pdSpecEst (version 1.2.4)

pdCART: Tree-structured trace thresholding of wavelet coefficients

Description

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 CvS17pdSpecEst or C18pdSpecEst. This function is primarily written for internal use in other functions and is typically not used as a stand-alone function.

Usage

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

Arguments

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.

Value

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.

Details

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 D97pdSpecEst, via a fast tree-pruning algorithm. By default, the penalty parameter in the optimization procedure is set equal to alpha times the universal threshold \(\sigma_w\sqrt(2\log(n))\), where \(\sigma_w^2\) 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.

References

See Also

WavTransf1D, InvWavTransf1D, WavTransf2D, InvWavTransf2D

Examples

# NOT RUN {
## 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
# }
# NOT RUN {
# }