Decomposing the trees into rating tables

Because of the constraints on the trees, the trained ensemble can be rewritten exactly as a functional ANOVA (fANOVA) decomposition: an intercept plus one table per main effect and per interaction.

The decomposition includes the following stages:

1. Grouping the trees by their features.

   A tree that uses the features \(u\) is a function of those features only: a lookup table on the grid formed by the borders it splits on. Adding up all the trees that use the same features gives one table per set of features, on the grid formed by all the borders the model uses for each feature.

2. Purification.

   A table on the features \(u\) can still contain parts that depend on fewer features. Each table is centred so that its weighted mean along every axis is zero under the reference measure, and the parts that are removed are pushed down to the tables of the subsets of \(u\), and finally to the intercept. This is done for the highest interaction order first.

The result is

\[ a(x) = f_0 + \sum\limits_{u \in U} f_{u}(x_{u}) \]

where every table \(f_u\) has zero weighted mean along each of its axes. The tables reproduce the model's predictions exactly, to floating-point tolerance, and no surrogate model is fitted. Each main effect carries as much of the signal as it can, and each interaction carries only what the main effects cannot express.

An effect whose dense table would be too large is kept in factored form: as the exact sum of the rank-one boxes the trees contributed, instead of as a dense table.

See Rating tables for how to export the tables and how an object finds its cell.