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Data preordering in generalized PAV algorithm for monotonic regression
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0003-1836-4200
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
2006 (English)In: Journal of Computational Mathematics, ISSN 0254-9409, Vol. 24, no 6, 771-790 p.Article in journal (Refereed) Published
Abstract [en]

Monotonic regression (MR) is a least distance problem with monotonicity constraints induced by a partially ordered data set of observations. In our recent publication [In Ser. {\sl Nonconvex Optimization and Its Applications}, Springer-Verlag, (2006) {\bf 83}, pp. 25-33], the Pool-Adjacent-Violators algorithm (PAV) was generalized from completely to partially ordered data sets (posets). The new algorithm, called GPAV, is characterized by the very low computational complexity, which is of second order in the number of observations. It treats the observations in a consecutive order, and it can follow any arbitrarily chosen topological order of the poset of observations. The GPAV algorithm produces a sufficiently accurate solution to the MR problem, but the accuracy depends on the chosen topological order. Here we prove that there exists a topological order for which the resulted GPAV solution is optimal. Furthermore, we present results of extensive numerical experiments, from which we draw conclusions about the most and the least preferable topological orders.

Place, publisher, year, edition, pages
2006. Vol. 24, no 6, 771-790 p.
National Category
URN: urn:nbn:se:liu:diva-36278Local ID: 30826OAI: diva2:257126
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2015-06-02
In thesis
1. Monotonic regression for large multivariate datasets
Open this publication in new window or tab >>Monotonic regression for large multivariate datasets
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Monoton regression för stora multivariata datamateriaI
Abstract [en]

Monotonic regression is a non-parametric statistical method that is designed especially for applications in which the expected value of a response variable increases or decreases in one or more explanatory variables. Such applications can be found in business, physics, biology, medicine, signal processing, and other areas. Inasmuch as many of the collected datasets can contain a very large number of multivariate observations, there is a strong need for efficient numerical algorithms. Here, we present new methods that make it feasible to fit monotonic functions to more than one hundred thousand data points. By simulation, we show that our algorithms have high accuracy and represent  considerable improvements with respect to computational time and memory requirements. In particular , we demonstrate how segmentation of a large-scale problem can greatly improve the performance of existing algorithms. Moreover, we show how the uncertainty of a monotonic regression model can be estimated. One of the procedures we developed can be employed to estimate the variance of the random error present in the observed response. Other procedures are based on resampling  techniques and can provide confidence intervals for the expected response at given levels of a set of predictors.

Abstract [sv]

Monoton regression är en icke-parametrisk statistisk metod som är utvecklad speciellt för tillämpningar i vilka det förväntade värdet aven responsvariabel ökar eller minskar med en eller flera förklaringsvariabler. Sådana tillämpningar finns inom företagsekonomi, fysik, biologi, medicin, signalbehandling och andra områden. Eftersom många insamlade datamaterial kan innehålla ett mycket stort antal multivariata observationer finns ett starkt behov av effektiva numeriska algoritmer. Här presenterar vi nya metoder som gör det möjligt att anpassa monotona funktioner till mer än 100000 datapunkter. Genom simulering visar vi. att våra algoritmer har hög noggrannhet och innebär betydande förbättringar med avseende på beräkningstid och krav på minnesutrymme. Speciellt visar vi hur segmentering av ett storskaligt problem starkt kan förbättra existerande algoritmer. Dessutom visar vi hur osäkerheten aven monoton regressions modell kan uppskattas. En av de metoder vi utvecklat kan användas för att uppskatta variansen för de slumpkomponenter som kan finnas i den observerade responsvariabeln. Andra metoder, baserade på s.k. återsampling, kan ge konfidensintervall för den förväntade responsen för givna värden på ett antal prediktorer.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2010. 75 p.
Linköping Studies in Statistics, ISSN 1651-1700 ; 11Linköping Studies in Arts and Science, ISSN 0282-9800 ; 514
National Category
Probability Theory and Statistics
urn:nbn:se:liu:diva-65349 (URN)978-91-7393-412-1 (ISBN)
Public defence
2010-04-16, Glashuset, Building B, Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Available from: 2011-02-04 Created: 2011-02-04 Last updated: 2012-11-08Bibliographically approved

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