`scipy.optimize._isotonic:isotonic_regression`

source: /scipy/optimize/_isotonic.py :15

Signature

   def     isotonic_regression (    y  :  npt.ArrayLike    ,  * ,     weights  :  npt.ArrayLike | None    =  None   ,    increasing  :  bool    =  True     )   →  OptimizeResult

Summary

Nonparametric isotonic regression.

Extended Summary

A (not strictly) monotonically increasing array x with the same length as y is calculated by the pool adjacent violators algorithm (PAVA), see ^[1]. See the Notes section for more details.

Parameters

y : (N,) array_like: Response variable.
weights : (N,) array_like or None: Case weights.
increasing : bool: If True, fit monotonic increasing, i.e. isotonic, regression. If False, fit a monotonic decreasing, i.e. antitonic, regression. Default is True.

Returns

res : OptimizeResult

The optimization result represented as a OptimizeResult object. Important attributes are:

x: The isotonic regression solution, i.e. an increasing (or decreasing) array of the same length than y, with elements in the range from min(y) to max(y).
weightsArray with the sum of case weights for each block (or pool) B.
blocks: Array of length B+1 with the indices of the start positions of each block (or pool) B. The j-th block is given by x[blocks[j]:blocks[j+1]] for which all values are the same.

Notes

Given data $y$ and case weights $w$ , the isotonic regression solves the following optimization problem:

argmin_{x_{i}} i \sum w_{i} (y_{i} - x_{i})^{2} subject to x_{i} \leq x_{j} whenever i \leq j .

For every input value $y_{i}$ , it generates a value $x_{i}$ such that $x$ is increasing (but not strictly), i.e. $x_{i} \leq x_{i + 1}$ . This is accomplished by the PAVA. The solution consists of pools or blocks, i.e. neighboring elements of $x$ , e.g. $x_{i}$ and $x_{i + 1}$ , that all have the same value.

Most interestingly, the solution stays the same if the squared loss is replaced by the wide class of Bregman functions which are the unique class of strictly consistent scoring functions for the mean, see ^[2] and references therein.

The implemented version of PAVA according to ^[1] has a computational complexity of O(N) with input size N.

Examples

This example demonstrates that ``isotonic_regression`` really solves a constrained optimization problem.

import numpy as np
from scipy.optimize import isotonic_regression, minimize
y = [1.5, 1.0, 4.0, 6.0, 5.7, 5.0, 7.8, 9.0, 7.5, 9.5, 9.0]
def objective(yhat, y):
    return np.sum((yhat - y)**2)
def constraint(yhat, y):
    # This is for a monotonically increasing regression.
    return np.diff(yhat)
result = minimize(objective, x0=y, args=(y,),
                  constraints=[{'type': 'ineq',
                                'fun': lambda x: constraint(x, y)}])

✓

result.x

✗

result = isotonic_regression(y)

✓

result.x

✗

The big advantage of ``isotonic_regression`` compared to calling ``minimize`` is that it is more user friendly, i.e. one does not need to define objective and constraint functions, and that it is orders of magnitudes faster. On commodity hardware (in 2023), for normal distributed input y of length 1000, the minimizer takes about 4 seconds, while ``isotonic_regression`` takes about 200 microseconds.