A. ROBINSON, and P. WILLETT. WILLETT, P. 1983. next = pi[j]; DUBES, R., and A. K. JAIN. ndistance[nj] =ndistance[nt]; 1982. ndistance[j] = D; Amsterdam: North-Holland (Handbook of Statistics, Vol. } BLASHFIELD, R. K., and M. S. ALDENDERFER. The SMART Retrieval System. 1975. However, two recent studies are noteworthy for their use of relatively large document collections for the evaluation of a variety of HACM (El-Hamdouchi and Willett 1989; Voorhees 1986a). /* clustering level */ "An Algorithm for Information Structuring and Retrieval." where mi is the number of objects in Di and d2ij is the squared Euclidean distance, given by: Advances in Information Systems Science, 8, 169-292. WILLETT, P. 1980. P. R. Krishnaiah and J. N. Kanal, pp. BENTLEY, J. L., B. W. WEIDE, and A. C. YAO. Scientometrics, 7, 391-409. The SMART Retrieval System. The answer to the first question can be found in evaluative studies of clustering methods, and to the second question, in validation techniques for clustering solutions. The reallocation methods operate by selecting some initial partition of the data set and then moving items from cluster to cluster to obtain an improved partition. Stop when only one point remains. next = pi [j]; /*consider the lastpoint in the tree for the NN list */ 1. It is important to consider the questions raised in section 16.1.2 regarding the potential application, selection of methods and algorithm and the parameters associated with it, and evaluation and validation of the results. ISBN: MILLIGAN, G. W., and M. C. COOPER. Advances in Information Systems Science, 8, 169-292. San Francisco: W. H. Freeman. The overlap test is applied to a set of documents for which query-relevance judgments are available. However, improvements in processing and storage capacity and the introduction of efficient algorithms for implementing some clustering methods and finding nearest neighbors have made it feasible to cluster increasingly large data sets. { Clustering Algorithms. /*iteratively add the remaining N-1 points to the hierarchy */ Voorhees found complete link most effective for larger collections, with complete and group average link comparable for smaller collections; single link hierarchies provided the worst retrieval performance. "Parallel Computations in Information Retrieval." Voorhees found complete link most effective for larger collections, with complete and group average link comparable for smaller collections; single link hierarchies provided the worst retrieval performance. In the stored matrix approach, an N N matrix containing all pairwise dissimilarity values is stored, and the Lance-Williams update formula makes it possible to recalculate the dissimilarity between cluster centers using only the stored values. The more complex hierarchical methods produce a nested data set in which pairs of items or clusters are successively linked until every item in the data set is connected. "Algorithm 422: Minimal Spanning Tree." Computer Journal, 26, 354-59. 267-84. A FORTRAN subroutine to effect the transformation is provided in Sibson's original paper. A. All the relevant-relevant (RR) and relevant-nonrelevant (RNR) interdocument similarities are calculated for a given query, and the overlap (the fraction of the RR and RNR distributions that is common to both) is calculated. New York: Halsted. 1989. For some methods, algorithms have been developed that are the optimal O (N2) in time and O (N) in space (Murtagh 1984). Communications of the ACM, 15, 273-74. Any isolated fragment (subset of an MST) can be connected to a nearest neighbor by a shortest available link. Since the large number of possible divisions of N items into M clusters make an optimal solution impossible, the nonhierarchical methods attempt to find an approximation, usually by partitioning the data set in some way and then reallocating items until some criterion is optimized. J. group average link: As the name implies, the group average link method uses the average values of the pairwise links within a cluster to determine similarity. ALDENDERFER, M. S., and R. K. BLASHFIELD. EL-HAMDOUCHI, A., and P. WILLETT. The goal of a hierarchical clustering process may be to partition the data set into some unknown number of clusters M (which may be visualized as drawing a horizontal line across the dendrogram at some clustering level). Dubes and Jain (1980) survey the approaches that have been used, categorizing them on their ability to answer four questions: is the data matrix random? Output may be based on retrieval of a single cluster, or the top-ranking clusters may be retrieved to produce a predetermined number of either documents or clusters; in the latter case, the documents retrieved may themselves be ranked against the query. Dubes and Jain (1980) survey the approaches that have been used, categorizing them on their ability to answer four questions: is the data matrix random? 2). 