[ENH] Implement DTW with Global alignment

[tanishy7777]

Fixes #444

Added DTW with global invariance in aeon

Changes

• Implentation of DTW with GI added
• Uses Numba for JIT compilation

TODO:

• Add Tests

Comparision of dtw_gi implentations:

first implementation is implented in aeon
second implementation taken from https://rtavenar.github.io/hdr/parts/01/dtw/dtw_gi.html

The cost of 2nd is square root of the 1st cost($18.95^2$ = 359.20) because tslearn uses the Euclidean_distance but aeon uses squared Euclidean_distance

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[chrisholder]

Generally this looks good but we need to add a couple things to make it compatible with the rest of our distances.

First you need to define a function for:

• dtw_gi_distance - this returns the distance between two time series using dtw_gi distance
• dtw_gi_cost_matrix - this returns the cost matrix used to produce the distance measure
• dtw_gi_pairwise_distance - this returns a pairwise distance matrix for a single dataset or between two dataset if y specified
• dtw_gi_alignment_path - this should return the optimal alignment path between two time series for dtw_gi

We spoke on slack earlier about this as well. While you mentioned it would introduce very little new logic it's still nice to have these as defined functions. For example DDTW essentially just returns the same thing as DTW see https://github.com/aeon-toolkit/aeon/blob/main/aeon/distances/elastic/_ddtw.py#L328. When adding the above look at the DTW distance measure and just copy and paste the code is my recommendation.

Once you have added the above you can add your distance to our DISTANCE list which will mean the tests will automatically run for this distances. See: https://github.com/aeon-toolkit/aeon/blob/main/aeon/distances/_distance.py#L688. Once the functions above are defined you will add this to the list:

{
    "name": "dtw_gi",
    "distance": dtw_gi_distance,
    "pairwise_distance": dtw_gi_pairwise_distance,
    "cost_matrix": dtw_gi_cost_matrix,
    "alignment_path": dtw_gi_alignment_path,
    "type": DistanceType.ELASTIC,
    "symmetric": True,
    "unequal_support": True,
}

Once you're at this point give me a ping as then we can start sorting some further tests out which is a bit more fiddly to do so I'll help you with that. You can also run some tests locally to ensure your implementation is equivalent to the original.

Looks good overall though!

[tanishy7777]

Once you have added the above you can add your distance to our DISTANCE list which will mean the tests will automatically run for this distances. See: https://github.com/aeon-toolkit/aeon/blob/main/aeon/distances/_distance.py#L688. Once the functions above are defined you will add this to the list:

@chrisholder Thanks for the detailed comment! I will start working on this first thing tomorrow. I was going through the paper for the MVM distance today and made some progress on that. I should be done with the PR for MVM soon(atleast the initial draft).

I'll follow your suggestions for DTW-GI, refer to the DDTW implementation, and ensure everything is properly added into the DISTANCE list.

[tanishy7777]

I checked equivalence with this source https://rtavenar.github.io/hdr/parts/01/dtw/dtw_gi.html for multivariate, univariate and unequal length

[chrisholder]

Great addition thanks for your hard work on this!