Lecture 6: Boolean to Vector

2011.02.07 - SLIDE 1IS 240 – Spring 2011

Prof. Ray Larson University of California, Berkeley

School of Information

Principles of Information Retrieval

Lecture 6: Boolean to Vector

2011.02.07 - SLIDE 2IS 240 – Spring 2011

Today

• Review – IR Models– The Boolean Model– Boolean implementation issues– Extended Boolean Approaches

• Vector Representation

• Term Weights

• Vector Matching

2011.02.07 - SLIDE 3IS 240 – Spring 2011

Review

• IR Models

• Extended Boolean

2011.02.07 - SLIDE 4IS 240 – Spring 2011

IR Models

• Set Theoretic Models– Boolean– Fuzzy– Extended Boolean

• Vector Models (Algebraic)

• Probabilistic Models (probabilistic)

2011.02.07 - SLIDE 5IS 240 – Spring 2011

Boolean Logic

A B

BABA

BABA

BAC

BAC

AC

AC

∩=∪

∪=∩

∪=∩=

=

=

:Law sDeMorgan'

2011.02.07 - SLIDE 6IS 240 – Spring 2011

Parse Result (Query Tree)

• Z39.50 queries…

Oper: AND

Title XXX and Subject YYY

Operand:Index = TitleValue = XXX

Operand:Index = SubjectValue = YYY

left right

2011.02.07 - SLIDE 7IS 240 – Spring 2011

Parse Results

• Subject XXX and (title yyy and author zzz)

Op: AND

Op: ANDOper:

Index: SubjectValue: XXX

Oper:Index: TitleValue: YYY

Oper:Index: AuthorValue: ZZZ

2011.02.07 - SLIDE 8IS 240 – Spring 2011

Boolean AND Algorithm

2578

152935

100135140155189190195198

28

15100135155189195

289

1215222850687784

100120128135138141150155188189195

AND =

2011.02.07 - SLIDE 9IS 240 – Spring 2011

Boolean OR Algorithm

2578

152935

100135140155189190195198

25789

12152228293550687784

100120128135138141150155188189190195198

289

1215222850687784

100120128135138141150155188189195

OR =

2011.02.07 - SLIDE 10IS 240 – Spring 2011

Boolean AND NOTAlgorithm

2578

152935

100135140155189190195198

57

152935

140190198

289

1215222850687784

100120128135138141150155188189195

AND NOT =

2011.02.07 - SLIDE 11IS 240 – Spring 2011

Boolean Summary

• Advantages– simple queries are easy to understand– relatively easy to implement

• Disadvantages– difficult to specify what is wanted, particularly

in complex situations– too much returned, or too little– ordering not well determined

• Dominant IR model in commercial systems until the WWW

2011.02.07 - SLIDE 12IS 240 – Spring 2011

Basic Concepts for Extended Boolean

• Instead of binary values, terms in documents and queries have a weight (importance or some other statistical property)

• Instead of binary set membership, sets are “fuzzy” and the weights are used to determine degree of membership.

• Degree of set membership can be used to rank the results of a query

2011.02.07 - SLIDE 13IS 240 – Spring 2011

Fuzzy Sets

• Introduced by Zadeh in 1965.

• If set {A} has value v(A) and {B} has value v(B), where 0 v 1

• v(AB) = min(v(A), v(B))

• v(AB) = max(v(A), v(B))

• v(~A) = 1-v(A)

2011.02.07 - SLIDE 14IS 240 – Spring 2011

Fuzzy Sets

• If we have three documents and three terms…– D1=(.4,.2,1), D2=(0,0,.8), D3=(.7, .4,0)

For search: t1t2 t3

v(D1) = max(.4, .2, 1) = 1v(D2) = max(0, 0, .8) = .8v(D3) = max(.7, .4, 0) = .7

For search: t1t2 t3

v(D1) = min(.4, .2, 1) = .2v(D2) = min(0, 0, .8) = 0v(D3) = min(.7, .4, 0) = 0

2011.02.07 - SLIDE 15IS 240 – Spring 2011

Fuzzy Sets

• Fuzzy set membership of term to document is f(A)[0,1]

• D1 = {(mesons, .8), (scattering, .4)}

• D2 = {(mesons, .5), (scattering, .6)}

• Query = MESONS AND SCATTERING

• RSV(D1) = MIN(.8,.4) = .4

• RSV(D2) = MIN(.5,.6) = .5

• D2 is ranked before D1 in the result set.

