Centroid clustering cambridge introductiontoinformationretrieval2008 readingversion
17.4 Centroid clustering
391 1 2 3 4 5 6 7 1 2 3 4 5 × d 1 × d 2 × d 3 × d 4 × d 5 × d 6 b c µ 1 b c µ 3 b c µ 2 ◮ Figure 17.11 Three iterations of centroid clustering. Each iteration merges the two clusters whose centroids are closest.17.4 Centroid clustering
In centroid clustering, the similarity of two clusters is defined as the similar- ity of their centroids: SIM - CENT ω i , ω j = ~ µ ω i · ~ µ ω j 17.5 = 1 N i ∑ d m ∈ω i ~ d m · 1 N j ∑ d n ∈ω j ~ d n = 1 N i N j ∑ d m ∈ω i ∑ d n ∈ω j ~ d m · ~ d n 17.6 Equation 17.5 is centroid similarity. Equation 17.6 shows that centroid similarity is equivalent to average similarity of all pairs of documents from different clusters. Thus, the difference between GAAC and centroid clustering is that GAAC considers all pairs of documents in computing average pair- wise similarity Figure 17.3 , d whereas centroid clustering excludes pairs from the same cluster Figure 17.3 , c. Figure 17.11 shows the first three steps of a centroid clustering. The first two iterations form the clusters { d 5 , d 6 } with centroid µ 1 and { d 1 , d 2 } with centroid µ 2 because the pairs h d 5 , d 6 i and h d 1 , d 2 i have the highest centroid similarities. In the third iteration, the highest centroid similarity is between µ 1 and d 4 producing the cluster { d 4 , d 5 , d 6 } with centroid µ 3 . Like GAAC, centroid clustering is not best-merge persistent and therefore Θ N 2 log N Exercise 17.6 . In contrast to the other three HAC algorithms, centroid clustering is not monotonic. So-called inversions can occur: Similarity can increase during INVERSION Preliminary draft c 2008 Cambridge UP 392 17 Hierarchical clustering 0 1 2 3 4 5 1 2 3 4 5 × × × b c d 1 d 2 d 3 − 4 − 3 − 2 − 1 d 1 d 2 d 3 ◮ Figure 17.12 Centroid clustering is not monotonic. The documents d 1 at 1 + ǫ , 1 , d 2 at 5, 1 , and d 3 at 3, 1 + 2 √ 3 are almost equidistant, with d 1 and d 2 closer to each other than to d 3 . The non-monotonic inversion in the hierarchical clustering of the three points appears as an intersecting merge line in the dendrogram. The intersection is circled. clustering as in the example in Figure 17.12 , where we define similarity as negative distance. In the first merge, the similarity of d 1 and d 2 is − 4 − ǫ . In the second merge, the similarity of the centroid of d 1 and d 2 the circle and d 3 is ≈ − cos π 6 × 4 = − √ 32 × 4 ≈ − 3.46 − 4 − ǫ . This is an example of an inversion: similarity increases in this sequence of two clustering steps. In a monotonic HAC algorithm, similarity is monotonically decreasing from iteration to iteration. Increasing similarity in a series of HAC clustering steps contradicts the fundamental assumption that small clusters are more coherent than large clusters. An inversion in a dendrogram shows up as a horizontal merge line that is lower than the previous merge line. All merge lines in Figures 17.1 and 17.5 are higher than their predecessors because single-link and complete- link clustering are monotonic clustering algorithms. Despite its non-monotonicity, centroid clustering is often used because its similarity measure – the similarity of two centroids – is conceptually simpler than the average of all pairwise similarities in GAAC. Figure 17.11 is all one needs to understand centroid clustering. There is no equally simple graph that would explain how GAAC works. ? Exercise 17.3 For a fixed set of N documents there are up to N 2 distinct similarities between clusters in single-link and complete-link clustering. How many distinct cluster similarities are there in GAAC and centroid clustering? Preliminary draft c 2008 Cambridge UP17.5 Optimality of HAC
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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