Fuzzy Cognitive Maps

Y. İlker TOPCU, Ph.D.

www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info

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• 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 < Wij ≤ 1)

• negative causality (-1 ≤ Wij < 0)

• no relationship (Wij = 0)

Aggregation

•

where Wija indicates the causality between concepts

Ci and Cj at the aggregated fuzzy cognitive map,

pk is the normalized priority of the expert k, and

Wijk is the causality between concepts Ci and Cj

according to expert k.

When experts are not prioritized pk should be 1/n.

FCM & Adjacency (Influence) Matrix

C1 C2 C3 C4 C5C1 0 W12 0 0 0C2 0 0 W23 0 0C3 0 W32 0 W34 0C4 W41 0 0 0 W45C5 W51 0 0 0 0

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

•

where Ai(k) is the activation level of concept Ci at

iteration step k,

Aj(k-1) is the activation level of the interconnected

concept Cj at iteration step k-1,

wji is the weight of interconnection between concepts Cj to Ci, and

f is a threshold function (activation function) used to reduce the activation level into a normalized range

Activation Functions

• (bivalent)• (trivalent)

• (logistic)

• (sigmoid)

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 3rd concept • The state values of the concepts converge at iteration 12

Iter

atio

n #

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

1 0.00 0.00 -1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 -0.81 0.00 0.00 0.14 -0.32 0.00 0.00 0.19 0.00 0.50 0.50 3 0.00 0.11 -0.62 0.10 0.11 0.51 -0.81 0.10 0.00 0.21 0.20 0.70 0.90 4 0.00 0.43 -0.48 0.19 0.21 0.82 -0.89 0.40 0.00 0.31 0.60 0.80 1.00 5 0.19 0.59 -0.31 0.37 0.43 0.93 -1.00 0.70 0.00 0.52 0.83 0.90 1.00 6 0.40 0.70 -0.19 0.64 0.62 1.00 -1.00 0.90 0.00 0.74 0.92 0.90 1.00 7 0.69 0.80 0.03 0.83 0.69 1.00 -1.00 0.90 0.00 0.83 0.91 1.00 1.00 8 0.81 0.80 0.24 0.91 0.81 1.00 -1.00 0.90 0.00 0.91 0.93 1.00 1.00 9 0.92 0.82 0.42 0.89 0.89 1.00 -1.00 1.00 0.00 0.91 0.92 1.00 1.00

10 0.91 0.82 0.56 0.88 0.89 1.00 -1.00 1.00 0.00 0.93 1.00 1.00 1.00 11 0.90 0.81 0.72 0.89 0.90 1.00 -1.00 1.00 0.00 0.93 1.00 1.00 1.00 12 0.90 0.81 0.72 0.89 0.90 1.00 -1.00 1.00 0.00 0.93 1.00 1.00 1.00

Example

• A decrease at C3:

• will increase the level of C6, C8, C11, C12, and C13;

• will decrease the level of C7;

• will increase the level of C1, C2, C4, C5, and C10 to some extent;

• In the long run, C3 has no effect on C9

Iter

atio

n #

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

12 0.90 0.81 0.72 0.89 0.90 1.00 -1.00 1.00 0.00 0.93 1.00 1.00 1.00