A generalized L1-type metric between fuzzy numbers for an approach to central tendency of fuzzy data

• Published in: Information sciences 2013-09, Vol.242, p.22-34
• Main Authors: Sinova, Beatriz, de la Rosa de Sáa, Sara, Gil, Marı́a Ángeles
• Format: Article
• Language: English
• Subjects: Characterization of a fuzzy number; Fuzzy numbers; L1 distance; Mean distance minimization; Random fuzzy numbers
• ISSN: 0020-0255; 1872-6291
• DOI: 10.1016/j.ins.2013.03.063

In dealing with data generated from a random experiment, L2 metrics are suitable for many statistical approaches and developments. To analyze fuzzy-valued experimental data a generalized L2 metric based on the mid/spread representation of fuzzy values has been stated, and a related methodology to conduct statistics with fuzzy data has been carried out. Most of the developed methods concern either explicitly or implicitly the mean values of the involved random mechanisms producing fuzzy data. Other statistical approaches and studies with experimental data consider L1 metrics, especially in dealing with errors or in looking for a more robust solution and intuitive interpretation. This paper aims to introduce a generalized L1 metric between fuzzy numbers based on a new characterization for them that will be referred to as the midβ/β-leftdev/β-rightdev characterization. More precisely, the metric will take into account both absolute differences in ‘location’ and absolute differences in ‘shape/imprecision’ of fuzzy numbers; moreover one can choose the weight of the influence of the second differences in contrast to the first one. After introducing the new characterization for fuzzy numbers, as well as the associated L1 metric, we will examine some properties. Finally, as an immediate application, the problem of minimizing the mean distance between a fuzzy number and the distribution of a random mechanism producing fuzzy number-valued data will be given, discussed and illustrated.