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An interval-based approach to fuzzy regression for fuzzy input-output data

An interval-based approach to fuzzy regression for fuzzy input-output data,10.1109/FUZZY.2011.6007457,Jalal Chachi,S. Mahmoud Taheri,H. Rezaei Pazhand

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An interval-based approach to fuzzy regression for fuzzy input-output data  
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A novel approach is introduced to construct a fuzzy regression model when the data available of independent and dependent variables are fuzzy numbers. The approach, consisting on the least-squares method, uses the α−level sets of fuzzy observations to estimate the crisp parameters of the model. A competitive study shows the performance and efficiency of the proposed approach with respect to some well-known methods. Identification and analysis of the functional relationship between a dependent and some independent variables have made great interest in statistical analysis. Based on such functional relationship, which is called regression model, one can describe and predict the values of the dependent variable using the observations of independent variables. However, when the data are reported as imprecise quantities we need to develop some soft methods to construct and ana- lyze regression models. Fuzzy set theory provides appropriate tools for modeling regression when imprecise data have been observed. Over the last decades, different approaches to re- gression modeling have been widely proposed in the statistical literature, focussing on the situation of fuzzy environment. We refer the reader to (2), (3), (4), (7), (11), (12), (14), (15), (16) for some recent works as well as some reviews of the main approaches to regression analysis in fuzzy environment. In the present work, we investigate the situation in which the data available of independent and dependent variables are fuzzy numbers. We introduce a novel approach to model such a fuzzy data set. In the proposed approach, we apply a squared distance on the space of real intervals. Then, consisting on minimizing the total least-squares of errors, a linear regression model is expressed in terms of interval arithmetic between the α-level sets of fuzzy input-output data at each level of α ∈ (0, 1). The errors are considered based on the differences between the interval-valued data obtained from the α-level sets of the observations and the estimations of the dependent vari- able. Finally, by aggregating the coefficients obtained from the interval regression model, we estimate the coefficients of the fuzzy regression model. The proposed method is investigated by two numerical competitive examples, which indicate the performance of the proposed model with respect to some well- known models. The rest of this paper is organized as follows. In the next section, some preliminaries of fuzzy arithmetic that will be used in the sequel are recalled. Section III presents a novel approach to fuzzy regression for fuzzy input output data. In this section, the method of estimation of the parameters of the model is explained. In Section IV, two numerical examples are given to clarify the proposed approach. Finally, in Section V some concluding remarks are made.
Conference: IEEE International Conference on Fuzzy Systems , pp. 2859-2863, 2011
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