"""
Elastic registration
====================

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

# Author: Pablo Marcos Manchón
# License: MIT

# sphinx_gallery_thumbnail_number = 5

# %%
# 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.

import matplotlib.pyplot as plt

from skfda.datasets import make_multimodal_samples

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

# %%
# 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.

from skfda.exploratory.stats import fisher_rao_karcher_mean

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

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

from skfda.preprocessing.registration import FisherRaoElasticRegistration

elastic_registration = FisherRaoElasticRegistration()

fd_align = elastic_registration.fit_transform(fd)

fd_align.plot()
plt.show()

# %%
# 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:

import numpy as np

from skfda.datasets import fetch_growth
from skfda.representation.interpolation import SplineInterpolation

growth = fetch_growth()

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

# Obtain velocity curves
fd.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()
plt.show()

# %%
# We now show the aligned curves:

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

fd_align.plot()
plt.show()

# %%
# * 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