Two approaches for handling different types of uncertainty in Fuzzy Cognitive Maps

Abstract

The world has recently experienced the relevance of having mathematical and computational models for simulating complex systems. The behavior of such systems is challenging to model because of the dependencies, uncertainty, nonlinearity, and feedback loops between variables. Overall, the modeling and simulation of complex systems allow experts to explore the outcome of hypothetical scenarios before being observed in reality—this translates into saving time and money or understanding the problem domain. Fuzzy Cognitive Maps (FCMs) [1] are recurrent neural networks used to design more complex machine learning solutions. Such knowledge-based networks use a collection of neural concepts that represent characteristics of the system with an activation value that expresses its degree of activation in the system at a particular time [2]. The cause-effect relationships between concepts are characterized by a numerical weight wij ∈ [−1, 1] indicating the strength of the association. One of the most attractive features lies in their graphical nature, transparency, adaptability, and ability to perform What-if simulations. These advantages have motivated the scientific community to use fuzzy cognitive mapping in various application domains, including social and political sciences, engineering, information technologies, robotics, expert systems, education, prediction, environments, medicine, etc. Most of these solutions have solid social and interdisciplinary scientific value. But, contrary to what might be expected, the actual FCMs do not include any mechanism to deal with the uncertain issues that may be present in the knowledge provided by human experts [3] [4]. When discussing uncertainty in the literature, we find the term overloaded, commonly used as a generic term for imperfection in data and as a particular form of imperfect knowledge of whether or not a statement is true [5]. Bearing that uncertainty can manifest itself in different ways and be induced by different causes [6], we use the word ”uncertainty” as a generic term in this research. We refer to the uncertainty that arises as imprecision when exiting a value that cannot be measured with suitable precision. Imprecision can be a linguistic value as ”Low” or can be an interval valued as ”[25 30]”. Therefore, this dissertation proposes to extend the action field of FCMs to deal with uncertainty environments, but with marked differences from the proposals found in the review of state of the art (see Section 1.3). The general scope is to develop models that allow symbolic reasoning using FCM in uncertainty environments. In order to investigate the general goal of this dissertation, we divide it into further research objectives. Such goals can be formalized as follows: 1. To develop models that handle uncertainty expressed as linguistic terms by combining FCM with Computing with words (CWW) [7]. 2. To develop a model to handle uncertainty expressed as interval values using 1.1 Fuzzy Cognitive Maps 3 FCMs and Grey System Theory (GST) [8]. 3. To develop learning algorithm variants for the model that handles uncertainty expressed as interval values. Chapter 2 groups four models proposed in this research that show good performance of the combination of the graphical nature of FCM with the natural language technique to express both the concepts’ activation values and weights, based on the methodology of Computing with words in different experimental studies. Also, the Chapter presents a transfer function for symbolic domains inspired by the sigmoid function to prevent the activation model from having negative values that may result from the aggregation process of the linguistic terms impacting the ith concept. Chapter 3 and Chapter 4 describe the model and the variants of nonsynaptic learning algorithms proposed as the second approach of this research, respectively. A merger of FCM and Grey System Theory is elaborated to deal with situations where the experts give crisp numbers but do not agree on these values. In some way, they hesitate about the variables’ exact values when designing the model. The proposal allows modeling the uncertainty in three components that affect the performance of the map: the inputs, the model activation of the nodes, and the weights.