# Overview - Stanford NLP Group

Introduction to Information Retrieval Introduction to Information Retrieval Hinrich Schtze and Christina Lioma Lecture 6: Scoring, Term Weighting, The Vector Space Model 1 Introduction to Information Retrieval Overview Recap Why ranked retrieval?

Term frequency tf-idf weighting The vector space model 2 Introduction to Information Retrieval Outline Recap Why ranked retrieval?

Term frequency tf-idf weighting The vector space model 3 Introduction to Information Retrieval Heaps law Vocabulary size M as a function of collection size T (number of tokens) for Reuters-RCV1. For these data, the dashed line

log10M = 0.49 log log10 T + 1.64 is the best least squares fit. Thus, M = 101.64T0.49 and k = 101.64 44 and b = 0.49. 4 Introduction to Information Retrieval Zipfs law The most frequent term (the) occurs cf1 times, the second most frequent term (of) occurs times, the third most frequent term (and) occurs times etc. 5 Introduction to Information Retrieval

Dictionary as a string 6 Introduction to Information Retrieval Gap encoding 7 Introduction to Information Retrieval Variable byte (VB) code Dedicate 1 bit (high bit) to be a continuation bit c. If the gap G fits within 7 bits, binary-encode it in the 7 available bits and set c = 1. Else: set c = 0, encode high-order 7 bits and then use one or more additional bytes to encode the lower order bits using the same algorithm. 8

Introduction to Information Retrieval Gamma codes for gap encoding Represent a gap G as a pair of length and offset. Offset is the gap in binary, with the leading bit chopped off. Length is the length of offset. Encode length in unary code The Gamma code is the concatenation of length and offset. 9 Introduction to Information Retrieval Compression of Reuters data structure

dictionary, fixed-width dictionary, term pointers into string , with blocking, k = 4 , with blocking & front coding collection (text, xml markup etc) collection (text) T/D incidence matrix postings, uncompressed (32-bit words) postings, uncompressed (20 bits) postings, variable byte encoded postings, encoded size in MB 11.2 7.6 7.1 5.9 3600.0 960.0 40,000.0 400.0 250.0

116.0 101.0 10 Introduction to Information Retrieval Take-away today Ranking search results: why it is important (as opposed to just presenting a set of unordered Boolean results) Term frequency: This is a key ingredient for ranking. Tf-idf ranking: best known traditional ranking scheme Vector space model: One of the most important formal models for information retrieval (along with Boolean and probabilistic models) 11 Introduction to Information Retrieval Outline

Recap Why ranked retrieval? Term frequency tf-idf weighting The vector space model 12 Introduction to Information Retrieval

Ranked retrieval Thus far, our queries have all been Boolean. Documents either match or dont. Good for expert users with precise understanding of their needs and of the collection. Also good for applications: Applications can easily consum 1000s of results. Not good for the majority of users Most users are not capable of writing Boolean queries . . . . . . or they are, but they think its too much work. Most users dont want to wade through 1000s of results. This is particularly true of web search. 13 Introduction to Information Retrieval Problem with Boolean search: Feast or famine Boolean queries often result in either too few (=0) or too many (1000s) results. Query 1 (boolean conjunction): [standard user dlink 650]

200,000 hits feast Query 2 (boolean conjunction): [standard user dlink 650 no card found] 0 hits famine In Boolean retrieval, it takes a lot of skill to come up with a query that produces a manageable number of hits. 14 Introduction to Information Retrieval Feast or famine: No problem in ranked retrieval With ranking, large result sets are not an issue. Just show the top 10 results Doesnt overwhelm the user

Premise: the ranking algorithm works: More relevant results are ranked higher than less relevant results. 15 Introduction to Information Retrieval Scoring as the basis of ranked retrieval We wish to rank documents that are more relevant higher than documents that are less relevant. How can we accomplish such a ranking of the documents in the collection with respect to a query? Assign a score to each query-document pair, say in [0, 1]. This score measures how well document and query match. 16 Introduction to Information Retrieval Query-document matching scores How do we compute the score of a query-document pair? Lets start with a one-term query.

If the query term does not occur in the document: score should be 0. The more frequent the query term in the document, the higher the score We will look at a number of alternatives for doing this. 17 Introduction to Information Retrieval Take 1: Jaccard coefficient A commonly used measure of overlap of two sets Let A and B be two sets Jaccard coefficient: JACCARD (A, A) = 1 JACCARD (A, B) = 0 if A B = 0 A and B dont have to be the same size. Always assigns a number between 0 and 1. 18

Introduction to Information Retrieval Jaccard coefficient: Example What is the query-document match score that the Jaccard coefficient computes for: Query: ides of March Document Caesar died in March JACCARD(q, d) = 1/6 19 Introduction to Information Retrieval Whats wrong with Jaccard? It doesnt consider term frequency (how many occurrences a term has). Rare terms are more informative than frequent terms. Jaccard does not consider this information. We need a more sophisticated way of normalizing for the length of a document. Later in this lecture, well use (cosine) . . .

. . . instead of |A B|/|A B| (Jaccard) for length normalization. 20 Introduction to Information Retrieval Outline Recap Why ranked retrieval? Term frequency tf-idf weighting

The vector space model 21 Introduction to Information Retrieval Binary incidence matrix Anthony Julius and Caesar Cleopatra ANTHONY BRUTUS CAESAR CALPURNIA CLEOPATRA MERCY WORSER ...

1 1 1 0 1 1 1 The Hamlet Tempest 1 1 1 1 0 0 0 0 0 0

0 0 1 1 Othello 0 1 1 0 0 1 1 Macbeth ... 0 0 1 0 0 1

1 1 0 1 0 0 1 0 Each document is represented as a binary vector {0, 1} {0, 1}|V|. 22 Introduction to Information Retrieval Binary incidence matrix Anthony Julius and Caesar Cleopatra ANTHONY

BRUTUS CAESAR CALPURNIA CLEOPATRA MERCY WORSER ... 157 4 232 0 57 2 2 73 157 227 10 0 0

0 The Hamlet Tempest 0 0 0 0 0 3 1 Othello 0 2 2 0 0 8 1

Macbeth ... 0 0 1 0 0 5 1 1 0 0 0 0 8 5 Each document is now represented as a count vector {0, 1} N|V|. 23

Introduction to Information Retrieval Bag of words model We do not consider the order of words in a document. John is quicker than Mary and Mary is quicker than John are represented the same way. This is called a bag of words model. In a sense, this is a step back: The positional index was able to distinguish these two documents. We will look at recovering positional information later in this course. For now: bag of words model 24 Introduction to Information Retrieval Term frequency tf The term frequency tft,d of term t in document d is defined as the number of times that t occurs in d. We want to use tf when computing query-document match scores.

But how? Raw term frequency is not what we want because: A document with tf = 10 occurrences of the term is more relevant than a document with tf = 1 occurrence of the term. But not 10 times more relevant. Relevance does not increase proportionally with term frequency. 25 Introduction to Information Retrieval Instead of raw frequency: Log frequency weighting The log frequency weight of term t in d is defined as follows tft,d wt,d : 0 0, 1 1, 2 1.3, 10 2, 1000 4, etc. Score for a document-query pair: sum over terms t in both q and d: tfmatching-score(q, d) = t {0, 1}qd (1 + log tft,d )

The score is 0 if none of the query terms is present in the document. 26 Introduction to Information Retrieval Exercise Compute the Jaccard matching score and the tf matching score for the following query-document pairs. q: [information on cars] d: all youve ever wanted to know about cars q: [information on cars] d: information on trucks, information on planes, information on trains q: [red cars and red trucks] d: cops stop red cars more often 27 Introduction to Information Retrieval Outline

Recap Why ranked retrieval? Term frequency tf-idf weighting The vector space model 28 Introduction to Information Retrieval

Frequency in document vs. frequency in collection In addition, to term frequency (the frequency of the term in the document) . . . . . .we also want to use the frequency of the term in the collection for weighting and ranking. 29 Introduction to Information Retrieval Desired weight for rare terms Rare terms are more informative than frequent terms. Consider a term in the query that is rare in the collection (e.g., ARACHNOCENTRIC). A document containing this term is very likely to be relevant. We want high weights for rare terms like ARACHNOCENTRIC. 30

Introduction to Information Retrieval Desired weight for frequent terms Frequent terms are less informative than rare terms. Consider a term in the query that is frequent in the collection (e.g., GOOD, INCREASE, LINE). A document containing this term is more likely to be relevant than a document that doesnt . . . . . . but words like GOOD, INCREASE and LINE are not sure indicators of relevance. For frequent terms like GOOD, INCREASE and LINE, we want positive weights . . . . . . but lower weights than for rare terms. 31 Introduction to Information Retrieval Document frequency We want high weights for rare terms like ARACHNOCENTRIC. We want low (positive) weights for frequent words like GOOD, INCREASE and LINE. We will use document frequency to factor this into

computing the matching score. The document frequency is the number of documents in the collection that the term occurs in. 32 Introduction to Information Retrieval idf weight dft is the document frequency, the number of documents that t occurs in. dft is an inverse measure of the informativeness of term t. We define the idf weight of term t as follows: (N is the number of documents in the collection.) idft is a measure of the informativeness of the term. [log N/dft ] instead of [N/dft ] to dampen the effect of idf Note that we use the log transformation for both term frequency and document frequency. 33

Introduction to Information Retrieval Examples for idf Compute idft using the formula: term calpurnia animal sunday fly under the dft 1 100 1000 10,000 100,000 1,000,000 idft 6

4 3 2 1 0 34 Introduction to Information Retrieval Effect of idf on ranking idf affects the ranking of documents for queries with at least two terms. For example, in the query arachnocentric line, idf weighting increases the relative weight of ARACHNOCENTRIC and decreases the relative weight of LINE. idf has little effect on ranking for one-term queries. 35 Introduction to Information Retrieval

Collection frequency vs. Document frequency word INSURANCE TRY collection frequency document frequency 10440 10422 3997 8760 Collection frequency of t: number of tokens of t in the collection Document frequency of t: number of documents t occurs in Why these numbers? Which word is a better search term (and should get a higher weight)? This example suggests that df (and idf) is better for

weighting than cf (and icf). 36 Introduction to Information Retrieval tf-idf weighting The tf-idf weight of a term is the product of its tf weight and its idf weight. tf-weight idf-weight Best known weighting scheme in information retrieval Note: the - in tf-idf is a hyphen, not a minus sign! Alternative names: tf.idf, tf x idf 37

Introduction to Information Retrieval Summary: tf-idf Assign a tf-idf weight for each term t in each document d: The tf-idf weight . . . . . . increases with the number of occurrences within a document. (term frequency) . . . increases with the rarity of the term in the collection. (inverse document frequency) 38 Introduction to Information Retrieval Exercise: Term, collection and document frequency Quantity Symbol Definition term frequency

tft,d document frequency dft collection frequency cft number of occurrences of t in d number of documents in the collection that t occurs in total number of occurrences of t in the collection Relationship between df and cf? Relationship between tf and cf? Relationship between tf and df? 39

Introduction to Information Retrieval Outline Recap Why ranked retrieval? Term frequency tf-idf weighting

The vector space model 40 Introduction to Information Retrieval Binary incidence matrix Anthony Julius and Caesar Cleopatra ANTHONY BRUTUS CAESAR CALPURNIA CLEOPATRA MERCY WORSER ... 1 1 1

