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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_elastic_registration.py"
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.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_elastic_registration.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_plot_elastic_registration.py:


Elastic registration
====================

Shows the usage of the elastic registration to perform a groupwise alignment.

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.. code-block:: Python


    # Author: Pablo Marcos Manchón
    # License: MIT

    # sphinx_gallery_thumbnail_number = 5

    import numpy as np

    import skfda
    from skfda.datasets import fetch_growth, make_multimodal_samples
    from skfda.exploratory.stats import fisher_rao_karcher_mean
    from skfda.preprocessing.registration import FisherRaoElasticRegistration








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In the example of pairwise alignment was shown the usage of
:class:`~skfda.preprocessing.registration.FisherRaoElasticRegistration` to
align a set of functional observations to a given template or a set of
templates.

In the groupwise alignment all the samples are aligned to the same template,
constructed to minimise some distance, generally a mean or a median. In the
case of the elastic registration, due to the use of the elastic distance in
the alignment, one of the most suitable templates is the karcher mean under
this metric.

We will create a synthetic dataset to show the basic usage of the
registration.

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.. code-block:: Python


    fd = make_multimodal_samples(n_modes=2, stop=4, random_state=1)
    fd.plot()




.. image-sg:: /auto_examples/images/sphx_glr_plot_elastic_registration_001.png
   :alt: plot elastic registration
   :srcset: /auto_examples/images/sphx_glr_plot_elastic_registration_001.png
   :class: sphx-glr-single-img


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 .. code-block:: none


    <Figure size 640x480 with 1 Axes>



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The following figure shows the
:func:`~skfda.exploratory.stats.fisher_rao_karcher_mean` of the
dataset and the cross-sectional mean, which correspond to the karcher-mean
under the :math:`\mathbb{L}^2` distance.

It can be seen how the elastic mean better captures the geometry of the
curves compared to the standard mean, since it is not affected by the
deformations of the curves.

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.. code-block:: Python



    fig = fd.mean().plot(label="L2 mean")
    fisher_rao_karcher_mean(fd).plot(fig=fig, label="Elastic mean")
    fig.legend()




.. image-sg:: /auto_examples/images/sphx_glr_plot_elastic_registration_002.png
   :alt: plot elastic registration
   :srcset: /auto_examples/images/sphx_glr_plot_elastic_registration_002.png
   :class: sphx-glr-single-img


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 .. code-block:: none


    <matplotlib.legend.Legend object at 0x7f52b3a8bfd0>



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In this case, the alignment completely reduces the amplitude variability
between the samples, aligning the maximum points correctly.

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.. code-block:: Python


    elastic_registration = FisherRaoElasticRegistration()

    fd_align = elastic_registration.fit_transform(fd)

    fd_align.plot()





.. image-sg:: /auto_examples/images/sphx_glr_plot_elastic_registration_003.png
   :alt: plot elastic registration
   :srcset: /auto_examples/images/sphx_glr_plot_elastic_registration_003.png
   :class: sphx-glr-single-img


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 .. code-block:: none


    <Figure size 640x480 with 1 Axes>



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In general these type of alignments are not possible, in the following
figure it is shown how it works with a real dataset.
The :func:`berkeley growth dataset<skfda.datasets.fetch_growth>`
contains the growth curves of a set children, in this case will be used only
the males. The growth curves will be resampled using cubic interpolation and
derived to obtain the velocity curves.

First we show the original curves:

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.. code-block:: Python


    growth = fetch_growth()

    # Select only one sex
    fd = growth['data'][growth['target'] == 0]

    # Obtain velocity curves
    fd.interpolation = skfda.representation.interpolation.SplineInterpolation(3)
    fd_derivative = fd.to_grid(np.linspace(*fd.domain_range[0], 200)).derivative()
    fd_derivative = fd_derivative.to_grid(np.linspace(*fd.domain_range[0], 50))
    fd_derivative.dataset_name = f"{fd.dataset_name} - derivative"
    fd_derivative.plot()




.. image-sg:: /auto_examples/images/sphx_glr_plot_elastic_registration_004.png
   :alt: Berkeley Growth Study - derivative
   :srcset: /auto_examples/images/sphx_glr_plot_elastic_registration_004.png
   :class: sphx-glr-single-img


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 .. code-block:: none


    <Figure size 640x480 with 1 Axes>



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We now show the aligned curves:

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.. code-block:: Python


    fd_align = elastic_registration.fit_transform(fd_derivative)
    fd_align.dataset_name = f"{fd.dataset_name} - derivative aligned"

    fd_align.plot()




.. image-sg:: /auto_examples/images/sphx_glr_plot_elastic_registration_005.png
   :alt: Berkeley Growth Study - derivative aligned
   :srcset: /auto_examples/images/sphx_glr_plot_elastic_registration_005.png
   :class: sphx-glr-single-img


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 .. code-block:: none


    <Figure size 640x480 with 1 Axes>



.. GENERATED FROM PYTHON SOURCE LINES 95-106

* Srivastava, Anuj & Klassen, Eric P. (2016). Functional and shape data
  analysis. In *Functional Data and Elastic Registration* (pp. 73-122).
  Springer.

* Tucker, J. D., Wu, W. and Srivastava, A. (2013). Generative Models for
  Functional Data using Phase and Amplitude Separation.
  Computational Statistics and Data Analysis, Vol. 61, 50-66.

* J. S. Marron, James O. Ramsay, Laura M. Sangalli and Anuj Srivastava
  (2015). Functional Data Analysis of Amplitude and Phase Variation.
  Statistical Science 2015, Vol. 30, No. 4


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

   **Total running time of the script:** (0 minutes 3.361 seconds)


.. _sphx_glr_download_auto_examples_plot_elastic_registration.py:

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        :alt: Launch binder
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      :download:`Download Jupyter notebook: plot_elastic_registration.ipynb <plot_elastic_registration.ipynb>`

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


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