2011.02.07 - SLIDE 16IS 240 – Spring 2011

Fuzzy Sets

• The set membership function can be, for example, the relative term frequency within a document, the IDF or any other function providing weights to terms

• This means that the fuzzy methods use sets of criteria for term weighting that are the same or similar to those used in other ranked retrieval methods (e.g., vector and probabilistic methods)

2011.02.07 - SLIDE 17IS 240 – Spring 2011

Robertson’s Critique of Fuzzy Sets

• D1 = {(mesons, .4), (scattering, .4)}• D2 = {(mesons, .39), (scattering, .99)}• Query = MESONS AND SCATTERING• RSV(D1) = MIN(.4,.4) = .4• RSV(D2) = MIN(.39,.99) = .39• However, consistent with the Boolean

model:– Query = t1t2t3…t100

– If D not indexed by t1 then it fails, even if D is indexed by t2,…,t100

2011.02.07 - SLIDE 18IS 240 – Spring 2011

Robertson’s critique of Fuzzy

• Fuzzy sets suffer from the same kind of lack of discrimination among the retrieval results almost to the same extent as standard Boolean

• The rank of a document depends entirely on the lowest or highest weighted term in an AND or OR operation

2011.02.07 - SLIDE 19IS 240 – Spring 2011

Other Fuzzy Approaches

• As described in the Modern Information Retrieval (optional) text, a keyword correlation matrix can be used to determine set membership values, and algebraic sums and products can be used in place of MAX and MIN

• Not clear how this approach works in real applications (or in tests like TREC) because the testing has been on a small scale

2011.02.07 - SLIDE 20IS 240 – Spring 2011

Extended Boolean (P-Norm)

• Ed Fox’s Dissertation work with Salton• Basic notion is that terms in a Boolean

query, and the Boolean Operators themselves can have weights assigned to them

• Binary weights means that queries behave like standard Boolean

• 0 < Weights < 1 mean that queries behave like a ranking system

• The system requires similarity measures

2011.02.07 - SLIDE 21IS 240 – Spring 2011

Probabilistic Inclusion of Boolean

• Most probabilistic models attempt to predict the probability that given a particular query Q and document D, that the searcher would find D relevant

• If we assume that Boolean criteria are to be ANDed with a probabilistic query…

€

P(R | Q,D) = P(R | Qbool ,D)P(R | Qprob ,D)

P(R | Qbool ,D) =1: if Boolean eval successful for D

0 : Otherwise

⎧ ⎨ ⎩

2011.02.07 - SLIDE 22IS 240 – Spring 2011

Rubric – Extended Boolean

• Scans full text of documents and stores them• User develops a hierarchy of concepts which

becomes the query• Leaf nodes of the hierarchy are combinations of

text patterns• A “fuzzy calculus” is used to propagate values

obtained at leaves up through the hierarchy to obtain a single retrieval status value (or “relevance” value)

• RUBRIC returns a ranked list of documents in descending order of “relevance” values.

