mvLSW (version 1.0)

AdjPositiveDef: Regularization of the Multivariate EWS to be Positive Definite

Description

Regularizes the multivariate evolutionary wavelet spectrum to ensure that each matrix for all locations and levels are positive definite.

Usage

AdjPositiveDef(object, tol = 1e-10)

Arguments

object

A mvLSW object containing the smoothed and bias corrected multivariate evolutionary wavelet spectrum estimate.

tol

Tolerance. At 1e-10 by default.

Value

An object of class mvLSW, invisibly.

Details

Given an multivariate evolutionary wavelet spectrum estimate, smoothed and bias corrected, there is no guarantee that the matrix $\hat{S}_{j,k}$ for any level j and location k is positive definite. This command assesses from the eigenvalues whether each matrix estimate is positive definite and if not attempts to regularize the estimate via modchol. If this fails to determine a positive definite estimate, then a threshold is applied to the eigenvalues of the orignal matrix at tol.

References

Park, T., Eckley, I. and Ombao, H.C. (2014) Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. Signal Processing, IEEE Transactions on 62(20) pp. 5240-5250.

See Also

mvEWS, mvLSW, eigen, modchol.

Examples

## Sample bivariate locally stationary time series
set.seed(100)
X <- matrix(rnorm(2 * 2^8), ncol = 2)
X[1:2^7, 2] <- 3 * (X[1:2^7, 2] + 0.95 * X[1:2^7, 1])
X[-(1:2^7), 2] <- X[-(1:2^7), 2] - 0.95 * X[-(1:2^7), 1]
X[-(1:2^7), 1] <- X[-(1:2^7), 1] * 4
X <- as.ts(X)

## Raw EWS using the Haar wavelet
RawEWS_Haar <- RawPeriodogram(X, format = TRUE)
plot(RawEWS_Haar, style = 2, info = 1)

## Smoothed EWS using kernel "daniell"
SmoothEWS_Haar <- Smooth_EWS(RawEWS_Haar, kernel.param = 4)
plot(SmoothEWS_Haar, style = 2, info = 1)

## Correct for the estimator bias
CorrectEWS_Haar <- CorrectBias(SmoothEWS_Haar)
plot(CorrectEWS_Haar, style = 2, info = 1)

## Adjust estimate for positive definite matrices
EWSest_Haar <- AdjPositiveDef(CorrectEWS_Haar)
plot(EWSest_Haar, style = 2, info = 1)

## As above, but using Daubechies least asymmetric wavelet with 10 vanishing moments
RawEWS_DLA10 <- RawPeriodogram(X, filter.number = 10, 
   family = "DaubLeAsymm", format = TRUE)
SmoothEWS_DLA10 <- Smooth_EWS(RawEWS_DLA10, kernel.param = 2)
CorrectEWS_DLA10 <- CorrectBias(SmoothEWS_DLA10)
EWSest_DLA10 <- AdjPositiveDef(CorrectEWS_DLA10)

plot(RawEWS_DLA10, style = 2, info = 1)
plot(SmoothEWS_DLA10, style = 2, info = 1)
plot(CorrectEWS_DLA10, style = 2, info = 1)
plot(EWSest_DLA10, style = 2, info = 1)