[ENH] TAR forecaster

[TonyBagnall]

Resolves #2817

Implements a numba TAR and AutoTAR forecaster. TAR is simple, if not a bit odd to me.

Given or find a threshold value. All points where previous delay point was below the threshold go in one series ("regime"), all values above in another. Fit separate AR models to each.

Also refactors TVPForecaster to just TVP for consistency

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[TonyBagnall]

comparing to our current optimised for speed ARIMA and current SETAR tree. MSE on simulated low order ARMA. Cheating a bit as giving TAR the correct AR order (doesnt fit MA though).

runtime

super fast TAR, but also super simple. Not trying to make the definitive version, if someone is a fan of the forecaster they might do so, Im only interested in potential components. Also shows we can optimise SETAR tree.

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[alexbanwell1]

Can't comment much on the actual correctness of the model, but it seems from the graph that it is doing the correct thing, but otherwise looks good

[TonyBagnall]

so this has got more complex. It now includes TAR: fixed parameters with optional defaults and AutoTAR, which finds the parameters. ChatGPT gave me a much more efficient way of auto fitting based on storing sum of squares (reminded me of shapelets). Its complex, so I have tested this against a grid search using the basic one, output identical (tested for each value of n), run times much faster (30x-70x),

For correctness, there are no proper TAR implementations in python to compare against and I will not go into R. It should be as good as AR/ARMA though, which gives us a core experiment

AutoTAR comparable in ARMA data, which you would hope.

Interestingly faster than ARMA, which has fixed p and q, which is the fastest ARMA I could find. much faster than our current SETAR, which I will optimise. Setting AR parameters with OLS is very fast