| Title: | Autocovariance Estimation via Difference-Based Methods |
|---|---|
| Description: | Provides methods for (auto)covariance/correlation function estimation in change point regression with stationary errors circumventing the pre-estimation of the underlying signal of the observations. Generic, first-order, (m+1)-gapped, difference-based autocovariance function estimator is based on M. Levine and I. Tecuapetla-Gómez (2023) <doi:10.48550/arXiv.1905.04578>. Bias-reducing, second-order, (m+1)-gapped, difference-based estimator is based on I. Tecuapetla-Gómez and A. Munk (2017) <doi:10.1111/sjos.12256>. Robust autocovariance estimator for change point regression with autoregressive errors is based on S. Chakar et al. (2017) <doi:10.3150/15-BEJ782>. It also includes a general projection-based method for covariance matrix estimation. |
| Authors: | Inder Tecuapetla-Gómez [aut, cre] |
| Maintainer: | Inder Tecuapetla-Gómez <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.2.8 |
| Built: | 2024-12-21 06:29:58 UTC |
| Source: | CRAN |
Difference-based (auto)covariance/correlation estimation in change point regression with stationary errors.
Provides bias-reducing methods for (auto)covariance-correlation
estimation in change point regression with stationary -dependent errors
without having to pre-estimate the underlying signal of the observations.
In the same spirit, provides a robust estimator of the autorregressive coefficient
in change point regression with stationary, errors.
It also includes a general projection-based method for covariance matrix estimation.
dbacf returns and plots by default (auto)covariance/correlation
estimates without pre-estimating the underlying not necessarily smooth
signal of observations with stationary -dependent errors. The corresponding
plot method plot.dbacf allows for adjusting graphical
parameters to users' liking. This method is based on plot.acf.
dbacf_AR1 returns (auto)covariance/correlation estimates while
circumventing the difficult estimation of the underlying change point regression
function from observations with stationary errors.
Given a matrix estimate, not necessarily positive definite, of
the covariance matrix of a stationary process,
nearPDToeplitz returns the nearest, in the Frobenius norm,
covariance matrix to the initial estimate. See projectToeplitz
for the projection of a given symmetric matrix onto the space of Toeplitz matrices.
See also symBandedToeplitz for creating a (stationary process'
large covariance) matrix by specifying its dimension and values of its
autocovariance function.
Tecuapetla-Gómez, I. [email protected]
Grigoriadis, K.M., Frazho, A., Skelton, R. (1994). Application of alternating convex projection methods for computation of positive Toeplitz matrices, IEEE Transactions on signal processing 42(7), 1873–1875.
N. Higham (2002). Computing the nearest correlation matrix - a problem from finance, Journal of Numerical Analysis 22, 329–343.
Tecuapetla-Gómez, I and Munk, A. (2017). Autocovariance
estimation in regression with a discontinuous signal and -dependent errors: A
difference-based approach. Scandinavian Journal of Statistics, 44(2), 346–368.
Levine, M. and Tecuapetla-Gómez, I. (2023). Autocovariance
function estimation via difference schemes for a semiparametric change point model
with -dependent errors. Submitted.
Computes and by default plots the (auto)covariance/correlation function estimate without pre-estimating the underlying piecewise constant signal of the observations. To that end, a class of second-order difference-based estimators is implemented according to Eqs.(2.5)-(2.6) of Tecuapetla-Gómez and Munk (2017). By default, this function computes a subclass of estimates with minimal bias according to Eqs.(2.12)-(2.14) of the aforementioned paper.
