A Non-Iterative Reasoning Algorithm for Fuzzy Cognitive Maps based on Type 2 Fuzzy Sets

Highlights

• New non-iterative reasoning algorithm for Fuzzy Cognitive Maps (FCMs) is developed.

• Weights are generated based on experts surveys and modelled using Type 2 Fuzzy Sets.

• Uncertainties in experts’ opinions are aggregated using Interval Agreement Approach.

• Sensitivity analysis provides insight into uncertainty propagation in FCMs.

Abstract

A Fuzzy Cognitive Map (FCM) is a causal knowledge graph connecting concepts using directional and weighted connections making it an effective approach for reasoning and decision making. However, the modelling and reasoning capabilities of a conventional FCM for real world problems in the presence of uncertain data is limited as it relies on Type 1 Fuzzy Sets (T1FSs). In this work, we extend the capability of FCMs for capturing greater uncertainties in the interrelations of the modelled concepts by introducing a new reasoning algorithm that uses Type 2 Fuzzy Sets based on z slices (zT2FSs) for the modelling of uncertain weights connecting FCM’s concepts. These Type 2 Fuzzy Sets are generated using interval valued data from surveyed participants and aggregated using the Interval Agreement Approach method. Our algorithm performs late defuzzification of the FCM’s values at the end of the reasoning process, preserving the uncertainty in values for as long as possible. The proposed algorithm is applied to the evaluation of the performance of modules of an undergraduate Mathematical programme. The results obtained show a greater correlation to domain experts’ subjective knowledge on the modules’ performance than both a corresponding FCM with weights modelled using T1FS and a statistical method currently used for evaluating the modules’ performance. Sensitivity analysis conducted demonstrates that the new algorithm effectively preserves the propagation of uncertainty captured from input data.

Introduction

Fuzzy Cognitive Maps (FCMs) are fuzzy directed graphs that represent information about causal relations among interrelated concepts of a modelled system. They were introduced by Bart Kosko [1] as an extension of Cognitive Maps [2] that introduced fuzziness in causal relations and concepts for modeling real world systems. The rationale for this was the existence of data uncertainties in assigning the values of concepts and causal relations, which could be handled more effectively using fuzzy sets. In the last few decades, there has been a noticeable research trend towards applying FCMs in various domain areas [3]. The literature reveals the success of FCMs in reasoning and modelling in Engineering [4], [5], Medicine [6], Business [7], Software Engineering [8] and Politics [9]. FCMs have gained momentum in these fields due to their simplicity and strong mathematical structure that enhances their prediction, analysis and reasoning capabilities.

Although the effectiveness of conventional FCMs were demonstrated in prior work, they had some drawbacks regarding their reasoning and modelling abilities. Conventional FCMs cannot be effectively used in applications requiring a high level of uncertainty and/or where relations within the modelled domain are nonlinear and/or non-monotonic. Typically, FCMs rely on crisp values generated by defuzzification of the fuzzy sets that represent weights between concepts. In this way, an FCMs’ ability to represent and control knowledge with high randomness and uncertainties between the concepts is hindered. Moreover, conventional FCMs cannot handle more than one relation between the concepts, and it cannot model a grey domain environment (an environment with multi-meaning). To overcome these shortcomings, several extensions to conventional FCMs have been proposed, which are classified into three categories, based on the types of drawbacks they aim to overcome [10]: 1) overcoming the drawbacks of modelling uncertainty and handling more relationships between the concepts, 2) dynamicity and 3) drawbacks related to a rule-based knowledge representation.

Most of these proposed extensions focused on improving the representation of the weight associated with causal relations between FCM concepts as the weights play a crucial role for knowledge representation and uncertainty propagations while reasoning. For example, Intuitionistic fuzzy sets (IFSs) [11] were used to model causal edges of a FCM in [12], [13] and thus the iterative reasoning algorithm was modified to be compatible with the introduced IFSs. In IFSs, the concept of hesitancy was modelled by assigning not just a degree of belonging of an element to an IFS, but the degree of the element’s non-belongingness to the set as well. Thus, Intuitionistic Fuzzy Cognitive Maps (iFCMs) were introduced to capture the hesitancy of experts in modelling of interrelations between the concepts using If-Then rules. Although, the iFCM was less affected by missing input data compared to the conventional FCM, incorporating IFSs made the reasoning process for each iteration much more complex.

