Evaluate the Approximate Confidence Interval of a mvEWS Estimate

Description

Evaluate the approximate confidence interval of a multivariate evolutionary wavelet spectrum.

Usage

  ApxCI(object, var = NULL, alpha = 0.05, ...)

Arguments

Details

The command evaluates the approximate Gaussian confidence intervals for the elements of the mvEWS estimate.

Value

Invisibly returns a list containing two mvLSW classed objects with names "L" and "U" that respectively identify the lower and upper interval estimates.

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. (2014) Wavelet Methods for Multivariate Nonstationary Time Series, PhD thesis, Lancaster University, pp. 91-111.

See Also

mvEWS, as.mvLSW, varEWS.

Examples

## Define evolutionary wavelet spectrum, structure only on level 2
Spec <- array(0, dim = c(3, 3, 8, 256))
Spec[1, 1, 2, ] <- 10
Spec[2, 2, 2, ] <- c(rep(5, 64), rep(0.6, 64), rep(5, 128))
Spec[3, 3, 2, ] <- c(rep(2, 128), rep(8, 128))
Spec[2, 1, 2, ] <- Spec[1, 2, 2, ] <- punif(1:256, 65, 192)
Spec[3, 1, 2, ] <- Spec[1, 3, 2, ] <- c(rep(-1, 128), rep(5, 128))
Spec[3, 2, 2, ] <- Spec[2, 3, 2, ] <- -0.5
EWS <- as.mvLSW(x = Spec, filter.number = 1, family = "DaubExPhase", 
  min.eig.val = NA)

## Sample time series and estimate the EWS.
set.seed(10)
X <- rmvLSW(Spectrum = EWS)
EWS_X <- mvEWS(X, kernel.name = "daniell", kernel.param = 20)

## Evaluate asymptotic spectral variance 
SpecVar <- varEWS(EWS_X)

## Plot Estimate & 95% confidence interval
CI <- ApxCI(object = EWS_X, var = SpecVar, alpha = 0.05)
plot(x = EWS_X, style = 2, info = 2, Interval = CI)

Wavelet Autocorrelation Inner Product Functions

Description

Inner product of cross-level wavelet autocorrelation functions.

Usage

  AutoCorrIP(J, filter.number = 1, family = "DaubExPhase",
    crop = TRUE)

Arguments

Details

Let \psi(x) denote the mother wavelet and the wavelet defined for level j as \psi_{j,k}(x) = 2^{j/2}\psi(2^{j}x-k). The wavelet autocorrelation function between levels j & l is therefore:

\Psi_{j,l}(\tau) = \sum_\tau \psi_{j,k}(0)\psi_{l,k-\tau}(0)

Here, integer \tau defines the offset of the latter wavelet function relative to the first.

The inner product of this wavelet autocorrelation function is defined as follows for level indices j, l & h and offset \lambda:

A^{\lambda}_{j,l,h} = \sum_{\tau} \Psi_{j,l}(\lambda - \tau) \Psi_{h,h}(\tau)

Value

A 4D array (invisibly returned) of order LxJxJxJ where L depends on the specified wavelet function. If crop=TRUE then L=2^{J+1}+1. The first dimension defines the offset \lambda, whilst the second to fourth dimensions identify the levels indexed by j, l & h respectively.

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.

Fryzlewicz, P. and Nason, G. (2006) HaarFisz estimation of evolutionary wavelet spectra. Journal of the Royal Statistical Society. Series B, 68(4) pp. 611-634.

See Also

ipndacw.

Examples

## Plot Haar autocorrelation wavelet functions inner product
AInnProd <- AutoCorrIP(J = 8, filter.number = 1, family = "DaubExPhase")
## Not run: 
MaxOffset <- 2^8
for(h in 6:8){
  x11()
  par(mfrow = c(3, 3))
  for(l in 6:8){
    for(j in 6:8){
      plot(-MaxOffset:MaxOffset, AInnProd[, j, l, h], type = "l", 
        xlab = "lambda", ylab = "Autocorr Inner Prod", 
        main = paste("j :", j, "- l :", l, "- h :", h))
    }
  }
}

## End(Not run)

## Special case relating to ipndacw function from wavethresh package
Amat <- matrix(NA, ncol = 8, nrow = 8)
for(j in 1:8) Amat[, j] <- AInnProd[2^8 + 1, j, j, ]
round(Amat, 5)
round(ipndacw(J = -8, filter.number = 1, family = "DaubExPhase"), 5)

Convert Between mvEWS and Transfer Function Matrices

Description

Convert between multivariate evolutionary wavelet spectrum and the set of transfer function matrices.

