A granular counterpart to the gradient operator

Abstract

 We introduce the linguistic gradient operator, a granular counterpart to the gradient operator in classical calculus. A multiple-input single-output Mamdani fuzzy rulebase is treated as a function in a linguistic state space, defined by the Cartesian product of the antecedent linguistic variables in the rulebase. The value of the consequent linguistic variable is treated as a function in this space, and thus represents the fuzzy rulebase. The linguistic gradient operator takes the consequent function and produces an N-vector of linguistic values, which are approximations to the N first partial derivatives of the consequent function. This vector points in the direction of the greatest local increase of the consequent function, and thus shows how a rule relates to its “neighboring” rules in rule space. Two Examples demonstrate that the linguistic gradient is a linguistic analogue of the classical gradient operator.