2011.02.07 - SLIDE 23IS 240 – Spring 2011

RUBRIC Rules for Concepts & Weights

• Team | event => World_Series• St._Louis_Cardinals | Milwaukee_Brewers =>

Team• “Cardinals” => St._Louis_Cardinals (0.7)• Cardinals_full_name => St._Louis_Cardinals

(0.9)• Saint & “Louis” & “Cardinals” =>

Cardinals_full_name• “St.” => Saint (0.9)• “Saint” => Saint• “Brewers” => Milwaukee_Brewers (0.5)

2011.02.07 - SLIDE 24IS 240 – Spring 2011

RUBRIC Rules for Concepts & Weights

• “Milwaukee Brewers” => Milwaukee_Brewers (0.9)

• “World Series” => event• Baseball_championship => event (0.9)• Baseball & Championship =>

Baseball_championship• “ball” => Baseball (0.5)• “baseball” => Baseball• “championship” => Championship (0.7)

2011.02.07 - SLIDE 25IS 240 – Spring 2011

RUBRIC combination methods

V(V1 or V2) = MAX(V1, V2)V(V1 and V2) = MIN(V1, V2)i.e., classic fuzzy matching,but with the addition…V(level n) = Cn*V(level n-1)

2011.02.07 - SLIDE 26IS 240 – Spring 2011

Rule Evaluation Tree

World_Series (0)

Event (0)

“World Series” Baseball_championship (0)

Baseball (0)

Championship (0)

St._Louis_Cardinals (0)

Team (0)

“Cardinals” (0)

Milwaukee_brewers (0)

Cardinals_full_name (0)

“Milwaukee Brewers” (0)“Brewers” (0)

Saint (0) “Louis” (0)

“Saint” (0)“St.” (0)

“Cardinals” (0)

“baseball” (0) “championship” (0)“ball” (0)0.9

0.90.7 0.90.5

0.9

0.50.7

2011.02.07 - SLIDE 27IS 240 – Spring 2011


World_Series (0)

Event (0)


Baseball (0)

Championship (0)


Team (0)

“Cardinals” (0)





“Saint” (0)“St.” (0)

“Cardinals” (0)

“baseball” (1.0)“championship” (1.0)“ball” (1.0)0.9

0.90.7 0.90.5

0.9

0.50.7

Document containing “ball”, “baseball” & “championship”

2011.02.07 - SLIDE 28IS 240 – Spring 2011


World_Series (0)

Event (0)


Baseball (1.0)

Championship (0.7)


Team (0)

“Cardinals” (0)





“Saint” (0)“St.” (0)

“Cardinals” (0)


0.90.7 0.90.5

0.9

0.50.7

2011.02.07 - SLIDE 29IS 240 – Spring 2011


World_Series (0)

Event (0)

“World Series” Baseball_championship (0.7)

Baseball (1.0)

Championship (0.7)


Team (0)

“Cardinals” (0)





“Saint” (0)“St.” (0)

“Cardinals” (0)


0.90.7 0.90.5

0.9

0.50.7

2011.02.07 - SLIDE 30IS 240 – Spring 2011


World_Series (0)

Event (0.63)


Baseball (1.0)

Championship (0.7)


Team (0)

“Cardinals” (0)





“Saint” (0)“St.” (0)

“Cardinals” (0)


0.90.7 0.90.5

0.9

0.50.7

2011.02.07 - SLIDE 31IS 240 – Spring 2011


World_Series (0.63)

Event (0.63)


Baseball (1.0)

Championship (0.7)


Team (0)

“Cardinals” (0)





“Saint” (0)“St.” (0)

“Cardinals” (0)


0.90.7 0.90.5

0.9

0.50.7

2011.02.07 - SLIDE 32IS 240 – Spring 2011

Today



• Term Weights

• Vector Matching

2011.02.07 - SLIDE 33IS 240 – Spring 2011

Non-Boolean IR

• Need to measure some similarity between the query and the document

• The basic notion is that documents that are somehow similar to a query, are likely to be relevant responses for that query

• We will revisit this notion again and see how the Language Modelling approach to IR has taken it to a new level

2011.02.07 - SLIDE 34IS 240 – Spring 2011

Similarity Measures (Set-based)

|)||,min(|

||

||||

||

||||

||||

||2

||

21

21

DQ

DQ

DQ

DQ

DQDQ

DQ

DQ

DQ

∩×

∩∪∩+∩

∩ Simple matching (coordination level match)