dbacf( data, m, d, type = c("covariance", "correlation"), order = c("second", "first"), plot = TRUE, ... )dbacf( data, m, d, type = c("covariance", "correlation"), order = c("second", "first"), plot = TRUE, ... )
data |
numeric vector or a univariate object of class
|
m |
integer scalar giving the underlying level of dependency. |
d |
numeric vector giving the weights used in difference-based
estimation method. Only pertinent when |
type |
character string specifying whether covariance (default) or correlation must be computed. |
order |
character specifying whether a |
plot |
logical. If |
... |
further arguments passed to |
An object of class "dbacf" containing:
acf |
numeric vector of length |
m |
integer giving underlying level of dependency. |
d |
numeric vector containing the weights used to estimate acf. |
acfType |
string indicating whether |
n |
integer giving |
series |
string with name of variable |
Although the theoretical properties of the methods implemented
in this function were derived for change point regression with stationary
Gaussian -dependent errors, these methods have proven robust against
non-normality of the errors and as efficient as other methods in which
pre-estimation of an underlying smooth signal is required. For further
details see Section 6 of Tecuapetla-Gómez and Munk (2017).
The first-order difference-based estimator was implemented following Eqs.(4)-(5) of Levine and Tecuapetla-Gómez (2023). For the robustness of this estimator see Section 4 of the just mentioned paper.
Tecuapetla-Gómez, I and Munk, A. (2017). Autocovariance
estimation in regression with a discontinuous signal and -dependent errors: A
difference-based approach. Scandinavian Journal of Statistics, 44(2), 346–368.
Levine, M. and Tecuapetla-Gómez, I. (2023). Autocovariance
function estimation via difference schemes for a semiparametric change point model
with -dependent errors. Submitted.
ma2 <- arima.sim(n = 50, model = list(ma = c(0.4, -0.4), order = c(0, 0, 2)), sd = 0.25) dbacf(data=ma2, m = 2) dbacf(data=ma2, m = 2, order="first")ma2 <- arima.sim(n = 50, model = list(ma = c(0.4, -0.4), order = c(0, 0, 2)), sd = 0.25) dbacf(data=ma2, m = 2) dbacf(data=ma2, m = 2, order="first")
In the context of change point regression with a stationary AR(1) error process, this function estimates the autoregressive coefficient along with the autocovariance/correlation function as a function of given lags.
dbacf_AR1(data, type = c("covariance", "correlation"), lags)dbacf_AR1(data, type = c("covariance", "correlation"), lags)
data |
numeric vector or a univariate object of class |
type |
character string specifying whether covariance (default) or correlation must be computed. |
lags |
numeric giving the number of lags to compute. |
An object of class "dbacf" containing:
acf numeric vector of length lags + 1 giving estimated (auto)covariance/correlation function
rho numeric, estimate of autoregressive coefficient
acfType string indicating whether covariance or correlation has been computed
n integer giving length(data)
Chakar, S. and Lebarbier, E. and Lévy-Leduc, C. and Robin, S. (2017). A robust approach for estimating change-points in the mean of an AR(1) process, Bernoulli, 23(2), 1408-1447
ar1 <- arima.sim(n = 50, model = list(ar = c(0.5), order = c(1, 0, 0)), sd = 0.25) dbacf_AR1(ar1, type="correlation", lags=10)ar1 <- arima.sim(n = 50, model = list(ar = c(0.5), order = c(1, 0, 0)), sd = 0.25) dbacf_AR1(ar1, type="correlation", lags=10)
Computes the nearest positive definite Toeplitz matrix to an initial approximation, typically a covariance (correlation) matrix of a stationary process. This function implements an alternating projection algorithm that combines Grigoriadis et al. (1994) and Higham (2002). For further details see Section 5 of Tecuapetla-Gómez and Munk (2017).
nearPDToeplitz( matrix, type = c("covariance", "correlation"), toleranceEigen = 1e-06, toleranceConvergence = 1e-06, tolerancePosDef = 1e-06, maxIterations = 100, doEigen = TRUE )nearPDToeplitz( matrix, type = c("covariance", "correlation"), toleranceEigen = 1e-06, toleranceConvergence = 1e-06, tolerancePosDef = 1e-06, maxIterations = 100, doEigen = TRUE )
matrix |
a symmetric matrix. |
type |
string indicating whether the elements of the main
diagonal must be all equal to 1 ( |
toleranceEigen |
defines relative positiveness of eigenvalues compared to largest one. |
toleranceConvergence |
numeric indicating convergence tolerance for alternating projection algorithm. |
tolerancePosDef |
tolerance for forcing positive definiteness (in the final step) of alternating projection algorithm. |
maxIterations |
integer giving maximum number of iterations
allowed in alternating projection algorithm; when
this number is exceeded without convergence
a warning is displayed and matrix computed in step
|
doEigen |
logical indicating whether finding the closest positive definite matrix -through a eigen step- should be applied to the result of the alternating projection algorithm. |
This function is based on an alternating projection algorithm which
involves projecting sequentially and iteratively the initial matrix into the
set of symmetric positive definite and into the space of Toeplitz matrices,
respectively. The iteration process will stop because either a criterion
of convergence is met or maxIterations has been exceeded (without convergence).
