Constructing the Nonlinear Regression Models on the Basis of Multivariate Normalizing Transformations

N.V. Prykhodko, PhD, Economics, S.B. Prykhodko, Dr.Sci.Tech.
Tech.Admiral Makarov National University of Shipbuilding(9, Heroes of Ukraine Ave., 54025, Mykolaiv, UkraineTel.: (0512) 424280, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.)

Èlektron. model. 2018, 40(6):101-110


The techniques to build the models, equations, confidence and prediction intervals of nonlinear regressions on the basis of multivariate normalizing transformations for non-Gaussian data are considered. The examples of application of the techniques for the four-dimensional non-Gaussian data set for two cases such as: univariate and multivariate normalizing transformations are given. The values of the multiple coefficient of determination such as: the mean magnitude of relative error and the percentage of prediction which are given are better for the nonlinear regression model for the Johnson multivariate transformation compared to the univariate one. The widths of the prediction interval of non-linear regression on the basis of the Johnson multivariate transformation
are less than following Johnson univariate transformation for 26 of 30 rows of data. Approximately the same results are obtained for confidence intervals of nonlinear regression. In general, when constructing the models, equations, confidence and prediction intervals of non-linear regressions for multivariate non-Gaussian data, one should use multivariate normalizing transformations. Normalizing data with univariate transformations instead of multivariate one may lead to increasing of width of the confidence and prediction intervals of non-linear regression.


non-linear regression model, prediction interval, normalizing transformation,non-linear regression model, prediction interval, normalizing transformation,multivariate non-Gaussian data.


1. Bates, D.M. and Watts, D.G. (1988), Nonlinear Regression Analysis and Its Applications, 2nd edition, John Wiley & Sons, New York, USA, DOI: 10.1002/9780470316757.
2. Chatterjee, S. and Simonoff, J.S. (2013), Handbook of Regression Analysis, John Wiley & Sons, New York, USA.
3. Drapper, N.R. and Smith, H. (1998), Applied Regression Analysis, John Wiley &Sons, New York, USA.
4. Freund, Rudolf J., Wilson, William J. and Sa, Ping (2006), Regression analysis: statistical modeling of a response variable, 2nd edition, Elsevier Academic Press, Burlington, MA, London, GB.
5. Prykhodko, S.B. (2016), Developing the software defect prediction models using regression analysis based on normalizing transformations, Modern Problems in Testing of the Applied Software: The Research and Practice Seminar (PTTAS-2016), Poltava, Ukraine, May 25-26, 2016, pp. 6-7.
6. Ryan, T.P. (1997), Modern regression methods, John Wiley & Sons, New York, USA, DOI: 10.1002/9780470382806.
7. Seber, G.A.F. and Wild, C.J. (1989), Nonlinear Regression, John Wiley & Sons, New York, USA, DOI: 10.1002/0471725315.
8. Duncan, G.T. (1978), An empirical study of jackknife constructed confidence regions in nonlinear regression, Technometrics, Vol. 20, no. 2, pp. 123-129, DOI: 10.2307/1268703.
9. Tan, H.B.K., Zhao, Y. and Zhang, H. (2006), Estimating LOC for information systems from their conceptual data models, Proceedings of the 28th International Conference on Software Engineering (ICSE ’06), Shanghai, China, May 20-28, 2006, pp. 321-330, DOI: 10.1145/1134285.1134331.
10. Tan, H.B.K., Zhao, Y. and Zhang, H. (2009), Conceptual data model-based software size estimation for information systems, Transactions on Software Engineering and Methodology, Vol. 19, Issue 2, October 2009, Article no. 4, DOI: 10.1145/1571629.1571630.
11. Stanfield, P.M., Wilson, J.R., Mirka, G.A., Glasscock, N.F., Psihogios, J.P. and Davis, J.R. (1996), Multivariate input modeling with Johnson distributions, Proceedings of the 28th Winter simulation conference WSC’96, Coronado, CA, USA, December 8-11, 1996, ed. S. Andradóttir, K.J. Healy, D.H. Withers and B.L. Nelson, IEEE Computer Society Washington, DC, USA, pp. 1457-1464.
12. Prykhodko, S., Prykhodko, N., Makarova, L. and Pukhalevych, A. (2018), Application of the Squared Mahalanobis Distance for Detecting Outliers in Multivariate Non-Gaussian Data, Proceedings of 14th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), Lviv-Slavske, Ukraine, February 20-24, 2018, pp. 962-965, DOI: 10.1109/TCSET.2018.8336353.
13. Prykhodko, S., Prykhodko, N., Makarova, L. and Pugachenko, K. (2017), Detecting Outliers in Multivariate Non-Gaussian Data on the basis of Normalizing Transformations, Proceedings of the 2017 IEEE First Ukraine Conference on Electrical and Computer Engineering (UKRCON) «Celebrating 25 Years of IEEE Ukraine Section», May 29 -June 2, 2017, Kyiv, Ukraine, pp. 846-849, DOI: 10.1109/UKRCON.2017.8100366.
14. Mardia, K.V. (1970), Measures of multivariate skewness and kurtosis with applications, Biometrika, Vol. 57, Issue 3, pp. 519-530, DOI: 10.1093/biomet/57.3.519.

Full text: PDF