Issue 757: Vectorise GP stan code

[seabbs]

Description

This PR builds on #741 and closes #757 by vectorising the spectral density functions, using a single matern function, and introducing periodic kernals. It uses ideas found in https://avehtari.github.io/casestudies/Motorcycle/motorcycle_gpcourse.html.

It takes advantage of this refactor to do some prior predictive checking for the priors related to the GP, and then leverages the improved performance to reduce stan settings and relax initialisation constraints (see the news section).

Initial submission checklist

• My PR is based on a package issue and I have explicitly linked it.
• I have tested my changes locally (using devtools::test() and devtools::check()).
• I have added or updated unit tests where necessary.
• I have updated the documentation if required and rebuilt docs if yes (using devtools::document()).
• I have followed the established coding standards (and checked using lintr::lint_package()).
• I have added a news item linked to this PR.

After the initial Pull Request

• I have reviewed Checks for this PR and addressed any issues as far as I am able.

[jamesmbaazam]

Thanks @seabbs. I've now reviewed the code aspects as much as I can. I am currently exploring the various gp kernels locally and will report if I find anything worth discussing.

[github-actions]

This is how benchmark results would change (along with a 95% confidence interval in relative change) if 7532c83 is merged into main:

• 🚀default: 46.5s -> 23.7s [-63.53%, -34.48%]
• 🚀no_delays: 52.7s -> 28.3s [-71.8%, -20.95%]
• ✔️random_walk: 9.7s -> 10.6s [-1.39%, +20.1%]
• 🚀stationary: 21.3s -> 15.6s [-41.99%, -10.88%]
• 🚀uncertain: 1.22m -> 37.5s [-63.74%, -33.68%]
  Further explanation regarding interpretation and methodology can be found in the documentation.