Random Forest regression, and out-of-bag error estimation

The problem, from first principles

Bootstrap training leaves some rows out of every tree. For regression, tree outputs are averaged rather than voted, and the trees that omitted a row form a free, honest evaluation set for that row. This exercise records those omitted indices and uses only them for out-of-bag error.

From theory to code

Implement bootstrap sampling with omitted indices, train a forest of (tree, oob_indices) pairs, average all tree predictions, and calculate OOB mean squared error.

Constraints

• Bootstrap arrays have the original shapes and share one replacement-drawn index array.
• oob_indices contains original positions never drawn in that tree's bag.
• Train exactly n_trees with one advancing seeded generator.
• Regression forest prediction averages across trees along axis 0.
• OOB error uses only a sample's own OOB trees; samples with zero OOB predictions are excluded.
• Return OOB error as a Python float.

Hints

Open one at a time. Each gives away a little more than the last.