mthreshC: Function to perform 'multiwavelet style' level-dependent soft...
In CNLTreg: Complex-Valued Wavelet Lifting for Signal Denoising

Description Usage Arguments Details Value Author(s) References See Also Examples

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

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

`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 `i`th 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.

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.

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.

Matt Nunes, Marina Knight

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.

cthresh, denoisepermC, denoisepermCh

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]