[ENH] Parameterise the cost function of dtw and adtw distance measures

[itsdivya1309]

Reference Issues/PRs

Closes: #1874

What does this implement/fix? Explain your changes.

As we discussed on the slack, I've implemented the parameterised DTW and ADTW as private functions to be used in the upcoming Proximity Forest 2.0.

Does your contribution introduce a new dependency? If yes, which one?

Any other comments?

PR checklist

For all contributions

• I've added myself to the list of contributors. Alternatively, you can use the @all-contributors bot to do this for you.
• The PR title starts with either [ENH], [MNT], [DOC], [BUG], [REF], [DEP] or [GOV] indicating whether the PR topic is related to enhancement, maintenance, documentation, bugs, refactoring, deprecation or governance.

For new estimators and functions

• I've added the estimator to the online API documentation.
• (OPTIONAL) I've added myself as a __maintainer__ at the top of relevant files and want to be contacted regarding its maintenance. Unmaintained files may be removed. This is for the full file, and you should not add yourself if you are just making minor changes or do not want to help maintain its contents.

For developers with write access

• (OPTIONAL) I've updated aeon's CODEOWNERS to receive notifications about future changes to these files.

[aeon-actions-bot]

Thank you for contributing to aeon

I did not find any labels to add based on the title. Please add the [ENH], [MNT], [BUG], [DOC], [REF], [DEP] and/or [GOV] tags to your pull requests titles. For now you can add the labels manually.
I have added the following labels to this PR based on the changes made: [ $\color{#5209C9}{\textsf{distances}}$ ]. Feel free to change these if they do not properly represent the PR.

The Checks tab will show the status of our automated tests. You can click on individual test runs in the tab or "Details" in the panel below to see more information if there is a failure.

If our pre-commit code quality check fails, any trivial fixes will automatically be pushed to your PR unless it is a draft.

Don't hesitate to ask questions on the aeon Slack channel if you have any.

PR CI actions

These checkboxes will add labels to enable/disable CI functionality for this PR. This may not take effect immediately, and a new commit may be required to run the new configuration.

• Run pre-commit checks for all files
• Run all pytest tests and configurations
• Run all notebook example tests
• Run numba-disabled codecov tests
• Stop automatic pre-commit fixes (always disabled for drafts)
• Push an empty commit to re-run CI checks

[hadifawaz1999]

maybe these can go in distance module as parameter to the original dtw based functions ? not saying in this PR as i know its only used for PF2 for now, but future perspective, what do you think @chrisholder ? however we should have a discussion on how it can affect BA stuff (some math related stuff if we go into this as a general parameter)

[MatthewMiddlehurst]

FYI there were some large conversations regarding this in the core_devs and distances Slack channels. It is not really compatible with the way those functions are currently set up, best to take a look at there.

For now we will do it like this just to move things forward.

[MatthewMiddlehurst]

I assume these are the same functions as your previous changes, but just in a new fine @itsdivya1309? If so I dont really see any issue for now.

[itsdivya1309]

I assume these are the same functions as your previous changes, but just in a new fine @itsdivya1309? If so I dont really see any issue for now.

Yes, that's right.

[MatthewMiddlehurst]

I think it looks fine for now, but I would put it in the same file as PF 2.0 rather than its own.

[MatthewMiddlehurst]

replaced by #1978