A Balanced Differential Learning algorithm in Fuzzy Cognitive Maps

Fuzzy Conceptual Maps have become an important means for describing a particular domain showing the concepts (variables) and the relationship between them. They have been used for several tasks like simulation processes, forecasting or decision support. In general, the task of creating Fuzzy Conceptual Maps is made by experts in a certain domain but it is very promising the automatic creation of Fuzzy Conceptual Maps from raw data. In this paper we present a new algorithm (the Balanced Differential Algorithm) to learn Fuzzy Conceptual Maps from data. We compare the results obtained from the proposed algorithm versus the results obtained from the Differential Hebbian algorithm. Based on the results we conclude that the algorithm proposed seems to be better to learn patterns and model a given domain than in the classical approach. Fuzzy Cognitive Maps Cognitive Maps (CM) also called causal maps are (apart from Bayesian Networks [Pearl 1988]) a useful model to represent concepts or variables in a given domain and their causal-effect relations. Cognitive Maps were first introduced by Axelrod [Axelrod 1976] in 1976 to model causal relations inferred between the concepts of a given domain. A CM is a directed graph, where nodes are the concepts of the given environment and arrows among nodes represent causal relations between concepts A special kind of CM are the Fuzzy Cognitive Maps (FCM) that were introduced by Kosko [Kosko 1986a] to represent fuzzy cause-effect relations instead of the crisp cause-effect relations represented in the original CM. FCMs are fuzzy-signed digraphs with feedback [Kosko 1986a][Kosko 1988]. Nodes in the graph that are Fuzzy Sets representing concepts. Directed edges (arrows) represent causal-effect relations between the concepts as in the case of generic CMs. Arrows can have positive or negative values, a positive value shows a positive causal connection. As is depicted in fig1 the value of concept B increases or decreases as concept A increases or decreases. Whereas in fig2 a negative causal connection causes the value of the concept B to decrease when the value of concept A increases, and also a negative causal connection causes the value of concept B to increase when the value of concept A decreases. FCMs have been used as an alternative to expert systems in several areas like economics, sociology or simulation. In the literature there are a lot of implementations of FCMs to model specific environments like decision making and policy-making [Carlsson and Fullér][Stylos and Groumpos]. The process to build a FCM is similar to the process used to create a knowledge base in an expert system. First, one or more domain experts identify the concepts and their causal relationships. They talk about if two concepts have strong, weak, null, etc... causal relation. This use of linguistic labels to explain the grade of causal relation is very used in Fuzzy Logic and has a direct translation in a FCM. The "degree-of-causality" values in the connecting edges indicate how much one concept causes another. Values can range from -1, indicating a strong negative impact, through 0, or no impact, to +1, a strong positive impact. An example is shown in fig3. We can see in the table the linguistic labels used for the domain experts and a possible automatic translation to weights in the causal web of the FCM. SYMBOLIC VALUES NUMERIC VALUES Affects a lot 1.0 Affects 0.5 Do not Affects 0.0 Affects negatively -0.5 Affects negatively a lot -1.0 fig3. Mapping between labels and values _ fig1.Positive causal weight + + A B