.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_ica_synth.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_auto_examples_plot_ica_synth.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_plot_ica_synth.py:


=================================================
Blind source separation using Picard and Picard-O
=================================================

The example runs the Picard algorithm proposed in:

Pierre Ablin, Jean-Francois Cardoso, Alexandre Gramfort
"Faster independent component analysis by preconditioning with Hessian approximations"
IEEE Transactions on Signal Processing, 2018
https://arxiv.org/abs/1706.08171

and Picard-O algorithm proposed in:

Pierre Ablin, Jean-François Cardoso, Alexandre Gramfort
"Faster ICA under orthogonal constraint"
ICASSP, 2018
https://arxiv.org/abs/1711.10873

.. GENERATED FROM PYTHON SOURCE LINES 21-33

.. code-block:: default


    # Author: Pierre Ablin <pierre.ablin@inria.fr>
    #         Alexandre Gramfort <alexandre.gramfort@inria.fr>
    # License: BSD 3 clause

    import numpy as np
    import matplotlib.pyplot as plt

    from picard import picard

    print(__doc__)








.. GENERATED FROM PYTHON SOURCE LINES 34-35

Generate sample data

.. GENERATED FROM PYTHON SOURCE LINES 35-54

.. code-block:: default

    np.random.seed(0)
    n_samples = 2000
    time = np.linspace(0, 8, n_samples)

    s1 = np.sin(2 * time) * np.sin(40 * time)
    s2 = np.sin(3 * time) ** 5
    s3 = np.random.laplace(size=s1.shape)

    S = np.c_[s1, s2, s3].T

    S /= S.std(axis=1)[:, np.newaxis]  # 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(A, S)  # Generate observations

    # Compute ICA
    _, _, Y_picard = picard(X, ortho=False, random_state=0)
    _, _, Y_picardo = picard(X, ortho=True, random_state=0)








.. GENERATED FROM PYTHON SOURCE LINES 55-56

Plot results

.. GENERATED FROM PYTHON SOURCE LINES 56-71

.. code-block:: default


    models = [X, S, Y_picard, Y_picardo]
    names = ['Observations (mixed signal)',
             'True Sources',
             'ICA recovered signals with Picard',
             'ICA recovered signals with Picard-O']
    colors = ['red', 'steelblue', 'orange']

    for ii, (model, name) in enumerate(zip(models, names), 1):
        fig, axes = plt.subplots(3, 1, figsize=(6, 4), sharex=True, sharey=True)
        plt.suptitle(name)
        for ax, sig, color in zip(axes, model, colors):
            ax.plot(sig, color=color)

    plt.show()



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_ica_synth_001.png
         :alt: Observations (mixed signal)
         :srcset: /auto_examples/images/sphx_glr_plot_ica_synth_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_ica_synth_002.png
         :alt: True Sources
         :srcset: /auto_examples/images/sphx_glr_plot_ica_synth_002.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_ica_synth_003.png
         :alt: ICA recovered signals with Picard
         :srcset: /auto_examples/images/sphx_glr_plot_ica_synth_003.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_examples/images/sphx_glr_plot_ica_synth_004.png
         :alt: ICA recovered signals with Picard-O
         :srcset: /auto_examples/images/sphx_glr_plot_ica_synth_004.png
         :class: sphx-glr-multi-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  0.615 seconds)


.. _sphx_glr_download_auto_examples_plot_ica_synth.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_ica_synth.py <plot_ica_synth.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_ica_synth.ipynb <plot_ica_synth.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_