Dice’s Coefficient

Jaccard’s Coefficient

Cosine Coefficient

Overlap Coefficient

Assuming that Q and D are the sets of terms associated with a Query and Document:

2011.02.07 - SLIDE 35IS 240 – Spring 2011

Document Vectors

• Documents are represented as “bags of words”

• Represented as vectors when used computationally– A vector is like an array of floating point– Has direction and magnitude– Each vector holds a place for every term in

the collection– Therefore, most vectors are sparse

2011.02.07 - SLIDE 36IS 240 – Spring 2011

Vector Space Model

• Documents are represented as vectors in “term space”– Terms are usually stems– Documents represented by binary or weighted vectors

of terms

• Queries represented the same as documents• Query and Document weights are based on

length and direction of their vector• A vector distance measure between the query

and documents is used to rank retrieved documents

2011.02.07 - SLIDE 37IS 240 – Spring 2011

Vector Representation

• Documents and Queries are represented as vectors

• Position 1 corresponds to term 1, position 2 to term 2, position t to term t

• The weight of the term is stored in each position

absent is terma if 0

,...,,

,...,,

21

21

=

=

=

w

wwwQ

wwwD

qtqq

dddi itii

2011.02.07 - SLIDE 38IS 240 – Spring 2011

Document Vectors + Frequency

ID nova galaxy heat h'wood film role diet furA 10 5 3B 5 10C 10 8 7D 9 10 5E 10 10F 9 10G 5 7 9H 6 10 2 8I 7 5 1 3

“Nova” occurs 10 times in text A“Galaxy” occurs 5 times in text A“Heat” occurs 3 times in text A(Blank means 0 occurrences.)

2011.02.07 - SLIDE 39IS 240 – Spring 2011



“Hollywood” occurs 7 times in text I“Film” occurs 5 times in text I“Diet” occurs 1 time in text I“Fur” occurs 3 times in text I

2011.02.07 - SLIDE 40IS 240 – Spring 2011



2011.02.07 - SLIDE 41IS 240 – Spring 2011

We Can Plot the Vectors

Star

Diet

Doc about astronomyDoc about movie stars

Doc about mammal behavior

2011.02.07 - SLIDE 42IS 240 – Spring 2011

Documents in 3D Space

Primary assumption of the Vector Space Model: Documents that are “close together” in space are similar in meaning

2011.02.07 - SLIDE 43IS 240 – Spring 2011

Vector Space Documents and Queries

docs t1 t2 t3 RSV=Q.DiD1 1 0 1 4D2 1 0 0 1D3 0 1 1 5D4 1 0 0 1D5 1 1 1 6D6 1 1 0 3D7 0 1 0 2D8 0 1 0 2D9 0 0 1 3D10 0 1 1 5D11 0 0 1 4Q 1 2 3

q1 q2 q3

D1D2

D3

D4

D5

D6

D7D8

D9

D10

D11

t2

t3

t1

Boolean term combinationsQ is a query – also represented as a vector

2011.02.07 - SLIDE 44IS 240 – Spring 2011

Documents in Vector Space

t1

t2

t3

D1

D2

D10

D3

D9

D4

D7

D8

D5

D11

D6

2011.02.07 - SLIDE 45IS 240 – Spring 2011

Document Space has High Dimensionality

• What happens beyond 2 or 3 dimensions?

• Similarity still has to do with how many tokens are shared in common.

• More terms -> harder to understand which subsets of words are shared among similar documents.

