Note
Go to the end to download the full example code.
ICA on 2D point clouds¶
This example illustrates visually in the feature space a comparison by results using two different component analysis techniques.
Representing ICA in the feature space gives the view of ‘geometric ICA’: ICA is an algorithm that finds directions in the feature space corresponding to projections with high non-Gaussianity. These directions need not be orthogonal in the original feature space, but they are orthogonal in the whitened feature space, in which all directions correspond to the same variance.
PCA, on the other hand, finds orthogonal directions in the raw feature space that correspond to directions accounting for maximum variance.
Here we simulate independent sources using a highly non-Gaussian process, 2 student T with a low number of degrees of freedom (top left figure). We mix them to create observations (top right figure). In this raw observation space, directions identified by PCA are represented by orange vectors. We represent the signal in the PCA space, after whitening by the variance corresponding to the PCA vectors (lower left). Running ICA corresponds to finding a rotation in this space to identify the directions of largest non-Gaussianity (lower right).
Note
This example is adapted from the Scikit-Learn documentation.
Generate sample data¶
import numpy as np
from amica import AMICA
from sklearn.decomposition import FastICA
rng = np.random.RandomState(42)
S = rng.standard_t(1.5, size=(20000, 2))
S[:, 0] *= 2.0
# Mix data
A = np.array([[1, 1], [0, 2]]) # Mixing matrix
X = np.dot(S, A.T) # Generate observations
transformer = AMICA(random_state=42)
S_amica_ = transformer.fit(X).transform(X)
fastica = FastICA(random_state=rng, whiten="arbitrary-variance")
S_fastica_ = fastica.fit(X).transform(X) # Estimate the sources
INFO | getting the mean ... - amica.linalg:pre_whiten
INFO | Getting the covariance matrix ... - amica.linalg:pre_whiten
INFO | doing eigenvalue decomposition for 2 features ... - amica.linalg:pre_whiten
INFO | minimum eigenvalues: [208.1693481] - amica.linalg:pre_whiten
INFO | maximum eigenvalues: [813.77043757 208.1693481 ] - amica.linalg:pre_whiten
INFO | num eigvals kept: 2 - amica.linalg:pre_whiten
INFO | Sphering the data... - amica.linalg:pre_whiten
INFO | numeigs = 2, nw = 2 - amica.linalg:pre_whiten
INFO | 1: block size = 20000 - amica.core:solve
INFO | Solving. (please be patient, this may take a while)... - amica.core:solve
INFO | Iteration 1, lrate = 0.05000, LL = -4.3389111, nd = 0.4896220, D = 0.00000 0.00000 took 0.01 seconds - amica.core:optimize
INFO | Iteration 2, lrate = 0.05000, LL = -3.4406895, nd = 0.4642117, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 3, lrate = 0.05000, LL = -3.0550419, nd = 0.3070402, D = 0.00013 0.00013 took 0.01 seconds - amica.core:optimize
INFO | Iteration 4, lrate = 0.05000, LL = -2.9823773, nd = 0.2908429, D = 0.00063 0.00063 took 0.19 seconds - amica.core:optimize
INFO | Iteration 5, lrate = 0.05000, LL = -2.9642010, nd = 0.2945571, D = 0.00158 0.00158 took 0.01 seconds - amica.core:optimize
INFO | Iteration 6, lrate = 0.05000, LL = -2.9522195, nd = 0.2960507, D = 0.00286 0.00286 took 0.09 seconds - amica.core:optimize
INFO | Iteration 7, lrate = 0.05000, LL = -2.9411480, nd = 0.2948532, D = 0.00434 0.00434 took 0.01 seconds - amica.core:optimize
INFO | Iteration 8, lrate = 0.05000, LL = -2.9304846, nd = 0.2891688, D = 0.00593 0.00593 took 0.10 seconds - amica.core:optimize
INFO | Iteration 9, lrate = 0.05000, LL = -2.9203678, nd = 0.2812223, D = 0.00750 0.00750 took 0.09 seconds - amica.core:optimize
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INFO | Likelihood decreasing! - amica.core:optimize
INFO | Iteration 31, lrate = 0.05000, LL = -2.8567384, nd = 0.1306010, D = 0.00000 0.00000 took 0.10 seconds - amica.core:optimize
INFO | Iteration 32, lrate = 0.05000, LL = -2.8568323, nd = 0.1304800, D = 0.00007 0.00007 took 0.10 seconds - amica.core:optimize
INFO | Likelihood decreasing! - amica.core:optimize
INFO | Iteration 33, lrate = 0.05000, LL = -2.8565917, nd = 0.1307377, D = 0.00000 0.00000 took 0.01 seconds - amica.core:optimize
INFO | Iteration 34, lrate = 0.05000, LL = -2.8566985, nd = 0.1306517, D = 0.00007 0.00007 took 0.09 seconds - amica.core:optimize
INFO | Likelihood decreasing! - amica.core:optimize
INFO | Iteration 35, lrate = 0.02500, LL = -2.8560296, nd = 0.0399911, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
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INFO | Starting Newton ... setting numdecs to 0 - amica.core:optimize
INFO | Iteration 51, lrate = 0.05000, LL = -2.8552796, nd = 0.0010573, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
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INFO | Iteration 292, lrate = 1.00000, LL = -2.8536467, nd = 0.0000970, D = 0.00001 0.00001 took 0.10 seconds - amica.core:optimize
INFO | Iteration 293, lrate = 1.00000, LL = -2.8536461, nd = 0.0000958, D = 0.00001 0.00001 took 0.10 seconds - amica.core:optimize
INFO | Iteration 294, lrate = 1.00000, LL = -2.8536455, nd = 0.0000934, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 295, lrate = 1.00000, LL = -2.8536449, nd = 0.0000922, D = 0.00001 0.00001 took 0.11 seconds - amica.core:optimize
