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Extrapolation methods

We will use the following interpolation to demonstrate the various extrapolation methods.

using DataInterpolations, Plots

u = [0.86, 0.65, 0.44, 0.76, 0.73]
t = [0.0022, 0.68, 1.41, 2.22, 2.46]
t_eval_left = range(-1, first(t), length = 25)
t_eval_right = range(last(t), 3.5, length = 25)
A = QuadraticSpline(u, t)
plot(A)
Example block output

Extrapolation behavior can be set left and right of the data simultaneously with the extrapolation keyword, or left and right separately with the extrapolation_left and extrapolation_right keywords respectively.

ExtrapolationType.None

This extrapolation type will throw an error when the input t is beyond the data in the specified direction.

ExtrapolationType.Constant

This extrapolation type extends the interpolation with the boundary values of the data u.

A = QuadraticSpline(u, t; extrapolation = ExtrapolationType.Constant)
plot(A)
plot!(t_eval_left, A.(t_eval_left); label = "extrapolation left")
plot!(t_eval_right, A.(t_eval_right); label = "extrapolation right")
Example block output

ExtrapolationType.Linear

This extrapolation type extends the interpolation with a linear continuation of the interpolation, making it $C^1$ smooth at the data boundaries.

A = QuadraticSpline(u, t; extrapolation = ExtrapolationType.Linear)
plot(A)
plot!(t_eval_left, A.(t_eval_left); label = "extrapolation left")
plot!(t_eval_right, A.(t_eval_right); label = "extrapolation right")
Example block output

ExtrapolationType.Extension

This extrapolation type extends the interpolation with a continuation of the expression for the interpolation at the boundary intervals for maximum smoothness.

A = QuadraticSpline(u, t; extrapolation = ExtrapolationType.Extension)
plot(A)
plot!(t_eval_left, A.(t_eval_left); label = "extrapolation down")
plot!(t_eval_right, A.(t_eval_right); label = "extrapolation up")
Example block output

ExtrapolationType.Periodic

this extrapolation type extends the interpolation such that A(t + T) == A(t) for all t, where the period is given by T = last(A.t) - first(A.t).

T = last(A.t) - first(A.t)
t_eval_left = range(first(t) - 2T, first(t), length = 100)
t_eval_right = range(last(t), last(t) + 2T, length = 100)
A = QuadraticSpline(u, t; extrapolation = ExtrapolationType.Periodic)
plot(A)
plot!(t_eval_left, A.(t_eval_left); label = "extrapolation down")
plot!(t_eval_right, A.(t_eval_right); label = "extrapolation up")
Example block output

ExtrapolationType.Reflective

this extrapolation type extends the interpolation such that A(t_ + t) == A(t_ - t) for all t_, t such that (t_ - first(A.t)) % T == 0 and 0 < t < T, where T = last(A.t) - first(A.t).

A = QuadraticSpline(u, t; extrapolation = ExtrapolationType.Reflective)
plot(A)
plot!(t_eval_left, A.(t_eval_left); label = "extrapolation down")
plot!(t_eval_right, A.(t_eval_right); label = "extrapolation up")
Example block output

Mixed extrapolation

You can also have different extrapolation types left and right of the data.

A = QuadraticSpline(u, t; extrapolation_left = ExtrapolationType.Reflective,
    extrapolation_right = ExtrapolationType.Periodic)
plot(A)
plot!(t_eval_left, A.(t_eval_left); label = "extrapolation left")
plot!(t_eval_right, A.(t_eval_right); label = "extrapolation right")
Example block output