Autocovariance function estimation via difference-based methods

Description

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 m-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, AR(1) errors. It also includes a general projection-based method for covariance matrix estimation.

Autocovariance Estimation

dbacf returns and plots by default (auto)covariance/correlation estimates without pre-estimating the underlying not necessarily smooth signal of observations with stationary m-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 AR(1) errors.

Covariance Matrix Estimation

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.

Author(s)

Tecuapetla-Gómez, I. itecuap@conabio.gob.mx

References

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 m-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 m-dependent errors. Submitted.

Difference-based (auto)covariance/correlation function estimation

Description

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.

Usage

dbacf(
  data,
  m,
  d,
  type = c("covariance", "correlation"),
  order = c("second", "first"),
  plot = TRUE,
  ...
)

Arguments

Value

An object of class "dbacf" containing:

Note

Although the theoretical properties of the methods implemented in this function were derived for change point regression with stationary Gaussian m-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.

References

Tecuapetla-Gómez, I and Munk, A. (2017). Autocovariance estimation in regression with a discontinuous signal and m-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 m-dependent errors. Submitted.

See Also

acf, plot.dbacf

Examples

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")
 

Robust dbacf in change point regression with AR(1) errors

Description

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.

Usage

dbacf_AR1(data, type = c("covariance", "correlation"), lags)

Arguments

Value

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)

References

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

Examples

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

Description

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).

Usage

nearPDToeplitz(
  matrix,
  type = c("covariance", "correlation"),
  toleranceEigen = 1e-06,
  toleranceConvergence = 1e-06,
  tolerancePosDef = 1e-06,
  maxIterations = 100,
  doEigen = TRUE
)

Arguments

Details

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 \lambda_k with respect to the largest one \lambda_1 and all eigenvalues \lambda_k are treated as zero if \lambda_k / \lambda_1 \leq \code{toleranceEigen}.

Value

A list containing:

References

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 m-dependent errors: A difference-based approach. Scandinavian Journal of Statistics, 44(2), 346–368.

See Also

nearPD, projectToeplitz, symBandedToeplitz, posdefify

Examples

# 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!

Plot autocovariance and autocorrelation functions

Description

This function returns the plot method for objects of class "dbacf".

Usage

## 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",
  ...
)

Arguments

Value

No return value

Note

dbacf documents the structure of objects of class "dbacf".

See Also

acf, dbacf.

Projection onto the set of symmetric Toeplitz matrices

Description

Computes the orthogonal projection onto the space of symmetric Toeplitz matrices as given in Grigoriadis et al. (1994).

Usage

projectToeplitz(matrix)

Arguments

Value

The computed projection matrix.

References

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

See Also

nearPDToeplitz

Examples

 
A <- matrix(c(2, 1, 1, 1, 2, 0, 1, 0, 0), byrow = 3, nrow = 3)
projectToeplitz(A)
 

Creates a symmetric banded Toeplitz matrix

Description

Creates a symmetric banded Toeplitz matrix

Usage

symBandedToeplitz(x, n)

Arguments

Value

An \code{n} \times \code{n} symmetric banded Toeplitz matrix whose entries in main band are given by object x.

See Also

nearPDToeplitz, bandSparse

Examples

alphas <- c(-2, 0.5, -4)
(true.acf <- ARMAacf(ma = alphas))
symBandedToeplitz(true.acf, n = 10)