CS276AText Information Retrieval, Mining, and

Exploitation

Lecture 415 Oct 2002

Recap of last time

Index size Index construction techniques Dynamic indices Real world considerations

Back of the envelope index size calculation

Number of docs = n = 40M Number of terms = m = 1M Use Zipf to estimate number of postings

entries: n + n/2 + n/3 + …. + n/m ~ n ln m = 560M

postings entries This is just a word-document index, not

one that includes positional information

Merge sort of 56 sorted runs

Merge tree of log256 ~ 6 layers. During each layer, read into memory runs

in blocks of 10M, merge, write back.

Disk

1

3 4

22

1

4

3

Merge sort of 56 sorted runs

How do you write back long merged runs? Wait to accumulate 10M-sized output

blocks before writing back. Thus amortize seek time over block

transfer.

Disk

1

3 4

2

2

1

4

3

Today’s topics

Ranking models The vector space model

Inverted indexes with term weighting Evaluation with ranking models

Ranking models in IR

Key idea: We wish to return in order the documents

most likely to be useful to the searcher To do this, we want to know which

documents best satisfy a query An obvious idea is that if a document talks

about a topic more then it is a better match A query should then just specify terms that

are relevant to the information need, without requiring that all of them must be present Document relevant if it has a lot of the terms

Binary term presence matrices

Record whether a document contains a word: document is binary vector in {0,1}v

What we have mainly assumed so far Idea: Query satisfaction = overlap

measure:

Antony and Cleopatra Julius Caesar The Tempest Hamlet Othello Macbeth

Antony 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

YX

Overlap matching

What are the problems with the overlap measure?

It doesn’t consider: Term frequency in document Term scarcity in collection (document

mention frequency) Length of documents

(And queries: score not normalized)

Overlap matching

One can normalize in various ways: Jaccard coefficient:

Cosine measure:

What documents would score best using Jaccard against a typical query? Does the cosine measure fix this problem?

YXYX /

YXYX /

Count term-document matrices

We haven’t considered frequency of a word

Count of a word in a document: Bag of words model Document is a vector in ℕv

Antony and Cleopatra Julius Caesar The Tempest Hamlet Othello Macbeth

Antony 157 73 0 0 0 0

Brutus 4 157 0 1 0 0

Caesar 232 227 0 2 1 1

Calpurnia 0 10 0 0 0 0

Cleopatra 57 0 0 0 0 0

mercy 2 0 3 5 5 1

worser 2 0 1 1 1 0

Normalization: Calpurnia

vs. Calphurnia

Weighting term frequency: tf

What is the relative importance of 0 vs. 1 occurrence of a term in a doc 1 vs. 2 occurrences 2 vs. 3 occurrences …

Unclear: but it seems that more is better, but a lot isn’t necessarily better than a few Can just use raw score Another option commonly used in practice:

0:log1?0 ,, dtdt tftf

Dot product matching

Match is dot product of query and document

[Note: 0 if orthogonal (no words in common)] Rank by match

It still doesn’t consider: Term scarcity in collection (document mention

frequency) Length of documents and queries

Not normalized

i diqi tftfdq ,,

Weighting should depend on the term overall

Which of these tells you more about a doc? 10 occurrences of hernia? 10 occurrences of the?

Suggest looking at collection frequency (cf) But document frequency (df) may be

better:Word cf dftry 10422 8760insurance 10440 3997

Document frequency weighting is only possible in known (static) collection.

tf x idf term weights

tf x idf measure combines: term frequency (tf)

measure of term density in a doc inverse document frequency (idf)

measure of informativeness of term: its rarity across the whole corpus

could just be raw count of number of documents the term occurs in (idfi = 1/dfi)

but by far the most commonly used version is:

See Kishore Papineni, NAACL 2, 2002 for theoretical justification

dfnidf

i

i log

Summary: tf x idf (or tf.idf)

Assign a tf.idf weight to each term i in each document d

Increases with the number of occurrences within a doc Increases with the rarity of the term across the whole corpus

)/log(,, ididi dfntfw

rmcontain te that documents ofnumber thedocuments ofnumber total

document in termoffrequency ,

idfn

jitf

i

di

What is the wtof a term thatoccurs in allof the docs?

Real-valued term-document matrices

Function (scaling) of count of a word in a document: Bag of words model Each is a vector in ℝv

Here log scaled tf.idf

Antony and Cleopatra Julius Caesar The Tempest Hamlet Othello Macbeth

Antony 13.1 11.4 0.0 0.0 0.0 0.0

Brutus 3.0 8.3 0.0 1.0 0.0 0.0

Caesar 2.3 2.3 0.0 0.5 0.3 0.3

Calpurnia 0.0 11.2 0.0 0.0 0.0 0.0

Cleopatra 17.7 0.0 0.0 0.0 0.0 0.0

mercy 0.5 0.0 0.7 0.9 0.9 0.3

worser 1.2 0.0 0.6 0.6 0.6 0.0

Documents as vectors

Each doc j can now be viewed as a vector of tfidf values, one component for each term

So we have a vector space terms are axes docs live in this space even with stemming, may have 20,000+

dimensions (The corpus of documents gives us a matrix,

which we could also view as a vector space in which words live – transposable data)

Why turn docs into vectors?

