Registration
============


We see often that variation in functional observations involves phase and
amplitude variation, which may hinder further analysis. That problem is treated
during the registration process. This module contains procedures for the
registration of the data.

Shift Registration
------------------

Many of the issues involved in registration can be solved by considering
the simplest case, a simple shift in the time scale. This often happens because
the time at which the recording process begins is arbitrary, and is unrelated
to the beginning of the interesting segment of the data. In the
:ref:`sphx_glr_auto_examples_plot_shift_registration.py` example
is shown the basic usage of this method.

.. autosummary::
   :toctree: autosummary

   skfda.preprocessing.registration.LeastSquaresShiftRegistration


Landmark Registration
---------------------

Landmark registration aligns features applying a transformation of the time that
takes all the times of a given feature into a common value.

The simplest case in which each sample presents a unique landmark can be solved
by performing a translation in the time scale. See the
:ref:`sphx_glr_auto_examples_plot_landmark_shift.py` example..

.. autosummary::
   :toctree: autosummary

   skfda.preprocessing.registration.landmark_shift_registration
   skfda.preprocessing.registration.landmark_shift_deltas


The general case of landmark registration may present multiple landmarks for
each sample and a non-linear transformation in the time scale should be applied.
See the :ref:`sphx_glr_auto_examples_plot_landmark_registration.py` example.

.. autosummary::
   :toctree: autosummary

   skfda.preprocessing.registration.landmark_elastic_registration
   skfda.preprocessing.registration.landmark_elastic_registration_warping


Elastic Registration
--------------------

The elastic registration is a novel approach to this problem that uses the
properties of the Fisher-Rao metric to perform the alignment of the curves.
In the examples of
:ref:`sphx_glr_auto_examples_plot_pairwise_alignment.py` and
:ref:`sphx_glr_auto_examples_plot_elastic_registration.py` is shown a brief
introduction to this topic along the usage of the corresponding functions.

.. autosummary::
   :toctree: autosummary

   skfda.preprocessing.registration.FisherRaoElasticRegistration


Validation
----------

This module contains several classes methods for the quantification and
validation of the registration procedure.

.. autosummary::
   :toctree: autosummary


   skfda.preprocessing.registration.validation.AmplitudePhaseDecomposition
   skfda.preprocessing.registration.validation.LeastSquares
   skfda.preprocessing.registration.validation.SobolevLeastSquares
   skfda.preprocessing.registration.validation.PairwiseCorrelation


Warping utils
-----------------

This module contains some functions related with the warping of functional
data.

.. autosummary::
   :toctree: autosummary

   skfda.preprocessing.registration.invert_warping
   skfda.preprocessing.registration.normalize_warping

References
----------

* Ramsay, J., Silverman, B. W. (2005). Functional Data Analysis. Springer.

* Kneip, Alois & Ramsay, James. (2008).  Quantifying amplitude and phase
  variation. Journal of the American Statistical Association.

* Ramsay, J., Hooker, G. & Graves S. (2009). Functional Data Analysis with
  R and Matlab. Springer.

* Srivastava, Anuj & Klassen, Eric P. (2016). Functional and shape data
  analysis. 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