1.15. Isotonic regression

The class IsotonicRegression fits a non-decreasing real function to 1-dimensional data. It solves the following problem:

minimize \(\sum_i w_i (y_i - \hat{y}_i)^2\)

subject to \(\hat{y}_i \le \hat{y}_j\) whenever \(X_i \le X_j\),

where the weights \(w_i\) are strictly positive, and both X and y are arbitrary real quantities.

The increasing parameter changes the constraint to \(\hat{y}_i \ge \hat{y}_j\) whenever \(X_i \le X_j\). Setting it to ‘auto’ will automatically choose the constraint based on Spearman’s rank correlation coefficient.

IsotonicRegression produces a series of predictions \(\hat{y}_i\) for the training data which are the closest to the targets \(y\) in terms of mean squared error. These predictions are interpolated for predicting to unseen data. The predictions of IsotonicRegression thus form a function that is piecewise linear: