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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: