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      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Blind source separation using FastICA\n\nAn example of estimating sources from noisy data.\n\n`ICA` is used to estimate sources given noisy measurements.\nImagine 3 instruments playing simultaneously and 3 microphones\nrecording the mixed signals. ICA is used to recover the sources\nie. what is played by each instrument. Importantly, PCA fails\nat recovering our `instruments` since the related signals reflect\nnon-Gaussian processes.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "# Authors: The scikit-learn developers\n# SPDX-License-Identifier: BSD-3-Clause"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Generate sample data\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\nfrom scipy import signal\n\nnp.random.seed(0)\nn_samples = 2000\ntime = np.linspace(0, 8, n_samples)\n\ns1 = np.sin(2 * time)  # Signal 1 : sinusoidal signal\ns2 = np.sign(np.sin(3 * time))  # Signal 2 : square signal\ns3 = signal.sawtooth(2 * np.pi * time)  # Signal 3: saw tooth signal\n\nS = np.c_[s1, s2, s3]\nS += 0.2 * np.random.normal(size=S.shape)  # Add noise\n\nS /= S.std(axis=0)  # Standardize data\n# Mix data\nA = np.array([[1, 1, 1], [0.5, 2, 1.0], [1.5, 1.0, 2.0]])  # Mixing matrix\nX = np.dot(S, A.T)  # Generate observations"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Fit ICA and PCA models\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "from sklearn.decomposition import PCA, FastICA\n\n# Compute ICA\nica = FastICA(n_components=3, whiten=\"arbitrary-variance\")\nS_ = ica.fit_transform(X)  # Reconstruct signals\nA_ = ica.mixing_  # Get estimated mixing matrix\n\n# We can `prove` that the ICA model applies by reverting the unmixing.\nassert np.allclose(X, np.dot(S_, A_.T) + ica.mean_)\n\n# For comparison, compute PCA\npca = PCA(n_components=3)\nH = pca.fit_transform(X)  # Reconstruct signals based on orthogonal components"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Plot results\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\n\nplt.figure()\n\nmodels = [X, S, S_, H]\nnames = [\n    \"Observations (mixed signal)\",\n    \"True Sources\",\n    \"ICA recovered signals\",\n    \"PCA recovered signals\",\n]\ncolors = [\"red\", \"steelblue\", \"orange\"]\n\nfor ii, (model, name) in enumerate(zip(models, names), 1):\n    plt.subplot(4, 1, ii)\n    plt.title(name)\n    for sig, color in zip(model.T, colors):\n        plt.plot(sig, color=color)\n\nplt.tight_layout()\nplt.show()"
      ]
    }
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      "file_extension": ".py",
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