mvLSW (version 1.1)

mvEWS: Multivariate Evolutionary Wavelet Spectrum

Description

Calculates the Evolutionary Wavelet Spectrum (EWS) of a multivariate locally stationary time series.

Usage

mvEWS(X, filter.number = 1, family = "DaubExPhase", 
    smooth = TRUE, type = "all", kernel.name = "daniell", 
    kernel.param = floor(sqrt(nrow(X))), optimize = FALSE, 
	smooth.Jset = NA, bias.correct = TRUE, tol = 1e-10, verbose = FALSE)

Arguments

X

A multivariate time series object of class ts, zoo, xts or matrix. The length of the time series must \(2^J\) for positive integer J.

filter.number

Integer number defining the number of vanishing moments of the wavelet function. By default, filter.number=1 and so defining the Haar wavelet.

family

Specifies the family of wavelet. Only two options are available, either "DaubExPhase" (default) or "DaubLeAsymm".

smooth

Logical, should EWS should be smoothed?

type

How should the smoothing be performed. If "all" (default) then the same smoothing kernel is applied to all levels, else if "by.level" then a different smoothing kernel is applied to each level.

kernel.name

Name of the smoothing kernel to be supplied to kernel(). Kernel "daniell" is defined by default.

kernel.param

Parameters to be passed to kernel(). This argument must be a vector if type="all", otherwise it must be a matrix with each column defining the kernel parameters for each log2(nrow(X)) levels from coarse to fine. If the kernel name is either "dirichlet" or "fejer" then kernel.param must have length 2 (or a matrix with 2 rows) which are supplied to kernel() as arguments m and r respectively. Note that the width of the kernel cannot be larger than the time series length. This is set by default as the square root of the length of the time series.

optimize

Logical, should the smoothing be optimized? If FALSE (default) then the smoothing is performed as specified with kernel.name and kernel.param. Otherwise, kernel.param defines the upper parameter bound and the optimal kernel is determined by minimising the generalized cross-validation gamma deviance criterion.

smooth.Jset

Integer vector indicating with levels to be used in calculating the generalized cross-validation gamma deviance criterion. This argument is only used if type="all" and is set as NA by default, implying that all levels should be used.

bias.correct

Logical, should the correction be applied to address the bias in the EWS estimator.

tol

Tolerance in applying matrix regularisation to ensure each EWS matrix to be strictly positive definite. If NA then the threshold is not applied. This is 1e-10 by default.

verbose

Logical. Controls the printing of messages whist the computations progress. Set as FALSE as default.

Value

An object of class mvLSW, invisibly.

Details

This command evaluates the evolutionary wavelet spectrum of a multivariate time series. The order of operations are as follows:

Calculate the non-decimated wavelet coefficients \(\{d^{p}_{j,k}\}\) for levels j = 1,…,J, locations k = 0,ldots,T-1 (T=\(2^J\)) and signals p = 1,…,P(=ncol(X)). The raw periodogram matrices are then evaluated by \(I^{(p,q)}_{j,k} = d^{p}_{j,k}d^{q}_{j,k}\) between any signal pair p & q.

The above estimator is inconsistent and so the matrix sequence is smoothed: \(\tilde{I}^{(p,q)}_{j,k} = \sum_i W_i I^{(p,q)}_{j,k+i}\). The kernel weights \(W_i\) are derived from the kernel command and satisfy \(W_i=W_{-i}\) and \(\sum_i W_i = 1\). The optimal parameter for the smoothing kernel is determined by minimising the generalized cross-validation gamma deviance criterion:

$$GCV = \sum_{p,j} GCV(p,j)$$ $$GCV(p,j) = (T(1-W_0)^2)^{-1} \sum_{k=0}^{T-1} q_k D^{(p,p)}_{j,k}$$ $$D^{(p,p)}_{j,k} = R^{(p,p)}_{j,k} - log(R^{(p,p)}_{j,k}) - 1$$ $$R^{(p,p)}_{j,k} = \tilde{I}^{(p,p)}_{j,k} / I^{(p,p)}_{j,k}$$

where \(q_0=q_{T-1}=0.5\) and \(q_k=1\) otherwise. Note that the criterion is not applicable on the wavelet cross-spectrum. A theshold is applied to \(\tilde{I}\) and \(I\) in order to produce a valid criterion estimate. In addition, the first summation is only applied over the signals if smoothing is to be applied on a by-level basis.

The raw wavelet periodogram is also a biased estimator. A correction is subsequently applied to the smoothed estimate as follows:

$$\hat{S}_{j,k} =\sum_{l=1}^{J} A_{j,l}^{-1} \hat{I}_{l,k}$$

Here, \(A_{j,l}\) denotes the wavelet autocorrelation inner product.

Finally, a threshold is applied to the eigenvalues of the EWS \(\hat{S}_{j,k}\) to ensure that the matrices are positive definite.

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

ts, wd, kernel, as.mvLSW, ipndacw.

Examples

# NOT RUN {
## 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)

## Haar wavelet, apply same smoothing to all levels & optimize
EWS <- mvEWS(X, kernel.name = "daniell", kernel.param = 20, optimize = TRUE)
summary(EWS)
plot(EWS, style = 2, info = 1)

## Over smoothed EWS
EWS_smooth <- mvEWS(X, filter.number = 10, family = "DaubLeAsymm",
  kernel.name = "modified.daniell", kernel.param = c(5, 5), optimize = FALSE)
summary(EWS_smooth)
plot(EWS_smooth, style = 2, info = 1)
# }