CS276A Text Information Retrieval, Mining, and Exploitation Lecture 4 15 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 log 2 56 ~ 6 layers. During each layer, read into memory runs in blocks of 10M, merge, write back. Disk

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

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:

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?

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 ℕ v 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:

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

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: Wordcfdf try insurance 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 (idf i = 1/df i ) but by far the most commonly used version is: See Kishore Papineni, NAACL 2, 2002 for theoretical justification

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 What is the wt of a term that occurs in all of 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 ℝ v Here log scaled tf.idf

Documents as vectors Each doc j can now be viewed as a vector of tf  idf 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. t1t1 d2d2 d1d1 d3d3 d4d4 d5d5 t3t3 t2t2 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 d 1 is near d 2, then d 2 is near d 1. If d 1 near d 2, and d 2 near d 3, then d 1 is not far from d 3. No doc is closer to d than d itself.

First cut Distance between vectors d 1 and d 2 is the length of the vector |d 1 – d 2 |. 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 d 1 and d 2 captured by the cosine of the angle x between them. Note – this is similarity, not distance t 1 d2d2 d1d1 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

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:

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

Example Docs: Austen's Sense and Sensibility, Pride and Prejudice; Bronte's Wuthering Heights cos(SAS, PAP) =.996 x x x 0.0 = cos(SAS, WH) =.996 x x x.254 = 0.929

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, 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