A fuzzy regression model based on distances and random variables with crisp input and fuzzy output data: a case study in biomass production

Least-squares technique is well-known and widely used to determine the coefficients of a explanatory model from observations based on a concept of distance. Traditionally, the observations consist of pairs of numeric values. However, in many real-life problems, the independent or explanatory variable can be observed precisely (for instance, the time) and the dependent or response variable is usually described by approximate values, such as “about $$\pounds300$$” or “approximately $500”, instead of exact values, due to sources of uncertainty that may affect the response. In this paper, we present a new technique to obtain fuzzy regression models that consider triangular fuzzy numbers in the response variable. The procedure solves linear and non-linear problems and is easy to compute in practice and may be applied in different contexts. The usefulness of the proposed method is illustrated using simulated and real-life examples.

[1]  Didier Dubois,et al.  Soft Methods for Handling Variability and Imprecision , 2008 .

[2]  Jung-Hsien Chiang,et al.  Support vector learning mechanism for fuzzy rule-based modeling: a new approach , 2004, IEEE Trans. Fuzzy Syst..

[3]  Jerome F. Saeman,et al.  Kinetics of Wood Saccharification - Hydrolysis of Cellulose and Decomposition of Sugars in Dilute Acid at High Temperature , 1945 .

[4]  Chiang Kao,et al.  Least-squares estimates in fuzzy regression analysis , 2003, Eur. J. Oper. Res..

[5]  Inés Couso,et al.  Diagnosis of dyslexia with low quality data with genetic fuzzy systems , 2010, Int. J. Approx. Reason..

[6]  Volker Krätschmer.  Strong consistency of least-squares estimation in linear regression models with vague concepts , 2006 .

[7]  Yizeng Chen,et al.  A non-linear possibilistic regression approach to model functional relationships in product planning , 2006 .

[8]  C. R. Bector,et al.  A simple method for computation of fuzzy linear regression , 2005, Eur. J. Oper. Res..

[9]  Ebrahim Nasrabadi,et al.  A mathematical-programming approach to fuzzy linear regression analysis , 2004, Appl. Math. Comput..

[10]  Reshma Khemchandani,et al.  Regularized least squares fuzzy support vector regression for financial time series forecasting , 2009, Expert Syst. Appl..

[11]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

[12]  Inés Couso,et al.  Higher order models for fuzzy random variables , 2008, Fuzzy Sets Syst..

[13]  Phil Diamond,et al.  Fuzzy least squares , 1988, Inf. Sci..

[14]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[15]  Rajkumar Roy,et al.  Advances in Soft Computing , 2018, Lecture Notes in Computer Science.

[16]  Pierpaolo D'Urso,et al.  Least squares estimation of a linear regression model with LR fuzzy response , 2006, Comput. Stat. Data Anal..

[17]  Ana Colubi,et al.  Statistical inference about the means of fuzzy random variables: Applications to the analysis of fuzzy- and real-valued data , 2009, Fuzzy Sets Syst..

[18]  Berlin Wu,et al.  A new approach to fuzzy regression models with application to business cycle analysis , 2002, Fuzzy Sets Syst..

[19]  G. Garrote,et al.  Kinetic study of the acid hydrolysis of sugar cane bagasse , 2002 .

[20]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[21]  Ana Colubi,et al.  Estimation of a simple linear regression model for fuzzy random variables , 2009, Fuzzy Sets Syst..

[22]  Dug Hun Hong,et al.  Support vector fuzzy regression machines , 2003, Fuzzy Sets Syst..

[23]  J. Lucas M. Barbosa,et al.  A preliminary discussion , 1986 .

[24]  Junzo Watada,et al.  Possibilistic linear regression analysis for fuzzy data , 1989 .

[25]  Wolfgang Näther.  Regression with fuzzy random data , 2006, Comput. Stat. Data Anal..

[26]  Ana Colubi,et al.  A linear regression model for imprecise response , 2010, Int. J. Approx. Reason..

[27]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[28]  Michael R. Lyu,et al.  Localized support vector regression for time series prediction , 2009, Neurocomputing.

[29]  Jorge Casillas,et al.  Genetic learning of fuzzy rules based on low quality data , 2009, Fuzzy Sets Syst..

[30]  Mathieu Serrurier,et al.  Imprecise Regression and Regression on Fuzzy Data - A Preliminary Discussion , 2006, 2006 IEEE International Conference on Fuzzy Systems.

[31]  Ram R. Bishu,et al.  Evaluation of fuzzy linear regression models by comparing membership functions , 1998, Fuzzy Sets Syst..

[32]  Liang-Hsuan Chen,et al.  Fuzzy Regression Models Using the Least-Squares Method Based on the Concept of Distance , 2009, IEEE Transactions on Fuzzy Systems.

[33]  Chia-Feng Juang,et al.  TS-fuzzy system-based support vector regression , 2009, Fuzzy Sets Syst..