# Background on Independent Component Analysis (ICA)¶

Many M/EEG signals including biological artifacts reflect non-Gaussian processes. Therefore PCA-based artifact rejection will likely perform worse at separating the signal from noise sources. MNE-Python supports identifying artifacts and latent components using temporal ICA. MNE-Python implements the mne.preprocessing.ICA class that facilitates applying ICA to MEG and EEG data. Here we discuss some basics of ICA.

## Concepts¶

ICA finds directions in the feature space corresponding to projections with high non-Gaussianity.

• not necessarily orthogonal in the original feature space, but orthogonal in the whitened feature space.

• In contrast, PCA finds orthogonal directions in the raw feature space that correspond to directions accounting for maximum variance.

• or differently, if data only reflect Gaussian processes ICA and PCA are equivalent.

Example: Imagine 3 instruments playing simultaneously and 3 microphones recording mixed signals. ICA can be used to recover the sources ie. what is played by each instrument.

ICA employs a very simple model: $$X = AS$$ where $$X$$ is our observations, $$A$$ is the mixing matrix and $$S$$ is the vector of independent (latent) sources.

The challenge is to recover $$A$$ and $$S$$ from $$X$$.

### First generate simulated data¶

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

from sklearn.decomposition import FastICA, PCA

np.random.seed(0)  # set seed for reproducible results
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: sawtooth 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


### Now try to recover the sources¶

# compute ICA
ica = FastICA(n_components=3)
S_ = ica.fit_transform(X)  # Get the estimated sources
A_ = ica.mixing_  # Get estimated mixing matrix

# compute PCA
pca = PCA(n_components=3)
H = pca.fit_transform(X)  # estimate PCA sources

plt.figure(figsize=(9, 6))

models = [X, S, S_, H]
names = ['Observations (mixed signal)',
'True Sources',
'ICA estimated sources',
'PCA estimated sources']
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()


$$\rightarrow$$ PCA fails at recovering our “instruments” since the related signals reflect non-Gaussian processes.

Total running time of the script: ( 0 minutes 1.821 seconds)

Estimated memory usage: 8 MB

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