# Fuzzy Cognitive Maps: Their Role in Explainable Artificial Intelligence
## 3 Materials and Methods
### Fuzzy Cognitive Maps
#### 3.1.2. FCMs in a Nutshell  
In their visual formation, FCMs consist of a signed direct graph, which includes all the input concepts (C\({}_{i}\)) and weighted arcs (Wij) to express their relationships [10]. There may be one or more output concepts. In their classical representation, the experts determine initial weight values (Wij), usually expressed verbally. For example, concept C\({}_{1}\) on concept C\({}_{2}\) may be specified as \"Very Strong\". This entails that a slight change in the value of C\({}_{1}\) would dramatically affect the value of C\({}_{2}\). Due to many interconnections, the computation of the concept values does not involve one forward procedure but requires many iterations until the system is stabilized. Figure 2 illustrates a graph display of an FCM.  
Figure 1: Machine learning and fuzzy cognitive map pipeline and tasks.  
Figure 2: Graph display of a fuzzy cognitive map.  
The mathematical operation formulates the influence of each concept on others and computes the updated values. The function, which determined the value of each concept at the end of each iteration, is usually described by the following expression [12; 19; 20]:  
\[C_{i}(t+1)=f\left[C_{i}(t)+\sum_{\begin{subarray}{c}j=1\\ j\
eq i\end{subarray}}^{N}C_{j}\left(t\right)\cdot W_{ji}\right] \tag{1}\]  
In Equation (1), \(f\) is the activation function, which normalizes the resulting mathematical operation and constraints within specific limits.  
The construction of an FCM is achieved as follows:  
1. The stakeholders are gathered, and they decide the input concepts, their potential initial state, and the output concepts;
2. The stakeholders determine the causal relationships among the concepts;
3. The engineers determine the concept value update function, the activation function, and the number of iterations. They also specify any other parameter involved.  
The following pseudo-code describes the steps of FCM modelling and functionality:  
1. Every input concept's verbal expression or potential values are transformed into numerical forms;
2. Every verbally assigned weight is transformed into a numerical form that usually lies within the interval [\(-\)1, 1];
3. The FMC is allowed to update the concept values using Equation (1) for a specific number of iterations;
4. The engineers and the stakeholders examine the final states of the input and the output concepts and explain the results.  
Recent advances in FCMs circumvent the static nature of the classical FCMs by introducing state concepts and their accompanying equations [21; 22] or learning algorithms [23; 24; 25]. In this way, FCMs are allowed to learn some of the interconnections, whereas the experts specify others. Therefore, FCMs may embody human knowledge and discover underlying patterns in data (Figure 3). The rigorous description of the system's inputs and outputs and the significance of their interactions are requirements for the operation of conventional FCMs [21]. Advanced FCM topologies allow some degrees of freedom to learn from the input data, giving a preassigned baseline weight value that represents the baseline causality between two concepts or freezing some weights that are known beforehand and allowing some other interconnections to be learned [26].