Set Theoretic Models 1

IR Models Non-Overlapping Lists Proximal Nodes Structured Models Retrieval: Adhoc Filtering Browsing U s e r T a s k Classic Models Boolean Vector Probabilistic Set Theoretic Fuzzy Extended Boolean Probabilistic Inference Network Belief Network Algebraic Generalized Vector Lat. Semantic Index Neural Networks Browsing Flat Structure Guided Hypertext 2

Set Theoretic Models  The Boolean model imposes a binary criterion for deciding relevance  The question of how to extend the Boolean model to accomodate partial matching and a ranking has attracted considerable attention in the past  Two set theoretic models for this:  Fuzzy Set Model  Extended Boolean Model 3

Fuzzy Set Model  Queries and docs represented by sets of index terms: matching is approximate from the start  This vagueness can be modeled using a fuzzy framework, as follows:  with each term is associated a fuzzy set  each doc has a degree of membership in this fuzzy set  This interpretation provides the foundation for many models for IR based on fuzzy theory  In here, we discuss the model proposed by Ogawa, Morita, and Kobayashi (1991) 4

Fuzzy Set Theory  Framework for representing classes whose boundaries are not well defined  Key idea is to introduce the notion of a degree of membership associated with the elements of a set  This degree of membership varies from 0 to 1 and allows modeling the notion of marginal membership  Thus, membership is now a gradual notion, contrary to the crispy notion enforced by classic Boolean logic 5

6 Fuzzy Set Theory  Model  A query term: a fuzzy set  A document: degree of membership in this test  Membership function  Associate membership function with the elements of the class  0: no membership in the test  1: full membership  0 ~1: marginal elements of the test documents

Fuzzy Set Theory  A fuzzy subset A of a universe of discourse U is characterized by a membership function µ A : U  [0,1] which associates with each element u of U a number µ A (u) in the interval [0,1] –complement: –union: –intersection: a class document collection for query term 7

Examples  Assume U={d 1, d 2, d 3, d 4, d 5, d 6 }  Let A and B be {d 1, d 2, d 3 } and {d 2, d 3, d 4 }, respectively.  Assume  A ={d 1 :0.8, d 2 :0.7, d 3 :0.6, d 4 :0, d 5 :0, d 6 :0} and  B ={d 1 :0, d 2 :0.6, d 3 :0.8, d 4 :0.9, d 5 :0, d 6 :0}  ={d 1 :0.2, d 2 :0.3, d 3 :0.4, d 4 :1, d 5 :1, d 6 :1}  = {d 1 :0.8, d 2 :0.7, d 3 :0.8, d 4 :0.9, d 5 :0, d 6 :0}  = {d 1 :0, d 2 :0.6, d 3 :0.6, d 4 :0, d 5 :0, d 6 :0} 8

Fuzzy Information Retrieval  basic idea –Expand the set of index terms in the query with related terms (from the thesaurus) such that additional relevant documents can be retrieved –A thesaurus can be constructed by defining a term-term correlation matrix c whose rows and columns are associated to the index terms in the document collection keyword connection matrix 9

Fuzzy Information Retrieval (Continued)  normalized correlation factor c i,l between two terms k i and k l (0~1)  In the fuzzy set associated to each index term k i, a document d j has a degree of membership µ i,j where n i is # of documents containing term k i n l is # of documents containing term k l n i,l is # of documents containing k i and k l 10

Fuzzy Information Retrieval (Continued)  physical meaning –A document d j belongs to the fuzzy set associated to the term k i if its own terms are related to k i, i.e.,  i,j =1. –If there is at least one index term k l of d j which is strongly related to the index k i, then  i,j  1. k i is a good fuzzy index –When all index terms of d j are only loosely related to k i,  i,j  0. k i is not a good fuzzy index 11

Example q = (k a  (k b   k c )) = (k a  k b  k c )  (k a  k b   k c )  (k a   k b   k c ) = cc 1 +cc 2 +cc 3 DaDa DbDb DcDc cc 3 cc 2 cc 1 D a : the fuzzy set of documents associated to the index k a d j  D a has a degree of membership  a,j > a predefined threshold K D a : the fuzzy set of documents associated to the index k a (the negation of index term k a ) 12

Example Query q=k a  (k b   k c ) disjunctive normal form q dnf =(1,1,1)  (1,1,0)  (1,0,0) (1) the degree of membership in a disjunctive fuzzy set is computed using an algebraic sum (instead of max function) more smoothly (2) the degree of membership in a conjunctive fuzzy set is computed using an algebraic product (instead of min function) Recall 13

Fuzzy Set Model –Q: “gold silver truck” D1:“Shipment of gold damaged in a fire” D2:“Delivery of silver arrived in a silver truck” D3:“Shipment of gold arrived in a truck” –IDF (Select Keywords) a = in = of = 0 = log 3/3 arrived = gold = shipment = truck = = log 3/2 damaged = delivery = fire = silver = = log 3/1 –8 Keywords (Dimensions) are selected arrived(1), damaged(2), delivery(3), fire(4), gold(5), silver(6), shipment(7), truck(8) 14

Fuzzy Set Model 15

Fuzzy Set Model 16

Fuzzy Set Model  Sim(q,d): Alternative 1 Sim(q,d 3 ) > Sim(q,d 2 ) > Sim(q,d 1 )  Sim(q,d): Alternative 2 Sim(q,d 3 ) > Sim(q,d 2 ) > Sim(q,d 1 ) 17

Extended Boolean Model Disadvantages of “Boolean Model” : No term weight is used Counterexample: query q=K x AND K y. Documents containing just one term, e,g, K x is considered as irrelevant as another document containing none of these terms. No term weight is used The size of the output might be too large or too small 18

Extended Boolean Model The Extended Boolean model was introduced in 1983 by Salton, Fox, and Wu The idea is to make use of term weight as vector space model. Strategy: Combine Boolean query with vector space model. Why not just use Vector Space Model? Advantages: It is easy for user to provide query. 19

Extended Boolean Model Each document is represented by a vector (similar to vector space model.) Remember the formula. Query is in terms of Boolean formula. How to rank the documents? 20

Fig. Extended Boolean logic considering the space composed of two terms k x and k y only. ky kx 21

Extended Boolean Model For query q=K x or K y, (0,0) is the point we try to avoid. Thus, we can use to rank the documents The bigger the better. 22

Extended Boolean Model For query q=K x and K y, (1,1) is the most desirable point. We use to rank the documents. The bigger the better. 23

Extend the idea to m terms q or =k 1  p k 2  p …  p K m q and =k 1  p k 2  p …  p k m 24

Properties The p norm as defined above enjoys a couple of interesting properties as follows. First, when p=1 it can be verified that Second, when p=  it can be verified that Sim(q or,d j )=max(x i ) Sim(q and,d j )=min(x i ) 25

Example For instance, consider the query q=(k 1  k 2 )  k 3. The similarity sim(q,d j ) between a document d j and this query is then computed as Any boolean can be expressed as a numeral formula. 26