A Causal Knowledge-Driven Inference Engine for Expert System

1. A Causal Knowledge-Driven Inference Engine for Expert System Presenters: Donna-Marie Anderson Oniel Charles

2. Introduction to Expert Systems • Consists of 3 major components • Dialogue Structure • Inference Engine – (CAKES) • Knowledge Base

3. Causal Knowledge Base • Syntactically similar to IF-THEN rule but semantically different • In most cases it is referred to as Fuzzy Cognitive Map (FCM) • Used mainly to represent political, economic or decision making problems

4. FCMs • Fuzzy signed directed graph with feedback • Model the world as a collection of concepts and causal relationship between the concepts • Concept is depicted as a node • Causal relation is represented as an edge Therefore, edge value between 2 concepts, i and j, written eij, indicates the causality value between the 2 concepts

5. FCMs • Causality values lies between [–1 1] • eij =0 means no causal relation between concepts i and j. • eij > 0 indicates causal increase (positive causality) • eij < 0 indicates causal decrease (negative causality) • Simple FCM has edge value in {-1, 0, 1}, i.e. if causality occurs it occurs at maximal positive or negative degree.

6. FCM Reasoning Technique • Uses modus ponens-based approximate reasoning method between the set of causal relationships between concepts. • If x is G y is F with causality exy then if x is Gi y is Fi with causality exy • Uses negation of the premises • If x is G y is F with causality exy then if x is not G y is not F with causality exy

7. Causal Knowledge Acquisition • Matrix Multiplication Inference Method • Define a concept node (row) vector, C, with the number of columns equal to the number of concepts, i.e., if there are L concepts then C = (C1,C2,C3,….,CL-1,CL) , where each Ci represents a concept. • Build an Adjacency matrix (FCM matrix), E, based on the causality value between the concepts.

8. Traditional Causal Knowledge-Based Inference • Forward-evolved inference • Use the adjacency matrix to test the effect of one of the concept on the others. • Get the row vector that represents the concept you wish to test. • Decide on a threshold value in the interval [-1,1], say .5 • Multiply the row vector by E. • Compare the values received to the threshold. If < then it becomes 0, otherwise it is 1.

9. Traditional Causal Knowledge-Based Inference • Forward-evolved inference • Repeat until you find a vector that after multiplying it by E and applying the threshold operation on the product you get back the vector. • At this point you you have obtained the set that the FCM has associatively inferred. • The set is given by the position in the resultant vector that contains a 1.

10. Traditional Causal Knowledge-Based Inference • Backward inference • Uses the transpose of E, Et, to yield a specific concept node value that should be accompanied with a given consequence.

11. CAsual Knowledge-based Expert system Shell (CAKES) • Core Menus • Concept Nodes Menu • Relationship Menu • Inference Menu

12. Fuzzy Causal Relationship • Definition of FCR- ‘A causally increases B” means that if A increases then B increases and if A decreases then B decreases. On the other hand if ‘A causally increases B” means that if A increases then B decreases and if A decreases then B increases. • In the concepts that constitute causal relationship, there must exist a quantitative elements that can increase or decrease.

13. Fuzzy Causal Relationship (FCR) • FCM fuzzy relations mean fuzzy causality. Causality can have a negative sign. The negative fuzzy relation between two concept nodes is the degree of relation with “negation” of a concept node. • Example: If the concept node Ci is noted as Cj the R(Ci,Cj)=-0.6 which means that R(Ci,~Cj)=0.6 conversely R(Ci,Cj)=0.6 the R(Ci,~Cj)=-0.6

14. Fuzzy Causal Relationship (example) Based on our Definition of FCR the following four FCRs are equivalent. Buying by institution investors -> Increase of composite stock price + Selling by institution investors -> Decrease composite stock price + Buying by institution investors -> Decrease of composite stock price - Selling by institution investors -> Increase composite stock price -

15. Fuzzy Causal Relationship (Causality Values) The following two FCRs with real valued causality are equivalent. Institute investors -> Composite stock price +0.9 Buying by Institute investors -> +0.1 Not Increase of composite stock price

16. Fuzzy Causal Relationship(Theorems) • There are 6 theorems that highlight the core principles of FCR. In the interest of time we will highlight only two. • Theorem 3. When fuzzy causal concepts Ci, and Cj, are given, the following FCRs are all equivalent. Ci -> Cj ~Ci -> ~Cj Ci -> ~Cj ~Ci -> Cj r r -r -r where –1<= r <=1

17. Fuzzy Causal Relationship(Theorems) • Theorem 6. When ~Ci is a negative concept of Ci and the dis-quantity fuzzy set of ~Ci is equal to the complement of Ci ’s quantity fuzzy set, then the following FCRs are all equivalent. Ci -> Cj implies Ci -> ~Cj Ci -> Cj 1 +0 -0 Ci -> Cj implies Ci -> ~Cj Ci -> Cj -1 -0 +0

18. Fuzzy Partially Causal Relationship • In reality, there exist many cases in which the definition of causality is not met. • For example; there might be a stock market situation in which institute investors buying causes increase of composite stock price but there selling cannot cause a decrease of composite stock price.

19. Fuzzy Partially Causal Relationship • Our FCR type discussed in the previous section should be adjusted to incorporate this situation. Hence :- Buying by institution investors ---> Increase of composite stock price +.09 Selling by institution investors ----> Decrease composite stock price + 0.9 where ---> means “partially causality”. This constitutes Fuzzy Partially Causal Relationship

20. Fuzzy Partially Causal Relationship (Example)

21. Fuzzy Partially Causal Relationship (Example)

22. CAKES Design and Implementation

23. Advanced Inference Mechanism • Refines FCM by using FCR and FPCR concepts • Principles • If two causal relationships support the same conclusion, then the addition of those 2 causality value is > each causality value. • If a causal relationship is connected consecutively to a causal relationship, then the absolute value of its additive value of the 2 causality values is <= the least of absolute value of the 2 causality

24. Advanced Inference Mechanism • Principles • The final additive value remains same irrespective of the order of addition of causality values of interest. • Both a +ve and a –ve causality value have the same amount of strength although they have the opposite direction with each other. • The final causality value lies in the interval [-1,1]

25. Conclusion • Improved FCM theory was needed for a more sophisticated representation of causal knowledge and to arrive at more logical conclusions. • Refined by using FCR and FPCR • The Expert System used to test these concepts was CAKES.