CMPS 561 Fuzzy Set Retrieval Ryan Benton September 1, 2010

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

Fuzzy Sets

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?

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

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]

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

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

Fuzzy Operations

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

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.

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.

Fuzzy Set Retrieval Model

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}

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>}

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,

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

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  )

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  ))

Processing Fuzzy Queries

Term-Document Incidence Matrix d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 d7d7 d8d8 t1t t2t t3t t4t t5t

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

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

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}

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

Set Issues

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’ = 

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 

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  

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 

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 =

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 =

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

Term-Document Incidence Matrix d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 d7d7 d8d8 t1t t2t t3t t4t t5t

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

Fuzzy example (Method 1) Max, Min q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d d d

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 d d

Differences DocumentMax, MinProb. Sum, Product d3d d6d d8d

-level Sets

-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’ (  ) }

-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’’ (  ) 

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

Fuzzy example Max, Min q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d d d

Fuzzy example Max, Min, *=0.3 q = (t 1   t 2 )  (t 2  t 3  t 4 ))  t1t1   t2t2   t2t2 t3t3 t4t4 d d d (0) (0) (0)

Questions?