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.