1. CMPS 561 Fuzzy Set Retrieval Ryan Benton September 1, 2010

2. Agenda Fuzzy Sets Fuzzy Operations Fuzzy Set Retrieval Processing Fuzzy Queries Set Issues -level Sets

3. Fuzzy Sets

4. Issue Problem – Boolean assumes documents Exist in 1 of 2 states States are Non-Overlapping – CRISP! – Other crisp states may exist Grades: A, B, C, D, F – Can be in one and not the other. – Often, exact boundaries are not known Hot, Warm, Luke-warm, Cool, Chilly, Cold – When does Hot become Warm?

5. Issue Fuzzy Sets – Informal Definition Fuzzy Set is a class with fuzzy boundaries. Alternatively, fuzzy set is a class with non-crisp boundaries.

6. Fuzzy Sets – Slightly More Formal Regular (Crisp) set – Let X = {x i } be the universe of objects of interest. Fuzzy Subset of X  A. – A is a set of order pairs A = { } –   (x)  Grade of membership of x in A –   : X  [0,1]

7. Example 1 Assume Crisp Set is – X = {x 1, x 2, x 3, x 4 } Fuzzy Subsets – A = {, } – B = { }

8. Example 2 Membership Functions 4’6” 6’ short mediumtall 1 0  1 0 .75.2 4’  tall  4’  medium (4’)=0.2,  short (4’)=.75 Crisp Fuzzy

9. Fuzzy Operations

10. Operations 1 Let – A = { } – B = { } Equality – A = B iff  A (x) =  B (x), for all x  X.

11. Operations 2 Containment – A  B iff  A (x)   B (x), for all x  X. Complement – Ā = { } iff  Ā (x) = 1 -  A (x), for all x  X.

12. Operations 3 Union – A  B = { } iff  A  B = max(  A (x),  B (x)), for all x  X. Intersection – A  B = { } iff  A  B = min(  A (x),  B (x)), for all x  X.

13. Fuzzy Set Retrieval Model

14. Document Representation Boolean Relation – D = { } –  D : D x T  {0,1} –  D (d , t i ) 1, if d  contains t i 0, otherwise D t = {d   D |  D (d , t) = 1} d  D d = {t i  T |  D (d, t i ) = 1}

15. Document Representation Fuzzy D = { }  D : D x T  [0,1]  D (d , t i ) – The degree of membership of t i to d  D t = { | d   D >} D d = { | t i   T>}

16. Processing Boolean Query (Method 2) D t1  t2 = {d   D |  D (d , t 1 )  D (d , t 2 ) = 1} D t1  t2 = {d   D |  D (d , t 1 )  D (d , t 2 ) = 1} D t = set of Documents containing term t T = {a,b,c,d,e} D a, D b, D c, D d, D e,

17. Processing Fuzzy Query (Method 2) D t1  t2 = { | d   D } D t1  t2 = { | d   D } D t = set of pairs containing term t T = {a, b, c, d, e} D a, D b, D c, D d, D e

18. Retrieval Function Boolean RSV  F RSV t (d  ) =  D (d , t) RSV  e (d  ) = 1 - RSV e (d  ) RSV e1  e2 (d  ) = RSV e1 (d  )  RSV e2 (d  ) RSV e1  e2 (d  ) = RSV e1 (d  )  RSV e2 (d  )

19. Retrieval Function Fuzzy RSV t (d  ) =  D (d , t) RSV  e (d  ) = 1 - RSV e (d  ) RSV e1  e2 (d  ) = min(RSV e1 (d  ),  RSV e2 (d  )) RSV e1  e2 (d  ) = max(RSV e1 (d  ),  RSV e2 (d  ))

20. Processing Fuzzy Queries

21. Term-Document Incidence Matrix d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 d7d7 d8d8 t1t1 0.10.70.60.40.80.60.30.6 t2t2 0.30.80.90.80.60.50.70.2 t3t3 0.810.20.60.80.20.80.5 t4t4 0.40.60.70.50.70.30.10.9 t5t5 0.70.20.80.40.60.20.80.4

22. Fuzzy example q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4

23. Fuzzy example (Method 1) q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d 3 0.6 d 6 0.6 d 8 0.6 0.9 0.5 0.2 0.9 0.5 0.2 0.5 0.7 0.3 0.9 0.1 0.5 0.8 0.5 0.1 0.5 0.6 0.7 0.3 0.2 0.7 0.5 0.6

24. Inverted File Processing (Method 2) D t = { |  D t (d  ) > 0} D  e = { | 1-RSV e (d  ) > 0} D e1  e2 = { | RSV e1  e2 (d  ) > 0} D e1  e2 = { | RSV e1  e2 (d  ) > 0}

