IDSS: Overview of Themes

1. AI • Introduction • Overview • IDT • Attribute-Value Rep. • Decision Trees • Induction • CBR • Introduction • Representation • Similarity • Adaptation • Rule-based Inference & Expert Systems • Computational Complexity • AI Method: Synthesis Tasks • AI Planning • Uncertainty (MDP, Utility, • Fuzzy logic) IDSS: Overview of Themes   • Applications to IDSS: • Analysis Tasks • Help-desk systems • Classification • Diagnosis • Prediction • Design • Textual CBR • Synthesis Tasks • KBPP • Configuration • Software Eng. • E-commerce • Knowledge Management          

2. Similarity in CBR Sources: Chapter 4 www.iiia.csic.es/People/enric/AICom.html www.ai-cbr.org

3. similarity similarity Computing Similarity • Similarity is a key (the key?) concept in CBR • We saw that a case consists of: • Problem • Solution • Adequacy • We saw that the CBR problem solving cycle consists of: • Retrieval • Reuse • Revise • Retain • We will distinguish between: • Meaning of similarity • Formal axioms capturing this meaning

4. Meaning of Similarity • Observation 1: Similarity always concentrates on one aspect or task: • There is no absolute similarity • Example: • Two cars are similar if they have similar capacity (two compact cars may be similar to each other but not to a full-size car) • Two cars are similar if they have similar price (a new compact car may be similar to an old full-size car but not to an old compact car) • When computing similarity we are doing some sort of abstraction of the cases

5. Meaning of Similarity (2) • Observation 2: Similarity is not always transitive: • Example: • I define similar to mean “within walking distance” • “Lehigh’s book store” is similar to “Café Havana” • “Café Havana” is similar to “Perkins” • “Perkins” is similar to “Monrovia book store” • … • But: “Lehigh’s book store” is not similar to “Best Buy” in Allentown ! • The problem is that the property “small difference” cannot be propagated

6. Meaning of Similarity (3) • Observation 3: Similarity is not always symmetric: • Example: • “Mike Tyson fights like a lion” • But do we really want to say that “a lion fights like Mike Tyson”? • The problem is that in general the distance from an element to a prototype of a category is larger than the other way around

7. Similarity and Utility in CBR • Utility: measure of the improvement in efficiency as a result of a body of knowledge (We’ll come back to this point) • The goal of the similarity is to select cases that can be easily adapted to solve a new problem Similarity = Prediction of the utility of the case • However: • The similarity is an a priori criterion • The utility is an a posteriori criterion • Ideal: Similarity makes a good prediction of the utility

8. Axioms for Similarity • There are 3 types of axioms: • Binary similarity predicate “x and y are similar” • Binary dissimilarity predicate “x and y are dissimilar” • Similarity as order relation: “x is at least as similar to y as it is to z” • Observation: • The first and the second are equivalent • The third provides more information: grade of similarity

9. Similarity Relations • We want to define a relation: • R(x,y,z) iff “x is at least as similar to y as it is to z” • First lets consider the following relation: • S(x,y,u,v) iff “x is at least as similar to y as u is similar to v” • Definition of R in terms of S: R(x,y,z) iff S(x,y,x,z)

10. Similarity Relations (2) • Possible requirements on the relation S: • S(x,x,u,v) • S(x,y,y,x) • S(x,y,u,v) & S(u,v,s,t)  S(x,y,s,t) • S(x,y,u,v) iff S(y,x,u,v) iff S (x,y,v,u)

11. Similarity Relations (3) • In CBR we have an object x fixed when computing similarity. Which x? The new problem • We are looking for a y such that y is the most similar to x. In terms of R this be seen as:  z: R(x,y,z) • Given a problem x we can define an ordering relation x as follows: • y x z iff R(x,y,z) • y >x z iff (y x z and ¬ z x y) • y ~x z iff (y x z and z x y)

12. Similarity Metric • We want to assign a number to indicate the similarity between a case and a problem • Definition: A similarity metric over a set M is a function: • sim: M  M  [0,1] • Such that: • For all x in M: sim(x,x) = 1 holds • For all x, y in M: sim(x,y) = sim(y,x) “ the closer the value of sim(x,y) to 1, the more similar is x to y”

13. sim provides a quantitative value for similarity: sim(x, yi) y1 y2 y3 y4 0 1 Thus y4 is more similar to x Similarity Metric (2) • Given a similarity metric: sim: M  M  [0,1], it induces a similarity relation Ssim (x,y,u,v) and xas follows: Ssim(x,y,u,v) iff sim(x,y)  sim(u,v) y xz iff sim(x,y)  sim(x,z)

14. Distance Metric • Definition: A distance function over a set M is a function: • d: M  M  [0,) • Such that: • For all x in M: d(x,x) = 0 holds • For all x, y in M: d(x,y) = d(y,x) • Definition: A distance function over a set M is a metric if: • For all x, y in M: d(x,y) = 0 holds then x = y • For all x, y, z in M: d(x,z) + d(z,y)  d(x,y)

15. Relation between Similarity and Distance Metric • Given a distance metric, d, it induces a similarity relation Sd(x,y,u,v), xas follows: • For all x, y, u, v: S(x,y,u,v) holds if • For all x, y, z: y x z if d(x,y)  d(u,v) d(x,y)  d(x,z) Definition: A similarity metric sim and a distance metric d are compatible iff: for all x,y, u, v: Sd(x,y,u,v) iff Ssim(x,y,u,v)

16. f(d(x,y)) > f(d(u,v)) Relation between Similarity and Distance Metric (2) • Property: Let • f: [0,)  (0,1] • Be a bijective and order inverting (if u< v then f(v) < f(u)) function such that: • f(0) = 1 • f(d(x,y)) = sim(x,y) • then d and sim are compatible If d(x,y) < d(u,v) then sim(x,y) > sim(u,v)

17. Relation between Similarity and Distance Metric (3) • F(x) can be used to construct sim giving d. Example of such a function is: • if you have the Euclidean distance: • d((x,y),(u,v)) = sqr((x-u)2 + (y-v)2) • Since f(x) = 1 – (x/(x+1)) meets the property before • Then: • sim((x,y),(u,v))) = f(d((x,y),(u,v))) • = 1 – (d((x,y),(u,v)) /(d((x,y),(u,v)) +1)) • is a similarity metric

18. Relation between Similarity and Distance Metric (3) • The function f(x) = 1 – (x/(x+1)) is a bijective function from [0,) into (0,1]: 1 0

19. Homework (Oct 23) • Find another order-inverting function and prove it