"""
Simulation of functional data
=============================

"""

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

# Load packages
import numpy as np

from FDApy.representation import DenseArgvals
from FDApy.simulation import KarhunenLoeve
from FDApy.visualization import plot

###############################################################################
# Two main features of functional data are noise and sparsity. The package provides 
# methods to simulate noisy and sparse functional data. In this example, we will 
# simulate noisy and sparse functional data using the Karhunen-Loève decomposition.


# Set general parameters
rng = 42
n_obs = 10

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


###############################################################################
# For one dimensional data
# ------------------------
#
# We simulate :math:`N = 10` curves on the one-dimensional observation grid
# :math:`\{0, 0.01, 0.02, \cdots, 1\}`, based on the first
# :math:`K = 5` B-splines basis functions on :math:`[0, 1]` and the variance of
# the scores random variables equal to :math:`1`.
kl = KarhunenLoeve(
    basis_name=name, n_functions=n_functions, argvals=argvals, random_state=rng
)
kl.new(n_obs=n_obs)

_ = plot(kl.data)

###############################################################################
# **Adding noise**
# ---
# We can generates a noisy version of the functional data by adding i.i.d.
# realizations of the random variable
# :math:`\varepsilon \sim \mathcal{N}(0, \sigma^2)` to the observation. In this
# example, we set :math:`\sigma^2 = 0.05`.

# Add some noise to the simulation.
kl.add_noise(0.05)

# Plot the noisy simulations
_ = plot(kl.noisy_data)

###############################################################################
# **Sparsification**
# ---
# We can generates a sparsified version of the functional data object by
# randomly removing a certain percentage of the sampling points. The percentage
# of retain samplings points can be supplied by the user. In this example, the
# retained number of observations will be different for each curve and be
# randomly drawn between :math:`0.45` and :math:`0.55` (percentage :math:`\pm` epsilon).

# Sparsify the data
kl.sparsify(percentage=0.5, epsilon=0.05)

_ = plot(kl.sparse_data)


###############################################################################
# For two dimensional data
# ------------------------
# We simulate :math:`N = 1` image on the two-dimensional observation grid
# :math:`\{0, 0.01, 0.02, \cdots, 1\} \times \{0, 0.01, 0.02, \cdots, 1\}`,
# based on the tensor product of the first :math:`K = 25` B-splines
# basis functions on :math:`[0, 1] \times [0, 1]` and the variance of
# the scores random variables equal to :math:`1`.

# Parameters of the basis
name = ("bsplines", "bsplines")
n_functions = (5, 5)
argvals = DenseArgvals(
    {"input_dim_0": np.linspace(0, 1, 101), "input_dim_1": np.linspace(0, 1, 101)}
)

kl = KarhunenLoeve(
    basis_name=name, n_functions=n_functions, argvals=argvals, random_state=rng
)
kl.new(n_obs=1)

_ = plot(kl.data)

###############################################################################
# **Adding noise**
# ---
# We can generates a noisy version of the functional data by adding i.i.d.
# realizations of the random variable
# :math:`\varepsilon \sim \mathcal{N}(0, \sigma^2)` to the observation. In this
# example, we set :math:`\sigma^2 = 0.05`.

# Add some noise to the simulation.
kl.add_noise(0.05)

# Plot the noisy simulations
_ = plot(kl.noisy_data)

###############################################################################
# **Sparsification**
# ---
# The sparsification is not implemented for two-dimensional (and higher) data.