# zbMATH — the first resource for mathematics

Uncertain Johnson-Schumacher growth model with imprecise observations and $$k$$-fold cross-validation test. (English) Zbl 1436.62368
Summary: Regression is a powerful tool to study how the response variables vary due to changes of explanatory variables. Unlike traditional statistics or mathematics where data are assumed fairly accurate, we notice that the real-world data are messy and obscure; thus, the uncertainty theory seems more appropriate. In this paper, we focus on the residual analysis of the Johnson-Schumacher growth model, with parameter estimation performed by the least squares method, followed by the prediction intervals for new explanatory variables. We also propose a $$k$$-fold cross-validation method for model selection with imprecise observations. A numerical example illustrates that our approach will achieve better prediction accuracy.
##### MSC:
 62J86 Fuzziness, and linear inference and regression 62J05 Linear regression; mixed models 62M20 Inference from stochastic processes and prediction
Full Text:
##### References:
 [1] Chen, X.; Ralescu, D., B-spline method of uncertain statistics with application to estimating distance, J Uncertain Syst, 6, 4, 256-262 (2012) [2] Edgeworth, Fy, On observations relating to several quantities, Hermathena, 6, 13, 279-285 (1887) [3] Edgeworth, Fy, A new method of reducing observations relating to several quantities, Philos Mag, 24, 147, 222-223 (1888) [4] Fang, L.; Hong, Y., Uncertain revised regression analysis with responses of logarithmic, square root and reciprocal transformations, Soft Comput (2019) [5] Galton, F., Regression towards mediocrity in hereditary stature, J Anthropol Inst, 15, 1, 246-263 (1885) [6] Gauss, Cf, Theory of the motion of the heavenly bodies moving about the Sun in Conic Sections (1809), Hamburg: Sumtibus Frid Perthes, Hamburg [7] Ishibuchi, H.; Tanaka, H., Fuzzy regression analysis using neural networks, Fuzzy Sets Syst, 50, 3, 257-265 (1992) [8] Johnson, No, A trend line for growth series, J Am Stat Assoc, 30, 192, 717 (1935) · JFM 61.1328.10 [9] Kao, C.; Chyu, C., Least-squares estimates in fuzzy regression analysis, Eur J Oper Res, 148, 2, 426-435 (2003) · Zbl 1045.62068 [10] Legendre, Am, New methods for the determination of the orbits of comets (1805), Paris: Firmin Didot, Paris [11] Lio, W.; Liu, B., Residual and confidence interval for uncertain regression model with imprecise observations, J Intell Fuzzy Syst, 35, 2, 2573-2583 (2018) [12] Liu, B., Uncertainty theory (2007), Berlin: Springer, Berlin [13] Liu, B., Some research problems in uncertainty theory, J Uncertain Syst, 3, 1, 3-10 (2009) [14] Liu, B., Uncertainty theory: a branch of mathematics for modeling human uncertainty (2010), Berlin: Springer, Berlin [15] Liu, B., Why is there a need for uncertainty theory, J Uncertain Syst, 6, 1, 3-10 (2012) [16] Liu, B., Uncertainty theory (2018), Berlin: Springer, Berlin [17] Schumacher, F., A new growth curve and its relation to timber yield studies, J Forest, 37, 1, 819-820 (1939) [18] Song, Y.; Fu, Z., Uncertain multivariable regression model, Soft Comput, 22, 17, 5861-5866 (2018) · Zbl 1398.62206 [19] Tanaka, H.; Uejima, S.; Asai, K., Fuzzy linear regression model, IEEE Trans Syst Man Cybern, 6, 2, 2933-2938 (1982) [20] Wang, X.; Guo, H., Uncertain variance of sample and its application, Inf Int Interdiscip J, 14, 1, 79-87 (2011) · Zbl 1243.62027 [21] Wang, X.; Peng, Z., Method of moments for estimating uncertainty distributions, J Uncertain Anal Appl, 2, 1 (2014) [22] Yang, Xiangfeng; Liu, Baoding, Uncertain time series analysis with imprecise observations, Fuzzy Optimization and Decision Making, 18, 3, 263-278 (2018) · Zbl 1427.62111 [23] Yang, Xf; Gao, J.; Ni, Y., Resolution principle in uncertain random environment, IEEE Trans Fuzzy Syst, 26, 1578-1588 (2018) [24] Yao, K., A formula to calculate the variance of uncertain variable, Soft Comput, 19, 10, 2947-2953 (2014) · Zbl 1379.60003 [25] Yao, K., Uncertain statistical inference models with imprecise observations, IEEE Trans Fuzzy Syst, 26, 1 (2017) [26] Yao, K.; Liu, B., Uncertain regression analysis: an approach for imprecise observations, Soft Comput, 22, 17, 5579-5582 (2018) · Zbl 1398.62207 [27] Yule, Gu, On the theory of correlation, J R Stat Soc, 60, 4, 812-854 (1897)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.