Add SmoothedConstantInterpolation

[SouthEndMusic]

Fixes #364.

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

using DataInterpolations
using Plots
using Random

N = 10
Random.seed!(8)
u = rand(N)
t = cumsum(rand(N))
A1 = ConstantInterpolation(u, t)

p = plot()
plot!(A1)

for d_max in 00.05:0.05:0.2
    A2 = SmoothedConstantInterpolation(u, t, cache_parameters = true; d_max)
    plot!(A2)
end

p

[SouthEndMusic]

Fun feature; if SmoothedConstantInterpolation is used with extrapolation = ExtrapolationType.Periodic, then the transitions are also smooth:

using DataInterpolations
using Plots
using Random

N = 4
Random.seed!(8)
u = rand(N)
t = cumsum(rand(N))
Δt = t[end] - t[1]
t_eval = range(first(t) - Δt, last(t) + Δt, length = 500)

A = SmoothedConstantInterpolation(u, t; extrapolation = ExtrapolationType.Periodic, d_max = 0.2)

plot(t_eval, A.(t_eval))
scatter!(t[1:end-1], u[1:end-1]; label = "data")
scatter!(t[1:end-1] .+ Δt, u[1:end-1]; label = "data one period forward")
scatter!(t[1:end-1] .- Δt, u[1:end-1]; label = "data one period back")

[ChrisRackauckas]

even for extrapolation?

[SouthEndMusic]

Yes, also for extrapolation

[sathvikbhagavan]

This line can now be removed/edited?

[ChrisRackauckas]

Conflicts?

[SouthEndMusic]

@visr I added the function get_transition_ts to know which tstops to pass to the integrator so that it doesn't jump over the smooth transitions:

[ChrisRackauckas]

What's the reason the end is ignored and it doesn't just constant extrapolate?

[SouthEndMusic]

The key properties of this interpolation type are that it is differentiable everywhere and sufficiently close to constant interpolation. I achieve the diffentiability by replacing the stepwise transition in each data point by a differentiable transition in an interval around each data point.

What that means for extrapolation is a bit murky. I already made it so that the extrapolation is smooth in the case of periodic extrapolation, which means that the interpolation around the edges of the domain has to account for that. I'd argue that for constant extrapolation, ignoring the last data point is also best. Maybe for extension extrapolation, the last data point can be used, but that feels a bit inconsistent.

[ChrisRackauckas]

I'd like to hear @sathvikbhagavan 's thoughts, because at least to me constant extrapolation with the last point feels natural

[sathvikbhagavan]

If I understand correctly, can't we do the same transition from second last to the last point after which it becomes constant? This is strictly not constant extrapolation as values would be different from the last point (for a small interval) as it needs to be C1 smooth everywhere

[SouthEndMusic]

I made it now so that both Constant and Extension extrapolation use u[end], e.g.

using DataInterpolations
using Plots
using Random

N = 4
Random.seed!(10)
u = rand(N)
t = cumsum(rand(N))
Δt = t[end] - t[1]
t_eval = range(first(t) - Δt, last(t) + Δt, length=500)

A = SmoothedConstantInterpolation(u, t; extrapolation=ExtrapolationType.Extension, d_max=0.2, cache_parameters=true)

plot(t_eval, A.(t_eval), label="interpolation with Extension extrapolation")

scatter!(t, u, label="Data")

[SouthEndMusic]

I don't understand why the derivative tests are failing 🤔

[SouthEndMusic]

fixed it 👍

[ChrisRackauckas]

I think I'm happy. @sathvikbhagavan ?

[sathvikbhagavan]

A couple of clarifications

[sathvikbhagavan]

This line can now be removed/edited?

[sathvikbhagavan]

comment about u[end] - it can be removed? When is u[end] ignored now?