| pdCART {pdSpecEst} | R Documentation |
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
(Chau and von
Sachs 2019) or (Chau 2018). 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.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. |
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 (Donoho 1997), 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.
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 |
D_w |
the list of thresholded wavelet coefficients, with each list component corresponding to an individual wavelet scale. |
Note
For thresholding of 1D wavelet coefficients, the noise
variance of the traces of the whitened wavelet coefficients is constant across scales as seen in (Chau and von
Sachs 2019). 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 (Chau 2018), 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.
References
Chau J (2018).
Advances in Spectral Analysis for Multivariate, Nonstationary and Replicated Time Series.
phdthesis, Universite catholique de Louvain.
Chau J, von
Sachs R (2019).
“Intrinsic wavelet regression for curves of Hermitian positive definite matrices.”
Journal of the American Statistical Association.
doi: 10.1080/01621459.2019.1700129.
Donoho D (1997).
“CART and best-ortho-basis: a connection.”
The Annals of Statistics, 25(5), 1870–1911.
See Also
WavTransf1D, InvWavTransf1D, WavTransf2D, InvWavTransf2D
Examples
## 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)