Add functionality for array-event-wise aggregation of dl1 image parameters

[LukasBeiske]

To define event types based on the quality of the direction reconstruction, machine learning models which predict the error of the direction reconstruction for array events are needed. Such models use the mean, standard deviation, maximum, and minimum value of some DL1 image parameters as input. This adds the functionality to compute these input features, while #2503 adds the reconstruction of the error itself.

To be more specific:

• This adds a new Component and a Tool for aggregating the telescope-wise dl1 image parameters for each array event.
• BaseStatisticsContainer is introduced which does not contain the higher order moments present in StatisticsContainer
• The helper functions for vectorizing numpy calculations in ctapipe.reco.stereo_combination are refactored and expanded in a new module called ctapipe.vectorization

TODO:

• [x] Add/ update unit tests

[codecov[bot]]

Codecov Report

Attention: Patch coverage is 91.04478% with 30 lines in your changes are missing coverage. Please review.

Project coverage is 92.47%. Comparing base (e2a848e) to head (0fcecaa). Report is 15 commits behind head on main.

:exclamation: Current head 0fcecaa differs from pull request most recent head 16f184e. Consider uploading reports for the commit 16f184e to get more accurate results

Files	Patch %	Lines
src/ctapipe/vectorization/aggregate.py	63.49%	23 Missing :warning:
src/ctapipe/image/statistics.py	91.07%	5 Missing :warning:
src/ctapipe/tools/aggregate_features.py	94.87%	2 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## main #2497 +/- ## ========================================== - Coverage 92.66% 92.47% -0.19% ========================================== Files 232 242 +10 Lines 20220 20268 +48 ========================================== + Hits 18736 18743 +7 - Misses 1484 1525 +41 ```

:umbrella: View full report in Codecov by Sentry.
:loudspeaker: Have feedback on the report? Share it here.

[kosack]

Can you explain better what this is and why is needed in the description? What does it mean to "Aggregate DL1 parameters"? and what is being aggregated over (over telescopes in an event, or over time, both?)

[Tobychev]

What does it mean to "Aggregate DL1 parameters"? and what is being aggregated over (over telescopes in an event, or over time, both?)

It looks like it groups things by trigger, so I guess it is for getting average intensity and width and such, maybe?

[kosack]

I don't quite understand how this will be used, but perhaps I need to see an example. I imagine when you say "mean", you really mean "weighted mean", weighted by the impact parameter and intensity, so basically the same as the "mean reduced scaled {width, length}" used in standard non-machine-learning gamma-hadron separation methods in HESS and VERITAS (but not MARS, if I understand correctly)? If that's true, we've basically come full-circle and re-invented those.

I only ask because the original reason we decided not to compute these mean-reduced-scaled parameters was that they were not needed, as using telescope-wise machine-learning models were better for predicting such a quantity (like "single-telescope reconstruction error"), which can then be merged in a weighted average like we do for energy and gammaness. So are we now undoing that design decision?

[LukasBeiske]

which can then be merged in a weighted average like we do for energy and gammaness. So are we now undoing that design decision?

AFAIK these averaged features are only to be used as input for the model estimating the error of the direction reconstruction. While the event type based on the background contamination will be using the averaged gammaness scores predicted by the telescope models. Therefore, it would also be possible to only compute these averaged features on-the-fly when training and applying this direction reconstruction error estimator, similar to the existing FeatureGenerator.

The separate tool to calculate and write these averaged features into files is only meant to keep this separate from the implementation of the direction reconstruction error estimator itself. If there really is no other use for these averaged features, the on-the-fly solution might be better.

[maxnoe]

@kosack this is part of the implementation of the event types based approach developed by @TarekHC and @JBernete.

It works by training a machine learning regressor on feature table for subarray events. Part of that feature table are multiplicities, the reconstructed values themselves but also these aggregated image features over the telescopes.

[kosack]

AFAIK these averaged features are only to be used as input for the model estimating the error of the direction reconstruction. While the event type based on the background contamination will be using the averaged gammaness scores predicted by the telescope models.

Yes, that I understand, the question is more why is being done using mean-scaled array parameters rather than using per-telescope estimators that are then combined, which is how we do the other machine learning methods. At least then all ML methods would be symmetric and there would be no need for this extra complexity.

In fact, I really expected to see something that leverages how the reconstruction algorithm actually works, for example computing some aggregate parameters not on the telescopes, but on the telescope pairs used in reconstruction. For example, computing the mean/std/min/max of the delta-phi angle between image pairs. Wouldn't that have much more power to estimate the uncertainty? To first order, the reconstruction error for a 2-telescope pair is just related to this angle: small is poorly reconstructed, and large is better reconstructed, which is why we weight by cos(delta_phi)--implicitly via the cross-product--in the algorithm itself, which leads to lower reconstruction uncertainty.

Anyhow, it might not be too hard to add pair-wise aggregation to this as well if needed.

[LukasBeiske]

codecov fails because of the new numba jitted code and the already existing numba functions in image.statistics

[LukasBeiske]

In fact, I really expected to see something that leverages how the reconstruction algorithm actually works, for example computing some aggregate parameters not on the telescopes, but on the telescope pairs used in reconstruction. For example, computing the mean/std/min/max of the delta-phi angle between image pairs. Wouldn't that have much more power to estimate the uncertainty? To first order, the reconstruction error for a 2-telescope pair is just related to this angle: small is poorly reconstructed, and large is better reconstructed, which is why we weight by cos(delta_phi)--implicitly via the cross-product--in the algorithm itself, which leads to lower reconstruction uncertainty.

Since this is specific to the reconstruction algorithm, it might be better to do something like this separately, as part of the reconstruction algorithm itself, or as part of the error estimator in #2503. In the latter case, delta_phi_{mean,std,max,min} could than be used as additional input features for the estimator and it then might be interesting to see how much more accurate the output of the estimator is compared to just using delta_phi_mean as error estimate.

I can't think of a way to calculate these angles efficiently on tabular data, so integrating it into the reconstructor might be best.

[LukasBeiske]

flake8 fails now because of already existing lines being to long. Is this something I should fix here, or would it be better to fix this everywhere in a separate PR?