• We will look in detail at ranking methods• Approaches to handling high

dimensionality: Clustering and LSI (later)

2011.02.07 - SLIDE 46IS 240 – Spring 2011

Today



• Term Weights

• Vector Matching

2011.02.07 - SLIDE 47IS 240 – Spring 2011

Assigning Weights to Terms

• Binary Weights

• Raw term frequency

• tf*idf– Recall the Zipf distribution– Want to weight terms highly if they are

• Frequent in relevant documents … BUT• Infrequent in the collection as a whole

• Automatically derived thesaurus terms

2011.02.07 - SLIDE 48IS 240 – Spring 2011

Binary Weights

• Only the presence (1) or absence (0) of a term is included in the vector

docs t1 t2 t3D1 1 0 1D2 1 0 0D3 0 1 1D4 1 0 0D5 1 1 1D6 1 1 0D7 0 1 0D8 0 1 0D9 0 0 1D10 0 1 1D11 1 0 1

2011.02.07 - SLIDE 49IS 240 – Spring 2011

Raw Term Weights

• The frequency of occurrence for the term in each document is included in the vector

docs t1 t2 t3D1 2 0 3D2 1 0 0D3 0 4 7D4 3 0 0D5 1 6 3D6 3 5 0D7 0 8 0D8 0 10 0D9 0 0 1D10 0 3 5D11 4 0 1

2011.02.07 - SLIDE 50IS 240 – Spring 2011

Assigning Weights

• tf*idf measure:– Term frequency (tf)– Inverse document frequency (idf)

• A way to deal with some of the problems of the Zipf distribution

• Goal: Assign a tf*idf weight to each term in each document

2011.02.07 - SLIDE 51IS 240 – Spring 2011

Simple tf*idf

)/log(* kikik nNtfw =

log

Tcontain that in documents ofnumber the

collection in the documents ofnumber total

in T termoffrequency document inverse

document in T termoffrequency

document in term

⎟⎠⎞⎜

⎝⎛=

=====

nNidf

CnCNCidf

DtfDkT

kk

kk

kk

ikik

ik

2011.02.07 - SLIDE 52IS 240 – Spring 2011

Inverse Document Frequency

• IDF provides high values for rare words and low values for common words

41

10000log

698.220

10000log

301.05000

10000log

010000

10000log

=⎟⎠

⎞⎜⎝

⎛

=⎟⎠

⎞⎜⎝

⎛

=⎟⎠

⎞⎜⎝

⎛

=⎟⎠

⎞⎜⎝

⎛

For a collectionof 10000 documents(N = 10000)

2011.02.07 - SLIDE 53IS 240 – Spring 2011

Word Frequency vs. Resolving Power

The most frequent words are not the most descriptive.

(from van Rijsbergen 79)

2011.02.07 - SLIDE 54IS 240 – Spring 2011

Weighting schemes

• We have seen something of– Binary– Raw term weights– TF*IDF

• There are many other possibilities– IDF alone– Normalized term frequency– etc.

2011.02.07 - SLIDE 55IS 240 – Spring 2011

tf x idf normalization

• Normalize the term weights (so longer documents are not unfairly given more weight)– normalize usually means force all values to

fall within a certain range, usually between 0 and 1, inclusive.

∑ =

=t

k kik

kikik

nNtf

nNtfw

1

22 )]/[log()(

)/log(

2011.02.07 - SLIDE 56IS 240 – Spring 2011

Vector space similarity

• Use the weights to compare the documents

terms.) thehting when weigdone tion was(Normaliza

product.inner normalizedor cosine, thecalled also is This

),(

:is documents twoof similarity theNow,

1∑=

∗=t

kjkikji wwDDsim

2011.02.07 - SLIDE 57IS 240 – Spring 2011

Vector Space Similarity Measure

• combine tf x idf into a measure

)()(

),(

:comparison similarity in the normalize could weotherwise

),( :normalized are weights term theif

absent is terma if 0 ...,,

,...,,

1

2

1

2

1

1

,21

21

∑∑

∑

∑

==

=

=

∗

∗=

∗=

==

=

t

jd

t

jqj

t

jdqj

i

t

jdqji

qtqq

dddi

ij

ij

ij

itii

ww

wwDQsim

wwDQsim

wwwwQ

wwwD

2011.02.07 - SLIDE 58IS 240 – Spring 2011

Weighting schemes

• We have seen something of– Binary– Raw term weights– TF*IDF

• There are many other possibilities– IDF alone– Normalized term frequency

2011.02.07 - SLIDE 59IS 240 – Spring 2011

Term Weights in SMART

• SMART is an experimental IR system developed by Gerard Salton (and continued by Chris Buckley) at Cornell.

