. .lr , ! ! e ELSEVIER Fuzzy Sets and Systems 105 (1999) 429-436 FU||Y sets and systems A general piecewise necessity regression analysis based on linear programming Jing-Rung Yu a, Gwo-Hshiung Tzeng b'*, Han-Lin Li a alnstitute of Information Management, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC bEnergy and Environmental Research Group, Institute of Traffic and Transportation, National Chiao Tung University, 114-4F, Sec. 1, Chung Hsiao W. Rd., Taipei 100, Taiwan, ROC Received November 1996; received in revised form June 1997 Abstract In possibilistic regression analysis proposed by Tanaka and lshibuchi (1992), linear programming (LP) formulation of necessity analysis has no feasible solution under the enormous variation of the given data. This work proposes a general piecewise necessity regression analysis based on LP rather than a non-linear interval model that they recommended to obtain the necessity area of the given data. In addition to maintaining a linear property, the proposed method prevents the necessity analysis from having no feasible solution. The problematic univariate example and a multivariate example with respect to different number of change-points are demonstrated by the general piecewise necessity regression. The proposed method characteristic is that, according to data distribution, practitioners can specify the number and the positions of change-points. The proposed method maintains the linear interval model and the order of necessity regression function does not need to be determined. (~ 1999 Elsevier Science B.V. All rights reserved. Keywords: Piecewise regression; Necessity area; Fuzzy linear regression 1. Introduction The possibility theory on possibility distribu- tions has been proposed by Zadeh [14] and ad- vanced by Dubois and Prade [1]. In the early 1980s, Tanaka et al. [11] have introduced a linear programming (LP) based regression method using a linear fuzzy model with symmetrics triangular fuzzy parameters. Since Tanaka et al. introduced *Corresponding author. Tel.: +88623146515; fax: +8862 3120082; e-mail: 48534806@cc.nctu.edu.tw. fuzzy linear regression, previous literature dealing with fuzzy linear regression has grown rapidly. Then possibility and necessity analyses have been clearly defined in Tanaka [8]. Recently, Sakawa and Yano [5, 6] have generalized the minimization, maximization and conjunction formulation de- veloped by Tanaka et al. [11], Tanaka [8] and Tanaka et al, [9], respectively. Tanaka and Ishibuchi [I0] proposed the possibility and necess- ity analyses on possibilistic regression analysis. However, a weakness of the fuzzy regression model has arised. In necessity analysis, the necessity area cannot be obtained owing to the large variation data [8, 10]. Tanaka [8] pointed out that if a 0165-0114/99/$ - see front matter ~;~ 1999 Elsevier Science B.V. All rights reserved. PII: S0 1 65-0 1 1 4(97)00223-6