Abstract
This paper introduces a nonlinear regression model to interval-valued data. The method extends the classical nonlinear regression model in order to manage interval-valued datasets. The parameter estimates of the nonlinear model considers some optimization algorithms aiming to identify which one presents the best accuracy and precision in the prediction task. A detailed prediction performance study comparing the proposed nonlinear method and other linear regression methods for interval variables is presented based on K-fold cross-validation scheme with synthetic interval-valued datasets generated on a Monte Carlo framework. Moreover, two suitable real interval-valued datasets are considered to illustrate the usefulness and the performance of the approaches presented in this paper. The results suggested that the use of the nonlinear method is suitable for real datasets, as well as in the Monte Carlo simulation study.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Draper NR, Smith H (1981) Applied regression analysis. Wiley, New York
Montgomery DC, Peck EA (1982) Introduction to linear regression analysis. Wiley, New York
Bates DM, Watts DG (1988) Nonlinear regression analysis and its applications. Wiley, New York
Seber GAF, Wild CJ (2003) Nonlinear regression. Wiley, New York
Colby E, Bair E (2013) Cross-validation for nonlinear mixed effects models. J Pharmacokinet Pharmacodyn 40:243–252
Rivas I, Personnaz L (1999) On cross-validation for model selection. Neural Comput 11:863–870
Shao J (1993) Linear model selection by cross-validation. J Am Stat Assoc 88:486–495
Li S, Cheng S, Li S, Liang Y (2012) Decentralized kinematic control of a class of collaborative redundant manipulators via recurrent neural networks. Neurocomputing 91:1–10
Li S, Liu B, Li Y (2013) Selective positive–negative feedback produces the winner-take-all competition in recurrent neural networks. IEEE Trans Neural Netw Learn Syst 24:301–309
Li S, Wang J, Liu B (2013) A nonlinear model to generate the winner-take-all competition. Commun Nonlinear Sci Numer Simul 18:435–442
Sheng W, Chen S, Fairhurst M, Xiao G, Mao J (2014) Multilocal search and adaptive niching based memetic algorithm with a consensus criterion for data clusterin. IEEE Trans Evol Comput 18(5):721–741
Billard L, Diday E (2006) Symbolic data analysis: conceptual statistics and data mining. Wiley, New York
Bock HH, Diday E (2000) Analysis of symbolic data, exploratory methods for extracting statistical information from complex data. Springer, Heidelberg
Diday E, Fraiture-Noirhomme M (2008) Symbolic data analysis and the SODAS software. Wiley, New York
Billard L, Diday E (2000) Regression analysis for interval-valued data. In: Data analysis, classification and related methods: proceedings of the 7th conference of the international federation of classification societies (IFCS’00). Springer, Belgium, pp 369–374
Brito P, Duarte Silva AP (2012) Modeling interval data with normal and skew-normal distributions. J Appl Stat 39(1):157–170
Blanco-Fernandez A, Corral N, Gonzlez-Rodrguez G (2011) Estimation of a flexible simple linear regression model for interval data based on set arithmetic. Comput Stat Data Anal 55:2568–2578
Lima Neto EA, De Carvalho FAT (2008) Centre and range method to fitting a linear regression model on symbolic interval data. Comput Stat Data Anal 52:1500–1515
Maia ALS, De Carvalho FAT (2008) Fitting a least absolute deviation regression model on interval-valued data, In: 6th Brazilian symposium on neural networks, advances in artificial intelligence, pp 3742–3747
Su ZG, Wang PH, Li YG, Zhou ZK (2015) Parameter estimation from interval-valued data using the expectation-maximization algorithm. J Appl Stat 85:1–19
Xu W (2010), Symbolic data analysis: interval-valued data regression. Ph.D. thesis, University of Georgia, Athens
Lima Neto EA, Cordeiro GM, De Carvalho FAT (2011) Bivariate symbolic regression models for interval-valued variables. J Stat Comput Simul 81:1727–1744
Lima Neto EA, Anjos UU (2015) Regression model for interval-valued variables based on copulas. J Appl Stat 42:2010–2029
Queiroz DCF, Souza RMCR, Cysneiros FJA (2011) Logistic regression-based pattern classifiers for symbolic interval data. Pattern Anal Appl 14:273–282
Lima Neto EA, De Carvalho FAT (2010) Constrained linear regression models for symbolic interval-valued variables. Comput Stat Data Anal 54:333–347
Judge GG, Takayama T (1966) Inequality restrictions in regression analysis. J Am Stat Assoc 61:166–181
Lawson CL, Hanson RJ (1974) Solving least squares problem. Prentice-Hall, New York
Nocedal J, Wight SJ (1999) Numerical optimization. Springer, Heidelberg
Chong EKP, Zak SH (2008) An introduction to optimization, 3rd edn. Wiley, New Jersey
Michaelis L, Menten M (1913) Die kinetik der invertinwirkung. Biochen Zeitung 49:333–369
Van Ness HC, Abbott M (1997) Perrys chemical engineers handbook. McGraw-Hill, New York
Lima Filho LMA, Silva JAA, Cordeiro GM, Ferreira RLC (2012) Modeling the growth of Eucalyptus clones using the Chapman–Richards model with different symmetrical error distributions. Ciência Florest 22(4):777–785
Acknowledgments
The authors are grateful to the anonymous referees for their careful revision, valuable suggestions, and comments which improved this paper. The authors also would like to thank CAPES (National Foundation for Post-Graduated Programs, Brazil) and CNPq (National Council for Scientific and Technological Development, Brazil) for their financial support. The second author would like to thank FACEPE (Research Agency from the State of Pernambuco, Brazil).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lima Neto, E.d.A., de Carvalho, F.d.A.T. Nonlinear regression applied to interval-valued data. Pattern Anal Applic 20, 809–824 (2017). https://doi.org/10.1007/s10044-016-0538-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10044-016-0538-y