Application of Isotonic Regression in Predicting Business Risk Scores

An isotonic regression model fits an isotonic function of the explanatory variables to estimate the expectation of the response variable. In other words, as the function increases, the estimated expectation of the response must be non-decreasing. With this characteristic, isotonic regression could be a suitable option to analyze and predict business risk scores. A current challenge of isotonic regression is the decrease of performance when the model is fitted in a large data set e.g. more than four or five dimensions. This paper attempts to apply isotonic regression models into prediction of business risk scores using a large data set – approximately 50 numeric variables and 24 million observations. Evaluations are based on comparing the new models with a traditional logistic regression model built for the same data set. The primary finding is that isotonic regression using distance aggregate functions does not outperform logistic regression. The performance gap is narrow however, suggesting that isotonic regression may still be used if necessary since isotonic regression may achieve better convergence speed in massive data sets

How to project onto the monotone extended second order cone

This paper introduce the monotone extended second order cone (MESOC), which is related to the monotone cone and the Lorentz cone. Some properties of MESOC are presented and its dual cone is computed. Formulas for projecting onto MESOC are also presented. In the most general case the formula for projecting onto MESOC depends on an equation for one real variable.Comment: 9 page

Sensitivity analysis in isotonic regression

AbstractThe isotonic regression problem is a specially structured quadratic programming problem which arises in various fields, such as production planning, inventory control, psychometry and statistics. The underlying graphical structure of the problem permits the development of easy and fast combinatorial solution algorithms. In this paper, we exploit this underlying structure, to develop efficient combinatorial methods for performing sensitivity analysis on the isotonic regression problem

Contrast Invariant SNR

We design an image quality measure independent of local contrast changes, which constitute simple models of illumination changes. Given two images, the algorithm provides the image closest to the first one with the component tree of the second. This problem can be cast as a specific convex program called isotonic regression. We provide a few analytic properties of the solutions to this problem. We also design a tailored first order optimization procedure together with a full complexity analysis. The proposed method turns out to be practically more efficient and reliable than the best existing algorithms based on interior point methods. The algorithm has potential applications in change detection, color image processing or image fusion. A Matlab implementation is available at http://www.math.univ-toulouse.fr/ ∼ weiss/PageCodes.html

Least Squares and Shrinkage Estimation under Bimonotonicity Constraints

In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r-by-s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j) in {1,...,r}x{1,...,s}. Various numerical examples illustrate our methods