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.
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Notes
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
Barlow RE, Bartholomew DJ, Bremner JM, Brunk HD (1972) Statistical inference under order restrictions. Wiley, New York
Birke M, Dette H (2007) Testing strict monotonicity in nonparametric regression. Math Meth Stat 16:110–123
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
Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2006) Testing constancy for isotonic regressions. Scand J Stat 33(3):463–475
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
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
Domínguez-Menchero JS, González-Rodríguez G (2007) Analyzing an extension of the isotonic regression problem. Metrika 66(1):19–30
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
Durot C (2003) A Kolmogorov-type test for monotonicity of regression. Stat Probab Lett 63:425–433
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
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
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
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
Qian S, Eddy WF (1996) An algorithm for isotonic regression on ordered rectangular grids. J Comput Graph Stat 5(3):225–235
Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York
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
Scott DW, Sain SR (2005) Multi-dimensional density estimation. Handb Stat Data Min Data Vis 24:229–261
Thierfelder T (1999) Empirical/statistical modeling of water quality in dimictic glacial/boreal lakes. J Hydrol 220:186–208
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
Žilinskas A, Žilinskas J (2010) Interval arithmetic based optimization in nonlinear regression. INFORMATICA 21(1):149–158
Zou DH (1995) Statistical regression applied to borehole strain measurements data analysis. Geotech Geol Eng 13(1):17–27
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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.
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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
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DOI: https://doi.org/10.1007/s11634-013-0131-9