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# Meta Model Contribution (MMC & BMC)

## What is MMC & BMC?

Meta Model Contribution (MMC) is the covariance of a model with the target, after its predictions have been neutralized to the Meta Model. Similarly, Benchmark Model Contribution (BMC) is the covariance of a model with the target, after its predictions have been neutralized to the stake-weighted Benchmark Models.
These metrics tell us how the unique component of a model contributes to the correlation of the Meta Model (or the Benchmark Models in the case of BMC). By neutralizing the model's predictions by the Meta Model or Benchmark Models, the remaining orthogonal component's covariance with the target is that model's contribution.

## Calculation

To calculate a user's MMC for a given round we
• Normalize the predictions in their submission
• Normalize the Meta Model
• Neutralize their submission to the Meta Model
• Find the covariance of the neutral submission with the target
```python
def contribution(
predictions: pd.DataFrame,
meta_model: pd.Series,
live_targets: pd.Series,
) -> pd.Series:
"""Calculate the contributive correlation of the given predictions
wrt the given meta model.
Then calculate contributive correlation by:
1. tie-kept ranking each prediction and the meta model
2. gaussianizing each prediction and the meta model
3. orthogonalizing each prediction wrt the meta model
4. multiplying the orthogonalized predictions and the targets
Arguments:
predictions: pd.DataFrame - the predictions to evaluate
meta_model: pd.Series - the meta model to evaluate against
live_targets: pd.Series - the live targets to evaluate against
Returns:
pd.Series - the resulting contributive correlation
scores for each column in predictions
"""
# filter and sort preds, mm, and targets wrt each other
meta_model, predictions = filter_sort_index(meta_model, predictions)
live_targets, predictions = filter_sort_index(live_targets, predictions)
live_targets, meta_model = filter_sort_index(live_targets, meta_model)
# rank and normalize meta model and predictions so mean=0 and std=1
p = gaussian(tie_kept_rank(predictions)).values
m = gaussian(tie_kept_rank(meta_model.to_frame()))[meta_model.name].values
# orthogonalize predictions wrt meta model
neutral_preds = orthogonalize(p, m)
# center the target
live_targets -= live_targets.mean()
# multiply target and neutralized predictions
# this is equivalent to covariance b/c mean = 0
mmc = (live_targets @ neutral_preds) / len(live_targets)
return pd.Series(mmc, index=predictions.columns)
```