.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_decomposition_plot_ica_vs_pca.py: ========================== FastICA on 2D point clouds ========================== This example illustrates visually in the feature space a comparison by results using two different component analysis techniques. :ref:`ICA` vs :ref:`PCA`. 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). .. image:: /auto_examples/decomposition/images/sphx_glr_plot_ica_vs_pca_001.png :alt: True Independent Sources, Observations, PCA recovered signals, ICA recovered signals :class: sphx-glr-single-img .. code-block:: default print(__doc__) # Authors: Alexandre Gramfort, Gael Varoquaux # License: BSD 3 clause import numpy as np import matplotlib.pyplot as plt from sklearn.decomposition import PCA, FastICA # ############################################################################# # Generate sample data rng = np.random.RandomState(42) S = rng.standard_t(1.5, size=(20000, 2)) S[:, 0] *= 2. # Mix data A = np.array([[1, 1], [0, 2]]) # Mixing matrix X = np.dot(S, A.T) # Generate observations pca = PCA() S_pca_ = pca.fit(X).transform(X) ica = FastICA(random_state=rng) S_ica_ = ica.fit(X).transform(X) # Estimate the sources S_ica_ /= S_ica_.std(axis=0) # ############################################################################# # Plot results 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: colors = ['orange', 'red'] for color, axis in zip(colors, axis_list): axis /= axis.std() x_axis, y_axis = axis # Trick to get legend to work plt.plot(0.1 * x_axis, 0.1 * y_axis, linewidth=2, color=color) plt.quiver((0, 0), (0, 0), x_axis, y_axis, zorder=11, width=0.01, scale=6, color=color) plt.hlines(0, -3, 3) plt.vlines(0, -3, 3) plt.xlim(-3, 3) plt.ylim(-3, 3) 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 = [pca.components_.T, ica.mixing_] plt.subplot(2, 2, 2) plot_samples(X / np.std(X), axis_list=axis_list) legend = plt.legend(['PCA', 'ICA'], loc='upper right') legend.set_zorder(100) plt.title('Observations') plt.subplot(2, 2, 3) plot_samples(S_pca_ / np.std(S_pca_, axis=0)) plt.title('PCA recovered signals') plt.subplot(2, 2, 4) plot_samples(S_ica_ / np.std(S_ica_)) plt.title('ICA recovered signals') plt.subplots_adjust(0.09, 0.04, 0.94, 0.94, 0.26, 0.36) plt.show() .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.282 seconds) .. _sphx_glr_download_auto_examples_decomposition_plot_ica_vs_pca.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: https://mybinder.org/badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.23.X?urlpath=lab/tree/notebooks/auto_examples/decomposition/plot_ica_vs_pca.ipynb :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_ica_vs_pca.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_ica_vs_pca.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_