0 1 1 1 The Hamlet Tempest 1 1 1 1 0 0 0 0 0 0 0 0 1

1 Othello 0 1 1 0 0 1 1 Macbeth ... 0 0 1 0 0 1 1 1

0 1 0 0 1 0 Each document is represented as a binary vector {0, 1} {0, 1}|V|. 41 Introduction to Information Retrieval Count matrix Anthony Julius and Caesar Cleopatra ANTHONY BRUTUS CAESAR CALPURNIA

CLEOPATRA MERCY WORSER ... 157 4 232 0 57 2 2 73 157 227 10 0 0 0 The

Hamlet Tempest 0 0 0 0 0 3 1 Othello 0 2 2 0 0 8 1 Macbeth ... 0

0 1 0 0 5 1 1 0 0 0 0 8 5 Each document is now represented as a count vector {0, 1} N|V|. 42 Introduction to Information Retrieval Binary count weight matrix

Anthony Julius and Caesar Cleopatra ANTHONY BRUTUS CAESAR CALPURNIA CLEOPATRA MERCY WORSER ... 5.25 1.21 8.59 0.0 2.85 1.51 1.37 3.18

6.10 2.54 1.54 0.0 0.0 0.0 The Hamlet Tempest 0.0 0.0 0.0 0.0 0.0 1.90 0.11 0.0 1.0 1.51 0.0

0.0 0.12 4.15 Othello 0.0 0.0 0.25 0.0 0.0 5.25 0.25 Macbeth ... 0.35 0.0 0.0 0.0 0.0 0.88 1.95

Each document is now represented as a real-valued vector of tf idf weights {0, 1} R|V|. 43 Introduction to Information Retrieval Documents as vectors Each document is now represented as a real-valued vector of tf-idf weights {0, 1} R|V|. So we have a |V|-dimensional real-valued vector space. Terms are axes of the space. Documents are points or vectors in this space. Very high-dimensional: tens of millions of dimensions when you apply this to web search engines Each vector is very sparse - most entries are zero. 44 Introduction to Information Retrieval Queries as vectors

Key idea 1: do the same for queries: represent them as vectors in the high-dimensional space Key idea 2: Rank documents according to their proximity to the query proximity = similarity proximity negative distance Recall: Were doing this because we want to get away from the youre-either-in-or-out, feast-or-famine Boolean model. Instead: rank relevant documents higher than nonrelevant documents 45 Introduction to Information Retrieval How do we formalize vector space similarity? First cut: (negative) distance between two points

( = distance between the end points of the two vectors) Euclidean distance? Euclidean distance is a bad idea . . . . . . because Euclidean distance is large for vectors of different lengths. 46 Introduction to Information Retrieval Why distance is a bad idea The Euclidean distance of and is large although the distribution of terms in the query q and the distribution of terms in the document d2 are very similar. Questions about basic vector space setup? 47 47 Introduction to Information Retrieval

Use angle instead of distance Rank documents according to angle with query Thought experiment: take a document d and append it to itself. Call this document d. d is twice as long as d. Semantically d and d have the same content. The angle between the two documents is 0, corresponding to maximal similarity . . . . . . even though the Euclidean distance between the two documents can be quite large. 48 Introduction to Information Retrieval From angles to cosines The following two notions are equivalent. Rank documents according to the angle between query and document in decreasing order Rank documents according to cosine(query,document) in increasing order

Cosine is a monotonically decreasing function of the angle for the interval [0, 180] 49 Introduction to Information Retrieval Cosine 50 Introduction to Information Retrieval Length normalization How do we compute the cosine? A vector can be (length-) normalized by dividing each of its components by its length here we use the L2 norm: This maps vectors onto the unit sphere . . . . . . since after normalization: As a result, longer documents and shorter documents have weights of the same order of magnitude. Effect on the two documents d and d (d appended to itself)

from earlier slide: they have identical vectors after lengthnormalization. 51 Introduction to Information Retrieval Cosine similarity between query and document qi is the tf-idf weight of term i in the query. di is the tf-idf weight of term i in the document. | | and | | are the lengths of and This is the cosine similarity of and . . . . . . or, equivalently, the cosine of the angle between and 52 Introduction to Information Retrieval Cosine for normalized vectors For normalized vectors, the cosine is equivalent to the dot product or scalar product.

