Davrot at 17:16, 17 October 2025

2025-10-17T17:16:54Z

← Older revision		Revision as of 17:16, 17 October 2025
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	~~== The goal ==~~
	Questions to [mailto:davrot@uni-bremen.de David Rotermund]		Questions to [mailto:davrot@uni-bremen.de David Rotermund]

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	The implementation is based on [https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#r44c805292efc-1 1].</blockquote><syntaxhighlight lang="python">fit(X, y=None)</syntaxhighlight><blockquote>Fit the model to X.		The implementation is based on [https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#r44c805292efc-1 1].</blockquote><syntaxhighlight lang="python">fit(X, y=None)</syntaxhighlight><blockquote>Fit the model to X.

	'''X''' : array-like of shape (n_samples, n_features)		'''X''': array-like of shape (n_samples, n_features)

	Training data, where n_samples is the number of samples and n_features is the number of features.</blockquote><syntaxhighlight lang="python">transform(X, copy=True)</syntaxhighlight><blockquote>Recover the sources from X (apply the unmixing matrix).		Training data, where n_samples is the number of samples and n_features is the number of features.</blockquote><syntaxhighlight lang="python">transform(X, copy=True)</syntaxhighlight><blockquote>Recover the sources from X (apply the unmixing matrix).

	'''X''' : array-like of shape (n_samples, n_features)		'''X''': array-like of shape (n_samples, n_features)

	Data to transform, where n_samples is the number of samples and n_features is the number of features.</blockquote><syntaxhighlight lang="python">import numpy as np		Data to transform, where n_samples is the number of samples and n_features is the number of features.</blockquote><syntaxhighlight lang="python">import numpy as np

Davrot: Created page with "== The goal == Questions to [mailto:davrot@uni-bremen.de David Rotermund] == Test data == We rotate the blue dots with a non-orthogonal rotation matrix into the red dots.import numpy as np import matplotlib.pyplot as plt rng = np.random.default_rng(1) a_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] a_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3 data_a = np.concatenate((a_x, a_y), axis=1) b_x = rng.normal(0.0, 1.0,..."

2025-10-17T15:35:38Z

Created page with "== The goal == Questions to [mailto:davrot@uni-bremen.de David Rotermund] == Test data == We rotate the blue dots with a non-orthogonal rotation matrix into the red dots.<syntaxhighlight lang="python">import numpy as np import matplotlib.pyplot as plt rng = np.random.default_rng(1) a_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] a_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3 data_a = np.concatenate((a_x, a_y), axis=1) b_x = rng.normal(0.0, 1.0,..."

New page

== The goal ==
Questions to [mailto:davrot@uni-bremen.de David Rotermund]

== Test data ==
We rotate the blue dots with a non-orthogonal rotation matrix into the red dots.<syntaxhighlight lang="python">import numpy as np
import matplotlib.pyplot as plt

rng = np.random.default_rng(1)

a_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
a_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
data_a = np.concatenate((a_x, a_y), axis=1)

b_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
b_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
data_b = np.concatenate((b_x, b_y), axis=1)

data = np.concatenate((data_a, data_b), axis=0)

angle_x = -0.3
angle_y = 0.3

roation_matrix = np.array(
[[np.cos(angle_x), -np.sin(angle_x)], [np.sin(angle_y), np.cos(angle_y)]]
)
data_r = data @ roation_matrix

plt.plot(data[:, 0], data[:, 1], "b.")
plt.plot(data_r[:, 0], data_r[:, 1], "r.")
plt.show()</syntaxhighlight>[[File:14 0.png]]<div class="figure">
</div>

== Train and use [https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA FastICA] ==
<syntaxhighlight lang="python">class sklearn.decomposition.FastICA(n_components=None, *, algorithm='parallel', whiten='unit-variance', fun='logcosh', fun_args=None, max_iter=200, tol=0.0001, w_init=None, whiten_solver='svd', random_state=None)</syntaxhighlight><blockquote>FastICA: a fast algorithm for Independent Component Analysis.

The implementation is based on [https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#r44c805292efc-1 1].</blockquote><syntaxhighlight lang="python">fit(X, y=None)</syntaxhighlight><blockquote>Fit the model to X.

'''X''' : array-like of shape (n_samples, n_features)

Training data, where n_samples is the number of samples and n_features is the number of features.</blockquote><syntaxhighlight lang="python">transform(X, copy=True)</syntaxhighlight><blockquote>Recover the sources from X (apply the unmixing matrix).

'''X''' : array-like of shape (n_samples, n_features)

Data to transform, where n_samples is the number of samples and n_features is the number of features.</blockquote><syntaxhighlight lang="python">import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import FastICA

rng = np.random.default_rng(1)

a_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
a_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
data_a = np.concatenate((a_x, a_y), axis=1)

b_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
b_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
data_b = np.concatenate((b_x, b_y), axis=1)

data = np.concatenate((data_a, data_b), axis=0)

angle_x = -0.3
angle_y = 0.3

roation_matrix = np.array(
[[np.cos(angle_x), -np.sin(angle_x)], [np.sin(angle_y), np.cos(angle_y)]]
)
data_r = data @ roation_matrix

# Train
ica = FastICA(n_components=2)
ica.fit(data_r)

# Use
transformed_data = ica.transform(data_r)

plt.plot(transformed_data[:, 0], transformed_data[:, 1], "k.")
plt.show()</syntaxhighlight>[[File:14 1.png]]<div class="figure">
</div>

== Use FastICA to transform the un-rotated data ==
<syntaxhighlight lang="python">inverse_transform(X, copy=True)</syntaxhighlight><blockquote>Transform the sources back to the mixed data (apply mixing matrix).