Criterion of convergence: if the Frobenius norm of the difference of the
projection matrices computed in the last two iterations of the algorithm
is smaller than toleranceConvergence, then the algorithm stops returning
the projection matrix computed in the last iteration.
When projecting onto the set of symmetric positive definite matrices, toleranceEigen
controls the relative magnitude of any eigenvalue with
respect to the largest one and all eigenvalues
are treated as zero if
.
A list containing:
projection |
a matrix, the computed symmetric positive definite Toeplitz matrix. |
normF |
Frobenius norm of the difference between original and projection matrix. |
iterations |
number of iterations used for alternating projection algorithm. |
relativeTolerance |
numeric giving relative error (in Frobenius norm) of final approximation with respect to original matrix. |
converged |
logical indicating if alternating projection algorithm converged. |
Grigoriadis, K.M., Frazho, A., Skelton, R. (1994). Application of alternating convex projection methods for computation of positive Toeplitz matrices, IEEE Transactions on signal processing 42(7), 1873–1875.
N. Higham (2002). Computing the nearest correlation matrix - a problem from finance, Journal of Numerical Analysis 22, 329–343.
Tecuapetla-Gómez, I and Munk, A. (2017). Autocovariance
estimation in regression with a discontinuous signal and -dependent errors: A
difference-based approach. Scandinavian Journal of Statistics, 44(2), 346–368.
nearPD, projectToeplitz,
symBandedToeplitz, posdefify
# Higham (2002), p. 334 (mat <- matrix(c(1, 1, 0, 1, 1, 1, 0, 1, 1), byrow = TRUE, ncol = 3)) matProj <- matrix(c(1, 0.7607, 0.1573, 0.7607, 1, 0.7607, 0.1573, 0.7607, 1), byrow = TRUE, ncol = 3) nrPDT.mat <- nearPDToeplitz(mat, type = "correlation") stopifnot( identical(unname(matProj), unname(round(as.matrix(nrPDT.mat$projection), digits=4) ) )) eigen(nrPDT.mat$projection)$values # Toeplitz banded matrix near to the covariance matrix of 100 realizations # of an MA(5) with following parameters: n <- 1e2 alphas <- c(-2, 0.5, -4, 0, 0.75) (true.acf <- ARMAacf(ma = alphas)) alphasMat <- symBandedToeplitz(true.acf, n = n) stopifnot( min(eigen(alphasMat)$values) > 0 ) # alphasMat is a positive definite matrix (l <- length(true.acf)) (acf.modified <- c(true.acf[-c(l - 1, l)], 0.25)) # modifying original acf x <- acf.modified acfMat <- symBandedToeplitz(x, n = n) # no. of non positive eigenvalues of acfMat (6) length( eigen(acfMat)$values[eigen(acfMat)$values < 0 ] ) # acfMat is a 100 x 100 symmetric banded Toeplitz matrix acfMat[1:15, 1:30] system.time(nrPDT.acfMat <- nearPDToeplitz(acfMat, type = "correlation")) y <- eigen(nrPDT.acfMat$projection)$values # no. of non positive eigenvalues of nrPDT.acfMat length( y[ y < 0 ] ) # none!