Note

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# 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
non-Gaussian processes.

## 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 FastICA, PCA
# Compute ICA
ica = FastICA(n_components=3)
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
```

```
/home/runner/work/scikit-learn/scikit-learn/sklearn/decomposition/_fastica.py:494: FutureWarning: Starting in v1.3, whiten='unit-variance' will be used by default.
warnings.warn(
```

## Plot results¶

```
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 0.215 seconds)