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.. _isotonic:
===================
Isotonic regression
===================
.. currentmodule:: sklearn.isotonic
The class :class:`IsotonicRegression` fits a non-decreasing real function to
1-dimensional data. It solves the following problem:
.. math::
\min \sum_i w_i (y_i - \hat{y}_i)^2
subject to :math:`\hat{y}_i \le \hat{y}_j` whenever :math:`X_i \le X_j`,
where the weights :math:`w_i` are strictly positive, and both `X` and `y` are
arbitrary real quantities.
The `increasing` parameter changes the constraint to
:math:`\hat{y}_i \ge \hat{y}_j` whenever :math:`X_i \le X_j`. Setting it to
'auto' will automatically choose the constraint based on `Spearman's rank
correlation coefficient
<https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient>`_.
:class:`IsotonicRegression` produces a series of predictions
:math:`\hat{y}_i` for the training data which are the closest to the targets
:math:`y` in terms of mean squared error. These predictions are interpolated
for predicting to unseen data. The predictions of :class:`IsotonicRegression`
thus form a function that is piecewise linear:
.. figure:: ../auto_examples/miscellaneous/images/sphx_glr_plot_isotonic_regression_001.png
:target: ../auto_examples/miscellaneous/plot_isotonic_regression.html
:align: center
.. rubric:: Examples
* :ref:`sphx_glr_auto_examples_miscellaneous_plot_isotonic_regression.py`
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