Usage

  Spectrum2Transfer(object, S2V = TRUE)

Arguments

Details

If the mvEWS is supplied, then the set of transfer function matrices are derived by the Choleski factorization of a real symmetric semi-positive definite square matrix. In the cases where the matrix is semi-definite, then the Choleski factorization is applied to the submatrix that is positive definite and the remaining lower triangular elements are populated such that the resulting matrix is a valid factorization.

Conversely, if the set of transfer function matrices are supplied, then the EWS are derived by squaring the matrices.

Value

A mvLSW object containing either the mvEWS or set of transfer function matrices depending on the specified transformation direction.

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

chol, as.mvLSW, mvEWS.

Examples

## Define evolutionary wavelet spectrum, structure only on level 2
Spec <- array(0, dim=c(3, 3, 8, 256)) ## Ensure all are positive def.
Spec[1, 1, 2, ] <- 10
Spec[2, 2, 2, ] <- c(rep(5, 64), rep(0.6, 64), rep(5, 128))
Spec[3, 3, 2, ] <- c(rep(2, 128), rep(8, 128))
Spec[2, 1, 2, ] <- Spec[1, 2, 2, ] <- punif(1:256, 65, 192)
Spec[3, 1, 2, ] <- Spec[1, 3, 2, ] <- c(rep(-1, 128), rep(5, 128))
Spec[3, 2, 2, ] <- Spec[2, 3, 2, ] <- -0.5

## Define EWS as mvLSW object
EWS <- as.mvLSW(x = Spec, filter.number = 1, family = "DaubExPhase",
  min.eig.val = NA)
plot(EWS, style = 2, info = 2)

## EWS to Transfer function matrices
Transfer <- Spectrum2Transfer(object = EWS, S2V = TRUE)

## Transfer function matrices to EWS
EWS2 <- Spectrum2Transfer(object = Transfer, S2V = FALSE)
plot(EWS2, style = 2, info = 2)

Multivariate Locally Stationary Wavelet Object

Description

Constructs a multivariate locally stationary wavelet (mvLSW) object.

Usage

  as.mvLSW(x, filter.number = 1, family = "DaubExPhase",
    smooth.type = "all", smooth.kernel = kernel("daniell", 0), 
    bias.correct = FALSE, min.eig.val = -Inf, names = NULL)
  
  is.mvLSW(object)

Arguments

Details

as.mvLSW constructs a multivariate locally stationary classed object that contains all information about the constructions of various multivariate wavelet estimates.

The command is.mvLSW checks that the supplied R object is a valid mvLSW object in that its structure and contents are as expected.

Value

The as.mvLSW command invisibly returns a list with the following items:

The list Information contains:

The command is.mvLSW returns TRUE if the supplied object is a valid mvLSW object as described above. Otherwise, the command returns FALSE.

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.

See Also

mvEWS, varEWS, kernel.

Examples

## Define evolutionary wavelet spectrum, structure only on level 2
Spec <- array(0, dim = c(3, 3, 8, 256))
Spec[1, 1, 2, ] <- 10
Spec[2, 2, 2, ] <- c(rep(5, 64), rep(0.6, 64), rep(5, 128))
Spec[3, 3, 2, ] <- c(rep(2, 128), rep(8, 128))
Spec[2, 1, 2, ] <- Spec[1, 2, 2, ] <- punif(1:256, 65, 192)
Spec[3, 1, 2, ] <- Spec[1, 3, 2, ] <- c(rep(-1, 128), rep(5, 128))
Spec[3, 2, 2, ] <- Spec[2, 3, 2, ] <- -0.5

## Define EWS as mvLSW object
EWS <- as.mvLSW(x = Spec, filter.number = 1, family = "DaubExPhase", 
  names = c("A", "B", "C"), min.eig.val = NA)
is.mvLSW(EWS)
plot(EWS, style = 2, info = 2)

Local Wavelet Coherence and Partial Coherence

Description

Wavelet coherence and partial coherence of an evolutionary wavelet spectrum.

Usage

  coherence(object, partial = FALSE)

Arguments

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.

Value

An object of class mvLSW, invisibly.

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

## 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

Multivariate Evolutionary Wavelet Spectrum

Description

Calculates the multivariate Evolutionary Wavelet Spectrum (mvEWS) 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

Details

This command evaluates the multivariate evolutionary wavelet spectrum of a multivariate locally stationary wavelet 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,...,T-1 (T=2^J) and channels 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 channel 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 (see Ombao et al., 2005).