A Fuzzy Grey Cognitive Map (FGCM), proposed in [14], was based on Grey Systems to represent domains with a high level of uncertainty, which included limited and incomplete data. In the FGCM, the weights of causal links are represented by grey intervals, which enhance the capability of the FGCM to represent the uncertain relationships of the domain. Grey Intervals measure both the intensity of existing causal relationships between two concepts and absent relations between any two concepts with partially or completely unknown intensity. Although the FGCM is adapted for handling uncertainty using the grey intervals, it is incapable of modelling dynamic and nonlinear relations. The conventional FCM was extended to the Triangular Fuzzy Cognitive Map in [15] by representing uncertain interrelations among the concepts using triangular fuzzy numbers. As a further step offering additional expressiveness and flexibility in representing values of concepts and weights of causal relationships, the approach proposed in [16] represented the concepts and edges of the FCM using intervals. In this approach, the membership degrees of an element’s belonging to the concept and the edge’s weight were given as intervals whose widths were understood as a measure of uncertainty. In [17], edges in FCMs were modelled using fuzzy If-Then rules that considered standard semantics of fuzzy sets, where the qualitative representation of the knowledge about causal relations was represented using T1FSs.

Although the former extensions of FCMs have achieved noticeable success in reasoning in the presence of imprecise or missed information, the weights of their causal relations rely on T1FSs, which limit their ability to capture higher levels of uncertainties associated with real world applications. This deficiency of the T1FS is due to its crisp membership function that hinders the modelling ability of the T1FS in domains that involve a high level of uncertainty. This shortcoming of the T1FS led to the extension of T1FSs through the introduction of a higher level of fuzzy sets, where the grade of the membership is uncertain.

Type 2 Fuzzy Sets (T2FSs) are fuzzy sets that represent the membership’s grades of elements, called the secondary membership grades, using T1FSs. The T2FSs were defined to handle higher levels of uncertainty that were present in many real-world problems [18] and to handle uncertain membership functions. Thus, T2FSs can model a high level of uncertainty, where determining the precise membership of the fuzzy set is difficult or even impossible. Although T2FSs are shown to outperform T1FSs in modelling uncertainties, the complexities of computation with the T2FSs are increased due to the existence of an additional two dimensions required for the T1FS membership grades. To overcome these complexities, some novel representations of T2FSs were proposed; for example, Interval Type 2 Fuzzy Sets (IT2FSs) were introduced in [18], where domains of fuzzy sets were intervals. IT2FSs were used in different applications, for example, to model the words and hence capture more linguistic uncertainties in [19], and [20], or to design a fuzzy fault detection filter for Markov jump systems in [21], [22].

Although the success of IT2FSs in the former applications was acknowledged, as they offered a wide scope for capturing and accurately representing input uncertainties, several studies have suggested the necessity of moving to an alternative representation of T2FS, where the third dimension can be exploited to capture more uncertainties. In that respect, zT2FS were defined to reduce complexities of representation and calculations for general T2FSs [23] to fully utilise their third dimension. Whilst in IT2FSs the degree of the secondary membership grade of each element is fixed to be 1 in the third dimension, in zT2FSs it is a value between 0 and 1, inclusively. The approach based on the zT2FS supports the smooth transition from a fuzzy logic system relying on IT2FSs to a fuzzy logic system relying on zT2FSs. zT2FSs were used in various applications, for example, in [24], [25], where the use of T1FSs and IT2FSs was not possible due to a high level of uncertainty; however, it was well captured by zT2FSs.

