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Ranked Retrieval

Thus far, our queries have all been Boolean.

Documents either match or don’t.

Good for expert users with precise understanding of their needs and the collection.

Also good for applications: Applications can easily consume 1000s of results.

Not good for the majority of users.

Most users incapable of writing Boolean queries (or they are, but they think it’s too much work).
Most users don’t want to wade through 1000s of results.

This is particularly true of web search.

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Problem with Boolean Search: feast or famine

Boolean queries often result in either too few (=0) or too many (1000s) results.
Query 1: “standard user dlink 650” → 200,000 hits
Query 2: “standard user dlink 650 no card found”: 0 hits
It takes a lot of skill to come up with a query that produces a manageable number of hits.
AND gives too few; OR gives too many

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Ranked Retrieval Models

Rather than a set of documents satisfying a query expression, in ranked retrieval, the system returns an ordering over the (top) documents in the collection for a query
Free text queries: Rather than a query language of operators and expressions, the user’s query is just one or more words in a human language
In principle, there are two separate choices here, but in practice, ranked retrieval has normally been associated with free text queries and vice versa

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feast or famine: not a problem in Ranked Retrieval

When a system produces a ranked result set, large result sets are not an issue
Indeed, the size of the result set is not an issue
We just show the top k ( ≈ 10) results
We don’t overwhelm the user

Premise: the ranking algorithm works

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Scoring as the basis of Ranked Retrieval

We wish to return in order the documents most likely to be useful to the searcher
How can we rank-order the documents in the collection with respect to a query?
Assign a score – say in [0, 1] – to each document
This score measures how well document and query “match”.

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Query-Document Matching Scores

We need a way of assigning a score to a query/document pair
Let’s 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 this.

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Jaccard Coefficient

A common measure of overlap of two sets A and B
jaccard(A,B) = |A ∩ B| / |A ∪ B|
jaccard(A,A) = 1
jaccard(A,B) = 0 if A ∩ B = 0
A and B don’t have to be the same size.
Always assigns a number between 0 and 1.

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Jaccard Coefficient: Scoring example

What is the query-document match score that the Jaccard coefficient computes for each of the two documents below?
Query: ides of march
Document 1: caesar died in march
Document 2: the long march

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Issue with Jaccard coefficients

It doesn’t consider term frequency (how many times a term occurs in a document)
Rare terms in a collection are more informative than frequent terms. Jaccard doesn’t consider this information
We need a more sophisticated way of normalizing for length

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Recall: Binary Term-Document Incidence Matrix

Each document is represented by a binary vector ∈ {0,1}|V|

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Term-Document Count Matrices

Consider the number of occurrences of a term in a document:

Each document is a count vector in ℕ^v: a column below

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Bag of Words Model

Vector representation doesn’t consider the ordering of words in a document
John is quicker than Mary and Mary is quicker than John have the same vectors
This is called the bag of words model.
In a sense, this is a step back: The positional index was able to distinguish these two documents.

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Term Frequency tf

The term frequency tf(t,d) of term t in document d is defined as the number of times that t occurs in d.

Note: Frequency means count in IR

We want to use tf when computing query-document match scores. But how?
Raw term frequency is not what we want:

A document with 10 occurrences of the term is more relevant than a document with 1 occurrence of the term.
But not 10 times more relevant.

Relevance does not increase proportionally with term frequency.

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Log-Frequency Weighting

The log frequency weight of term t in d is

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:

score

The score is 0 if none of the query terms is present in the document.

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Rare terms are more informative

Rare terms are more informative than frequent terms

Recall stop words

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 to the query arachnocentric
→ We want a high weight for rare terms like arachnocentric

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Collection vs. Document Frequency

Collection frequency of t is the number of occurrences of t in the collection
Document frequency of t is the number of documents in which t occurs
Example:

Which word is for better search (gets higher weight)

Word	Collection Frequency	Document Frequency
Insurance	10440	3997
Try	10422	8760

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idf Weight

dft is the document frequency of t: the number of documents that contain t
dft is an inverse measure of the informativeness of t
dft ≤ N
We define the idf (inverse document frequency) of t by

We use log (N/dft) instead of N/dft to “dampen” the effect of idf.

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idf Weight, suppose N = 1 million

There is one idf value for each term t in a collection.

term	df_t	idf_t
calpurnia	1	6
animal	100	4
sunday	1,000	3
fly	10,000	2
under	100,000	1
the	1,000,000	0

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Effect of idf on Ranking

Does idf have an effect on ranking for one-term queries, like

iPhone

idf has no effect on ranking one term queries

idf affects the ranking of documents for queries with at least two terms

For the query capricious person, idf weighting makes occurrences of capricious count for much more in the final document ranking than occurrences of person.

