Description Usage Arguments Details Value Note References See Also Examples
pdCART
performs hard treestructured thresholding of the Hermitian matrixvalued wavelet coefficients obtained with
WavTransf1D
or WavTransf2D
based on the trace of the whitened wavelet coefficients, as explained in
\insertCiteCvS17pdSpecEst or \insertCiteC18pdSpecEst. This function is primarily written for internal use in other functions and
is typically not used as a standalone function.
1 
D 
a list of wavelet coefficients as obtained from the 
D.white 
a list of whitened wavelet coefficients as obtained from the 
order 
the order(s) of the intrinsic 1D or 2D AI refinement scheme as in 
alpha 
tuning parameter specifying the penalty/sparsity parameter as 
tree 
logical value, if 
... 
additional arguments for internal use. 
Depending on the structure of the input list of arrays D
the function performs 1D or 2D treestructured thresholding of wavelet coefficients.
The optimal tree of wavelet coefficients is found by minimization of the complexity penalized residual sum of squares (CPRESS) criterion
in \insertCiteD97pdSpecEst, via a fast treepruning algorithm. By default, the penalty parameter in the optimization procedure is set equal to
alpha
times the universal threshold σ_w√(2\log(n)), where σ_w^2 is the noise variance of the traces of the whitened
wavelet coefficients determined from the finest wavelet scale and n is the total number of coefficients. By default, alpha = 1
,
if alpha = 0
, the penalty parameter is zero and the coefficients remain untouched.
Returns a list with two components:

a list of logical values specifying which coefficients to keep, with each list component
corresponding to an individual wavelet scale starting from the coarsest wavelet scale 

the list of thresholded wavelet coefficients, with each list component corresponding to an individual wavelet scale. 
For thresholding of 1D wavelet coefficients, the noise
variance of the traces of the whitened wavelet coefficients is constant across scales as seen in \insertCiteCvS17pdSpecEst. For thresholding of 2D
wavelet coefficients, there is a discrepancy between the constant noise variance of the traces of the whitened wavelet coefficients at the first
abs(J1  J2)
scales and the remaining scales, as discussed in Chapter 5 of \insertCiteC18pdSpecEst, where J_1 = \log_2(n_1) and
J_2 = \log_2(n_2) with n_1 and n_2 the dyadic number of observations in each marginal direction of the 2D rectangular tensor grid.
The reason is that the variances of the traces of the whitened coefficients are not homogeneous between: (i) scales at which the 1D wavelet refinement
scheme is applied and (ii) scales at which the 2D wavelet refinement scheme is applied. To correct for this discrepancy, the variances of the coefficients
at the 2D wavelet scales are normalized by the noise variance determined from the finest wavelet scale. The variances of the coefficients at the 1D wavelet
scales are normalized using the analytic noise variance of the traces of the whitened coefficients for a grid of complex random Wishart matrices, which
corresponds to the asymptotic distributional behavior of the HPD periodogram matrices obtained with e.g., pdPgram2D
. Note that if the
timefrequency grid is square, i.e., n_1 = n_2, the variances of the traces of the whitened coefficients are again homogeneous across all wavelet scales.
WavTransf1D
, InvWavTransf1D
, WavTransf2D
, InvWavTransf2D
1 2 3 4 5 6 7 8 9 10 11 12  ## 1D treestructured trace thresholding
P < rExamples1D(2^8, example = "bumps")$P
Coeffs < WavTransf1D(P)
pdCART(Coeffs$D, Coeffs$D.white, order = 5)$w ## logical tree of nonzero coefficients
## Not run:
## 2D treestructured trace thresholding
P < rExamples2D(c(2^6, 2^6), 2, example = "tvar")$P
Coeffs < WavTransf2D(P)
pdCART(Coeffs$D, Coeffs$D.white, order = c(3, 3))$w
## End(Not run)

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