• Designed for laboratory experiments in IR – Easy to mix and match different weighting

methods– Really terrible user interface– Intended for use by code hackers

2011.02.07 - SLIDE 60IS 240 – Spring 2011

Term Weights in SMART

• In SMART weights are decomposed into three factors:

norm

collectfreqw kkd

kd

∗=

2011.02.07 - SLIDE 61IS 240 – Spring 2011

SMART Freq Components

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+

+=

1)ln(

)max(2

1

2

1)max(

}1,0{

kd

kd

kd

kd

kd

kd

freq

freqfreq

freq

freq

freq

Binary

maxnorm

augmented

log

2011.02.07 - SLIDE 62IS 240 – Spring 2011

Collection Weighting in SMART

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

k

k

k

k

k

k

Doc

Doc

DocNDocDoc

NDoc

Doc

NDoc

collect

1

log

log

log

2

Inverse

squared

probabilistic

frequency

2011.02.07 - SLIDE 63IS 240 – Spring 2011

Term Normalization in SMART

( )⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=∑∑∑

jvector

vectorj

vectorj

vectorj

w

w

w

w

norm

max

4

2

sum

cosine

fourth

max

2011.02.07 - SLIDE 64IS 240 – Spring 2011

Lucene Algorithm

• The open-source Lucene system is a vector based system that differs from SMART-like systems in the ways the TF*IDF measures are normalized

2011.02.07 - SLIDE 65IS 240 – Spring 2011

Lucene

• The basic Lucene algorithm is:

• Where is the length normalized query

– and normd,t is the term normalization (square root of the number of tokens in the same document field as t)

– overlap(q,d) is the proportion of query terms matched in the document

– boostt is a user specified term weight enhancement

∑ ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅⋅

⋅=

tt

td

ttd

q

ttq

q

dqoverlapboost

norm

idftf

norm

idftfdqScore

),(),(

,

,,

q

ttq

norm

idftf ⋅,

2011.02.07 - SLIDE 66IS 240 – Spring 2011

How To Process a Vector Query

• Assume that the database contains an inverted file like the one we discussed earlier…– Why an inverted file?– Why not a REAL vector file?

• What information should be stored about each document/term pair?– As we have seen SMART gives you choices

about this…

2011.02.07 - SLIDE 67IS 240 – Spring 2011

Simple Example System

• Collection frequency is stored in the dictionary

• Raw term frequency is stored in the inverted file postings list

• Formula for term ranking

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=

⋅=∑=

kkk

M

kikqki

n

Ntfw

wwDQsim

log

),(1

2011.02.07 - SLIDE 68IS 240 – Spring 2011

Processing a Query

• For each term in the query– Count number of times the term occurs – this

is the tf for the query term– Find the term in the inverted dictionary file

and get:• nk : the number of documents in the collection with

this term• Loc : the location of the postings list in the inverted

file• Calculate Query Weight: wqk • Retrieve nk entries starting at Loc in the postings

file

2011.02.07 - SLIDE 69IS 240 – Spring 2011

Processing a Query

• Alternative strategies…– First retrieve all of the dictionary entries

before getting any postings information• Why?

– Just process each term in sequence

• How can we tell how many results there will be? – It is possible to put a limitation on the number

of items returned• How might this be done?

2011.02.07 - SLIDE 70IS 240 – Spring 2011

Processing a Query

• Like Hashed Boolean OR:– Put each document ID from each postings list into hash table

• If match increment counter (optional) – If first doc, set a WeightSUM variable to 0

• Calculate Document weight wik for the current term

• Multiply Query weight and Document weight and add it to WeightSUM

• Scan hash table contents and add to new list – including document ID and WeightSUM

• Sort by WeightSUM and present in sorted order

2011.02.07 - SLIDE 71IS 240 – Spring 2011

Next time

• More on Vector Space

• Document Clustering