Fuzzy Cognitive Maps Y. İlker TOPCU, Ph.D twitter.com/yitopcu

Causal cognitive mapping is a method that captures the diverse mental models of the experts in simple directed graphs where concepts are represented by nodes and relations between concepts are represented by an arc from the affecting concept to affected concept. The relation is positive if there is an increase at the affected concept when affecting concept increases. If there is a decrease at the affected concept when affecting concept increases, the relation is negative. Cognitive Maps

By interviews with experts or by examining published reports or studies, the related concepts and interrelations among them can be revealed. These beliefs and judgments are brought together to have an aggregated cognitive map. Qualitative analyses can be conducted on this map. Cognitive Maps

If there are even number of concepts affecting a concept C, half of the relations are positive and others are negative in the long run, it cannot be determined whether C will increase, decrease, or remain same Drawbacks

FCM To predict the overall system behavior of the concepts in the cognitive map, a formal analysis can be conducted  Fuzzy Cognitive Map (FCM) Cognitive Maps + Quantitative Values + Time = Fuzzy Cognitive Maps FCM is based on Fuzzy Set Theory Theory of Neural Networks

Fuzziness Causal Cognitive Maps are limited by the trivalent values [-1, 0, 1] which is binding in the Fuzzy Causality Environment At FCMs, the strength of impacts are assessed During the causality gathering stage: verbal phrases such as little, partially, usually, a lot, strong, weak, etc are used. values at the interval of [-1, +1] are assigned: positive causality (0 < W ij ≤ 1) negative causality (-1 ≤ W ij < 0) no relationship (W ij = 0)

Aggregation

FCM & Adjacency (Influence) Matrix

Neuron Firing FCM is regarded as a simple form of Recursive Neural Network, with concepts being equivalent to neurons; however, concepts are not either off or on (0 or 1), but can take values in-between [0, 1]. Fuzzy concepts are non-linear functions that transform the path-weighted activations directed towards them into a value.

Neuron Firing When a concept changes its value (a neuron fires), it affects all concepts that are causally dependent upon it. Depending on the sign of the relation and the strength of it, the affected concepts subsequently may change their values as well (further concepts are activated at the network). For this purpose an iterative process is conducted

Iterative Process A given state vector with degree of activation values of -1, 0, or 1 for the concepts (i.e. the value of a concept will be -1 if DMs let it decrease and it would be 1 if it is let to increase) is multiplied with adjacency (influence) matrix (a nxn square matrix representing the fuzzy causal relations among n concepts) in each iteration to come up with an updated state vector.

Activation Function

Activation Functions

Simulation After determining the type of the threshold function and creating different initial state vectors for several scenarios, the simulations are run to observe and analyze the dynamic behavior of the system under consideration. A simulation ends after a certain number of iterations or when the values of concepts converge to a fixed form (stable state).

Results The values of concepts at the final state vector: to which extent which concept will increase and which one will decrease in the long run.

Example An illustrative fuzzy cognitive map with 13 concepts A hypothetical scenario indicating a decrease at the 3 rd concept The state values of the concepts converge at iteration 12

Example A decrease at C 3 : will increase the level of C 6, C 8, C 11, C 12, and C 13 ; will decrease the level of C 7 ; will increase the level of C 1, C 2, C 4, C 5, and C 10 to some extent; In the long run, C 3 has no effect on C 9