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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download__generated_examples_plot_basic_fit.py>`
        to download the full example code.

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

.. _sphx_glr__generated_examples_plot_basic_fit.py:


Basic AMICA Fit
===============

Fit a single-model AMICA (equivalent to Infomax ICA) on synthetic data.
Inspects the raw input, convergence via the log-likelihood curve, and
confirms that the sphering matrix correctly whitens the data.

.. GENERATED FROM PYTHON SOURCE LINES 9-16

.. code-block:: Python


    import numpy as np
    import torch
    import matplotlib.pyplot as plt

    from pyamica import AMICA








.. GENERATED FROM PYTHON SOURCE LINES 17-20

Generate Synthetic Data
-----------------------
Mix 8 independent Laplacian sources with a random matrix.

.. GENERATED FROM PYTHON SOURCE LINES 20-29

.. code-block:: Python


    rng = np.random.default_rng(0)
    n_ch, T = 8, 4000

    sources = rng.laplace(0, 1, (n_ch, T))
    A_true  = rng.standard_normal((n_ch, n_ch))
    data    = (A_true @ sources).T.astype("float64")   # (T, n_ch)
    X       = torch.from_numpy(data)








.. GENERATED FROM PYTHON SOURCE LINES 30-34

Raw Data
--------
A look at the first 4 channels.  Because the sources are Laplacian,
the mixture has occasional large spikes.

.. GENERATED FROM PYTHON SOURCE LINES 34-46

.. code-block:: Python


    fig, axes = plt.subplots(4, 1, figsize=(10, 5), sharex=True)
    t = np.arange(T)
    for i, ax in enumerate(axes):
        ax.plot(t, data[:, i], lw=0.5, color="steelblue")
        ax.set_ylabel(f"Ch {i}")
        ax.set_yticks([])
    axes[-1].set_xlabel("Sample")
    fig.suptitle("Raw input data (4 of 8 channels)")
    fig.tight_layout()
    plt.show()




.. image-sg:: /_generated/examples/images/sphx_glr_plot_basic_fit_001.png
   :alt: Raw input data (4 of 8 channels)
   :srcset: /_generated/examples/images/sphx_glr_plot_basic_fit_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 47-49

Fit AMICA (M=1)
---------------

.. GENERATED FROM PYTHON SOURCE LINES 49-56

.. code-block:: Python


    model = AMICA(n_models=1, max_iter=100, verbose=False)
    model.fit(X)

    print(f"Iterations:           {model.n_iter_}")
    print(f"Final log-likelihood: {model.ll_history()[-1]:.6f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Iterations:           100
    Final log-likelihood: -2.081794




.. GENERATED FROM PYTHON SOURCE LINES 57-60

Log-Likelihood Curve
--------------------
The LL should increase monotonically and flatten as the model converges.

.. GENERATED FROM PYTHON SOURCE LINES 60-69

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 3))
    ax.plot(model.ll_history().numpy(), color="steelblue")
    ax.set_xlabel("Iteration")
    ax.set_ylabel("Log-likelihood")
    ax.set_title("AMICA M=1: convergence")
    fig.tight_layout()
    plt.show()




.. image-sg:: /_generated/examples/images/sphx_glr_plot_basic_fit_002.png
   :alt: AMICA M=1: convergence
   :srcset: /_generated/examples/images/sphx_glr_plot_basic_fit_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 70-78

Sphering Matrix
---------------
AMICA pre-whitens the data with a ZCA (symmetric) sphering matrix
``S = V D^{-1/2} V^T`` (eigendecomposition of the sample covariance).

The defining property of a whitening matrix is that the sphered data
has identity covariance: :math:`S \cdot \text{Cov}(X) \cdot S^T = I`.
Verified directly below.

.. GENERATED FROM PYTHON SOURCE LINES 78-93

.. code-block:: Python


    S    = model.sphere_.numpy()              # (n_ch, n_ch)
    X_c  = data - data.mean(axis=0)          # centre
    Xs   = X_c @ S.T                         # (T, n_ch) - sphered data
    cov_sphered = Xs.T @ Xs / T              # should be identity

    fig, ax = plt.subplots(figsize=(4, 4))
    im = ax.imshow(cov_sphered, vmin=-0.05, vmax=1.05, cmap="RdBu_r")
    fig.colorbar(im, ax=ax, shrink=0.8)
    ax.set_title("Covariance of sphered data\n(should be identity)")
    fig.tight_layout()
    plt.show()

    residual = np.abs(cov_sphered - np.eye(n_ch)).max()
    print(f"Max deviation from identity: {residual:.2e}  (expect < 1e-12)")



.. image-sg:: /_generated/examples/images/sphx_glr_plot_basic_fit_003.png
   :alt: Covariance of sphered data (should be identity)
   :srcset: /_generated/examples/images/sphx_glr_plot_basic_fit_003.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Max deviation from identity: 1.38e-14  (expect < 1e-12)





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

   **Total running time of the script:** (0 minutes 4.029 seconds)


.. _sphx_glr_download__generated_examples_plot_basic_fit.py:

.. only:: html

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

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

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

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

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

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

      :download:`Download zipped: plot_basic_fit.zip <plot_basic_fit.zip>`


.. only:: html

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

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