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tf-idf Weighting

The tf-idf weight of a term is the product of its tf weight and its 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

Increases with the number of occurrences within a document
Increases with the rarity of the term in the collection

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Score for a Document given a Query

There are many variants

How “tf” is computed (with/without logs)
Whether the terms in the query are also weighted
…

score(q.d)=

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Binary → Count → Weight Matrix

Each document is now represented by a real-valued vector of tf-idf weights ∈ R|V|

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Documents as Vectors

So we have a |V|-dimensional 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 a web search engine
These are very sparse vectors - most entries are zero.

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Queries as Vectors

Key idea 1: Do the same for queries: represent them as vectors in the space
Key idea 2: Rank documents according to their proximity to the query in this space
proximity = similarity of vectors
proximity ≈ inverse of distance

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Formalizing Vector Space Proximity

First cut: 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.

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Why Distance is a Bad Idea

The Euclidean distance between q and d2 is large even though the distribution of terms in the query q and the distribution of terms in the document d2 are very similar.

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Use Angle instead of Distance

Thought experiment: take a document d and append it to itself. Call this document d′.
“Semantically” d and d′ have the same content
The Euclidean distance between the two documents can be quite large
The angle between the two documents is 0, corresponding to maximal similarity.

Key idea: Rank documents according to angle with query.

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from Angles to Cosines

The following two notions are equivalent.

Rank documents in decreasing order of the angle between query and document
Rank documents in increasing order of cosine(query,document)

Cosine is a monotonically decreasing function for the interval [0 degree, 180 degree]

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Length Normalization

A vector can be (length-) normalized by dividing each of its components by its length – for this we use the L2 norm:

Dividing a vector by its L2 norm makes it a unit (length) vector (on surface of unit hypersphere)
Effect on the two documents d and d′ (d appended to itself) from earlier slide: they have identical vectors after length-normalization.

Long and short documents now have comparable weights

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Cosine Similarity

qi is the weight of term i in the query
di is the weight of term i in the document
cos(q,d) is the cosine similarity of q and d … or,
equivalently, the cosine of the angle between q and d

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Cosine Similarity Illustrated

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Cosine Similarity amongst 3 Documents

How similar are the novels?

SaS: Sense and Sensibility

PaP: Pride and Prejudice

WH: Wuthering Heights?

term	SaS	PaP	WH
affection	115	58	20
jealous	10	7	11
gossip	2	0	6
wuthering	0	0	38

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Cosine Similarity amongst 3 Documents

Log frequency weighting

dot(SaS,PaP) ≈ 12.1

dot(SaS,WH) ≈ 13.4

dot(PaP,WH) ≈ 10.1

After length normalization

cos(SaS,PaP) ≈ 0.94

cos(SaS,WH) ≈ 0.79

cos(PaP,WH) ≈ 0.69

term	SaS	PaP	WH
affection	3.06	2.76	2.30
jealous	2.00	1.85	2.04
gossip	1.30	0	1.78
wuthering	0	0	2.58

term	SaS	PaP	WH
affection	0.789	0.832	0.524
jealous	0.515	0.555	0.465
gossip	0.335	0	0.405
wuthering	0	0	0.588

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Computing Cosine Similarity

D1=(1,2,2)

D2=(2,1,2)

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Cosine Similarity: Term-at-a-Time Algorithm

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Computing Cosine Similarity

Algorithm can be adapted to scoring document-at-a-time (DAAT)

Storing wt,d in each posting could be expensive

…because we’d have to store a floating point number

For tf-idf scoring, it suffices to store tft,d in the posting and idft in the head of the postings list

Extracting the top K items can be done with a priority queue (e.g., a heap)

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Variants of TF-IDF

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Weighting in Queries vs Documents

Many search engines allow for different weightings for queries vs. documents

SMART Notation: denotes the combination in use in an engine, with the notation ddd.qqq, using the acronyms from the previous table

A very standard weighting scheme is: lnc.ltc

Document: logarithmic tf (l as first character), no idf and cosine normalization

Query: logarithmic tf (l in leftmost column), idf (t in second column), cosine normalization …

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Calculating TF-IDF: lnc.ltc

Document: car insurance auto insurance

Query: best car insurance

Doc length =

Score = 0+0+0.27+0.53 = 0.8

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Summary: Vector Space Ranking

Represent the query as a weighted tf-idf vector

Represent each document as a weighted tf-idf vector

Compute the cosine similarity score for the query vector and each document vector

Rank documents with respect to the query by score

Return the top K (e.g., K = 10) to the user

Code

Code 2

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