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Updated: Mar 19, 2026

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Published on: December 10, 2014
Projections Onto Order Simplexes and Isotonic Regression
1Mathematical and Computational Science Division, National Institute of Standards and Technology, Gaithersburg, MD 20899.
Abstract:
Isotonic regression is the problem of fitting data to order constraints. This problem can be solved numerically in an efficient way by successive projections onto order simplex constraints. An algorithm for solving the isotonic regression using successive projections onto order simplex constraints was originally suggested and analyzed by Grotzinger and Witzgall. This algorithm has been employed repeatedly in a wide variety of applications. In this paper we briefly discuss the isotonic regression problem and its solution by the Grotzinger-Witzgall method. We demonstrate that this algorithm can be appropriately modified to run on a parallel computer with substantial speed-up. Finally we illustrate how it can be used to pre-process mass spectral data for automatic high throughput analysis.
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In regression analysis, a regression equation is determined based on the line of best fit– a line that best fits the data points plotted in a graph. This line is also called the regression line. The algebraic equation for the regression line is called the regression equation. It is represented as:
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If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...