Why ranked retrieval? Term frequency tf-idf weighting The vector space model
PV211: Introduction to Information Retrieval
https://www.fi.muni.cz/~sojka/PV211
IIR 6: Scoring, term weighting, the vector space model
Handout version
Petr Sojka, Hinrich Schütze et al.
Faculty of Informatics, Masaryk University, Brno
Center for Information and Language Processing, University of Munich
2024-03-14
(compiled on 2024-03-08 15:05:57)
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Overview
1 Why ranked retrieval?
2 Term frequency
3 tf-idf weighting
4 The vector space model
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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)
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Ranked retrieval
Thus far, our queries have been Boolean.
Documents either match or do not.
Good for expert users with precise understanding of their
needs and of the collection.
Also good for applications: Applications can easily consume
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 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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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Feast or famine: No problem in ranked retrieval
With ranking, large result sets are not an issue.
Just show the top 10 results
Doesn’t overwhelm the user
Premise: the ranking algorithm works: More relevant results
are ranked higher than less relevant results.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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”.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Query-document matching scores
How do we compute the score of 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 doing this.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Take 1: Jaccard coefficient
A commonly used measure of overlap of two sets
Let A and B be two sets
Jaccard coefficient:
jaccard(A, B) =
|A ∩ B|
|A ∪ B|
(A = ∅ or 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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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
What’s wrong with Jaccard?
It doesn’t 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, we’ll use |A ∩ B|/ |A ∪ B| (cosine) . . .
. . . instead of |A ∩ B|/|A ∪ B| (Jaccard) for length
normalization.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Binary incidence matrix
Anthony Julius The Hamlet Othello Macbeth . . .
and Caesar Tempest
Cleopatra
Anthony 1 1 0 0 0 1
Brutus 1 1 0 1 0 0
Caesar 1 1 0 1 1 1
Calpurnia 0 1 0 0 0 0
Cleopatra 1 0 0 0 0 0
mercy 1 0 1 1 1 1
worser 1 0 1 1 1 0
. . .
Each document is represented as a binary vector ∈ {0, 1}dim(V ).
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Count matrix
Anthony Julius The Hamlet Othello Macbeth . . .
and Caesar Tempest
Cleopatra
Anthony 157 73 0 0 0 1
Brutus 4 157 0 2 0 0
Caesar 232 227 0 2 1 0
Calpurnia 0 10 0 0 0 0
Cleopatra 57 0 0 0 0 0
mercy 2 0 3 8 5 8
worser 2 0 1 1 1 5
. . .
Each document is now represented as a count vector ∈ Ndim(V ).
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Instead of raw frequency: Log frequency weighting
The log frequency weight of term t in d is defined as follows
wt,d =
1 + log10 tft,d if tft,d > 0
0 otherwise
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:
tf-matching-score(q, d) = t∈q∩d (1 + log tft,d)
The score is 0 if none of the query terms is present in the
document.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Exercise
Compute the Jaccard matching score and the tf matching
score for the following query-document pairs.
q: [information on cars] d: “all you’ve 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”
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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 doesn’t . . .
. . . 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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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:
idft = log10
N
dft
(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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Examples for idf
Compute idft using the formula: idft = log10
1,000,000
dft
term dft idft
calpurnia 1 6
animal 100 4
sunday 1000 3
fly 10,000 2
under 100,000 1
the 1,000,000 0
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Collection frequency vs. Document frequency
word collection frequency document frequency
insurance 10440 3997
try 10422 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”).
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
tf-idf weighting
The tf-idf weight of a term is the product of its tf weight and
its idf weight.
wt,d = (1 + log tft,d) · log
N
dft
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
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Summary: tf-idf
Assign a tf-idf weight for each term t in each document d:
wt,d = (1 + log tft,d ) · log N
dft
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)
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Exercise: Term, collection and document frequency
Quantity Symbol Definition
term frequency tft,d number of occurrences of t in
d
document frequency dft number of documents in the
collection that t occurs in
collection frequency cft total number of occurrences of
t in the collection
Relationship between df and cf?
Relationship between tf and cf?
Relationship between tf and df?
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Binary incidence matrix
Anthony Julius The Hamlet Othello Macbeth . . .
and Caesar Tempest
Cleopatra
Anthony 1 1 0 0 0 1
Brutus 1 1 0 1 0 0
Caesar 1 1 0 1 1 1
Calpurnia 0 1 0 0 0 0
Cleopatra 1 0 0 0 0 0
mercy 1 0 1 1 1 1
worser 1 0 1 1 1 0
. . .
Each document is represented as a binary vector ∈ {0, 1}dim(V ).
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Count matrix
Anthony Julius The Hamlet Othello Macbeth . . .
and Caesar Tempest
Cleopatra
Anthony 157 73 0 0 0 1
Brutus 4 157 0 2 0 0
Caesar 232 227 0 2 1 0
Calpurnia 0 10 0 0 0 0
Cleopatra 57 0 0 0 0 0
mercy 2 0 3 8 5 8
worser 2 0 1 1 1 5
. . .
Each document is now represented as a count vector ∈ Ndim(V ).
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Binary → count → weight matrix
Anthony Julius The Hamlet Othello Macbeth . . .
and Caesar Tempest
Cleopatra
Anthony 5.25 3.18 0.0 0.0 0.0 0.35
Brutus 1.21 6.10 0.0 1.0 0.0 0.0
Caesar 8.59 2.54 0.0 1.51 0.25 0.0
Calpurnia 0.0 1.54 0.0 0.0 0.0 0.0
Cleopatra 2.85 0.0 0.0 0.0 0.0 0.0
mercy 1.51 0.0 1.90 0.12 5.25 0.88
worser 1.37 0.0 0.11 4.15 0.25 1.95
. . .