INFO | Iteration 296, lrate = 1.00000, LL = -2.8536443, nd = 0.0000909, D = 0.00001 0.00001 took 0.19 seconds - amica.core:optimize
INFO | Iteration 297, lrate = 1.00000, LL = -2.8536437, nd = 0.0000896, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 298, lrate = 1.00000, LL = -2.8536431, nd = 0.0000883, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 299, lrate = 1.00000, LL = -2.8536425, nd = 0.0000874, D = 0.00001 0.00001 took 0.19 seconds - amica.core:optimize
INFO | Iteration 300, lrate = 1.00000, LL = -2.8536420, nd = 0.0000863, D = 0.00001 0.00001 took 0.02 seconds - amica.core:optimize
INFO | Iteration 301, lrate = 1.00000, LL = -2.8536414, nd = 0.0001328, D = 0.00001 0.00001 took 0.29 seconds - amica.core:optimize
INFO | Iteration 302, lrate = 1.00000, LL = -2.8536409, nd = 0.0001484, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 303, lrate = 1.00000, LL = -2.8536403, nd = 0.0001549, D = 0.00001 0.00001 took 0.30 seconds - amica.core:optimize
INFO | Iteration 304, lrate = 1.00000, LL = -2.8536398, nd = 0.0001583, D = 0.00001 0.00001 took 0.19 seconds - amica.core:optimize
INFO | Iteration 305, lrate = 1.00000, LL = -2.8536393, nd = 0.0001601, D = 0.00001 0.00001 took 0.10 seconds - amica.core:optimize
INFO | Iteration 306, lrate = 1.00000, LL = -2.8536387, nd = 0.0001614, D = 0.00001 0.00001 took 0.10 seconds - amica.core:optimize
INFO | Iteration 307, lrate = 1.00000, LL = -2.8536382, nd = 0.0000730, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 308, lrate = 1.00000, LL = -2.8536376, nd = 0.0000908, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 309, lrate = 1.00000, LL = -2.8536371, nd = 0.0000839, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 310, lrate = 1.00000, LL = -2.8536365, nd = 0.0000839, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 311, lrate = 1.00000, LL = -2.8536360, nd = 0.0000825, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 312, lrate = 1.00000, LL = -2.8536357, nd = 0.0000617, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 313, lrate = 1.00000, LL = -2.8536355, nd = 0.0000657, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 314, lrate = 1.00000, LL = -2.8536354, nd = 0.0000655, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 315, lrate = 1.00000, LL = -2.8536352, nd = 0.0000640, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 316, lrate = 1.00000, LL = -2.8536351, nd = 0.0000617, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 317, lrate = 1.00000, LL = -2.8536350, nd = 0.0000591, D = 0.00001 0.00001 took 0.10 seconds - amica.core:optimize
INFO | Iteration 318, lrate = 1.00000, LL = -2.8536348, nd = 0.0000563, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 319, lrate = 1.00000, LL = -2.8536347, nd = 0.0000536, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 320, lrate = 1.00000, LL = -2.8536346, nd = 0.0000508, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 321, lrate = 1.00000, LL = -2.8536345, nd = 0.0000483, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 322, lrate = 1.00000, LL = -2.8536344, nd = 0.0000456, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 323, lrate = 1.00000, LL = -2.8536343, nd = 0.0000438, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 324, lrate = 1.00000, LL = -2.8536343, nd = 0.0000415, D = 0.00001 0.00001 took 0.01 seconds - amica.core:optimize
INFO | Iteration 325, lrate = 1.00000, LL = -2.8536342, nd = 0.0000394, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Iteration 326, lrate = 1.00000, LL = -2.8536341, nd = 0.0000375, D = 0.00001 0.00001 took 0.09 seconds - amica.core:optimize
INFO | Exiting because likelihood increasing by less than 1e-07 for more than 5 iterations ... - amica.core:optimize
INFO | Finished in 24.82 seconds - amica.core:optimize
Plot results¶
import matplotlib.pyplot as plt
def plot_samples(S, axis_list=None):
plt.scatter(
S[:, 0], S[:, 1], s=2, marker="o", zorder=10, color="steelblue", alpha=0.5
)
if axis_list is not None:
for axis, color, label in axis_list:
x_axis, y_axis = axis / axis.std()
plt.quiver(
(0, 0),
(0, 0),
x_axis,
y_axis,
zorder=11,
width=0.01,
scale=6,
color=color,
label=label,
)
plt.hlines(0, -5, 5, color="black", linewidth=0.5)
plt.vlines(0, -3, 3, color="black", linewidth=0.5)
plt.xlim(-5, 5)
plt.ylim(-3, 3)
plt.gca().set_aspect("equal")
plt.xlabel("x")
plt.ylabel("y")
plt.figure()
plt.subplot(2, 2, 1)
plot_samples(S / S.std())
plt.title("True Independent Sources")
axis_list = [(transformer.mixing_, "orange", "AMICA"), (fastica.mixing_, "red", "FastICA")]
plt.subplot(2, 2, 2)
plot_samples(X / np.std(X), axis_list=axis_list)
legend = plt.legend(loc="upper left")
legend.set_zorder(100)
plt.title("Observations")
plt.subplot(2, 2, 3)
plot_samples(S_amica_ / np.std(S_amica_))
plt.title("AMICA recovered signals")
plt.subplot(2, 2, 4)
plot_samples(S_fastica_ / np.std(S_fastica_))
plt.title("FastICA recovered signals")
plt.subplots_adjust(0.09, 0.04, 0.94, 0.94, 0.26, 0.36)
plt.tight_layout()
plt.show()
Total running time of the script: (0 minutes 25.491 seconds)