[Feature Request] Integration procedures for Interpolation and LinearRegression

[bvenn]

Integrals can easily be calculated for step functions, linear splines, polynomials and splines. These procedures are missing for most of the interpolation and regression modules.

Important: unit tests should cover both, negative x and negative y values!

Interpolation

• step interpolation
• akima interpolation
• hermite interpolation
• polynomial interpolation
• cubic spline interpolation

Regression

• simple linear regression
• polynomial regression
• smoothing spline
• (optional) robust simple linear regression

[github-actions]

🤖 This is an automated response from Repo Assist.

I've created a draft PR implementing several of the integration functions from this checklist. Here's the current status:

Interpolation

• Step interpolation — implemented in this PR
• Akima interpolation — Akima.integrate already existed ✓
• Hermite interpolation — still missing
• Polynomial interpolation — Polynomial.integrate already existed ✓
• Cubic spline interpolation — implemented in this PR

I also added LinearSpline.integrate (not in the original checklist but a natural addition).

Regression — these remain unimplemented (simple linear regression, polynomial regression, smoothing spline).

The PR uses closed-form antiderivatives for all three new functions and includes 5 tests against known analytical results (1198/1198 pass). The Hermite case is more involved since it requires expressing the integral in terms of the cubic Hermite basis functions; happy to tackle that in a follow-up if useful.

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[annacwe]

PR #375 added functions to calculate definite integrals for LinearSpline, Step, and CubicSpline. Akima also has definite integrals already.
I also think that for Polynomial interpolation, both the integral function as well as the definite integrals have been added in 6a9038e. Unit tests for that seem to be missing though.

@bvenn Just to better my understanding going forward, is this issue specifically asking to implement calculations for definite integrals or for integral functions (or both)?