object.
new()Class constructor.
Toeplitz$new(N, acf)
NSize of Toeplitz matrix.
acfAutocorrelation vector of length N.
A Toeplitz object.
print()Print method.
Toeplitz$print()
size()Get the size of the Toeplitz matrix.
Toeplitz$size()
Size of the Toeplitz matrix. ncol(), nrow(), and dim() methods for Toeplitz objects also work as expected.
set_acf()Set the autocorrelation of the Toeplitz matrix.
Toeplitz$set_acf(acf)
acfAutocorrelation vector of length N.
get_acf()Get the autocorrelation of the Toeplitz matrix.
Toeplitz$get_acf()
The autocorrelation vector of length N.
has_acf()Check whether the autocorrelation of the Toeplitz matrix has been set.
Toeplitz$has_acf()
Logical; TRUE if Toeplitz$set_acf() has been called.
prod()Toeplitz matrix-matrix product.
Toeplitz$prod(x)
xVector or matrix with N rows.
The matrix product Tz %*% x. Tz %*% x and x %*% Tz also work as expected.
solve()Solve a Toeplitz system of equations.
Toeplitz$solve(x, method = c("gschur", "pcg"), tol = 1e-10)xOptional vector or matrix with N rows.
methodSolve method to use. Choices are: gschur for a modified version of the Generalized Schur algorithm of Ammar & Gragg (1988), or pcg for the preconditioned conjugate gradient method of Chen et al (2006). The former is faster and obtains the log-determinant as a direct biproduct. The latter is more numerically stable for long-memory autocorrelations.
tolTolerance level for the pcg method.
The solution in z to the system of equations Tz %*% z = x. If x is missing, returns the inverse of Tz. solve(Tz, x) and solve(Tz, x, method, tol) also work as expected.
log_det()Calculate the log-determinant of the Toeplitz matrix.
Toeplitz$log_det()
The log-determinant log(det(Tz)). determinant(Tz) also works as expected.
trace_grad()Computes the trace-gradient with respect to Toeplitz matrices.
Toeplitz$trace_grad(acf2)
acf2Length-N autocorrelation vector of the second Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
Computes the trace of
solve(Tz, toeplitz(acf2)).
This is used in the computation of the gradient of log(det(Tz(theta))) with respect to theta.
trace_hess()Computes the trace-Hessian with respect to Toeplitz matrices.
Toeplitz$trace_hess(acf2, acf3)
acf2Length-N autocorrelation vector of the second Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
acf3Length-N autocorrelation vector of the third Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
Computes the trace of
solve(Tz, toeplitz(acf2)) %*% solve(Tz, toeplitz(acf3)).
This is used in the computation of the Hessian of log(det(Tz(theta))) with respect to theta.
clone()The objects of this class are cloneable with this method.
Toeplitz$clone(deep = FALSE)
deepWhether to make a deep clone.
An N x N Toeplitz matrix Tz is defined by its length-N "autocorrelation" vector acf, i.e., first row/column Tz. Thus, for the function stats::toeplitz(), we have Tz = toeplitz(acf).
It is assumed that acf defines a valid (i.e., positive definite) variance matrix. The matrix multiplication methods still work when this is not the case but the other methods do not (return values typically contain NaNs).
as.Toeplitz(x) attempts to convert its argument to a Toeplitz object by calling Toeplitz$new(acf = x). is.Toeplitz(x) checks whether its argument is a Toeplitz object.
# NOT RUN {
# construct a Toeplitz matrix
acf <- exp(-(1:5))
Tz <- Toeplitz$new(acf = acf)
# alternatively, can allocate space first
Tz <- Toeplitz$new(N = length(acf))
Tz$set_acf(acf = acf)
# basic methods
Tz$get_acf() # extract the acf
dim(Tz) # == c(nrow(Tz), ncol(Tz))
Tz # print method
# linear algebra methods
X <- matrix(rnorm(10), 5, 2)
Tz %*% X
t(X) %*% Tz
solve(Tz, X)
determinant(Tz) # log-determinant
# }
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