Note
Go to the end to download the full example code.
Blind Source Separation with AMICA & FastICA¶
An example of estimating sources from noisy mixtures.
ICA separates independent sources given only mixed microphone recordings. Imagine three instruments playing simultaneously and three microphones recording the mixtures. ICA recovers the instrument tracks because the sources are non-Gaussian. PCA, by contrast, fails in this setting.
Note
This example is adapted from the Scikit-Learn documentation.
Generate sample data¶
import numpy as np
from scipy import signal
rng = np.random.default_rng(0)
n_samples = 2000
time = np.linspace(0, 8, n_samples)
s1 = np.sin(2 * time) # Sinusoidal
s2 = np.sign(np.sin(3 * time)) # Square wave
s3 = signal.sawtooth(2 * np.pi * time) # Sawtooth
S = np.c_[s1, s2, s3]
S += 0.2 * rng.standard_normal(S.shape) # Add noise
S /= S.std(axis=0) # Standardize
A = np.array([[1, 1, 1],
[0.5, 2, 1.0],
[1.5, 1.0, 2.0]]) # Mixing matrix
X = S @ A.T # Observed mixtures
Run AMICA and FastICA¶
from amica import AMICA
from sklearn.decomposition import FastICA
models = {}
labels = {}
# AMICA
# We instantiate amica.AMICA and call fit..
ica = AMICA(n_components=3, whiten="zca", random_state=0)
models["AMICA"] = ica.fit_transform(X)
labels["AMICA"] = "AMICA recovered signals"
INFO | getting the mean ... - amica.linalg:pre_whiten
INFO | Getting the covariance matrix ... - amica.linalg:pre_whiten
INFO | doing eigenvalue decomposition for 3 features ... - amica.linalg:pre_whiten
INFO | minimum eigenvalues: [0.05201252] - amica.linalg:pre_whiten
INFO | maximum eigenvalues: [14.0572034 1.35729453 0.05201252] - amica.linalg:pre_whiten
INFO | num eigvals kept: 3 - amica.linalg:pre_whiten
INFO | Sphering the data... - amica.linalg:pre_whiten
INFO | numeigs = 3, nw = 3 - amica.linalg:pre_whiten
INFO | 1: block size = 2000 - amica.core:solve
INFO | Solving. (please be patient, this may take a while)... - amica.core:solve
INFO | Iteration 1, lrate = 0.05000, LL = -1.4208643, nd = 0.1965180, D = 0.00003 0.00003 took 0.00 seconds - amica.core:optimize
INFO | Iteration 2, lrate = 0.05000, LL = -1.3636536, nd = 0.1723261, D = 0.00006 0.00006 took 0.00 seconds - amica.core:optimize
INFO | Iteration 3, lrate = 0.05000, LL = -1.3238623, nd = 0.2980232, D = 0.00034 0.00034 took 0.00 seconds - amica.core:optimize
INFO | Iteration 4, lrate = 0.05000, LL = -1.2865584, nd = 0.4412654, D = 0.00188 0.00188 took 0.01 seconds - amica.core:optimize
INFO | Iteration 5, lrate = 0.05000, LL = -1.2385767, nd = 0.5302712, D = 0.00651 0.00651 took 0.01 seconds - amica.core:optimize
INFO | Iteration 6, lrate = 0.05000, LL = -1.1797772, nd = 0.5185603, D = 0.01541 0.01541 took 0.09 seconds - amica.core:optimize
INFO | Iteration 7, lrate = 0.05000, LL = -1.1225606, nd = 0.4013239, D = 0.02655 0.02655 took 0.01 seconds - amica.core:optimize
INFO | Iteration 8, lrate = 0.05000, LL = -1.0865829, nd = 0.2305540, D = 0.03484 0.03484 took 0.19 seconds - amica.core:optimize
INFO | Iteration 9, lrate = 0.05000, LL = -1.0738310, nd = 0.1660310, D = 0.03542 0.03542 took 0.39 seconds - amica.core:optimize
INFO | Iteration 10, lrate = 0.05000, LL = -1.0678697, nd = 0.1539337, D = 0.03093 0.03093 took 0.02 seconds - amica.core:optimize
INFO | Iteration 11, lrate = 0.05000, LL = -1.0633096, nd = 0.1482986, D = 0.02599 0.02599 took 0.29 seconds - amica.core:optimize
