Regression trees

Regression trees

Classic regression trees

In a classical regression tree, we follow a decison process as before, but the outcome is real number.

Within each leaf, all inputs are assigned that same number.

Training

With a regression problem we cannot split nodes the same way as we did for classification.

Instead by split by the residual sum of squares.

Training decision trees with Mean Squared Error (MSE)

Mixed regression trees

In classical trees all items in a leaf are assigned the same values. In this model, all are given \(\theta \) for a parametric model.

This makes the resulting trees smoother.

We have some \(\hat y_i = f(\mathbf x_i, \theta ) + \epsilon \)

The approach generalises classic regression trees. There the estimate was \(\bar y\). Here it’s a regression.

Training

At each node we do OLS. If the \(R^2\) of the model is less than some constant, we find a split which maximises the minimum of the two new \(R^2\).

Classifying with probabilistic decision trees

Previously our decision tree classifier was binary.

We can instead adapt the mixed tree model and using a probit model at each leaf.