||This article provides insufficient context for those unfamiliar with the subject. (February 2012)|
Such constraints define partial order or total order and can be represented as a directed graph , where N is the set of variables involved, and E is the set of pairs (i, j) for each constraint . Thus, the IR problem corresponds to the following quadratic program (QP):
In the case when is a total order, a simple iterative algorithm for solving this QP is called the pool adjacent violators algorithm (PAVA). Best and Chakravarti (1990) have studied the problem as an active set identification problem, and have proposed a primal algorithm in O(n), the same complexity as the PAVA, which can be seen as a dual algorithm.
IR has applications in statistical inference, for example, to fit of an isotonic curve to mean experimental results when an order is expected. A benefit of isotonic regression is that it does not assume any form for the target function, such as linearity assumed by linear regression.
Another application is nonmetric multidimensional scaling (Kruskal, 1964), where a low-dimensional embedding for data points is sought such that order of distances between points in the embedding matches order of dissimilarity between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order.
Isotonic regression is also sometimes referred to as monotonic regression. Correctly speaking, isotonic is used when the direction of the trend is strictly increasing, while monotonic could imply a trend that is either strictly increasing or strictly decreasing.
Isotonic Regression under the for is defined as follows:
Simply ordered case 
To illustrate the above, let , and , and .
The isotonic estimator, , minimizes the weighted least squares-like condition:
Where is the unknown function we are estimating, and is a known function.
Software has been developed in the R statistical package for computing isotone (monotonic) regression.
- Best, M.J.; & Chakravarti N. (1990). "Active set algorithms for isotonic regression; a unifying framework". Mathematical Programming 47: 425–439. doi:10.1007/BF01580873.
- De Leeuw, Jan; K. Hornik, P. Mair (2009). "Isotone Optimization in R: Pool-Adjacent-Violators Algorithm (PAVA) and Active Set Methods". Journal of statistical software 32 (5): 1.
- Robertson, T.; Wright, F.T.; & Dykstra, R.L. (1988). Order restricted statistical inference. New York: Wiley, 1988. ISBN 0-471-91787-7
- Barlow, R. E.; Bartholomew, D.J.; Bremner, J. M.; & Brunk, H. D. (1972). Statistical inference under order restrictions; the theory and application of isotonic regression. New York: Wiley, 1972. ISBN 0-471-04970-0.
- Kruskal, J. B. (1964). "Nonmetric Multidimensional Scaling: A numerical method". Psychometrika 29 (2): 115–129. doi:10.1007/BF02289694.
- Shively, T.S., Sager, T.W., Walker, S.G. (2009). "A Bayesian approach to non-parametric monotone function estimation". Journal of the Royal Statistical Society, Series B 71 (1): 159–175. doi:10.1111/j.1467-9868.2008.00677.x.