INFO | Iteration 12, lrate = 0.05000, LL = -1.0596034, nd = 0.1333072, D = 0.02200 0.02200 took 0.22 seconds - amica.core:optimize
INFO | Iteration 13, lrate = 0.05000, LL = -1.0565292, nd = 0.1328886, D = 0.01825 0.01825 took 0.18 seconds - amica.core:optimize
INFO | Iteration 14, lrate = 0.05000, LL = -1.0539693, nd = 0.1170593, D = 0.01554 0.01554 took 0.21 seconds - amica.core:optimize
INFO | Iteration 15, lrate = 0.05000, LL = -1.0519178, nd = 0.1299362, D = 0.01269 0.01269 took 0.19 seconds - amica.core:optimize
INFO | Iteration 16, lrate = 0.05000, LL = -1.0503348, nd = 0.1265691, D = 0.01106 0.01106 took 0.20 seconds - amica.core:optimize
INFO | Iteration 17, lrate = 0.05000, LL = -1.0493665, nd = 0.1684180, D = 0.00879 0.00879 took 0.10 seconds - amica.core:optimize
INFO | Iteration 18, lrate = 0.05000, LL = -1.0491577, nd = 0.2095800, D = 0.00809 0.00809 took 0.10 seconds - amica.core:optimize
INFO | Iteration 19, lrate = 0.05000, LL = -1.0505280, nd = 0.2933917, D = 0.00606 0.00606 took 0.20 seconds - amica.core:optimize
INFO | Likelihood decreasing! - amica.core:optimize
INFO | Iteration 20, lrate = 0.05000, LL = -1.0537696, nd = 0.4207022, D = 0.00620 0.00620 took 0.20 seconds - amica.core:optimize
INFO | Likelihood decreasing! - amica.core:optimize
INFO | Iteration 21, lrate = 0.05000, LL = -1.0631841, nd = 0.5297685, D = 0.00406 0.00406 took 0.09 seconds - amica.core:optimize
INFO | Likelihood decreasing! - amica.core:optimize
INFO | Iteration 22, lrate = 0.02500, LL = -1.0491921, nd = 0.2979483, D = 0.00452 0.00452 took 0.10 seconds - amica.core:optimize
INFO | Iteration 23, lrate = 0.02500, LL = -1.0464746, nd = 0.2140679, D = 0.00406 0.00406 took 0.20 seconds - amica.core:optimize
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INFO | Iteration 50, lrate = 0.02500, LL = -1.0393096, nd = 0.0095508, D = 0.00170 0.00170 took 0.10 seconds - amica.core:optimize
INFO | Starting Newton ... setting numdecs to 0 - amica.core:optimize
INFO | Iteration 51, lrate = 0.05000, LL = -1.0392460, nd = 0.0094513, D = 0.00170 0.00170 took 0.10 seconds - amica.core:optimize
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INFO | Iteration 178, lrate = 1.00000, LL = -1.0368079, nd = 0.0000067, D = 0.00219 0.00219 took 0.09 seconds - amica.core:optimize
INFO | Iteration 179, lrate = 1.00000, LL = -1.0368078, nd = 0.0000067, D = 0.00219 0.00219 took 0.01 seconds - amica.core:optimize
INFO | Exiting because likelihood increasing by less than 1e-07 for more than 5 iterations ... - amica.core:optimize
INFO | Finished in 12.32 seconds - amica.core:optimize
We can prove that the ICA model applies by reverting the unmixing.
X_rec = models["AMICA"] @ ica.mixing_.T + ica.mean_
np.testing.assert_allclose(X, X_rec)
# FastICA
fastica = FastICA(n_components=3, whiten="arbitrary-variance", random_state=0)
models["FastICA"] = fastica.fit_transform(X)
labels["FastICA"] = "FastICA recovered signals"
Plot results¶
import matplotlib.pyplot as plt
# Merge dictionaries into one mapping title -> data
to_plot = {
"Observed mixtures": X,
"True sources": S,
}
to_plot.update({ labels[k]: v for k, v in models.items() })
colors = ["red", "steelblue", "orange"]
fig, axes = plt.subplots(len(to_plot), 1, figsize=(8, 6), sharex=True)
for ax, (title, model) in zip(axes, to_plot.items()):
ax.set_title(title)
for sig, color in zip(model.T, colors):
ax.plot(sig, color=color, lw=1)
plt.tight_layout()
plt.show()
Total running time of the script: (0 minutes 12.854 seconds)