Heaps’ law: Estimating the number of terms Zipf’s law: Modeling the distribution of terms
5.1.1 Heaps’ law: Estimating the number of terms
A better way of getting a handle on M is Heaps’ law, which estimates vocab- H EAPS ’ LAW ulary size as a function of collection size: M = kT b 5.1 where T is the number of tokens in the collection. Typical values for the parameters k and b are: 30 ≤ k ≤ 100 and b ≈ 0.5. The motivation for Heaps’ law is that the simplest possible relationship between collection size and vocabulary size is linear in log-log space and the assumption of linearity is usually borne out in practice as shown in Figure 5.1 for Reuters-RCV1. In this case, the fit is excellent for T 10 5 = 100,000, for the parameter values b = 0.49 and k = 44. For example, for the first 1,000,020 tokens Heaps’ law predicts 38,323 terms: 44 × 1,000,020 0.49 ≈ 38,323 Preliminary draft c 2008 Cambridge UP5.1 Statistical properties of terms in information retrieval
89 The actual number is 38,365 terms, very close to the prediction. The parameter k is quite variable because vocabulary growth depends a lot on the nature of the collection and how it is processed. Case-folding and stemming reduce the growth rate of the vocabulary, whereas including numbers and spelling errors will increase it. Regardless of the values of the parameters for a particular collection, Heaps’ law suggests that: i the dic- tionary size will continue to increase with more documents in the collection, rather than a maximum vocabulary size being reached, and ii the size of the dictionary will be quite large for large collections. These two hypotheses have been empirically shown to be true of large text collections Section 5.4 . So dictionary compression is important for an effective information retrieval system.5.1.2 Zipf’s law: Modeling the distribution of terms
We also want to understand how terms are distributed across documents. This will help us characterize the properties of the algorithms for compress- ing postings lists in Section 5.3 . A commonly used model of the distribution of terms in a collection is Zipf’s Z IPF ’ S LAW law . It states that, if t 1 is the most common term in the collection, t 2 the next most common etc, then the collection frequency cf i of the ith most common term is proportional to 1i: cf i ∝ 1 i 5.2 So if the most frequent term occurs cf 1 times, then the second most frequent term has half as many occurrences, the third most frequent term a third as many occurrences, etc. The intuition is that frequency decreases very rapidly with rank. Equation 5.2 is one of the simplest ways of formalizing such a rapid decrease and it has been found to be a reasonably good model. Equivalently, we can write Zipf’s law as cf i = ci k or as log cf i = log c + k log i where k = − 1 and c is a constant to be defined in Section 5.3.2 . It is therefore a power law with exponent k = − 1. See Chapter 19 , page 426 , POWER LAW for another power law, a law characterizing the distribution of links on web pages. The log-log graph in Figure 5.2 plots the collection frequency of a term as a function of its rank for Reuters-RCV1. A line with slope -1, corresponding to the Zipf function log cf i = log c − log i, is also shown. The fit of the data to the law is not particularly good, but good enough to serve as a model for term distributions in our calculations in Section 5.3 . Preliminary draft c 2008 Cambridge UP 90 5 Index compression 1 2 3 4 5 6 7 1 2 3 4 5 6 7 log10 rank log10 cf ◮ Figure 5.2 Zipf’s law for Reuters-RCV1. Frequency is plotted as a function of frequency rank for the terms in the collection. The line is the distribution predicted by Zipf’s law weighted least squares fit, intercept is 6.95. ? Exercise 5.1 [ ⋆ ] Assuming one machine word per posting, what is the size of the uncompressed non- positional index for different tokenizations based on Table 5.1 ? How do these num- bers compare with Table 5.6 ?5.2 Dictionary compression
Parts
» cambridge introductiontoinformationretrieval2008 readingversion
» An example information retrieval problem
» A first take at building an inverted index
» Processing Boolean queries cambridge introductiontoinformationretrieval2008 readingversion
» The extended Boolean model versus ranked retrieval
» References and further reading
» Obtaining the character sequence in a document
» Tokenization Determining the vocabulary of terms
» Dropping common terms: stop words
» Normalization equivalence classing of terms
» Stemming and lemmatization Determining the vocabulary of terms
» Faster postings list intersection via skip pointers
» Biword indexes Positional postings and phrase queries
» Positional indexes Positional postings and phrase queries
» Combination schemes Positional postings and phrase queries
» Search structures for dictionaries
» General wildcard queries Wildcard queries
» k-gram indexes for wildcard queries
» Implementing spelling correction Spelling correction
» Forms of spelling correction Edit distance
» k-gram indexes for spelling correction
» Context sensitive spelling correction
» Phonetic correction cambridge introductiontoinformationretrieval2008 readingversion
» Hardware basics Blocked sort-based indexing
» Single-pass in-memory indexing cambridge introductiontoinformationretrieval2008 readingversion
