Wavelet Reconstruction of Time Series

Wavelet reconstruction of time series

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:

wavelet package Inverse DWT fails to reconstruct series?

Following the help file ?dwt, you can modify your script, such as:

library(wavelets)
set.seed(42)
x <- rnorm(100)
y <- idwt(dwt(x, n.levels=3, boundary="reflection", fast=FALSE))
plot(x, y)
abline(0,1)

Multilevel partial wavelet reconstruction with pyWavelets

I managed to write my own version of the wrcoef function which appears to work as expected:

import pywt
import numpy as np

def wrcoef(X, coef_type, coeffs, wavename, level):
    N = np.array(X).size
    a, ds = coeffs[0], list(reversed(coeffs[1:]))

    if coef_type =='a':
        return pywt.upcoef('a', a, wavename, level=level)[:N]
    elif coef_type == 'd':
        return pywt.upcoef('d', ds[level-1], wavename, level=level)[:N]
    else:
        raise ValueError("Invalid coefficient type: {}".format(coef_type))

level = 4
X = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
coeffs = pywt.wavedec(X, 'db1', level=level)
A4 = wrcoef(X, 'a', coeffs, 'db1', level)
D4 = wrcoef(X, 'd', coeffs, 'db1', level)
D3 = wrcoef(X, 'd', coeffs, 'db1', 3)
D2 = wrcoef(X, 'd', coeffs, 'db1', 2)
D1 = wrcoef(X, 'd', coeffs, 'db1', 1)
print A4 + D4 + D3 + D2 + D1

# Results:
[  9.99200722e-16   1.00000000e+00   2.00000000e+00   3.00000000e+00
   4.00000000e+00   5.00000000e+00   6.00000000e+00   7.00000000e+00
   8.00000000e+00   9.00000000e+00   1.00000000e+01   1.10000000e+01
   1.20000000e+01   1.30000000e+01   1.40000000e+01   1.50000000e+01
   1.60000000e+01   1.70000000e+01]

---

Related Topics