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^{-1})_{j,l} \hat{I}_{l,k}

Here, A denotes the wavelet autocorrelation inner product matrix.

If chosen to, the mvEWS matrices at each level and location, \hat{S}_{j,k}, is regularised to ensure positive definiteness.

Value

An object of class mvLSW, invisibly.

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.

Ombao, H., von Sachs, R. and Guo, W. (2005) SLEX analysis of multivariate nonstationary time series. Journal of the American Statistical Association 100(470) pp.519-531.

See Also

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

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)

## 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)

Multivariate, Locally Stationary Wavelet Process Estimation

Description

The mvLSW package provides an implementation of the multivariate locally stationary time series modelling approach proposed by Park, Eckley and Ombao (2014).

The approach extends the locally stationary wavelet time series work of Nason, von Sachs and Kroisandt (2000) to a multivariate setting, introducing wavelet-based measures of local coherence and local partial coherence. The package implements the estimation scheme by Park et al. (2014) for such processes. Note that mvLSW should be used in conjunction with the wavethresh package developed by Nason (2016).

Details

Package: mvLSW

Type: Package

Version: 1.2.3

Date: 2019-08-05

License: GPL(>=3)

Author(s)

Simon Taylor, <s.taylor2@lancaster.ac.uk>

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

Nason, G.P., von Sachs, R. and Kroisandt, G. (2000) Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum Journal of the Royal Statistical Society B 62, pp. 271–292.

Nason, G. (2016) wavethresh: Wavelets Statistics and Transforms. R package version 4.6.8.

https://CRAN.R-project.org/package=wavethresh

See Also

mvEWS, coherence, rmvLSW, as.mvLSW, mvEWS

Examples

#
# See examples in individual help pages
#

Plot mvLSW Object

Description

Plot the data contained within a mvLSW object based on the requested format.

Usage

  ## S3 method for class 'mvLSW'
plot(x, style = 1, info = NULL, Interval = NULL, 
    diag = TRUE, sub = "Spectrum", ...)

Arguments

Details

This command plots the data contained within the mvLSW based on requested plotting style.

Plotting style style=1 with information info=c(p,q,j) generates a single plot for a specified channel pair p & q and level j.

Plotting style style=2 with information info=j creates a set of plots from x for all channel pairs in a lower-triangular panel corresponding to the specified level j. If diag=FALSE then the plots along the diagonal are suppressed, which is ideal when x contain coherence estimates.

Plotting style style=3 with information info=c(p,q) creates a set of plots from x for all levels (from fine to coarse) for channel pair p and q.

Finally, the plotting style style=4 with information info=c(p,q) presents the same information as for the previous case, but in a compact matrix format. Please refer to image.plot from the fields library for additional information on this plotting style.

The argument Interval must be supplied in order to draw a polygon depicting the pointwise interval. See ApxCI for deriving an approximate confidence interval for the evolutionary wavelet spectrum estimate. This argument is ignored in the case style=4.

Value

Generates a plot. No data is returned.

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.

See Also

plot.default, image.plot, as.mvLSW, mvEWS, coherence, ApxCI.

Examples

## Define evolutionary wavelet spectrum, structure only on level 2
Spec <- array(0, dim=c(3, 3, 8, 256))
Spec[1, 1, 2, ] <- 10
Spec[2, 2, 2, ] <- c(rep(5, 64), rep(0.6, 64), rep(5, 128))
Spec[3, 3, 2, ] <- c(rep(2, 128), rep(8, 128))
Spec[2, 1, 2, ] <- Spec[1, 2, 2, ] <- punif(1:256, 65, 192)
Spec[3, 1, 2, ] <- Spec[1, 3, 2, ] <- c(rep(-1, 128), rep(5, 128))
Spec[3, 2, 2, ] <- Spec[2, 3, 2, ] <- -0.5
EWS <- as.mvLSW(x = Spec, filter.number = 1, family = "DaubExPhase",
  min.eig.val = NA)

## Sample time series and estimate the EWS and coherence.
set.seed(10)
X <- rmvLSW(Spectrum = EWS)
EWS_X <- mvEWS(X, kernel.name = "daniell", kernel.param = 20)
RHO_X <- coherence(EWS_X, partial = FALSE)

