"""
One-dimensional Basis
=====================

"""

# Author: Steven Golovkine <steven_golovkine@icloud.com>
# License: MIT

# Load packages
import numpy as np

from FDApy.representation import Basis
from FDApy.representation import DenseArgvals
from FDApy.visualization import plot

###############################################################################
# The package include different basis functions to represent functional data. In this section, we are showing the building blocks of the representation of basis functions. To define a :class:`~FDApy.representation.Basis` object, we need to specify the name of the basis, the number of functions in the basis and the sampling points. The sampling points are defined as a :class:`~FDApy.representation.DenseArgvals`.


###############################################################################
# We will show the basis functions for the Fourier, B-splines and Wiener basis. The number of functions in the basis is set to :math:`5` and the sampling points are defined as a :class:`~FDApy.representation.DenseArgvals` object with a hundred points between :math:`0` and :math:`1`.

# Parameters
n_functions = 5
argvals = DenseArgvals({"input_dim_0": np.linspace(0, 1, 101)})


###############################################################################
# Fourier basis
# -------------
# First, we will show the basis functions for the Fourier basis. The basis functions consist of the sine and cosine functions with a frequency that increases with the number of the function. Note that the first function is a constant function. This basis may be used to represent periodic functions.

basis = Basis(name="fourier", n_functions=n_functions, argvals=argvals)

_ = plot(basis)

###############################################################################
# B-splines basis
# ---------------
# Second, we will show the basis functions for the B-splines basis. The basis functions are piecewise polynomials that are smooth at the knots. The number of knots is equal to the number of functions in the basis minus :math:`2`. This basis may be used to represent smooth functions.

basis = Basis(name="bsplines", n_functions=n_functions, argvals=argvals)

_ = plot(basis)

###############################################################################
# Wiener basis
# ------------
# Third, we will show the basis functions for the Wiener basis. The basis functions are the eigenfunctions of a Brownian process. This basis may be used to represent rough functions.

basis = Basis(name="wiener", n_functions=n_functions, argvals=argvals)

_ = plot(basis)