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"""
.. _ica:
==================================================
Background on Independent Component Analysis (ICA)
==================================================
.. contents:: Contents
:local:
:depth: 2
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 :class:`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: :math:`X = AS` where :math:`X` is our observations, :math:`A` is the mixing matrix and :math:`S` is the vector of independent (latent) sources.
The challenge is to recover :math:`A` and :math:`S` from :math:`X`.
First generate simulated data
-----------------------------
""" # noqa: E501
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()
###############################################################################
# :math:`\rightarrow` PCA fails at recovering our "instruments" since the
# related signals reflect non-Gaussian processes.
|
