Allow user to set precision in CWT, increase default to 12

[OverLordGoldDragon]

Addresses this Issue. Detailed explanation here; a summary:

1. Low precision distorts CWT, and heavily at high scales.
2. With the L2-norm used, scales beyond 64 are increasingly distorted.
3. Zipper-like artifacts can be seen in computed heatmaps
4. precision=10 is low, and precision=12 does not add much computation cost, while considerably remedying 1-3.

At the least, the user should get to decide precision instead of carving it in stone. And a doc/comment should be made about these caveats.

---

Improvement comparison

Code

import numpy as np
import matplotlib.pyplot as plt

from pywt._extensions._pywt import DiscreteContinuousWavelet
from pywt._functions import integrate_wavelet
#%%##########################################################################
def l2(x):
    return np.sqrt(np.sum(np.abs(x ** 2)))

scales = np.power(2 ** (1 / 32), np.arange(1, 320 + 1))
wavelet = DiscreteContinuousWavelet('morl')
#%%##########################################################################
l2res = []
for precision in (10, 12, 14):
    l2res.append([])
    for scale in scales:
        int_psi, x = integrate_wavelet(wavelet, precision=precision)
        step = x[1] - x[0]
        linrange = max(x) - min(x)

        j = np.arange(scale * linrange + 1) / (scale * step)
        j = j.astype(int)  # floor
        if j[-1] >= int_psi.size:
            j = np.extract(j < int_psi.size, j)
        int_psi_scale = int_psi[j][::-1].real
        l2res[-1].append(l2(np.diff(int_psi_scale) * np.sqrt(scale)))
    
#%%##########################################################################
plt.plot(np.log2(scales), l2res[0])
plt.plot(np.log2(scales), l2res[1])
plt.plot(np.log2(scales), l2res[2])
plt.show()

[grlee77]

Thanks, I am fine with exposing this parameter.

I will take a look at the info you found soon and review it a little closer.

[OverLordGoldDragon]

@grlee77 Some interesting findings here (see bottom); to my surprise, pywt seems to outperform scipy on low scales. Also found a caveat2 (bottom) on integrated wavelet.

I'm curious on rationale behind integrated wavelet, which isn't documented on the site you found; perhaps it can be improved further.

[meiyasan]

hi @grlee77 is that PR still planned ?

[OverLordGoldDragon]

I've meant to write about this a while back, and I think I still intend to, but in short, I strongly discourage pywt's CWT, some info here.

[OverLordGoldDragon]

*write to the team

[rgommers]

Wow, that's a detailed analysis @OverLordGoldDragon, thanks for sharing. Does this PR help improve the behavior demonstrated in that StackExchange post?

[meiyasan]

Thanks @OverLordGoldDragon for the work.

May I ask also ask about your ssqueeze package. what would it cost to merge with pywavelet? I believe this address most of the drawback in your stack post.
It would definitely be great to have both into one single package and including also synchrosqueezing.