101 lines
3.1 KiB
Python
101 lines
3.1 KiB
Python
"""
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==========================
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FastICA on 2D point clouds
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==========================
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Illustrate visually the results of :ref:`ICA` vs :ref:`PCA` in the
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feature space.
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Representing ICA in the feature space gives the view of 'geometric ICA':
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ICA is an algorithm that finds directions in the feature space
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corresponding to projections with high non-Gaussianity. These directions
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need not be orthogonal in the original feature space, but they are
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orthogonal in the whitened feature space, in which all directions
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correspond to the same variance.
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PCA, on the other hand, finds orthogonal directions in the raw feature
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space that correspond to directions accoutning for maximum variance.
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Here we simulate independent sources using a highly non-Gaussian
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process, 2 student T with a low number of degrees of freedom (top left
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figure). We mix them to create observations (top right figure).
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In this raw observation space, directions identified by PCA are
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represented by green vectors. We represent the signal in the PCA space,
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after whitening by the variance corresponding to the PCA vectors (lower
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left). Running ICA corresponds to finding a rotation in this space to
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identify the directions of largest non-Gaussianity (lower right).
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"""
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print __doc__
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# Authors: Alexandre Gramfort, Gael Varoquaux
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# License: BSD
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import numpy as np
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import pylab as pl
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from scikits.learn.pca import PCA
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from scikits.learn.fastica import FastICA
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###############################################################################
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# Generate sample data
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S = np.random.standard_t(1.5, size=(2, 10000))
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S[0] *= 2.
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# Mix data
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A = [[1, 1], [0, 2]] # Mixing matrix
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X = np.dot(A, S) # Generate observations
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pca = PCA()
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S_pca_ = pca.fit(X.T).transform(X.T).T
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ica = FastICA()
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S_ica_ = ica.fit(X).transform(X) # Estimate the sources
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S_ica_ /= S_ica_.std(axis=1)[:,np.newaxis]
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###############################################################################
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# Plot results
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def plot_samples(S, axis_list=None):
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pl.scatter(S[0], S[1], s=2, marker='o', linewidths=0, zorder=10)
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if axis_list is not None:
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colors = [(0, 0.6, 0), (0.6, 0, 0)]
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for color, axis in zip(colors, axis_list):
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axis /= axis.std()
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x_axis, y_axis = axis
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# Trick to get legend to work
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pl.plot(0.1*x_axis, 0.1*y_axis, linewidth=2, color=color)
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# pl.quiver(x_axis, y_axis, x_axis, y_axis, zorder=11, width=0.01,
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pl.quiver(0, 0, x_axis, y_axis, zorder=11, width=0.01,
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scale=6, color=color)
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pl.hlines(0, -3, 3)
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pl.vlines(0, -3, 3)
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pl.xlim(-3, 3)
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pl.ylim(-3, 3)
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pl.xlabel('$x$')
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pl.ylabel('$y$')
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pl.subplot(2, 2, 1)
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plot_samples(S / S.std())
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pl.title('True Independant Sources')
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axis_list = [pca.components_, ica.get_mixing_matrix()]
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pl.subplot(2, 2, 2)
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plot_samples(X / np.std(X), axis_list=axis_list)
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pl.legend(['PCA', 'ICA'], loc='upper left')
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pl.title('Observations')
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pl.subplot(2, 2, 3)
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plot_samples(S_pca_ / np.std(S_pca_, axis=-1)[:, np.newaxis])
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pl.title('PCA scores')
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pl.subplot(2, 2, 4)
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plot_samples(S_ica_ / np.std(S_ica_))
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pl.title('ICA estimated sources')
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pl.subplots_adjust(0.09, 0.04, 0.94, 0.94, 0.26, 0.26)
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pl.show()
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