pdCART: Tree-structured trace thresholding of wavelet coefficients
In JorisChau/pdSpecEst: An Analysis Toolbox for Hermitian Positive Definite Matrices

Description Usage Arguments Details Value Note References See Also Examples

pdCART performs hard tree-structured thresholding of the Hermitian matrix-valued 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 stand-alone function.

1	pdCART(D, D.white, order, alpha = 1, tree = TRUE, ...)

`D`	a list of wavelet coefficients as obtained from the `$D` component of `WavTransf1D` or `WavTransf2D` .
`D.white`	a list of whitened wavelet coefficients as obtained from the `$D.white` component of `WavTransf1D` or `WavTransf2D`.
`order`	the order(s) of the intrinsic 1D or 2D AI refinement scheme as in `WavTransf1D` and `WavTransf2D`.
`alpha`	tuning parameter specifying the penalty/sparsity parameter as `alpha` times the universal threshold.
`tree`	logical value, if `tree = TRUE` performs tree-structured thresholding, otherwise performs non-tree-structured hard thresholding of the coefficients.
`...`	additional arguments for internal use.

Depending on the structure of the input list of arrays D the function performs 1D or 2D tree-structured 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 tree-pruning 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:

`w`	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 `j = 0`.
`D_w`	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 time-frequency 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.

\insertAllCited

WavTransf1D, InvWavTransf1D, WavTransf2D, InvWavTransf2D

## 1D tree-structured trace thresholding
P <- rExamples1D(2^8, example = "bumps")$P
Coeffs <- WavTransf1D(P)
pdCART(Coeffs$D, Coeffs$D.white, order = 5)$w ## logical tree of non-zero coefficients

## Not run: 
## 2D tree-structured 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)