mthreshC {CNLTreg}

Function to perform 'multiwavelet style' level-dependent soft thresholding for complex-valued wavelet coefficients

Description

This function uses chi^2 statistics similar to Barber and Nason (2004) to threshold wavelet coefficients based on their magnitude

Usage

mthreshC(coeffv, Sigma, rl, po, ali, verbose = FALSE)

Arguments

coeffv

A matrix of complex-valued wavelet coefficients (columns are real and imaginary parts of the coefficients respectively).

Sigma

An array of dimension 2 x 2 x n describing the covariance between real and imaginary parts of the wavelet coefficients. In particular, Sigma[,,i] represents the covariance between real and imaginary parts of the ith lifted wavelet coefficient (see rl argument).

rl

The removelist (trajectory of lifted points) corresponding to a forward lifting transform.

po

A vector of indices describing the unlifted scaling coefficients in a forward lifting transform.

ali

A list of indices of observations, each entry corresponding to an 'artificial level' (finest to coarsest), see artlev for more details.

verbose

Indicates whether helpful messages should be printed to the console during the procedure.

Details

The procedure in Downie and Silverman (1998) or Barber and Nason (2004) makes use of the magnitude of wavelet coefficients to threshold them. In particular, the covariance between the components of the wavelet coefficients (contained in Sigma is taken into account to compute a thresholding statistic, the distribution of which is chi-squared_2 distributed, see cthresh for more details. These statistics are then compared with level-dependent universal thresholds computed by counting the numnber of coefficients in specific artificial levels.

Value

A list with the following components:

chi

the vector of chi-squared statistics used in the thresholding procedure.

coeffvt

the matrix of thresholded coefficients, columns representing the real and imaginary components respectively.

Author(s)

Matt Nunes, Marina Knight

References

Hamilton, J., Knight, M. I., Nunes, M. A. and Fryzlewicz (2018) Complex-valued wavelet lifting and applications. Technometrics, 60 (1), 48-60,DOI 10.1080/00401706.2017.1281846. Barber, S. and Nason, G. P. (2004) Real nonparametric regression using complex wavelets. J. Roy. Stat. Soc. B 66 (4), 927–939. Downie, T. R. and Silverman, B. W. (1998) The discrete multiple wavelet tranform and thresholding methods. IEEE Trans. Sig. Proc. 46 2558–2561.

See Also

cthresh, denoisepermC, denoisepermCh

Examples

library(adlift)

set.seed(100)

# construct an (irregular) sampling structure:

x<-sort(runif(200))

g<-make.signal2("bumps",x=x)

# generate IID noise with a particular sd
noise<-rnorm(200,0,sd=0.5)

f<-g+noise

# perform forward complex lifting transform

out<-fwtnppermC(x,f,LocalPred=LinearPred,neigh=1)

# have a look at some of the coefficients

out$coeffv[1:10]

# extract lifting matrix and induced lifting variances
W <- out$W

Gpre<-tcrossprod(W,Conj(W))

indsd<-sqrt(diag(Gpre))

# now estimate noise sd using the first artificial level:

al<-artlev(out$lengthsremove,out$removelist)

fine<-(out$coeffv/indsd)[al[[1]]]

varest<-mad(Re(fine))^2

# now compute coefficient covariance structure, see
# Hamilton et al. (2018), Appendix B

C = varest * tcrossprod(W)
    G = varest * Gpre
    P = Conj(G) - t(Conj(C)) 
    Sigma <- array(0, dim = c(2, 2, length(out$coeffv)))
    Sigma[1, 1, ] <- diag(Re(G + C)/2)
    Sigma[2, 2, ] <- diag(Re(G - C)/2)
    Sigma[1, 2, ] <- -diag(Im(G - C)/2)
    Sigma[2, 1, ] <- diag(Im(G + C)/2)

# now threshold complex coefficients according to this structure:

coeff.thresh<-mthreshC(out$coeffv,Sigma,out$removelist,out$pointsin,al)

# have a look at some of these coefficients

coeff.thresh$coeffv[1:10]