Advanced approaches to fuzzy cognitive mapping for enhancing convergence, learning, and prediction

Abstract

Fuzzy Cognitive Maps (FCMs) are widely used for modeling complex systems and analyzing their dynamic behaviors. However, traditional FCMs rely on classical reasoning mechanisms, typically based on nonlinear activation functions such as sigmoid or hyperbolic tangent, that often converge to a steady state regardless of initial conditions. While this behavior may be acceptable for certain control-oriented problems, it limits the applicability of FCMs in both prediction and learning-based contexts where modeling non-equilibrium or cyclic behaviors is essential. Additionally, the common reliance on expert-defined relationships introduces subjectivity and inconsistency, which can compromise the reliability of the model, especially when interdependencies are intricate or partially hidden. This study addresses these limitations by proposing an alternative reasoning mechanism that incorporates indirect relationships through higher-order dependencies. By considering the powers of the relationship matrix, the mechanism captures the deeper structure of causal influence among system concepts. This provides a more comprehensive framework for modeling, predicting and learning complex behaviors. This thesis is structured around three key parts, each addressing a distinct aspect of the research. The first part introduces the conceptual foundation of the proposed mechanism, focusing on how the inclusion of both direct and indirect relationships enhances the assessment of concept importance in a system. This is demonstrated through an application in a decision-making context under uncertainty, where interrelated criteria are analyzed using fuzzy representations. The results show that accounting for hidden causal relationships improves the model's depth and interpretability. The second part elaborates the theoretical structure and implementation method. Unlike traditional approaches that rely on nonlinear activation functions, the proposed mechanism uses successive powers of the relationship matrix to represent indirect interrelationships. Each resulting matrix is normalized and aggregated to compute new activation levels. This approach enables dynamic transitions that generate solutions sensitive to initial conditions rather than always converging to a single fixed state. A case study on supply chain vulnerability analysis is presented to demonstrate how the method captures different behavioral patterns under various disruption scenarios. The third part presents a comprehensive experimental evaluation within the context of FCM learning. Different versions of the reasoning framework, using varied normalization techniques, cumulative and non-cumulative aggregation strategies, and three fitness functions (RMSE, Max-RMSE, and Heaviside), are tested on randomly initialized FCM models. The learning process is supported by the Particle Swarm Optimization (PSO) algorithm, which is used to fine-tune relationship weights based on historical data, reducing reliance on subjective expert inputs. Performance is compared against classical reasoning mechanisms and a recent exponential normalized reasoning approach. The results show that the proposed mechanism consistently provides higher prediction accuracy. Notably, the Heaviside function yields the best results across most setups. To ensure an unbiased evaluation, synthetic data were not generated using any of the tested mechanisms. Instead, the experiments utilized a real data about marine ecosystem involving large aquatic species such as dolphins, inspired by early pioneering work in the field. This approach ensured independence of test data and preserved objectivity in performance comparisons. Statistical validation using the Wilcoxon signed-rank test and effect size calculations confirmed the proposed method's superiority in most scenarios. In conclusion, this thesis presents an effective reasoning framework that extends the learning and prediction capabilities of FCMs. By moving beyond conventional activation-based reasoning and integrating higher-order structural knowledge, the proposed mechanism improves the accuracy of FCMs. It provides valuable tools for modeling complex dynamic systems in domains such as social analysis, ecological management, and strategic planning.