## Evaluate asymptotic spectral variance 
SpecVar <- varEWS(EWS_X)

## Evaluate 95% approximate confidence interval
CI <- ApxCI(object = EWS_X, var = SpecVar, alpha=0.05)

## Plot mvEWS between channels 1 & 3 at level 2
plot(x = EWS_X, style = 1, info = c(1, 3, 2), Interval = CI)

## Plot coherence between channels 1 & 3 at level 2
plot(x = RHO_X, style = 1, info = c(1, 3, 2), ylab = "Coherence")

## mvEWS panel plot for level 2
plot(x = EWS_X, style = 2, info = 2, Interval = CI)

## Panel plot of coherence for level 2
plot(x = RHO_X, style = 2, info = 2, diag = FALSE, ylab = "Coherence")

## Plot mvEWS for channel pair 1 & 3 at all levels
plot(x = EWS_X, style = 3, info = c(1, 3), Interval = CI)

## Plot coherence for channel pair 1 & 3 at all levels
plot(x = RHO_X, style = 3, info = c(1, 3), ylab = "Coherence")

## Image plot for coherence between channels 1 & 3
plot(x = RHO_X, style = 4, info = c(1, 3), sub = "Coherence")

Sample a Multivariate Locally Stationary Wavelet Process

Description

Sample a multivariate locally stationary wavelet process.

Usage

rmvLSW(Transfer = NULL, Spectrum = NULL, noiseFN = rnorm, ...)  

## S3 method for class 'mvLSW'
simulate(object, nsim = 1, seed = NULL, ...)

Arguments

Details

Samples a single multivariate locally stationary wavelet time series for the given set of transfer function matrices. These are assumed to be lower-triangular (including diagonal) matrices. If the mvEWS is supplied instead, then this is pre-processed by Spectrum2Transfer() to obtain the transfer function matrices.

The Transfer and Spectrum are both mvLSW objects and therefore contain information about defining the wavelet function.

The innovation process is assumed to be second order stationary with expectation zero, orthogonal and unit variance. The first argument of noiseFN must be n and define the number of samples to generate. The function must also return a numerical vector of length n.

The simulate command implements rmvLSW under default arguments unless specified via ....

Value

A ts matrix object of a multivariate locally stationary time series. The columns of the matrix correspond to different channels and the rows identify the time axis.

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

mvLSW, Spectrum2Transfer, rnorm, AvBasis, ts.

Examples

## Define evolutionary wavelet spectrum, structure only on level 2
Spec <- array(0, dim = c(3, 3, 8, 256))
Spec[1, 1, 2, ] <- 10
Spec[2, 2, 2, ] <- c(rep(5, 64), rep(0.6, 64), rep(5, 128))
Spec[3, 3, 2, ] <- c(rep(2, 128), rep(8, 128))
Spec[2, 1, 2, ] <- Spec[1, 2, 2, ] <- punif(1:256, 65, 192)
Spec[3, 1, 2, ] <- Spec[1, 3, 2, ] <- c(rep(-1, 128), rep(5, 128))
Spec[3, 2, 2, ] <- Spec[2, 3, 2, ] <- -0.5

## Define Haar wavelet function and create mvLSW object
EWS <- as.mvLSW(x = Spec, filter.number = 1, family = "DaubExPhase",
  min.eig.val = NA)
plot(EWS, style = 2, info = 2)

## Sample with Gaussian innovations
set.seed(10)
X <- rmvLSW(Spectrum = EWS)
plot(X)

## Alternatively:
X1 <- simulate(object = EWS)
plot(X1)

## Define smoother wavelet function and create mvLSW object
EWS2 <- as.mvLSW(x = Spec, filter.number = 10, family = "DaubExPhase")

## Sample with logistic innovations
set.seed(10)
X2 <- rmvLSW(Spectrum = EWS2, noiseFN = rlogis, scale = sqrt(3)/pi)
plot(X2)

Print a Summary of mvLSW Object

Description

Prints a summary of the information contained within a mvLSW classed object.

Usage

  ## S3 method for class 'mvLSW'
summary(object, ...)
  ## S3 method for class 'mvLSW'
print(x, ...)

Arguments

Details

The command prints to screen a summary of the information contained within a mvLSW object. Information printed includes: dimensions, wavelet function, the smoothing regime applied, smoothing kernel(s), generalized cross-validation gamma deviance criteria score, application of the bias correction and minimum eigenvalue from across all spectral matrices.

Value

This command returns nothing, only prints a summary to the console.

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.

