.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/decomposition/plot_ica_blind_source_separation.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. or to run this example in your browser via JupyterLite or Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_decomposition_plot_ica_blind_source_separation.py: ===================================== Blind source separation using FastICA ===================================== An example of estimating sources from noisy data. :ref:`ICA` is used to estimate sources given noisy measurements. Imagine 3 instruments playing simultaneously and 3 microphones recording the mixed signals. ICA is used to recover the sources ie. what is played by each instrument. Importantly, PCA fails at recovering our `instruments` since the related signals reflect non-Gaussian processes. .. GENERATED FROM PYTHON SOURCE LINES 16-20 .. code-block:: Python # Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause .. GENERATED FROM PYTHON SOURCE LINES 21-23 Generate sample data -------------------- .. GENERATED FROM PYTHON SOURCE LINES 23-43 .. code-block:: Python import numpy as np from scipy import signal np.random.seed(0) 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: saw tooth 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 .. GENERATED FROM PYTHON SOURCE LINES 44-46 Fit ICA and PCA models ---------------------- .. GENERATED FROM PYTHON SOURCE LINES 46-61 .. code-block:: Python from sklearn.decomposition import PCA, FastICA # Compute ICA ica = FastICA(n_components=3, whiten="arbitrary-variance") S_ = ica.fit_transform(X) # Reconstruct signals A_ = ica.mixing_ # Get estimated mixing matrix # We can `prove` that the ICA model applies by reverting the unmixing. assert np.allclose(X, np.dot(S_, A_.T) + ica.mean_) # For comparison, compute PCA pca = PCA(n_components=3) H = pca.fit_transform(X) # Reconstruct signals based on orthogonal components .. GENERATED FROM PYTHON SOURCE LINES 62-64 Plot results ------------ .. GENERATED FROM PYTHON SOURCE LINES 64-86 .. code-block:: Python import matplotlib.pyplot as plt plt.figure() models = [X, S, S_, H] names = [ "Observations (mixed signal)", "True Sources", "ICA recovered signals", "PCA recovered signals", ] 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() plt.show() .. image-sg:: /auto_examples/decomposition/images/sphx_glr_plot_ica_blind_source_separation_001.png :alt: Observations (mixed signal), True Sources, ICA recovered signals, PCA recovered signals :srcset: /auto_examples/decomposition/images/sphx_glr_plot_ica_blind_source_separation_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.396 seconds) .. _sphx_glr_download_auto_examples_decomposition_plot_ica_blind_source_separation.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/decomposition/plot_ica_blind_source_separation.ipynb :alt: Launch binder :width: 150 px .. container:: lite-badge .. image:: images/jupyterlite_badge_logo.svg :target: ../../lite/lab/index.html?path=auto_examples/decomposition/plot_ica_blind_source_separation.ipynb :alt: Launch JupyterLite :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_ica_blind_source_separation.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_ica_blind_source_separation.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_ica_blind_source_separation.zip ` .. include:: plot_ica_blind_source_separation.recommendations .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_