Representation of univariate and dense functional data

Examples of representation of univariate and dense functional data.

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

# Load packages
import numpy as np

from FDApy import DenseFunctionalData
from FDApy.representation import DenseArgvals, DenseValues
from FDApy.visualization import plot

The representation of dense functional data

We are showing the building blocks of the representation of univariate and dense functional data. To define a FunctionalData object, we need a set of argvals (the sampling points of the curves) and a set of values (the observed points of the curves). The sampling points of the data are defined as a dictionary where each entry is a npt.NDArray that represents an input dimension (one entry corresponds to curves, two entries correspond to surface, …). The shape of the array of the first dimension would be \((m_1,)\), the shape of the array of the second dimension would be \((m_2,)\) and so on. Curves will thus be sampled on \(m_1\) points, surface will be sampled on \(m_1 \times m_2\), etc. The values of the functional data are defined as an npt.NDArray. The shape of the array is \((n, m_1, m_2, \dots)\) where \(n\) is the number of curves in the sample.

For unidimensional functional data

First, we will define unidimensional functional data. We consider a hundred sampling points and two observations.

argvals = np.linspace(0, np.pi, num=100)
X = np.array([np.sin(2 * np.pi * argvals), np.cos(2 * np.pi * argvals)])

fdata = DenseFunctionalData(
    argvals=DenseArgvals({"input_dim_0": argvals}), values=DenseValues(X)
)

_ = plot(fdata)

For two-dimensional functional data

Second, we will defined two-dimensional functional data. We consider a hundred sampling points from both dimensions and two observations.

argvals = np.linspace(0, np.pi, num=100)
X = np.array(
    [
        np.outer(np.sin(argvals), np.cos(argvals)),
        np.outer(np.sin(-argvals), np.cos(argvals)),
    ]
)

fdata = DenseFunctionalData(
    argvals=DenseArgvals({"input_dim_0": argvals, "input_dim_1": argvals}),
    values=DenseValues(X),
)

_ = plot(fdata)

We could use this representation for whatever data dimension. The plotting is however not easy for data with dimension higher than two.