# Higham (2002), p. 334 (mat <- matrix(c(1, 1, 0, 1, 1, 1, 0, 1, 1), byrow = TRUE, ncol = 3)) matProj <- matrix(c(1, 0.7607, 0.1573, 0.7607, 1, 0.7607, 0.1573, 0.7607, 1), byrow = TRUE, ncol = 3) nrPDT.mat <- nearPDToeplitz(mat, type = "correlation") stopifnot( identical(unname(matProj), unname(round(as.matrix(nrPDT.mat$projection), digits=4) ) )) eigen(nrPDT.mat$projection)$values # Toeplitz banded matrix near to the covariance matrix of 100 realizations # of an MA(5) with following parameters: n <- 1e2 alphas <- c(-2, 0.5, -4, 0, 0.75) (true.acf <- ARMAacf(ma = alphas)) alphasMat <- symBandedToeplitz(true.acf, n = n) stopifnot( min(eigen(alphasMat)$values) > 0 ) # alphasMat is a positive definite matrix (l <- length(true.acf)) (acf.modified <- c(true.acf[-c(l - 1, l)], 0.25)) # modifying original acf x <- acf.modified acfMat <- symBandedToeplitz(x, n = n) # no. of non positive eigenvalues of acfMat (6) length( eigen(acfMat)$values[eigen(acfMat)$values < 0 ] ) # acfMat is a 100 x 100 symmetric banded Toeplitz matrix acfMat[1:15, 1:30] system.time(nrPDT.acfMat <- nearPDToeplitz(acfMat, type = "correlation")) y <- eigen(nrPDT.acfMat$projection)$values # no. of non positive eigenvalues of nrPDT.acfMat length( y[ y < 0 ] ) # none!
This function returns the plot method for objects of class "dbacf".
## S3 method for class 'dbacf' plot( x, type = "h", xlab = "Lag", ylab = paste("ACF", ifelse(x$acfType == "covariance", "(cov)", " ")), xlim = c(0, x$m + 1), main = paste("Series", x$series), ltyZeroLine = 3, colZeroLine = "blue", ... )## S3 method for class 'dbacf' plot( x, type = "h", xlab = "Lag", ylab = paste("ACF", ifelse(x$acfType == "covariance", "(cov)", " ")), xlim = c(0, x$m + 1), main = paste("Series", x$series), ltyZeroLine = 3, colZeroLine = "blue", ... )
x |
an object of class "dbacf". |
type |
what type of plot should be drawn. For possible types see
|
xlab |
the x label of the plot. |
ylab |
the y label of the plot. |
xlim |
numeric vector of length 2 giving the |
main |
an overall title for the plot. |
ltyZeroLine |
type of line used to draw horizontal line passing at 0. |
colZeroLine |
string indicating color of horizontal line passing at 0. |
... |
extra arguments to be passed to plot. |
No return value
dbacf documents the structure of objects of class "dbacf".
Computes the orthogonal projection onto the space of symmetric Toeplitz matrices as given in Grigoriadis et al. (1994).
projectToeplitz(matrix)projectToeplitz(matrix)
matrix |
a symmetric matrix. |
The computed projection matrix.
Grigoriadis, K.M., Frazho, A., Skelton, R. (1994). Application of alternating convex projection methods for computation of positive Toeplitz matrices, IEEE Transactions on signal processing 42(7), 1873–1875
A <- matrix(c(2, 1, 1, 1, 2, 0, 1, 0, 0), byrow = 3, nrow = 3) projectToeplitz(A)A <- matrix(c(2, 1, 1, 1, 2, 0, 1, 0, 0), byrow = 3, nrow = 3) projectToeplitz(A)
Creates a symmetric banded Toeplitz matrix
symBandedToeplitz(x, n)symBandedToeplitz(x, n)
x |
numeric vector or an object of class |
n |
integer specifying number of columns (rows) of banded matrix. |
An symmetric banded Toeplitz
matrix whose entries in main band are given by object x.
alphas <- c(-2, 0.5, -4) (true.acf <- ARMAacf(ma = alphas)) symBandedToeplitz(true.acf, n = 10)alphas <- c(-2, 0.5, -4) (true.acf <- ARMAacf(ma = alphas)) symBandedToeplitz(true.acf, n = 10)