"""
Shift Registration
==================

Shows the use of shift registration applied to a sinusoidal
process represented in a Fourier basis.
"""

# Author: Pablo Marcos Manchón
# License: MIT

# sphinx_gallery_thumbnail_number = 3

# %%
# In this example we will use a
# :func:`sinusoidal process <skfda.datasets.make_sinusoidal_process>`
# synthetically generated. This dataset consists in a sinusoidal wave with
# fixed period which contanis phase and amplitude variation with gaussian
# noise.
#
# In this example we want to register the curves using a translation
# and remove the phase variation to perform further analysis.

import matplotlib.pyplot as plt

from skfda.datasets import make_sinusoidal_process

fd = make_sinusoidal_process(random_state=1)
fd.plot()
plt.show()

# %%
# We will smooth the curves using a basis representation, which will help us
# to remove the gaussian noise. Smoothing before registration
# is essential due to the use of derivatives in the optimization process.
# Because of their sinusoidal nature we will use a Fourier basis.

from skfda.representation.basis import FourierBasis

fd_basis = fd.to_basis(FourierBasis(n_basis=11))
fd_basis.plot()
plt.show()

# %%
# We will use the
# :func:`~skfda.preprocessing.registration.LeastSquaresShiftRegistration`
# transformer, which is suitable due to the periodicity of the dataset and
# the small amount of amplitude variation.
#
# We can observe how the sinusoidal pattern is easily distinguishable
# once the alignment has been made.

from skfda.preprocessing.registration import LeastSquaresShiftRegistration

# sphinx_gallery_start_ignore
# isort: split
from skfda import FDataBasis

shift_registration: LeastSquaresShiftRegistration[FDataBasis]
# sphinx_gallery_end_ignore

shift_registration = LeastSquaresShiftRegistration()
fd_registered = shift_registration.fit_transform(fd_basis)

fd_registered.plot()
plt.show()

# %%
# We will plot the mean of the original smoothed curves and the registered
# ones, and we will compare with the original sinusoidal process without
# noise.
#
# We can see how the phase variation affects to the mean of the original
# curves varying their amplitude with respect to the original process,
# however, this effect is mitigated after the registration.

# sinusoidal process without variation and noise
sine = make_sinusoidal_process(
    n_samples=1,
    phase_std=0,
    amplitude_std=0,
    error_std=0,
)

fig = fd_basis.mean().plot()
fd_registered.mean().plot(fig)
sine.plot(fig, linestyle="dashed")

fig.axes[0].legend(["original mean", "registered mean", "sine"])
plt.show()

# %%
# The values of the shifts :math:`\delta_i`, stored in the attribute `deltas_`
# may be relevant for further analysis, as they may be considered as nuisance
# or random effects.

print(shift_registration.deltas_)