Description Usage Arguments Details Value References See Also Examples

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

1 |

`object` |
A |

`ACWIP` |
4D array containing the wavelet autocorrelation
inner product functions. Set to |

`verbose` |
Logical. Controls the printing of messages whist
the computations progress. Set as |

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

Here, *A^{λ}_{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.

Invisibly returns a `mvLSW`

object containing the asymptotic variance
of the multivariate evolutionary wavelet spectrum.

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.

`ipndacw`

, `AutoCorrIP`

,
`as.mvLSW`

, `mvEWS`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
## 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)
``` |

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