(if and are length-normalized). 53 Introduction to Information Retrieval Cosine similarity illustrated 54 Introduction to Information Retrieval Cosine: Example term frequencies (counts) How similar are these novels? SaS: Sense and

Sensibility PaP: Pride and Prejudice WH: Wuthering Heights term AFFECTION JEALOUS GOSSIP WUTHERING SaS 115 10 2 0 PaP 58 7 0

0 WH 20 11 6 38 55 Introduction to Information Retrieval Cosine: Example term frequencies (counts) term SaS PaP WH AFFECTION 115 58 20 JEALOUS 10 7 11

GOSSIP 2 0 6 WUTHERING 0 0 38 log frequency weighting term AFFECTION JEALOUS GOSSIP WUTHERING SaS PaP 3.06 2.76 2.0 1.85 1.30 0 0 0

WH 2.30 2.04 1.78 2.58 (To simplify this example, we don't do idf weighting.) 56 Introduction to Information Retrieval Cosine: Example log frequency weighting SaS PaP WH log frequency weighting &

cosine normalization term SaS PaP WH AFFECTION 3.06 JEALOUS 2.0 GOSSIP 1.30 WUTHERING 0 2.76 1.85 0 0 2.30 2.04 1.78 2.58

AFFECTION JEALOUS GOSSIP WUTHERING term 0.789 0.515 0.335 0.0 0.832 0.555 0.0 0.0 0.524 0.465 0.405 0.588

cos(SaS,PaP) 0.789 log 0.832 + 0.515 log 0.555 + 0.335 log 0.0 + 0.0 log 0.0 0.94. cos(SaS,WH) 0.79 cos(PaP,WH) 0.69 Why do we have cos(SaS,PaP) > cos(SAS,WH)? 57 Introduction to Information Retrieval Computing the cosine score 58 Introduction to Information Retrieval Components of tf-idf weighting 59 Introduction to Information Retrieval

tf-idf example We often use different weightings for queries and documents. Notation: ddd.qqq Example: lnc.ltn document: logarithmic tf, no df weighting, cosine normalization query: logarithmic tf, idf, no normalization Isnt it bad to not idf-weight the document? Example query: best car insurance Example document: car insurance auto insurance 60

Introduction to Information Retrieval tf-idf example: Inc.Itn Query: best car insurance. Document: car insurance auto insurance. Key to columns: tf-raw: raw (unweighted) term frequency, tf-wght: logarithmically weighted term frequency, df: document frequency, idf: inverse document frequency, weight: the final weight of the term in the query or document, nlized: document weights after cosine normalization, product: the product of final query weight and final document weight 1/1.92 0.52 1.3/1.92 0.68 Final similarity score between query and document: i wqi wdi = 0 + 0 + 1.04 + 2.04 = 3.08 Questions? 61 Introduction to Information Retrieval Summary: Ranked retrieval in the vector space model Represent the query as a weighted tf-idf vector

Represent each document as a weighted tf-idf vector Compute the cosine similarity between the query vector and each document vector Rank documents with respect to the query Return the top K (e.g., K = 10) to the user 62 Introduction to Information Retrieval Take-away today Ranking search results: why it is important (as opposed to just presenting a set of unordered Boolean results) Term frequency: This is a key ingredient for ranking. Tf-idf ranking: best known traditional ranking scheme Vector space model: One of the most important formal models for information retrieval (along with Boolean and probabilistic models) 63 Introduction to Information Retrieval

Resources Chapters 6 and 7 of IIR Resources at http://ifnlp.org/ir Vector space for dummies Exploring the similarity space (Moffat and Zobel, 2005) Okapi BM25 (a state-of-the-art weighting method, 11.4.3 of IIR) 64

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