[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

An isotonic trivariate statistical regression method

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

Abstract

The present research work outlines the main ideas behind statistical regression by a two-independent-variates and one-dependent-variate model based on the invariance of measures in probabilistic spaces. The principle of probabilistic measure invariance, applied under the assumption that the model be isotonic, leads to a system of differential equations. Such differential system is reformulated in terms of an integral equation that affords an iterative numerical solution. Numerical tests performed on the devised statistical regression procedure illustrate its features.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. Acrylamide (Marstokk et al. 2000) is a chemical compound with chemical formula C\(_3\)H\(_5\)NO. It is a white odorless crystalline solid, soluble in water, ethanol, ether and chloroform. Acrylamide decomposes in the presence of acids, bases, oxidizing agents, iron and iron salts. It decomposes non-thermally to form ammonia, while its thermal decomposition produces carbon monoxide, carbon dioxide and oxides of nitrogen.

References

  • Ahuja S, Lakshminarayana A, Shukla SK (2012) Statistical regression based power models. In: Low power design with high-level power estimation and power-aware synthesis. Springer, New York, pp 59–70

  • Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10(4):1100–1120

    Article  MathSciNet  MATH  Google Scholar 

  • Barlow RE, Bartholomew DJ, Bremner JM, Brunk HD (1972) Statistical inference under order restrictions. Wiley, New York

    MATH  Google Scholar 

  • Birke M, Dette H (2007) Testing strict monotonicity in nonparametric regression. Math Meth Stat 16:110–123

    Google Scholar 

  • Chen MJ, Hsu HT, Lin CL, Ju WY (2012) A statistical regression model for the estimation of acrylamide concentrations in French fries for excess lifetime cancer risk assessment. Food Chem Toxicol 50(10):3867–3876

    Article  Google Scholar 

  • Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2006) Testing constancy for isotonic regressions. Scand J Stat 33(3):463–475

    Article  MATH  Google Scholar 

  • Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2007) A test for constancy of isotonic regressions using the \(L_2\)-Norm. Stat Sinica 17:713–724

    MATH  Google Scholar 

  • Cortez P, Cerdeira A, Almeida F, Matos T, Reis J (2009) Modeling wine preferences by data mining from physicochemical properties. Decis Support Syst 47:547–553

    Article  Google Scholar 

  • Domínguez-Menchero JS, González-Rodríguez G (2007) Analyzing an extension of the isotonic regression problem. Metrika 66(1):19–30

    Article  MathSciNet  Google Scholar 

  • Domínguez-Menchero JS, González-Rodríguez G, López-Palomo MJ (2005) An \(L_2\) point of view in testing monotone regression. J Nonparametric Stat 17(2):135–153

    Article  MATH  Google Scholar 

  • Durot C (2003) A Kolmogorov-type test for monotonicity of regression. Stat Probab Lett 63:425–433

    Article  MathSciNet  MATH  Google Scholar 

  • Fiori S (2011) Statistical nonparametric bivariate isotonic regression by look-up-table-based neural networks. In: Lu B-L, Zhang L, Kwok J (eds) Proceedings of the 2011 international conference on neural information processing (ICONIP 2011, Shanghai (China), Part III, LNCS 7064, Springer, Heidelberg, pp 365–372

  • Fiori S (2012) Fast statistical regression in presence of a dominant independent variable, Neural Computing and Applications (Springer). (Special issue of the 2011 International Conference on Neural Information Processing—ICONIP’2011). Accepted for publication (available online)

  • Forrest DR, Hetland RD, DiMarco SF (2011) Multivariable statistical regression models of the areal extent of hypoxia over the Texas-Louisiana continental shelf. Environ Res Lett 6(4):045002

    Article  Google Scholar 

  • Kulkarni MA, Patil S, Rama GV, Sen PN (2008) Wind speed prediction using statistical regression and neural network. J Earth Syst Sci 117(4):457–463

    Article  Google Scholar 

  • Li X, Liu HZ (2008) Statistical regression for efficient high-dimensional modeling of analog and mixed-signal performance variations. In: Proceedings of the 45th ACM/IEEE design automation conference. DAC 2008, Anaheim Convention Center, California, pp 38–43

  • Liu J, Li H (2010) Application research of a statistical regression algorithm in the IVR system. In: Proceedings of the 2010 international conference on educational and network technology. ICENT, Qinhuangdao (China), pp 358–360

  • Liu S, Gao RX, He Q, Staudenmayer J, Freedson P (2009) Development of statistical regression models for ventilation estimation. In: Proceedings of the 31st annual international conference of the IEEE engineering in medicine and biology society. EMBC, Minneapolis (Minnesota, USA), pp 1266–1269

  • Maheshwari N, Balaji C, Ramesh A (2011) A nonlinear regression based multi-objective optimization of parameters based on experimental data from an IC engine fueled with biodiesel blends. Biomass Bioenergy 35:2171–2183

    Article  Google Scholar 

  • Marstokk KM, Møllendal H, Samdal S (2000) Microwave spectrum, conformational equilibrium, \(^{14}{\rm {N}}\) quadrupole coupling constants, dipole moment, vibrational frequencies and quantum chemical calculations for acrylamide. J Mol Struct 524(13):69–85

    Google Scholar 

  • Qian S, Eddy WF (1996) An algorithm for isotonic regression on ordered rectangular grids. J Comput Graph Stat 5(3):225–235

    MathSciNet  Google Scholar 

  • Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York

    MATH  Google Scholar 

  • Roelant R, Constales D, Van Keer R, Marin GB (2008) Second-order statistical regression and conditioning of replicate transient kinetic data. Chem Eng Sci 63(7):1850–1865

    Article  Google Scholar 

  • Scott DW, Sain SR (2005) Multi-dimensional density estimation. Handb Stat Data Min Data Vis 24:229–261

    Article  Google Scholar 

  • Thierfelder T (1999) Empirical/statistical modeling of water quality in dimictic glacial/boreal lakes. J Hydrol 220:186–208

    Article  Google Scholar 

  • Velikova MV (2006) Monotone models for prediction in data mining. Ph.D Dissertation, Dutch graduate school for information and knowledge systems and graduate school of the faculty of economics and business administration of Tilburg University, Tilburg

  • White Vugrin K, Painton Swiler L, Roberts RM, Stucky-Mack NJ, Sullivan SP (2005) Confidence region estimation techniques for nonlinear regression: three case studies. Sandia report SAND2005-6893 (Unlimited Release), Sandia National Laboratories, Carlsbad

  • Woodhouse R (2003) Statistical regression line-fitting in the oil and gas industry. PennWell Books, Tulsa

    Google Scholar 

  • Žilinskas A, Žilinskas J (2010) Interval arithmetic based optimization in nonlinear regression. INFORMATICA 21(1):149–158

    MathSciNet  MATH  Google Scholar 

  • Zou DH (1995) Statistical regression applied to borehole strain measurements data analysis. Geotech Geol Eng 13(1):17–27

    Article  Google Scholar 

Download references

Acknowledgments

The author wishes to gratefully thank the anonymous referees and the associate editor who coordinated the review of the present paper for their thorough and stimulating comments and suggestions that helped improving and enriching the presentation of the technical content conveyed by the present manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Simone Fiori.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fiori, S. An isotonic trivariate statistical regression method. Adv Data Anal Classif 7, 209–235 (2013). https://doi.org/10.1007/s11634-013-0131-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-013-0131-9

Keywords

Mathematics Subject Classification (2000)

Navigation