7
$\begingroup$

For a given time series which is n timestamps in length, we can take Discrete Wavelet Transform (using 'Haar' wavelets), then we get (for an example, in Python) -

>>> import pywt
>>> ts = [2, 56, 3, 22, 3, 4, 56, 7, 8, 9, 44, 23, 1, 4, 6, 2]
>>> (ca, cd) = pywt.dwt(ts,'haar')
>>> ca
array([ 41.01219331,  17.67766953,   4.94974747,  44.54772721,
        12.02081528,  47.37615434,   3.53553391,   5.65685425])
>>> cd
array([-38.18376618, -13.43502884,  -0.70710678,  34.64823228,
        -0.70710678,  14.8492424 ,  -2.12132034,   2.82842712])

where ca and cd are approximation and detailed coefficients. Now if I use all of them I can construct my original time series back using inverse DWT. But instead I want to use a fewer coefficients (like in Fourier Transform if we use only first few coefficients, we can approximately reconstruct the original time series). If I just use ca or just use cd I don't get the desired results. If I use only we coefficients from each of them (like first 4), I get only half of the time series.

How should I select the coefficients (from ca and cd) so that I can approximately create the original signal from them (i.e. most of its energy)?

$\endgroup$
9
$\begingroup$

I think it is kind'a similar to soft and hard thresholding using in wavelet de-noising. Have you come across this topic? pywt has already an in-built function for this purpose. Please take a closer look at this code and try to play with it:

import pywt
import matplotlib.pyplot as plt
import numpy as np

ts = [2, 56, 3, 22, 3, 4, 56, 7, 8, 9, 44, 23, 1, 4, 6, 2]

(ca, cd) = pywt.dwt(ts,'haar')

cat = pywt.thresholding.soft(ca, np.std(ca)/2)
cdt = pywt.thresholding.soft(cd, np.std(cd)/2)

ts_rec = pywt.idwt(cat, cdt, 'haar')

plt.close('all')

plt.subplot(211)
# Original coefficients
plt.plot(ca, '--*b')
plt.plot(cd, '--*r')
# Thresholded coefficients
plt.plot(cat, '--*c')
plt.plot(cdt, '--*m')
plt.legend(['ca','cd','ca_thresh', 'cd_thresh'], loc=0)
plt.grid('on')

plt.subplot(212)
plt.plot(ts)
plt.hold('on')
plt.plot(ts_rec, 'r')
plt.legend(['original signal', 'reconstructed signal'])
plt.grid('on')
plt.show()

This will produce following - hope this will guide you.

enter image description here

$\endgroup$
  • $\begingroup$ @theharshest: Glad to hear that. Good luck! $\endgroup$ – jojek Apr 24 '14 at 16:00

protected by jojek Apr 24 '15 at 6:08

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.