Each document is now represented as a real-valued vector of tf-idf weights
∈ R|V |.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Documents as vectors
Each document is now represented as a real-valued vector of
tf-idf weights ∈ 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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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: We’re doing this because we want to get away from
the you’re-either-in-or-out, feast-or-famine Boolean model.
Instead: rank relevant documents higher than nonrelevant
documents.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Why distance is a bad idea
0 1
0
1
rich
poor
q:[rich poor]
d1:Ranks of starving poets swell
d2:Rich poor gap grows
d3:Record baseball salaries in 2010
The Euclidean distance of q and d2 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?
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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◦]
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Cosine
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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:
||x||2 = i x2
i
This maps vectors onto the unit sphere . . .
. . . since after normalization: ||x||2 = i x2
i = 1.0
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
length-normalization.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Cosine similarity between query and document
cos(q, d) = sim(q, d) =
q · d
|q||d|
=
|V |
i=1 qi di
|V |
i=1 q2
i
|V |
i=1 d2
i
qi is the tf-idf weight of term i in the query.
di is the tf-idf weight of term i in the document.
|q| and |d| are the lengths of q and d.
This is the cosine similarity of q and d . . . . . . or, equivalently,
the cosine of the angle between q and d.
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Cosine for normalized vectors
For normalized vectors, the cosine is equivalent to the dot
product or scalar product.
cos(q, d) = q · d = i qi · di
(if q and d are length-normalized).
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Cosine similarity illustrated
0 1
0
1
rich
poor
v(q)
v(d1)
v(d2)
v(d3)
θ
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Cosine: Example
How similar are
these novels?
SaS: Sense and
Sensibility
PaP: Pride and
Prejudice
WH: Wuthering
Heights
term frequencies (counts)
term SaS PaP WH
affection 115 58 20
jealous 10 7 11
gossip 2 0 6
wuthering 0 0 38
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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 SaS PaP WH
affection 3.06 2.76 2.30
jealous 2.0 1.85 2.04
gossip 1.30 0 1.78
wuthering 0 0 2.58
(To simplify this example, we don’t do idf weighting.)
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Cosine: Example
log frequency weighting
term SaS PaP WH
affection 3.06 2.76 2.30
jealous 2.0 1.85 2.04
gossip 1.30 0 1.78
wuthering 0 0 2.58
log frequency weighting
& cosine normalization
term SaS PaP WH
affection 0.789 0.832 0.524
jealous 0.515 0.555 0.465
gossip 0.335 0.0 0.405
wuthering 0.0 0.0 0.588
cos(SaS,PaP) ≈
0.789 ∗ 0.832 + 0.515 ∗ 0.555 + 0.335 ∗ 0.0 + 0.0 ∗ 0.0 ≈ 0.94.
cos(SaS,WH) ≈ 0.79
cos(PaP,WH) ≈ 0.69
Why do we have cos(SaS,PaP) > cos(SAS,WH)?
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Computing the cosine score
CosineScore(q)
1 float Scores[N] = 0
2 float Length[N]
3 for each query term t
4 do calculate wt,q and fetch postings list for t
5 for each pair(d, tft,d ) in postings list
6 do Scores[d]+ = wt,d × wt,q
7 Read the array Length
8 for each d
9 do Scores[d] = Scores[d]/Length[d]
10 return Top K components of Scores[]
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Components of tf-idf weighting
Term frequency Document frequency Normalization
n (natural) tft,d n (no) 1 n (none)
1
l (logarithm) 1 + log(tft,d) t (idf) log N
dft
c (cosine) 1√
w2
1 +w2
2 +...+w2
M
a (augmented) 0.5 +
0.5×tft,d
maxt (tft,d )
p (prob idf) max{0, log N−dft
dft
} u (pivoted
unique)
1/u
b (boolean)
1 if tft,d > 0
0 otherwise
b (byte size) 1/CharLengthα
,
α < 1
L (log ave)
1+log(tft,d )
1+log(avet∈d (tft,d ))
Best known combination of weighting options
Default: no weighting
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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
Isn’t it bad to not idf-weight the document?
Example query: “best car insurance”
Example document: “car insurance auto insurance”
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
tf-idf example: lnc.ltn
Query: “best car insurance”. Document: “car insurance auto insurance”.
word query document product
tf-raw tf-wght df idf weight tf-raw tf-wght weight n’lized
auto 0 0 5000 2.3 0 1 1 1 0.52 0
best 1 1 50000 1.3 1.3 0 0 0 0 0
car 1 1 10000 2.0 2.0 1 1 1 0.52 1.04
insurance 1 1 1000 3.0 3.0 2 1.3 1.3 0.68 2.04
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, n’lized: document weights after cosine
normalization, product: the product of final query weight and final document weight
√
12 + 02 + 12 + 1.32 ≈ 1.92
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?
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
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)
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Why ranked retrieval? Term frequency tf-idf weighting The vector space model
Resources
Chapter 6 of IIR
Resources at https://www.fi.muni.cz/~sojka/PV211/
and http://cislmu.org, materials in MU IS and FI MU
library
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)
Sojka, IIR Group: PV211: Scoring, term weighting, the vector space model 56 / 56