Vague Correlation Coefficient of Interval Vague Sets

Various attempts have been made by researchers on the study of vagueness of data through intuitionistic fuzzy sets and vague sets, and also it was shown that vague sets are intuitionistic fuzzy sets. But there are algebraic and graphical differences between vague sets and intuitionistic fuzzy sets. In this paper an attempt is made to define the correlation coefficient of interval vague sets lying in the interval [0, 1], and a new method for computing the correlation coefficient of interval vague sets lying in the interval [-1, 1] using α-cuts over the vague degrees through statistical confidence intervals is presented by an example. The new method proposed in this paper produces a correlation coefficient in the form of an interval. DOI: 10.4018/ijfsa.2012010102 International Journal of Fuzzy System Applications, 2(1), 18-34, January-March 2012 19 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. relation coefficient of fuzzy numbers lying in the interval [-1, 1]. Buckley (2004), using the approach of fuzzy probabilities, defined the fuzzy correlation coefficient. Zeng and Li (2007) focused on the probability spaces to define a new kind of correlation for intuitionistic fuzzy sets. Gerstenkon and Manko (1991) defined the correlation of intuitionistic fuzzy sets. Hong and Hwang (1995) defined it on probability spaces. Hung and Wu (2002) introduced the concepts of positively and negatively correlated results based on the concept of centroid for intuitionistic fuzzy sets lying in the interval [-1, 1]. Hong (1998) studied the correlation coefficient of interval-valued intuitionistic fuzzy sets in probability spaces. Mitchell (2004) adopted a statistical view point to interpret intuitionistic fuzzy sets as an ensemble of ordinary fuzzy set, and defined the correlation coefficient of intuitionistic fuzzy sets by using the correlation coefficient of two ordinary fuzzy sets and a mean aggregation function. At present, the researches of vague set are focused on the operation rules of the interval Vague set, correlation degree (Bustince & Burillo, 1995; Hong, 1998; Robinson & Amirtharaj, 2011), and topological structures. This paper focuses on interval vague sets. Gau and Buehrer (1993), Xu and Chen (2007), Li and Rao (2001), and Piede Liu (2009) have presented in detail the essential operations of interval vague sets. Interval vague set is one of the higher order fuzzy sets that are now in the literature and are being applied into many specialized application domains. The notions of truth membership function, false membership function and uncertainty function in interval vague sets can describe the objective world more realistic and practical. Interval vague sets can reflect people’s understanding comprehensively in three aspects: Support degree, Negative degree and Uncertainty degree. Combining the interval-valued fuzzy sets and vague sets, Ma et al. (2001) introduced the concept of interval valued vague sets. Many of the previously defined correlation coefficients for imprecise data lie in the interval [0, 1], except a very few authors who tried to make it fall in the interval [-1, 1]. In this paper the correlation coefficient of interval vague sets is defined on the usual interval [0, 1] and then a new computational method is proposed which allows the correlation coefficient to fall in the interval [-1, 1]. The novelty in this paper is that the correlation coefficient of interval vague sets is obtained in the form of an interval rather than a single point measure. Statistical confidence intervals and α-cuts are used to construct a kind of trapezoidal fuzzy number derived from the vague degrees which is used for the computational purposes. The differences in deriving the correlation by two different methods are studied. The paper is organized as follows: basic concepts of vague sets and interval vague sets are introduced in the first part. The second part leads to the defining of correlation coefficient of interval vague sets lying in the interval [0, 1], with a simple example. In the third part the uses of statistical confidence intervals in stretching the vague degrees into trapezoidal fuzzy number through α-cuts is presented. The fourth part is the application of the above techniques to calculate the vague correlation coefficient for the same example. Finally some conclusions are drawn based on the newly proposed method of confidence intervals for correlation coefficient of IVSs. VAGUE SETS AND INTERVAL VAGUE SETS Consider the domain of discourse X= { x1, x2, ..., xn }. A vague set A is described by the true membership function tA and the false membership function fA in the domain of discourse X. The true membership function tA and the false membership function fA are defined as follows. tA: X → [0, 1], fA: X→ [0, 1], where tA(xi) is the lower bound that affirms the membership exported by the evidence that support xi, and fA(xi) is the lower bound that negates the membership exported by the evidence that support xi, and tA (xi) + fA (xi) ≤ 1. The membership of the element xi in a vague set A is defined by a subinterval [tA (xi), 1fA (xi)] in the interval [0, 15 more pages are available in the full version of this document, which may be purchased using the "Add to Cart" button on the product's webpage: www.igi-global.com/article/vague-correlation-coefficientinterval-vague/63353?camid=4v1 This title is available in InfoSci-Journals, InfoSci-Journal Disciplines Computer Science, Security, and Information Technology. Recommend this product to your librarian: www.igi-global.com/e-resources/libraryrecommendation/?id=2