First application: Query-by-example Given a doc d, find others “like” it.

Now that d is a vector, find vectors (docs) “near” it.

Intuition

Postulate: Documents that are “close together” in vector space talk about the same things.

t1

d2

d1

d3

d4

d5

t3

t2

θ

φ

The vector space model

Query as vector: We regard query as short document We return the documents ranked by the

closeness of their vectors to the query, also represented as a vector.

Developed in the SMART system (Salton, c. 1970) and standardly used by TREC participants and web IR systems

Desiderata for proximity

If d1 is near d2, then d2 is near d1. If d1 near d2, and d2 near d3, then d1 is not

far from d3. No doc is closer to d than d itself.

First cut

Distance between vectors d1 and d2 is the length of the vector |d1 – d2|. Euclidean distance

Why is this not a great idea? We still haven’t dealt with the issue of

length normalization Long documents would be more similar to

each other by virtue of length, not topic However, we can implicitly normalize by

looking at angles instead

Cosine similarity

Distance between vectors d1 and d2 captured by the cosine of the angle x between them.

Note – this is similarity, not distance

t 1

d2

d1

t 3

t 2

θ

Cosine similarity

Cosine of angle between two vectors The denominator involves the lengths of

the vectors So the cosine measure is also known as the

normalized inner product

n

i ki

n

i ji

n

i kiji

kj

kjkj

ww

ww

dd

ddddsim

1

2,1

2,

1 ,,),(

n

i jij wd1

2,Length

Cosine similarity exercises

Exercise: Rank the following by decreasing cosine similarity: Two docs that have only frequent words

(the, a, an, of) in common. Two docs that have no words in common. Two docs that have many rare words in

common (wingspan, tailfin).

Normalized vectors

A vector can be normalized (given a length of 1) by dividing each of its components by the vector's length

This maps vectors onto the unit circle:

Then, Longer documents don’t get more weight For normalized vectors, the cosine is

simply the dot product:

11 ,

n

i jij wd

kjkj dddd

),cos(

Exercise

Euclidean distance between vectors: Euclidean distance:

Show that, for normalized vectors, Euclidean distance gives the same closeness ordering as the cosine measure

n

i kijikj wwdd1

2,,

Example

Docs: Austen's Sense and Sensibility, Pride and Prejudice; Bronte's Wuthering Heights

cos(SAS, PAP) = .996 x .993 + .087 x .120 + .017 x 0.0 = 0.999 cos(SAS, WH) = .996 x .847 + .087 x .466 + .017 x .254 =

0.929

SaS PaP WHaffection 115 58 20jealous 10 7 11gossip 2 0 6

SaS PaP WHaffection 0.996 0.993 0.847jealous 0.087 0.120 0.466gossip 0.017 0.000 0.254

Digression: spamming indices

This was all invented before the days when people were in the business of spamming web search engines: Indexing a sensible passive document

collection vs. An active document collection, where

people (and indeed, service companies) are trying to shape documents in an attempt to achieve ranking function maximization

Digression: ranking in Machine Learning

Our problem is: Given document collection D and query q,

return a ranking of D according to relevance to q.

Such ranking problems have been much less studied in machine learning than classification/regression problems

But much more interest recently, e.g., W.W. Cohen, R.E. Schapire, and Y. Singer.

Learning to order things. Journal of Artificial Intelligence Research, 10:243–270, 1999.

And subsequent research

Digression: ranking in Machine Learning

Many “WWW” applications are ranking (aka ordinal regression) problems: Text information retrieval Image similarity search (QBIC) Book/movie recommendations

Collaborative filtering Meta-search engines

Summary: What’s the real point of using vector spaces?

Key: A user’s query can be viewed as a (very) short document.

Query becomes a vector in the same space as the docs.

Can measure each doc’s proximity to it. Natural measure of scores/ranking – no

longer Boolean.

Evaluation II

Evaluation of ranked results: You can return any number of results

ordered by similarity By taking various numbers of documents

(levels of recall), you can produce a precision-recall curve

Precision-recall curves

Interpolated precision

If you can increase precision by increasing recall, then you should get to count that…

Evaluation

There are various other measures Precision at fixed recall

This is perhaps the most appropriate thing for web search: all people want to know is how many good matches there are in the first one or two pages of results

11-point interpolated average precision The standard measure in the TREC competitions:

you take the precision at 11 levels of recall varying from 0 to 1 by tenths of the documents, using interpolation (the value for 0 is always interpolated!), and average them

We’ll use more notions from linear algebra next lecture

Matrix, vector Transpose and product Rank Eigenvalues and eigenvectors.

Resources, and beyond

MG 4.4–4.5, MIR 2.5. Next steps

Computing cosine similarity efficiently. Dimensionality reduction. Probabilistic approaches to IR