[ENH] - Naive variance prediction estimator

[IlyasMoutawwakil]

Reference Issues/PRs

#1687

What does this implement/fix? Explain your changes.

@fkiraly
As part of the probabilistic forecasting API, this will implement the _predict_var method to forecasters

What should a reviewer concentrate their feedback on?

STEP probabilistic forecasting API

PR checklist

• add _predict_var and its corresponding tag to the forecasting extension template.
• add NaiveVariance composite class which implements the 'simple strategy' for variance prediction.
• implement _predict_var and its corresponding in NaiveVariance.
• implement _predict_quantiles in NaiveVariance where quantiles are inferred from a normal distribution.
• implement covariance prediction (cov=True)

For all contributions

• I've added unit tests and made sure they pass locally.

[IlyasMoutawwakil]

@fkiraly I have created the naive variance estimator and tested it with some forecasters but there some cases where it won't work:

• if the forecaster can't do in sample forecasts on all of self._y.index, which can be solved (won't be able to compute variance for in sample forecasts)
• if there's not enough training points (self._y.index[-1] < fh[-1])

[fkiraly]

Good start!

Regarding the design:

• you should also implement predict_var (without underscore) in the BaseForecaster. Boilerplate methods like check_fh should go there - see how the other predict_etc methods are implemented.
• I think it´s better to return not a numpy.ndarray, but a pd.DataFrame (so the index is preserved)

I also left comments in the code.

Some bugs not mentioned there:

• your code does not work if the forecaster is multivariate
• I think your _predict_quantiles is incorrect
• is your _predict_var implementing the formula we discussed? I think not? I also recommend to put the math in the docstring.

[fkiraly]

if the forecaster can't do in sample forecasts on all of self._y.index, which can be solved (won't be able to compute variance for in sample forecasts)

I´m slightly confused - are your predictions not all out of sample, with a rolling window?

[IlyasMoutawwakil]

I´m slightly confused - are your predictions not all out of sample, with a rolling window?

they are all out of sample now. I actually made it compute in sample as well so that it would be possible to compute in sample variance but I removed it now.

[IlyasMoutawwakil]

Finally found why predict quantiles and intervals tests were failing. It was because I didn't use pd.MultiIndex to index quantiles and instead used plain indexes and columns

[fkiraly]

I made some minor changes to the base class docstrings, extension template, and signature, to ensure everything is coherent after the batch of recent merges.

[fkiraly]

Thanks for implementing!

Excellent, does not seem to require any additional changes imo.

I took the liberty of ensuring that this consistently merges with a number of other PR on probabilistic forecasting recently merged. Mostly docstring changes, but I also made sure that the signatures are consistent.