mvLSW (version 1.2.5)

coherence: Local Wavelet Coherence and Partial Coherence

Description

Wavelet coherence and partial coherence of an evolutionary wavelet spectrum.

Usage

coherence(object, partial = FALSE)

Arguments

object

Multivariate evolutionary wavelet spectrum as a mvLSW object.

partial

Logical, should the partial coherence be calculated. Set as FALSE by default.

Value

An object of class mvLSW, invisibly.

Details

Given the evolutionary wavelet spectrum of a multivariate locally stationary time series, denoted by the matrix sequence \(S_{j,k}\), then the coherence matrix for level j and location k is:

$$R_{j,k} = D_{j,k} S_{j,k} D_{j,k}$$

where \(D_{j,k} = diag\{ (S^{(p,p)}_{j,k})^{-0.5} : p=1,\ldots,P \}\). This measures the linear cross-dependence between different channels at a particular level.

Notate the inverse spectrum matrix as \(G_{j,k} = S^{-1}_{j,k}\), then the partial coherence matrix for level j and location k is derived as follows:

$$\Gamma_{j,k} = -H_{j,k} G_{j,k} H_{j,k}$$

where \(H_{j,k} = diag\{ (G^{(p,p)}_{j,k})^{-0.5} : p=1,\ldots,P \}\). This measures the coherence between channels after removing the linear effects if all other channels and so enable the distinction between direct and indirect linear dependency between channels.

For valid calculations of (partial) coherence, values within [-1,1], it is important that the spectral matrices are positive definite.

References

Taylor, S.A.C., Park, T.A. and Eckley, I. (2019) Multivariate locally stationary wavelet analysis with the mvLSW R package. Journal of statistical software 90(11) pp. 1--16, doi: 10.18637/jss.v090.i11.

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

as.mvLSW, mvEWS.

Examples

# NOT RUN {
## Sample tri-variate time series
##  Series 2 & 3 are dependent indirectly via Series 1
set.seed(100)
X <- matrix(rnorm(3 * 2^8), ncol = 3)
X[1:192, 2] <- X[1:192, 2] + 0.95 * X[1:192, 1]
X[65:256, 3] <- X[65:256, 3] - 0.95 * X[65:256, 1]
X <- as.ts(X)

## Evolutionary Wavelet Spectrum
EWS <- mvEWS(X, filter.number = 4, kernel.name = "daniell", 
  kernel.param = 20)

## Coherence
RHO <- coherence(EWS, partial = FALSE)
plot(RHO, style = 2, info = 1, ylab = "Coherence", diag = FALSE)

## Partial Coherence
PRHO <- coherence(EWS, partial = TRUE)
plot(PRHO, style = 2, info = 1, ylab = "P. Coh.", diag = FALSE) 
#series 2&3 are closer to 0
# }