Prediction contributions

A vector of contributions of each rating table to the prediction for every input object, and the base value of the model (its intercept).

• \(base\_value\) is the intercept \(f_0\) of the model.
• The contribution of table \(u\) to object \(x\) is \(f_u(x_u)\), the value of the cell the object falls in.

For a given object the sum of the base value and the contributions is equal to the raw score of the model on this object, and applying the inverse link gives the prediction:

\[ base\_value + \sum\limits_{u \in U} f_{u}(x_{u}) = a(x), \qquad prediction = link^{-1}(a(x)) \]

The decomposition is exact, because the deployed model is its rating tables: the contributions are not estimated. For a model with a log link, the exponentials of the contributions are the multiplicative relativities of the object.

Contributions by feature

With split_interactions=True, each interaction is shared equally among its features, giving one contribution per input feature:

\[ \phi_{j}(x) = \sum\limits_{u \ni j} \frac{f_{u}(x_{u})}{|u|} \]

Because every table is purified, these are exact interventional Shapley values of the raw score: no sampling and no approximation is involved.

Exposure and offset

exposure and offset passed to predict_contributions appear as contributions of their own (log(exposure) and the offset), so the identity holds for the raw score including them. With the exposure, the prediction is the expected total for the object rather than the rate per unit of exposure.

Usage examples

records = model.predict_contributions(test_data.head(5))
row = records[0]
print(row["base_value"], row["prediction_value"])
for term in row["contributions"]:
    print(term["term"], term["feature_value"], term["contribution"])

See predict_contributions for the parameters and the output formats. The output matches rustystats' GLMModel.predict_contributions.