Introduction

The cir-model library implements the Centered Isotonic Regression (CIR) model.[1] CIR is a variant of Isotonic Regression (IR), which is a nonparametric regression model which only assumes that the data generating function is monotonically increasing or decreasing. The function can take any monotonic form that minimizes the sum of squared errors. scikit-learn has implemented Isotonic Regression as IsotonicRegression.[2]

In some scenarios, the assumption of monotonicity is too weak. IR can result in a function that has constant, non-decreasing intervals. In some applications, like dose-response curve fitting or in cumululative distribution function estimation, it can be safe to make the additional assumption of strict monotonicity: at any point the function should be increasing, not just non-decreasing (or vice versa: decreasing, not just non-increasing). This additional requirement results in a function with the desirable property that the inverse function is a unique mapping, because there are no longer ranges of constant intervals.

Centered Isotonic Regression (CIR) is similar to IR, but with the additional assumption of strict monotonicity. The algorithm is described in detail in [1]. The plot below shows a comparison of IR (in blue) and CIR (in red) fitted to the same datapoints (in green).

This Python library, cir-model, implements CenteredIsotonicRegression in line with the algorithm in [1]. cir-model uses the IsotonicRegression implementation of scikit-learn. Therefore, CenteredIsotonicRegression takes the same parameters as IsotonicRegression and is completely compatible with scikit-learn. You can use it for example in scikit-learn pipelines.

You can install cir-model using pip:

pip install -U cir-model

Examples

To fit a CIR model:

>>> from cir_model import CenteredIsotonicRegression
>>> x = [1, 2, 3, 4]
>>> y = [1, 37, 42, 5]
>>> model = CenteredIsotonicRegression().fit(x, y)
>>> model.transform(x)
array([ 1. , 14.5, 28. , 28. ])

Finding the inverse of the CIR model above, for example for the value x for which y=25:

>>> from scipy import optimize
>>> optimize.newton(lambda x: model.transform([x]) - 25, 2)
array([2.77777778])

References

1. Centered Isotonic Regression: Point and Interval Estimation for Dose-Response Studies, Assaf P. Oron & Nancy Flournoy, Statistics in Biopharmaceutical Research, Volume 9, Issue 3, 258-267, 2017
2. Scikit-learn: Machine Learning in Python, Fabian Pedregosa et al., JMLR 12, 2825-2830, 2011