As demonstrated, the weights of the casual relations in FCMs are crucial for the propagation of uncertainty and T2FSs and their representations, such as IT2FSs and zT2FSs, are more capable than T1FSs for capturing more uncertainties and hence enhancing the decision modelling. Some methods were proposed to enhance FCMs by introducing T2FSs to the modelling of their edges’ weights. For example, in [26], two Interval Type 2 Fuzzy Cognitive Maps (IT2FCMs) for a flight control system were proposed, one for stabilizing the attitude and altitude dynamics and one for tracking trajectories. Weights of the causal relations between the concepts were represented by IT2FSs to capture the inter uncertainty of the experts’ opinions about the assigned weights. Subsequently, these fuzzy weights were defuzzified to scalar values to be used in the FCM’s reasoning algorithm. A comparison between the IT2FCM and the FCM which relied on T1FSs weights (the T1FCM), showed that the IT2FCM performed better than the T1FCM in the presence of noise and imperfect conditions, as the IT2FSs capture more uncertainties compared to the T1FSs.

In [27], a methodology for modelling weights of a FCM edges using zT2FSs was proposed, creating a zT2FCM for an autism diagnosis in toddlers. The proposed zT2FCM had 20 concepts that had casual relations, with one decision concept that predicted the autistic disorder. Weights were generated using the Interval Agreement Approach (IAA) [28] to capture high level uncertainties in the presence of imprecise interval valued data acquired from different doctors in a hospital. The results presented were more accurate and consistent with doctors’ opinions compared with those of a conventional FCM presented in [29], using the same data.

In both these studies, the reasoning method applied in the proposed IT2FCM and zT2FCM, respectively, was iterative, where the value of a decision concept was calculated using an arithmetic function and then defuzzified into a scalar value in each iteration. This suggests a loss of information captured in the weights pertaining to the Type 2 Footprint of Uncertainties [18] and secondary membership values (z slices), that can no longer propagate to influence the selected decision concept.

The research presented in this paper is motivated by two important complex issues identified from the previous works. Firstly, representation of weights of FCM’s edges using a particular type of fuzzy set is crucial for the knowledge capture, modelling and its propagation through the FCM. Initially, the weights were represented using T1FSs. However, to overcome some of the identified drawbacks when using this modelling tool, several other approaches were proposed, such as zT2FSs. Second, the reasoning algorithm used in all the previous work was iterative and required defuzzification of the weights in the FCMs that could inevitably have led to the loss of some information.

Therefore, this research proposes an extension of FCMs and their reasoning algorithm to overcome the drawbacks of modelling and reasoning in the presence of uncertainty in relationships between the concepts. The main contributions of this research are as follows:

1) Interval Agreement Approach (IAA) is applied to aggregate both intra and inter uncertainty of experts to model the FCM’s weights using zT2FSs that are kept in the reasoning algorithm in their fuzzy set forms until the end of the reasoning process.

2) A new non-iterative reasoning algorithm is developed that uses zT2FSs rather than their defuzzified values; the output value of a decision concept is only defuzzified at the very end of the reasoning process. This late defuzzification reduces the chance of losing information and uncertainties captured in the modelled system compared to using the conventional reasoning process.

3) The accuracy of the proposed reasoning algorithm is examined using the case study of a real-world problem of evaluating the performance of 30 modules taught on an undergraduate Mathematical programme. A novel FCM is generated for this problem by collecting experts’ opinions. Results obtained by applying the new reasoning algorithm are compared with those generated by a method based on statistics, currently used in the Department of Mathematics and Applied Sciences under consideration, a standard FCM with weights represented using T1FSs and subjective opinions of lecturers in the Department. The comparison is performed using a statistical measure of correlation.

4) Sensitivity analysis carried out provides new insights into the impact of an FCM structure on uncertainty propagation. Experiments demonstrate that the new reasoning algorithm preserves the propagation of uncertainty across weighted connections in the FCM during the reasoning process.

The rest of this paper is organised as follows. Section 2 provides an essential background about the main concepts related to this paper, including FCMs, zT2FSs and IAA that generates z slices. Section 3 presents the new proposed reasoning algorithm. Section 4 describes a case study related to the evaluation of module performance using different methods and reasoning algorithms, including the reasoning algorithm proposed. Section 5 presents results obtained using these different methods and their comparison. The sensitivity analysis is presented in Section 6. Finally, Section 7 includes the conclusion and future work.