A not so successful attempt to perform single channel ICA using Wavelet Transform.

single_channel_ica.py

import itertools
from scipy import signal

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

from sklearn.decomposition import FastICA

# #############################################################################
# Generate sample data
np.random.seed(0)
n_samples = 2_000
time = np.linspace(0, 8, n_samples)

s1 = np.sin(2 * time)  # Signal 1 : sinusoidal signal
s2 = signal.sawtooth(2 * np.pi * time)   # Signal 2 : sinusoidal signal

# Mix data as sum not through mixing matrix - is it valid?
S = s1 + s2
coeffs = pywt.swt(S, 'db4', level=4)
decomposition = tuple(a for a, d in coeffs)

# X = np.c_[S, decomposition]
X = np.c_[tuple([decomposition[0], decomposition[1]])]
# Compute ICA
ica = FastICA(n_components=2, whiten=False)
S_ = ica.fit_transform(X)  # Reconstruct signals
A_ = ica.mixing_  # Get estimated mixing matrix

# We can `prove` that the ICA model applies by reverting the unmixing.
# assert np.allclose(X, np.dot(S_, A_.T) + ica.mean_)

# #############################################################################
# Plot results

plt.figure()

recovered_signals = pywt.iswt([S_], "db4")
models = [S[:, np.newaxis], np.c_[s1, s2], recovered_signals, recovered_signals.sum(axis=1)[:, np.newaxis]]
names = [
    'Observations (mixed signal)',
    'True Sources',
    'ICA recovered sources',
    'ICA recovered (mixed signal)'
]
colors = ['red', 'steelblue']

for ii, (model, name) in enumerate(zip(models, names), 1):
    plt.subplot(4, 1, ii)
    plt.title(name)
    for sig, color in zip(model.T, colors):
        plt.plot(sig, color=color)

plt.tight_layout()
plt.show()