| 1. |
- Sysoev, Oleg, et al.
(författare)
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A segmentation-based algorithm for large-scale partially ordered monotonic regression
- 2011
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Ingår i: Computational Statistics & Data Analysis. - : Elsevier Science B.V., Amsterdam.. - 0167-9473 .- 1872-7352. ; 55:8, s. 2463-2476
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Tidskriftsartikel (refereegranskat)abstract
- Monotonic regression (MR) is an efficient tool for estimating functions that are monotonic with respect to input variables. A fast and highly accurate approximate algorithm called the GPAV was recently developed for efficient solving large-scale multivariate MR problems. When such problems are too large, the GPAV becomes too demanding in terms of computational time and memory. An approach, that extends the application area of the GPAV to encompass much larger MR problems, is presented. It is based on segmentation of a large-scale MR problem into a set of moderate-scale MR problems, each solved by the GPAV. The major contribution is the development of a computationally efficient strategy that produces a monotonic response using the local solutions. A theoretically motivated trend-following technique is introduced to ensure higher accuracy of the solution. The presented results of extensive simulations on very large data sets demonstrate the high efficiency of the new algorithm.
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| 2. |
- Sysoev, Oleg, et al.
(författare)
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Bootstrap estimation of the variance of the error term in monotonic regression models
- 2013
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Ingår i: Journal of Statistical Computation and Simulation. - : Taylor & Francis Group. - 0094-9655 .- 1563-5163. ; 83:4, s. 625-638
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Tidskriftsartikel (refereegranskat)abstract
- The variance of the error term in ordinary regression models and linear smoothers is usually estimated by adjusting the average squared residual for the trace of the smoothing matrix (the degrees of freedom of the predicted response). However, other types of variance estimators are needed when using monotonic regression (MR) models, which are particularly suitable for estimating response functions with pronounced thresholds. Here, we propose a simple bootstrap estimator to compensate for the over-fitting that occurs when MR models are estimated from empirical data. Furthermore, we show that, in the case of one or two predictors, the performance of this estimator can be enhanced by introducing adjustment factors that take into account the slope of the response function and characteristics of the distribution of the explanatory variables. Extensive simulations show that our estimators perform satisfactorily for a great variety of monotonic functions and error distributions.
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| 3. |
- Sysoev, Oleg
(författare)
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Monotonic regression for large multivariate datasets
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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