» Distributed indexing cambridge introductiontoinformationretrieval2008 readingversion
» Dynamic indexing cambridge introductiontoinformationretrieval2008 readingversion
» References and further reading Exercises
» Heaps’ law: Estimating the number of terms Zipf’s law: Modeling the distribution of terms
» Blocked storage Dictionary compression
» Variable byte codes γ Postings file compression
» Exercises cambridge introductiontoinformationretrieval2008 readingversion
» Weighted zone scoring Parametric and zone indexes
» Learning weights Parametric and zone indexes
» The optimal weight Parametric and zone indexes
» Inverse document frequency Term frequency and weighting
» Tf-idf weighting Term frequency and weighting
» Dot products The vector space model for scoring
» Queries as vectors Computing vector scores
» Sublinear tf scaling Maximum tf normalization
» Document and query weighting schemes Pivoted normalized document length
» Inexact top Efficient scoring and ranking
» Index elimination Champion lists
» Static quality scores and ordering
» Impact ordering Efficient scoring and ranking
» Cluster pruning Efficient scoring and ranking
» Tiered indexes Query-term proximity
» Designing parsing and scoring functions
» Vector space scoring and query operator interaction
» Information retrieval system evaluation
» Standard test collections cambridge introductiontoinformationretrieval2008 readingversion
» Evaluation of unranked retrieval sets
» Evaluation of ranked retrieval results
» Critiques and justifications of the concept of relevance
» Results snippets cambridge introductiontoinformationretrieval2008 readingversion
» The Rocchio algorithm for relevance feedback
» Probabilistic relevance feedback When does relevance feedback work?
» Relevance feedback on the web
» Evaluation of relevance feedback strategies
» Pseudo relevance feedback Indirect relevance feedback
» Vocabulary tools for query reformulation Query expansion
» Basic XML concepts cambridge introductiontoinformationretrieval2008 readingversion
» A vector space model for XML retrieval
» Text-centric vs. data-centric XML retrieval
» Review of basic probability theory
» The 10 loss case The Probability Ranking Principle
» Deriving a ranking function for query terms
» Probability estimates in theory
» Probability estimates in practice
» Probabilistic approaches to relevance feedback
» An appraisal of probabilistic models
» Tree-structured dependencies between terms Okapi BM25: a non-binary model
» Finite automata and language models
» Multinomial distributions over words
» Using query likelihood language models in IR
» Estimating the query generation probability
» Ponte and Croft’s Experiments
» Language modeling versus other approaches in IR
» Extended language modeling approaches
» The text classification problem
» The Bernoulli model cambridge introductiontoinformationretrieval2008 readingversion
» Mutual information Feature selection
» Frequency-based feature selection Feature selection for multiple classifiers
» Evaluation of text classification
» Document representations and measures of relatedness in vec-
» Rocchio classification cambridge introductiontoinformationretrieval2008 readingversion
» Time complexity and optimality of kNN
» Linear versus nonlinear classifiers
» Classification with more than two classes
» The bias-variance tradeoff cambridge introductiontoinformationretrieval2008 readingversion
» Support vector machines: The linearly separable case
» Soft margin classification Extensions to the SVM model
» Multiclass SVMs Nonlinear SVMs
» Experimental results Extensions to the SVM model
» Choosing what kind of classifier to use
» Improving classifier performance Issues in the classification of text documents
» A simple example of machine-learned scoring
» Result ranking by machine learning
» Clustering in information retrieval
» Evaluation of clustering cambridge introductiontoinformationretrieval2008 readingversion
» Cluster cardinality in K-means
» Model-based clustering cambridge introductiontoinformationretrieval2008 readingversion
» Centroid clustering cambridge introductiontoinformationretrieval2008 readingversion
» Optimality of HAC cambridge introductiontoinformationretrieval2008 readingversion
» Divisive clustering cambridge introductiontoinformationretrieval2008 readingversion
» Cluster labeling cambridge introductiontoinformationretrieval2008 readingversion
» Implementation notes cambridge introductiontoinformationretrieval2008 readingversion
» Term-document matrices and singular value decompositions
» Low-rank approximations cambridge introductiontoinformationretrieval2008 readingversion
» Latent semantic indexing cambridge introductiontoinformationretrieval2008 readingversion
» Background and history cambridge introductiontoinformationretrieval2008 readingversion
» The web graph Web characteristics
» Advertising as the economic model
» Near-duplicates and shingling cambridge introductiontoinformationretrieval2008 readingversion
» Features a crawler Features a crawler
» Distributing indexes cambridge introductiontoinformationretrieval2008 readingversion
» Connectivity servers cambridge introductiontoinformationretrieval2008 readingversion
» Anchor text and the web graph
» The PageRank computation PageRank
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