.. 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 <sphx_glr_download_auto_examples_decomposition_plot_ica_blind_source_separation.py>`
        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_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 0.329 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:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_ica_blind_source_separation.ipynb <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 <plot_ica_blind_source_separation.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_ica_blind_source_separation.zip <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 <https://sphinx-gallery.github.io>`_