On the Convergence of Sigmoidal Fuzzy Grey Cognitive Maps

Abstract

Fuzzy cognitive maps (FCMs) are recurrent neural networks applied for modelling complex systems using weighted causal relations. In FCM-based decision-making, the inference about the modelled system is provided by the behaviour of an iteration. Fuzzy grey cognitive maps (FGCMs) are extensions of fuzzy cognitive maps, applying uncertain weights between the concepts. This uncertainty is expressed by the so-called grey numbers. Similarly as in FCMs, the inference is determined by an iteration process which may converge to an equilibrium point, but limit cycles or chaotic behaviour may also turn up. In this paper, based on the grey connections between the concepts and the parameters of the sigmoid threshold function, we give sufficient conditions for the existence and uniqueness of fixed points of sigmoid FGCMs.

Modeling Implicit Bias with Fuzzy Cognitive Maps

“…The existence and uniqueness of fixed points of sigmoid FCMs was firstly discussed by Boutalis et al [5] when the transfer function is given by f (x) = 1/(1 + e −x ) . Other theoretical results about this topic are the ones reported in Knight et al [22], and Harmati et al [19]. As a generalization of the stability and instability properties of FCM-based models, in [30] we introduced the definitions of E-stability and E-instability, and four sufficient conditions to fulfill these properties.…”

The Dynamic Extensions of Fuzzy Grey Cognitive Maps

“…To improve the uncertainty modeling and dynamic modeling ability are two main research fields for FCM extensions [19]- [21]. To tackle the uncertainty modeling problem, the Fuzzy Grey Cognitive Map (FGCM) is an impressive model that combines the Grey System Theory (GST) and FCM [22], it makes the FCM can deal with the data with uncertainty like interval values [23].…”

“…FGCMs also have their activation functions like sigmoid and tanh in the form of IGN [25]. The convergence characteristic of FGCM is similar to the counterpart of FCM: a fixed point, a limited circle or chaos [19], [26], [27]. Compared with other efficient and excellent methods that also deals with the uncertainty data, such as complex mass function [28]- [31] in the field of Dempster-Shafer (D-S) evidence theory, the FGCMs and their extensions try to exploit all information from the uncertainty data rather than choose the maximum likelihood value from data.…”