Rough set theory (RST), since its introduction in Pawlak (1982), continues to develop as an effective tool in classification problems and decision support. In the majority of applications using RST based methodologies, there is the construction of ‘if .. then ..’ decision rules that are used to describe the results from an analysis. The variation of applications in management and decision making, using RST, recently includes discovering the operating rules of a Sicilian irrigation purpose reservoir (Barbagallo, Consoli, Pappalardo, Greco, & Zimbone, 2006), feature selection in customer relationship management (Tseng & Huang, 2007) and decisions that insurance companies make to satisfy customers’ needs (Shyng, Wang, Tzeng, & Wu, 2007). As a nascent symbolic machine learning technique, the popularity of RST is a direct consequence of its set theoretical operational processes, mitigating inhibiting issues associated with traditional techniques, such as within-group probability distribution assumptions (Beynon & Peel, 2001). Instead, the rudiments of the original RST are based on an indiscernibility relation, whereby objects are grouped into certain equivalence classes and inference taken from these groups. Characteristics like this mean that decision support will be built upon the underlying RST philosophy of “Let the data speak for itself” (Dunstch & Gediga, 1997). Recently, RST was viewed as being of fundamental importance in artificial intelligence and cognitive sciences, including decision analysis and decision support systems (Tseng & Huang, 2007). One of the first developments on RST was through the variable precision rough sets model (VPRSß), which allows a level of mis-classification to exist in the classification of objects, resulting in probabilistic rules (see Ziarko, 1993; Beynon, 2001; Li and Wang, 2004). VPRSß has specifically been applied as a potential decision support system with the UK Monopolies and Mergers Commission (Beynon & Driffield, 2005), predicting bank credit ratings (Griffiths & Beynon, 2005) and diffusion of medicaid home care programs (Kitchener, Beynon, & Harrington, 2004). Further developments of RST include extended variable precision rough sets (VPRSl,u), which infers asymmetric bounds on the possible classification and mis-classification of objects (Katzberg & Ziarko, 1996), dominance-based rough sets, which bases their approach around a dominance relation (Greco, Matarazzo, & Slowinski, 2004), fuzzy rough sets, which allows the grade of membership of objects to constructed sets (Greco, Inuiguchi, & Slowinski, 2006), and probabilistic bayesian rough sets model that considers an appropriate certainty gain function (Ziarko, 2005). A literal presentation of the diversity of work on RST can be viewed in the annual volumes of the Transactions on Rough Sets (most recent year 2006), also the annual conferences dedicated to RST and its developments (see for example, RSCTC, 2004). In this article, the theory underlying VPRSl,u is described, with its special case of VPRSß used in an example analysis. The utilisation of VPRSl,u, and VPRSß, is without loss of generality to other developments such as those referenced, its relative simplicity allows the non-proficient reader the opportunity to fully follow the details presented.
Matroidal Structure of Generalized Rough Sets Based on Tolerance Relations
