Parameter analysis for sigmoid and hyperbolic transfer functions of fuzzy cognitive maps

Abstract

Fuzzy cognitive maps (FCM) have recently gained ground in many engineering applications, mainly because they allow stakeholder engagement in reduced-form complex systems representation and modelling. They provide a pictorial form of systems, consisting of nodes (concepts) and node interconnections (weights), and perform system simulations for various input combinations. Due to their simplicity and quasi-quantitative nature, they can be easily used with and by non-experts. However, these features come with the price of ambiguity in output: recent literature indicates that changes in selected FCM parameters yield considerably different outcomes. Furthermore, it is not a priori known whether an FCM simulation would reach a fixed, unique final state (fixed point). There are cases where infinite, chaotic, or cyclic behaviour (non-convergence) hinders the inference process, and literature shows that the primary culprit lies in a parameter determining the steepness of the most common transfer functions, which determine the state vector of the system during FCM simulations. To address ambiguity in FCM outcomes, we propose a certain range for the value of this parameter, $${\uplambda }$$ λ , which is dependent on the FCM layout, for the case of the log-sigmoid and hyperbolic tangent transfer functions. The analysis of this paper is illustrated through a novel software application, In-Cognitive, which allows non-experts to define the FCM layout via a Graphical User Interface and then perform FCM simulations given various inputs. The proposed methodology and developed software are validated against a real-world energy policy-related problem in Greece, drawn from the literature.

An Optimal Distributed RES Sizing Strategy in Hybrid Low Voltage Networks Focused on EVs’ Integration

“…where Cj is the value of feature j, wj is its corresponding weight and wbias,i is its bias weight. As for the activation function of the graph nodes, the sigmoid function is adopted (14) since it is commonly used in other FCMs [38], [39].…”

A stakeholder-informed modelling study of Greece's energy transition amidst an energy crisis: The role of natural gas and climate ambition

“…Upon FCM simulation, we optimise and normalise the value of parameter λ to ensure that the FCM simulation process converges, as shown in [90]. The FCM design and simulation process is presented in Section 4.…”

“…Contrary to common practice in FCM research, where FCMs are simulated for policy (strategy) shocks (e.g., [84,87,88]), we instead stimulate the FCM explicitly based on uncertainty shocks, to explore which of the eight identified bottlenecks impact the system's decarbonisation process the most. As discussed in Section 2.3, we use the tool and method discussed in [90], by activating one bottleneck while keeping all others inactive (i.e., eight FCM simulations, one for each bottleneck). Results are presented in Fig.…”

Characterizing the Social-Ecological System for Inland Freshwater Salinization using Fuzzy Cognitive Maps: Implications for Collective Management

“…where λ is a real positive number that determines the steepness of the transfer function and therefore the degree to which it squashes activation values at each iterative step. 28,102,103 The same value for λ was used for all dynamic simulations (λ = 0.3). Selection of this value was informed by Koutsellis et al 103 Brie y, the upper bound of λ that guarantees a unique simulation state was calculated for each aggregate FCM,…”

“…28,102,103 The same value for λ was used for all dynamic simulations (λ = 0.3). Selection of this value was informed by Koutsellis et al 103 Brie y, the upper bound of λ that guarantees a unique simulation state was calculated for each aggregate FCM,…”