Fuzzy Cognitive Maps
Ø Motivation
Ø Cognitive maps and fuzzy cognitive maps
Ø Fuzzy cognitive maps and learning
Ø Limitations of differential Hebbian learning
Ø A methodology for differential hebbian learning
Ø Application of the methodology
Ø Conclusions and further work
Motivation
Fuzzy cognitive maps (FCMs) can model dynamic systems with interacting concepts
Can be a useful tool in decision support
Learning in FCMs can improve their performance
Differential Hebbian Learning (DHL) proposed as a learning method
But no formal DHL methodology so far
A new methodology has been developed and applied
Cognitive maps as a basis for FCMs
Cognitive maps (Axelrod 1976) are signed directed graphs used for decision analysis
Effect of node A on node B is given by number of negative edges forming the path between A and B
- effect positive if the path has an even number of negative edges, and negative otherwise.
Drawbacks of cognitive maps
· Suffers from possible ‘imbalance problem’ with more than one paths between nodes
· Cognitive maps are static – do not evolve with time
Representation of fuzzy cognitive maps
An FCM is represented by a matrix storing the causal link (edge) values
A snapshot of events at any point in time is represented by the state vector
Eg,
the event “Migration into city” is represented by the vector
[0.0 1.0 0.0 0.0 0.0 0.0 0.0]
Operation and use of an FCM
An FCM behaves like a recurrent neural network
The inputs to each node, multiplied by corresponding edge strengths, are summed
Outputs of each node is produced by a non-linear transform function
Application of a stimulus state vector with a thresholding transformation function results in the FCM stabilising to:
1. A fixed state
2. A “limit cycle” – a cycle of repeating states
Assuming the FCM accurately represents a problem domain, it can
1. provide answers to “what-if” questions”
2. simulate evolution of a system over time
FCMs as adaptive systems
FCMs can be combined
For a particular problem domain, experts can differ
A number of FCMs related to the same domain can be merged
FCMs can learn from examples
One approach uses differential Hebbian learning (DHL)
Where the causal link between two nodes is strengthened (weakend) if they both change in the same direction
Fuzzy Cognitive Maps and Differential Hebbian Learning (DHL)
A form of unsupervised learning
Correlates the changes of two concepts
If concepts i and j both increase or decrease, the edge strength between the two concepts is increased; otherwise, it is decreased:
where DCi(t) = Ci(t) - Ci(t – 1)
The learning coefficient mt decreases slowly over time.
Known pair of succeeding state vectors Ci(t), Ci(t + 1) can be used train the FCM
Proposed methodology for Differential Hebbian Learning
1. Create a crisp cognitive map - edge values in {-1,0,1}
2. Identify groups of events that happen simultaneously, or in a sequence.
3. Fuzzify crisp cognitive map by encoding identified sequence using DHL
4. Revise the resulting FCM
The final step involves detailed examination of the trained FCM by the domain expert to identify and correct any flaws.
Application of proposed learning methodology
An example using an FCM to model a freeway scenario (Tsadiras & Margaritis 1994)
The FCM matrix for the crisp cognitive map
The stimulus vector [ 1 1 0 0 0 0 ] representing the events
bad weather and freeway congestion,
leads to the equilibrium state [ 1 1 1 1 0 1 ]
Inference:
Bad weather and freeway congestion eventually lead to
decreased driving speed, increased driver caution level, and increased number of accidents.
[ 1 1 1 1 0 1 ]
Step 2: Identification of event sequences for encoding
Given the observation:
bad weather, congestion and police patrol can form a set of simultaneous events following on from no (significant) events taking place previously, the following sequence of events is selected for encoding:
0 0 0 0 0 0
1 1 0 0 1 0 (seq1)
Similarly considering accidents, congestion and police patrol as simultaneous events:
0 0 0 0 0 0
0 1 0 0 1 1 (seq2)
Congestion and police patrol leads to reduced driving speed, giving
0 0 0 0 0 0
0 1 0 0 1 0
0 1 1 0 1 0 (seq3)
Step 3: Encoding Sequences with Differential Hebbian Learning
Training the FCM with the three sequences gives the following edge matrix
Step 4: Trained FCM revision
1. All diagonal values set to zero to prevent self-feedback.
2. No concepts should affect the weather. Accordingly, all values in the first column of E representing effects on bad weather due to all other nodes are set to zero.
The final version of the FCM edge matrix:
Test results for the trained FCM
Application of the stimulus vector C1 [ 1 1 0 0 0 0 ] (representing events bad weather and freeway congestion ) led to the following sequence of FCM states:
C1´E = C2 = [ 1 1 1 0 1 1 ]
C2´E = C3 = [ 1 1 1 1 0 0 ]
C3´E = C2 = [ 1 1 1 0 1 1 ]
which is a limit cycle (C2-C3-C2)
Interpretation of the FCM behaviour:
The incidence of bad weather and freeway congestion leads to a situation where driving speed stays reduced, but driver caution level, police patrol frequency and incidence of accidents keep fluctuating.
Conclusions
The idea of using differential Hebbian learning to implement adaptive FCMs is not new
This has been an attempt to formalise a methodology for its application
A proposed methodology has been tried on an FCM modelling a dynamic system with feedback
The results agree with an intuitive understanding of the scenario modelled
Adaptive FCMs can facilitate development of FCMs by relieving human experts of the task of specifying fuzzy causal link strengths
The DHL methodology can be used convert crisp cognitive maps into FCMs
Further work
The applicability of the methodology to real-world industrial problems remains the subject of further investigations
Of particular interest is the development of a decision support systems (DSS) framework around adaptive FCMs
FCM transparencies.doc 2