Using trapezoids for representing granular objects: Applications to learning and OWA aggregation

We discuss the role and benefits of using trapezoidal representations of granular information. We focus on the use of level sets as a tool for implementing many operations on trapezoidal sets. We point out the simplification that the linearity of the trapezoid brings by requiring us to perform operations on only two level sets. We investigate the classic learning algorithm in the case when our observations are granule objects represented as trapezoidal fuzzy sets. An important issue that arises is the adverse effect that very uncertain observations have on the quality of our estimates. We suggest an approach to addressing this problem using the specificity of the observations to control its effect. We next consider the OWA aggregation of information represented as trapezoids. An important problem that arises here is the ordering of the trapezoidal fuzzy sets needed for the OWA aggregation. We consider three approaches to accomplish this ordering based on the location, specificity and fuzziness of the trapezoids. From these three different approaches three fundamental methods of ordering are developed. One based on the mean of the 0.5 level sets, another based on the length of the 0.5 level sets and a third based on the difference in lengths of the core and support level sets. Throughout this work particular emphasis is placed on the simplicity of working with trapezoids while still retaining a rich representational capability. (c) 2007 Elsevier Inc. All rights reserved.