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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:

min \sum_{i} w_{i} (y_{i} - {\hat{y}}_{i})^{2}

subject to ${\hat{y}}_{i} \leq {\hat{y}}_{j}$ whenever $X_{i} \leq X_{j}$ , where the weights $w_{i}$ are strictly positive, and both X and y are arbitrary real quantities.

The increasing parameter changes the constraint to ${\hat{y}}_{i} \geq {\hat{y}}_{j}$ whenever $X_{i} \leq X_{j}$ . 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:

../_images/sphx_glr_plot_isotonic_regression_001.png