25. Processing Fuzzy Query (Method 2) Output D t D e1 D e2 D e1  D e2 D e1  D e2 D\D e1 Query t e1 e2 e1  e2 e1  e2  e1

26. Set Issues

27. Set Identities A  B = B  A (A  B)  C = A  (B  C) A  B  C) = (A  B)   C) A  = A A  A’ = S A  B = B  A (A  B)  C = A  (B  C) A  B  C) = (A  B)   C) A  S = A A  A’ = 

28. Definitions of Union and Intersection NameUnionIntersection max, min  max, product  · probabilistic sum, product· Einstein bold Hamacher Yager Schweizer-Sklard ⊥p⊥p ⊤p⊤p ℇ    +  · +  v  v 

29. Operations – Union & Intersection a · b = ab a b = a + b – ab a b = (a + b) / (1 + ab) a b = (ab) / ( 1 + (1-a)(1-b) a b = min(a + b, 1) a b = max(0, a + b - 1) +  ℇ  

30. Operations – Union & Intersection a b = (a + b – (1 –  )ab) / (  1- ab  –  ∞) a b = (ab) / (  a+b  –  ∞) a b = min(1, (a v + b v ) (1/v) ) – v  ∞) a b = 1 - min(1, [(1-a) v + (1-b) v ] (1/v) ) – v  ∞)  +  · v  v 

31. Operations – Union & Intersection 1 – ((1-a) -p + (1-b) -p -1) (1/p) p > 0 a bp = 0 1 – ((1-a) -p + (1-b) -p -1) (1/p) p < 0, (1-a) -p + (1-b) -p >1 1p < 0, (1-a) -p +(1-b) -p ≤ 1 a T p b =

32. Operations – Union & Intersection (a -p + b -p -1) (1/p) p > 0 abp = 0 (a -p + b -p -1) (1/p) p < 0, a -p + b -p >1 0p < 0, a -p +b -p ≤ 1 a T p b =

33. Example of Set Operations Assume – Union = Probabilistic Sum  = a b = a + b – ab – Intersection = Product  = a · b = ab

34. Term-Document Incidence Matrix d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 d7d7 d8d8 t1t1 0.10.70.60.40.80.60.30.6 t2t2 0.30.80.90.80.60.50.70.2 t3t3 0.810.20.60.80.20.80.5 t4t4 0.40.60.70.50.70.30.10.9 t5t5 0.70.20.80.40.60.20.80.4

35. Fuzzy example q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4

36. Fuzzy example (Method 1) Max, Min q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d 3 0.6 d 6 0.6 d 8 0.6 0.9 0.5 0.2 0.9 0.5 0.2 0.5 0.7 0.3 0.9 0.1 0.5 0.8 0.5 0.1 0.5 0.6 0.7 0.3 0.2 0.7 0.5 0.6

37. Fuzzy example (Method 1) probabilistic sum and product q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d 3 0.6 d 6 0.6 d 8 0.6 0.9 0.5 0.2 0.9 0.5 0.2 0.5 0.7 0.3 0.9 0.1 0.5 0.8 0.5 0.06 0.3 0.48 0.504 0.12 0.09 0.5316 0.384 0.4998

38. Differences DocumentMax, MinProb. Sum, Product d3d3 0.70.5316 d6d6 0.50.384 d8d8 0.60.4998

39. -level Sets

40. -level Sets - Definitions -level Set – D(  ) = { |  D (d , t) ≥  } – D t (  ) = { d |  D (  )} Then meaning of D t(  ) of descriptor t  T  T* – D t(  ) = { | d  D t (  )} Note: T* has same meaning as E used in Boolean IR If t’  T*, then meaning of D t(  ) of descriptor t =  t’ – D t(  ) = { | 1-  D t’(  ) (d) > ,d  D t’ (  ) }

41. -level Sets - Definitions If t’, t’’  T*, then meaning of D t(  ) of descriptor t = t’  t’’ – D t(  ) = { |d  D t’ (  )  D t’’ (  )  If t’, t’’  T*, then meaning of D t(  ) of descriptor t = t’  t’’ – D t(  ) = { |d  D t’ (  )  D t’’ (  ) 

42. Fuzzy example q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4

43. Fuzzy example Max, Min q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d 3 0.6 d 6 0.6 d 8 0.6 0.9 0.5 0.2 0.9 0.5 0.2 0.5 0.7 0.3 0.9 0.1 0.5 0.8 0.5 0.1 0.5 0.6 0.7 0.3 0.2 0.7 0.5 0.6

44. Fuzzy example Max, Min, *=0.3 q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d 3 0.6 d 6 0.6 d 8 0.6 0.9 0.5 0.2 (0) 0.9 0.5 0.0 0.2 (0) 0.5 0.7 0.3 0.9 0.1 (0) 0.5 1.0 0.5 0.0 0.5 0.6 0.7 0.3 0.0 0.7 0.5 0.6

45. Questions?