Abstract
Fuzzy cognitive maps (FCMs) play an important role in high-level reasoning but are limited in their ability to model complex systems with singularities. We are interested in systems that exhibit discontinuous behaviors as one or more of their internal node states approach a threshold. In a new approach to FCM dynamics, we define general classes of aggregation functions which “jump” to a boundary value when any input crosses a threshold, or when all inputs do. The threshold value is a context-dependent parameter which can be readily understood by subject matter experts. Aggregation functions are applied separately to positively and negatively causal antecedents to each node then combined to form the nodal state. This modeling is applied in Computing with Words (CWW) settings, in which link strengths and activation levels are elicited using vocabulary words represented by interval type-2 fuzzy membership functions. We illustrate the behaviors of these novel FCM systems in comparison with their nonsingularity versions.
Highlights
F UZZY cognitive maps (FCM) [1]–[6] are fuzzy signed digraphs in which the nodes represent high-level descriptive concepts and the links represent positively or negatively causal influences between concepts, along with the corresponding strengths of these influences.In Kosko’s original formulations of Fuzzy cognitive maps (FCMs) as qualitative models of social systems [7], the node activations and link strengths were described in linguistic terms, but in most subsequent work, they are described using scalar values
FCMs have been applied in numerous fields to model first-order feedback relationships in complex systems, with the general objective of predicting the activations of certain key nodes resulting from the fixed activations of one or more “exogenous” nodes [8]–[10]
The state xi(k) ∈ X of the ith node of an FCM at time k is conventionally defined to be a weighted average of the input activation levels at time k − 1 followed by “squashing” with a transfer function f : R → X from the real line R into the set of valid activation values [12]
Summary
F UZZY cognitive maps (FCM) [1]–[6] are fuzzy signed digraphs in which the nodes represent high-level descriptive concepts and the links represent positively or negatively causal influences between concepts, along with the corresponding strengths of these influences. WPMs are an example of a more general class of operators called quasi-means or Kolmogorov–Nagumo means [23] in which inputs are operated on by an invertible function prior to being aggregated as a weighted sum, which is transformed by the inverse of the initial function With this form of aggregation, one can feasibly compute aggregations of fuzzy sets even when link strengths/weights are modeled as fuzzy sets. This article takes this work further and applies it to FCMs. In Section II, we present a generalized node structure inspired by mean operators and describe its application when link strengths and activation levels are intervals, and, by iterative application, when they are type-1 and type-2 fuzzy sets.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
