.. _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.



.. code-block:: python

    print(__doc__)

    import numpy as np
    import matplotlib.pyplot as plt
    from scipy import signal

    from sklearn.decomposition import FastICA, PCA







Generate sample data


.. code-block:: python

    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

    # Compute ICA
    ica = FastICA(n_components=3)
    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







Plot results


.. code-block:: python


    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.subplots_adjust(0.09, 0.04, 0.94, 0.94, 0.26, 0.46)
    plt.show()



.. image:: /auto_examples/decomposition/images/sphx_glr_plot_ica_blind_source_separation_001.png
    :align: center




**Total running time of the script:**
(0 minutes 0.299 seconds)



.. container:: sphx-glr-download

    **Download Python source code:** :download:`plot_ica_blind_source_separation.py <plot_ica_blind_source_separation.py>`


.. container:: sphx-glr-download

    **Download IPython notebook:** :download:`plot_ica_blind_source_separation.ipynb <plot_ica_blind_source_separation.ipynb>`