Blind source separation using FastICA¶
An example of estimating sources from noisy data.
Independent component analysis (ICA) is used to estimate sources given noisy measurements.
Imagine 3 instruments playing simultaneously and 3 microphones
recording the mixed signals. ICA is used to recover the sources
ie. what is played by each instrument. Importantly, PCA fails
at recovering our
instruments since the related signals reflect
Generate sample data¶
import numpy as np from scipy import signal np.random.seed(0) n_samples = 2000 time = np.linspace(0, 8, n_samples) s1 = np.sin(2 * time) # Signal 1 : sinusoidal signal s2 = np.sign(np.sin(3 * time)) # Signal 2 : square signal s3 = signal.sawtooth(2 * np.pi * time) # Signal 3: saw tooth signal S = np.c_[s1, s2, s3] S += 0.2 * np.random.normal(size=S.shape) # Add noise S /= S.std(axis=0) # Standardize data # Mix data A = np.array([[1, 1, 1], [0.5, 2, 1.0], [1.5, 1.0, 2.0]]) # Mixing matrix X = np.dot(S, A.T) # Generate observations
Fit ICA and PCA models¶
from sklearn.decomposition import PCA, FastICA # Compute ICA ica = FastICA(n_components=3, whiten="arbitrary-variance") 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_) # For comparison, compute PCA pca = PCA(n_components=3) H = pca.fit_transform(X) # Reconstruct signals based on orthogonal components
import matplotlib.pyplot as plt plt.figure() models = [X, S, S_, H] names = [ "Observations (mixed signal)", "True Sources", "ICA recovered signals", "PCA recovered signals", ] colors = ["red", "steelblue", "orange"] 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()
Total running time of the script: (0 minutes 0.404 seconds)