Smoothing of dense two-dimensional 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

The package includes different smoothing methods to smooth functional data. In this section, we are showing the building blocks of the smoothing of dense two-dimensional functional data. First, we simulate functional data using the Karhunen-Loève decomposition using B-splines basis functions. We then add some noise to the simulation.

# Set general parameters
rng = 42
n_obs = 4

# Parameters of the basis
name = ("bsplines", "bsplines")
n_functions = (5, 5)

argvals = DenseArgvals({
    "input_dim_0": np.linspace(0, 1, 51),
    "input_dim_1": np.linspace(0, 1, 51)
})


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

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

Smoothing two-dimensional functional data is similar to smoothing one-dimensional functional data. The main difference is that the smoothing is done in two dimensions. In this example, we will smooth the noisy data using the smooth() function. This function allows to smooth the data using different methods such as local polynomials and P-splines. In this example, we will use the local polynomials smoothing method with an Epanechnikov kernel and a bandwidth of \(0.5\). We plot the smoothed data.

# Smooth the data
points = DenseArgvals({
    "input_dim_0": np.linspace(0, 1, 11),
    "input_dim_1": np.linspace(0, 1, 11)
})
kernel_name = "epanechnikov"
bandwidth = 0.5
degree = 1

data_smooth = kl.noisy_data.smooth(
    points=points,
    method="LP",
    kernel_name=kernel_name,
    bandwidth=bandwidth,
    degree=degree,
)

_ = plot(data_smooth)