Toeplitz: Constructor and methods for Toeplitz matrix objects.
In SuperGauss: Superfast Likelihood Inference for Stationary Gaussian Time Series
Toeplitz R Documentation
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
is.Toeplitz(x)
as.Toeplitz(x)
## S3 method for class 'Toeplitz'
dim(x)
Arguments
x
An R object.
Details
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$new(acf = x). is.Toeplitz(x) checks whether its argument is a Toeplitz object.
Methods
Public methods
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Method new()
Class constructor.
Usage
Toeplitz$new(N, acf)
Arguments
NSize of Toeplitz matrix.
acfAutocorrelation vector of length N.
Returns
A Toeplitz object.
Method print()
Print method.
Usage
Toeplitz$print()
Method size()
Get the size of the Toeplitz matrix.
Usage
Toeplitz$size()
Returns
Size of the Toeplitz matrix. ncol(), nrow(), and dim() methods for Toeplitz objects also work as expected.
Method set_acf()
Set the autocorrelation of the Toeplitz matrix.
Usage
Toeplitz$set_acf(acf)
Arguments
acfAutocorrelation vector of length N.
Method get_acf()
Get the autocorrelation of the Toeplitz matrix.
Usage
Toeplitz$get_acf()
Returns
The autocorrelation vector of length N.
Method has_acf()
Check whether the autocorrelation of the Toeplitz matrix has been set.
Usage
Toeplitz$has_acf()
Returns
Logical; TRUE if Toeplitz$set_acf() has been called.
Method prod()
Toeplitz matrix-matrix product.
Usage
Toeplitz$prod(x)
Arguments
xVector or matrix with N rows.
Returns
The matrix product Tz %*% x. Tz %*% x and x %*% Tz also work as expected.
Method solve()
Solve a Toeplitz system of equations.
Usage
Toeplitz$solve(x, method = c("gschur", "pcg"), tol = 1e-10)
Arguments
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.
Returns
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.
Method log_det()
Calculate the log-determinant of the Toeplitz matrix.
Usage
Toeplitz$log_det()
Returns
The log-determinant log(det(Tz)). determinant(Tz) also works as expected.
Method trace_grad()
Computes the trace-gradient with respect to Toeplitz matrices.
Usage
Toeplitz$trace_grad(acf2)
Arguments
acf2Length-N autocorrelation vector of the second Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
Returns
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.
Method trace_hess()
Computes the trace-Hessian with respect to Toeplitz matrices.
Usage
Toeplitz$trace_hess(acf2, acf3)
Arguments
acf2Length-N autocorrelation vector of the second Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
acf3Length-N autocorrelation vector of the third Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
Returns
Computes the trace of
solve(Tz, toeplitz(acf2)) %*% solve(Tz, toeplitz(acf3)).
This is used in the computation of the Hessian of log(det(Tz(theta))) with respect to theta.
Method clone()
The objects of this class are cloneable with this method.
Usage
Toeplitz$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Examples
# 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
SuperGauss documentation built on March 18, 2022, 6:35 p.m.
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| Toeplitz | R Documentation |
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
is.Toeplitz(x) as.Toeplitz(x) ## S3 method for class 'Toeplitz' dim(x)
Arguments
x |
An R object. |
Details
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$new(acf = x). is.Toeplitz(x) checks whether its argument is a Toeplitz object.
Methods
Public methods
Method new()
Class constructor.
Usage
Toeplitz$new(N, acf)
Arguments
NSize of Toeplitz matrix.
acfAutocorrelation vector of length
N.
Returns
A Toeplitz object.
Method print()
Print method.
Usage
Toeplitz$print()
Method size()
Get the size of the Toeplitz matrix.
Usage
Toeplitz$size()
Returns
Size of the Toeplitz matrix. ncol(), nrow(), and dim() methods for Toeplitz objects also work as expected.
Method set_acf()
Set the autocorrelation of the Toeplitz matrix.
Usage
Toeplitz$set_acf(acf)
Arguments
acfAutocorrelation vector of length
N.
Method get_acf()
Get the autocorrelation of the Toeplitz matrix.
Usage
Toeplitz$get_acf()
Returns
The autocorrelation vector of length N.
Method has_acf()
Check whether the autocorrelation of the Toeplitz matrix has been set.
Usage
Toeplitz$has_acf()
Returns
Logical; TRUE if Toeplitz$set_acf() has been called.
Method prod()
Toeplitz matrix-matrix product.
Usage
Toeplitz$prod(x)
Arguments
xVector or matrix with
Nrows.
Returns
The matrix product Tz %*% x. Tz %*% x and x %*% Tz also work as expected.
Method solve()
Solve a Toeplitz system of equations.
Usage
Toeplitz$solve(x, method = c("gschur", "pcg"), tol = 1e-10)Arguments
xOptional vector or matrix with
Nrows.methodSolve method to use. Choices are:
gschurfor a modified version of the Generalized Schur algorithm of Ammar & Gragg (1988), orpcgfor 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
pcgmethod.
Returns
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.
Method log_det()
Calculate the log-determinant of the Toeplitz matrix.
Usage
Toeplitz$log_det()
Returns
The log-determinant log(det(Tz)). determinant(Tz) also works as expected.
Method trace_grad()
Computes the trace-gradient with respect to Toeplitz matrices.
Usage
Toeplitz$trace_grad(acf2)
Arguments
acf2Length-
Nautocorrelation vector of the second Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
Returns
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.
Method trace_hess()
Computes the trace-Hessian with respect to Toeplitz matrices.
Usage
Toeplitz$trace_hess(acf2, acf3)
Arguments
acf2Length-
Nautocorrelation vector of the second Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.acf3Length-
Nautocorrelation vector of the third Toeplitz matrix. This matrix must be symmetric but not necessarily positive definite.
Returns
Computes the trace of
solve(Tz, toeplitz(acf2)) %*% solve(Tz, toeplitz(acf3)).
This is used in the computation of the Hessian of log(det(Tz(theta))) with respect to theta.
Method clone()
The objects of this class are cloneable with this method.
Usage
Toeplitz$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Examples
# 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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