'''X''': array-like of shape (n_samples, n_components)

Sources, where n_samples is the number of samples and n_components is the number of components.</blockquote><syntaxhighlight lang="python">import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import FastICA

rng = np.random.default_rng(1)

a_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
a_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
data_a = np.concatenate((a_x, a_y), axis=1)

b_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
b_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
data_b = np.concatenate((b_x, b_y), axis=1)

data = np.concatenate((data_a, data_b), axis=0)

angle_x = -0.3
angle_y = 0.3

roation_matrix = np.array(
[[np.cos(angle_x), -np.sin(angle_x)], [np.sin(angle_y), np.cos(angle_y)]]
)
data_r = data @ roation_matrix

# Train
ica = FastICA(n_components=2)
ica.fit(data_r)

# Use
transformed_data = ica.inverse_transform(data)

plt.plot(transformed_data[:, 0], transformed_data[:, 1], "k.")
plt.show()</syntaxhighlight>[[File:14 2.png]]<div class="figure">
</div>

== Inspect the extracted coordinate system ==
<blockquote>'''components_''': ndarray of shape (n_components, n_features)

The linear operator to apply to the data to get the independent sources. This is equal to the unmixing matrix when whiten is False, and equal to np.dot(unmixing_matrix, self.whitening_) when whiten is True.</blockquote>'''Be aware that the sign of any axis can switch !!!''' Like it happend in this example:<syntaxhighlight lang="python">import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import FastICA

rng = np.random.default_rng(1)

a_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
a_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
data_a = np.concatenate((a_x, a_y), axis=1)

b_x = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis] ** 3
b_y = rng.normal(0.0, 1.0, size=(5000))[:, np.newaxis]
data_b = np.concatenate((b_x, b_y), axis=1)

data = np.concatenate((data_a, data_b), axis=0)

angle_x = -0.3
angle_y = 0.3

roation_matrix = np.array(
[[np.cos(angle_x), -np.sin(angle_x)], [np.sin(angle_y), np.cos(angle_y)]]
)
data_r = data @ roation_matrix

# Train
ica = FastICA(n_components=2)
ica.fit(data_r)

plt.plot([-ica.components_.max(), ica.components_.max()], [0, 0], "k")
plt.plot([0, 0], [-ica.components_.max(), ica.components_.max()], "k")

plt.plot(
[-ica.components_[0, 0], ica.components_[0, 0]],
[-ica.components_[0, 1], ica.components_[0, 1]],
"m",
)

plt.plot(
[-ica.components_[1, 0], ica.components_[1, 0]],
[-ica.components_[1, 1], ica.components_[1, 1]],
"c",
)

plt.show()</syntaxhighlight>[[File:14 3.png]]<div class="figure">
</div>

== Fast ICA Methods ==
{| class="wikitable"
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.fit fit](X[, y])
|Fit the model to X.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.fit_transform fit_transform](X[, y])
|Fit the model and recover the sources from X.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.get_feature_names_out get_feature_names_out]([input_features])
|Get output feature names for transformation.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.get_metadata_routing get_metadata_routing]()
|Get metadata routing of this object.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.get_params get_params]([deep])
|Get parameters for this estimator.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.inverse_transform inverse_transform](X[, copy])
|Transform the sources back to the mixed data (apply mixing matrix).
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.set_inverse_transform_request set_inverse_transform_request](*[, copy])
|Request metadata passed to the inverse_transform method.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.set_output set_output](*[, transform])
|Set output container.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.set_params set_params](**params)
|Set the parameters of this estimator.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.set_transform_request set_transform_request](*[, copy])
|Request metadata passed to the transform method.
|-
|[https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA.transform transform](X[, copy])
|Recover the sources from X (apply the unmixing matrix).
|}

== Fast ICA Attributes ==
<blockquote>'''components_''': ndarray of shape (n_components, n_features)

The linear operator to apply to the data to get the independent sources. This is equal to the unmixing matrix when whiten is False, and equal to np.dot(unmixing_matrix, self.whitening_) when whiten is True.</blockquote><blockquote>'''mixing_''': ndarray of shape (n_features, n_components)

The pseudo-inverse of components_. It is the linear operator that maps independent sources to the data.</blockquote><blockquote>'''mean_''': ndarray of shape(n_features,)

The mean over features. Only set if self.whiten is True.</blockquote><blockquote>'''n_features_in_''': int

Number of features seen during fit.</blockquote><blockquote>'''feature_names_in_''': ndarray of shape (n_features_in_,)

Names of features seen during fit. Defined only when X has feature names that are all strings.</blockquote><blockquote>'''n_iter_''': int

If the algorithm is “deflation”, n_iter is the maximum number of iterations run across all components. Else they are just the number of iterations taken to converge.</blockquote><blockquote>'''whitening_''': ndarray of shape (n_components, n_features)

Only set if whiten is ‘True’. This is the pre-whitening matrix that projects data onto the first n_components principal components.</blockquote>

FastICA - Revision history

Davrot at 17:16, 17 October 2025