Services Customization Using Web Technologies by Kardaras Dimitris

Author:Kardaras, Dimitris...
Language: eng
Format: epub
Published: 2014-04-24T18:23:05.878000+00:00

The analysis of a fuzzy cognitive map starts with the manipulation of the signs of the causal paths. Axelrod (1976) suggests two rules that apply to paths of any length, in order to determine the direction of the effect due to changes in cause variables. The following rules, which are similar to the rules presented earlier in cognitive maps analysis, apply to the two-valued system, however there are also rules proposed by Axelrod (1976) for the eight-valued system as well.

Rule 1: The indirect effect of a path from a cause variable (X) to an effect variable (Y) I(x,y), is positive if the path has an even number of negative arrows, and it is negative if it has as odd number of negative arrows. The indirect effect is defined as the multiplication of the signs of the causal relationships that form the path from the cause variable to the effect variable. In fuzzy cognitive maps the fuzzy weight of the indirect effect is determined by the minimum of all fuzzy weights in the path (Kosko, 1986).

Rule 2: The total effect of a cause variable (X) on an effect variable (Y) is denoted by T(x,y), and is the sum of all the indirect effects from the cause variable X to the effect variable Y. The total effect is defined only if all indirect effect reaching the effect variable (Y) have the same sign, positive or negative. In such a case the common sign specifies the sign of the total effect while the fuzzy weight of the total effect is the maximum of all indirect fuzzy weights reaching the effect variable (Kosko, 1986). In the case that the total effect cannot be determined when it is the result of negative and positive indirect effects, then there is an indeterminacy problem of the total effect (Axelrod, 1976). Other approaches should be applied in order to resolve the indeterminacy problem, e.g. the one discussed in Kosko (1986).

The FCM approach is, therefore, an inferential mechanism allowing the existence of fuzzy causal relations between concepts and the monitoring of their effects (Lee & Han, 2000). In a similar way as with CMs, if certain nodes are stimulated, the change will be conveyed through positive or negative weighted links throughout the map until equilibrium is reached. Equilibrium is reached when the system does not evolve any further, i.e. it does not produce any changes.

A fuzzy cognitive map can also be represented by using an NxN matrix (E), where N is the number of nodes-concepts contained in the graph. Every value of this matrix represents the strength and direction of causality between various concepts. As already stated, the value of causality eij is assigned values from the interval [-1, +1]. It will thus be (Schneider et al., 1998):

eij > 0 indicates a causal increase or positive causality from node i to j.

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