Creating a new interpolation or extrapolation strategy

Shows how to add new interpolation and extrapolation strategies.

# Author: Carlos Ramos Carreño
# License: MIT

In this example, we want to showcase how it is possible to make new interpolation and extrapolation strategies. These are Python callables with the following prototype:

def evaluator_prototype(fdata, eval_points, *, aligned):
    """Prototype of a extrapolation/interpolation strategy."""

Here, fdata is a FData object, èval_points is a NumPy array with the points at which this object will be evaluated, and aligned is a boolean parameter indicating if the points are common for all samples or different for each.

For example, lets consider for illustration purposes only an interpolation/extrapolation strategy that uses the Lagrange polynomial between the points of a FDataGrid. This is in general a bad idea, as when the number of points is high this polynomial has rapid oscillations between the measured points. Moreover, the implementation here is not vectorized and has low performance.

import numpy as np
from scipy.interpolate import lagrange


def evaluator_lagrange(fdata, eval_points, *, aligned):
    """Lagrange interpolation, for 1D FDataGrid only."""
    grid_points = fdata.grid_points[0]
    result = []

    for i, data in enumerate(fdata.data_matrix):
        polynomial = lagrange(grid_points, data)

        if aligned:
            # Same points for all observations.
            # The dimension is n_points x dim_domain (1 in this case).
            result.append(polynomial(eval_points))
        else:
            # Different points for every observation.
            # The dimension is n_samples x n_points x dim_domain.
            result.append(polynomial(eval_points[i]))

    return np.array(result)

We can now create a new FDataGrid and plot it. Note that the plot uses the specified interpolation between the measured points. Note also that is not necessary to specify the extrapolation, as by default for FDataGrid it already calls the interpolation if no extrapolation is defined.

import matplotlib.pyplot as plt

from skfda.representation import FDataGrid

X = FDataGrid(
    [[0, 1, 2], [0, 1, 8], [0, 0, 0]],
    grid_points=[0, 1, 2],
    interpolation=evaluator_lagrange,
)

fig = X.plot()
X.scatter(fig=fig)
plt.show()

We can try to evaluate the function at different points, including some between measurements or outside the original range, to test the interpolation and extrapolation.

X([-1, 0, 0.5, 1, 2, 3])

array([[[-1.  ],
        [ 0.  ],
        [ 0.5 ],
        [ 1.  ],
        [ 2.  ],
        [ 3.  ]],

       [[ 5.  ],
        [ 0.  ],
        [-0.25],
        [ 1.  ],
        [ 8.  ],
        [21.  ]],

       [[ 0.  ],
        [ 0.  ],
        [ 0.  ],
        [ 0.  ],
        [ 0.  ],
        [ 0.  ]]])

We can also try to evaluate each observation at different points to test this behavior.

X([[-1, 0], [0.5, 1], [2, 3]], aligned=False)

array([[[-1.  ],
        [ 0.  ]],

       [[-0.25],
        [ 1.  ]],

       [[ 0.  ],
        [ 0.  ]]])