Wavelet reconstruction of time series

Question

I\'m trying to reconstruct the original time series from a Morlet\'s wavelet transform. I\'m working in R, package Rwave, function cwt. The result of this function is a mat

Answer 1

I have been working on this topic currently, using the same paper. I show you code using an example dataset, detailing how I implemented the procedure of wavelet decomposition and reconstruction.

# Lets first write a function for Wavelet decomposition as in formula (1):
mo<-function(t,trans=0,omega=6,j=0){
  dial<-2*2^(j*.125)
  sqrt((1/dial))*pi^(-1/4)*exp(1i*omega*((t-trans)/dial))*exp(-((t-trans)/dial)^2/2)
}

# An example time series data: 
y<-as.numeric(LakeHuron)

From my experience, for correct reconstruction you should do two things: first subject the mean to get a zero-mean dataset. I then increase the maximal scale. I mostly use 110 (although the formula in the Torrence and Compo suggests 71)

# subtract mean from data:
y.m<-mean(y)
y.madj<-y-y.m

# increase the scale:
J<-110
wt<-matrix(rep(NA,(length(y.madj))*(J+1)),ncol=(J+1))

# Wavelet decomposition:
for(j in 0:J){
for(k in 1:length(y.madj)){
  wt[k,j+1]<-mo(t=1:(length(y.madj)),j=j,trans=k)%*%y.madj
  }
}

#Extract the real part for the reconstruction:
wt.r<-Re(wt)

# Reconstruct as in formula (11):
dial<-2*2^(0:J*.125)
rec<-rep(NA,(length(y.madj)))
for(l in 1:(length(y.madj))){
  rec[l]<-0.2144548*sum(wt.r[l,]/sqrt(dial))
}
rec<-rec+y.m

plot(y,type="l")
lines(rec,col=2)

As you can see in the plot, it looks like a perfect reconstruction: