On the convergence of Sigmoid Fuzzy Cognitive Maps

Abstract

Fuzzy Cognitive Maps (FCM) are Recurrent Neural Networks that are used for modeling complex dynamical systems using causal relations. Similarly to other recurrent models, a FCM-based system repeatedly propagates an initial activation vector over the causal network until either the map converges to a fixed-point or a maximal number of cycles is reached. The former scenario leads to a hidden pattern, whereas the latter implies that cyclic or chaotic configurations may be produced. It should be highlighted that FCM equipped with discrete transfer functions never exhibit chaotic states, but this premise cannot be ensured for systems having continuous neurons. Recently, a few studies dealing with convergence on continuous FCM have been introduced. However, such methods are not suitable for FCM-based systems used in pattern classification environments. In this paper, we first review a new heuristic procedure called Stability based on sigmoid Functions, which allows to improve the convergence on sigmoid FCM, without modifying the weights configuration. Afterwards, we examine some drawbacks that affect the algorithm performance and introduce solutions to enhance its performance in pattern classification environments. Additionally, we formalize several definitions which were omitted in the original research. These solutions lead to accurate classifiers and prevent specific scenarios where the method may fail. Towards the end, we conduct numerical simulations across six FCM-based classifiers with unstable features in order to evaluate the proposed improvements in pattern classification environments.