See Also

mvEWS, as.mvLSW.

Examples

## Generate a bivariate 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)
print(EWS)
plot(EWS, style = 2, info = 1)

Asymptotic Variance of the mvEWS Estimate

Description

Calculates the asymptotic variance of a multivariate evolutionary wavelet spectrum estimate.

Usage

  varEWS(object, ACWIP = NULL, verbose = FALSE)

Arguments

Details

The varEWS commands evaluate the asymptotic variance of a multivariate evolutionary wavelet spectrum (mvEWS) estimate. Note, the variance is only applicable when the mvEWS is smoothed consistently across all levels with list item smooth.type="all". This can be written in terms of the smoothed periodogram relating to the bias correction of the mvEWS estimate, where A_{j,k} is the inner product matrix of the wavelet autocorrelation function:

Var( \hat{S}^{(p,q)}_{j,k} ) = \sum_{l_1,l_2=1}^{J} (A^{-1})_{j,l_1} (A^{-1})_{j,l_2} Cov( \tilde{I}^{(p,q)}_{l_1,k}, \tilde{I}^{(p,q)}_{l_1,k})

The covariance between elements of the smoothed periodogram can also be expressed in terms of the raw wavelet periodogram:

Cov( \tilde{I}^{(p,q)}_{l_1,k}, \tilde{I}^{(p,q)}_{l_1,k}) = \sum_{m_1,m_2} W_{m_1} W_{m_2} Cov( I^{(p,q)}_{l_1,m_1}, I^{(p,q)}_{l_2,m_2} )

The weights W_i, for integer i, define the smoothing kernel function that is evaluated by the kernel command. Note that W_i = W_{-i} and \sum_i W_i = 1.

The final step is to derive the covariance of the raw periodogram. This has a long derivation, which can be concisely calculated by:

Cov( I^{(p,q)}_{j,k}, I^{(p,q)}_{l,m} ) = E(p,j,k,q,l,m)^2 + E(p,j,k,p,l,m)E(q,j,k,q,l,m)

where

E(p,j,k,q,l,m) = \sum_{h=1}^{J} A^{k-m}_{j,l,h} S^{(p,q)}_h((k+m)/2T)

Here, A^{\lambda}_{j,l,h} defines the autocorrelation wavelet inner product function and S^{(p,q)}_{j}(k/T) is the true spectrum of the process between channels p & q, level j and location k. The true spectrum is not always available and so this may be substituted with the smoothed and bias corrected mvEWS estimate. For practical purposes, if k+m is odd then the average between the available spectrum values at neighbouring locations are substituted.

For efficiency purpose, if the varEWS command is going to be called multiple times then it is highly recommended that the autocorrelation wavelet inner product should be evaluated beforehand by AutoCorrIP and supplied via the ACWIP argument.

Value

Invisibly returns a mvLSW object containing the asymptotic variance of the multivariate evolutionary wavelet spectrum.

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. (2014) Wavelet Methods for Multivariate Nonstationary Time Series, PhD thesis, Lancaster University, pp. 91-111.

See Also

ipndacw, AutoCorrIP, as.mvLSW, mvEWS

Examples

## Define evolutionary wavelet spectrum, structure only on level 2
Spec <- array(0, dim=c(3, 3, 8, 256))
Spec[1, 1, 2, ] <- 10
Spec[2, 2, 2, ] <- c(rep(5, 64), rep(0.6, 64), rep(5, 128))
Spec[3, 3, 2, ] <- c(rep(2, 128), rep(8, 128))
Spec[2, 1, 2, ] <- Spec[1, 2, 2, ] <- punif(1:256, 65, 192)
Spec[3, 1, 2, ] <- Spec[1, 3, 2, ] <- c(rep(-1, 128), rep(5, 128))
Spec[3, 2, 2, ] <- Spec[2, 3, 2, ] <- -0.5
EWS <- as.mvLSW(x = Spec, filter.number = 1, family = "DaubExPhase",
  min.eig.val = NA)

## Sample time series and estimate the EWS.
set.seed(10)
X <- rmvLSW(Spectrum = EWS)
EWS_X <- mvEWS(X, kernel.name = "daniell", kernel.param = 20)

## Evaluate asymptotic spectral variance 
SpecVar <- varEWS(EWS_X)

## Plot Estimate & 95% confidence interval
CI <- ApxCI(object = EWS_X, var = SpecVar, alpha = 0.05)
plot(x = EWS_X, style = 2, info = 2, Interval = CI)