16.6.2 Validation Can and Ozkarahan (1989) review the approaches that have been taken for cluster maintenance and propose a strategy for dynamic cluster maintenance based on their cover coefficient concept. Algorithms for Clustering Data. CLINK is efficient, requiring O (N2) time, O (N) space, but it does not seem to generate an exact hierarchy and has given unsatisfactory results in some information retrieval experiments (El-Hamdouchi and Willett 1989). Englewood Cliffs, N.J.: Prentice Hall. GORDON, A. D. 1987. SIBSON, R. 1973. ROMESBURG, H. C. 1984. The single link method merges at each stage the closest previously unlinked pair of points in the data set. In certain cases, update of the cluster structure is implicit in the clustering algorithm. With improved processing capability and more efficient hierarchical algorithms, the HACMs are now usually preferred in practice, and the nonhierarchical methods will not be considered further in this chapter. Most of the early work used approximate clustering methods, or the least demanding of the HACM, the single link method, a restriction imposed by limited processing resources. Chichester: Ellis Horwood. In this case, the similarity between a cluster centroid and any document is equal to the mean similarity between the document and all the documents in the cluster. J. GRIFFITHS, A., L . The data set need not be stored and the similarity matrix is processed serially, which minimizes disk accesses. : Addison-Wesley. CROFT, W. B. 1988. Ph.D. thesis, Cornell University. /* grow the tree an object at a time */ When two points Di and Dj are clustered, the increase in variance Iij is given by: Document clustering has been studied because of its potential for improving the efficiency of retrieval, for improving the effectiveness of retrieval, and because it provides an alternative to Boolean or best match retrieval. All of the hierarchical agglomerative clustering methods can be described by a general algorithm: { JAIN, A. K., and R. C. DUBES. Willett (1988) has reviewed the application of validation methods to clustering of document collections, primarily the application of the random graph hypothesis and the use of distortion measures. The thoughtful user of a clustering method or algorithm must answer two questions: (i) Which clustering method is appropriate for a particular data set? KAUFMAN, L. 1990. Scientometrics, 7, 391-409. for (j = 0; j < i-1; j++) SALTON, G., ed. LANCE, G. N., and W. T. WILLIAMS. 1963. The large number of zero-valued similarities in a typical document collection make it more efficient than its worst case O (N3) time, O (N2) storage requirement would suggest; however it is still very demanding of resources, and El-Hamdouchi and Willett found it impractical to apply it to the largest of the collections they studied. 1963. 1987. Computer Journal, 20, 364-66. EVERITT, B. A FORTRAN version of the Prim-Dijkstra algorithm is provided by Whitney (1972). Where the application uses a partitioned data set (from a nonhierarchical or hierarchical method), new items may simply be added to the most similar partition until the cluster structure becomes distorted and it is necessary to regenerate it. Multivariate Behavioral Research, 13, 271-95. The density test is defined as the total number of postings in the document collection divided by the product of the number of documents and the number of terms that have been used for the indexing of those documents. Cosine coefficient: "The Problem of the Shortest Subspanning Tree." All the relevant-relevant (RR) and relevant-nonrelevant (RNR) interdocument similarities are calculated for a given query, and the overlap (the fraction of the RR and RNR distributions that is common to both) is calculated. A number of algorithms for the single link method have been reviewed by Rohlf (1982), including related minimal spanning tree algorithms. It can be shown that all the information required to generate a single link hierarchy for a set of points is contained in their MST (Gower and Ross 1969). The large number of zero-valued similarities in a typical document collection make it more efficient than its worst case O (N3) time, O (N2) storage requirement would suggest; however it is still very demanding of resources, and El-Hamdouchi and Willett found it impractical to apply it to the largest of the collections they studied. CROUCH, D. B. 1983. As Dubes and Jain (1980, p. 179) point out: The data set need not be stored and the similarity matrix is processed serially, which minimizes disk accesses. npoint[j] =lastpoint; CAN, F., and E. A. OZKARAHAN. This requires a sorted list of document-document similarities, and a means of counting the number of similarities seen between any two active clusters. San Francisco: W. H. Freeman. 16.5.1 General Algorithm for the HACM The cluster center for a pair of points Di and Dj is given by: 1980. BENTLEY, J. L., B. W. WEIDE, and A. C. YAO. 1973. Multidimensional Clustering Algorithms. J. Information Retrieval. Ph.D. thesis, University of Sherfield. The most commonly used hierarchical agglomerative clustering methods and their characteristics are: "Document Clustering Using an Inverted File Approach." Computer Journal, 26, 354-59. D = calculate_distance(lastpoint, notintree[j]); VAN RIJSBERGEN. nt = n-1; Since it is difficult to adequately represent the clusters in the very large top-level clusters, a useful modification is to eliminate the top-level clusters by applying a threshold clustering level to the hierarchy to obtain a partition, and using the best of these mid-level clusters as the starting point for the top-down search. KAUFMAN, L. 1990. This is designated the Single Cluster Algorithm since it carries out one agglomeration per iteration; a Multiple Cluster Algorithm, suitable for parallel processing, has also been proposed (Murtagh 1985). This is designated the Single Cluster Algorithm since it carries out one agglomeration per iteration; a Multiple Cluster Algorithm, suitable for parallel processing, has also been proposed (Murtagh 1985). SNEATH, P. H. A., and R. R. Sokal. A single cluster is retrieved when the search is terminated. The hierarchical clustering methods previously discussed are presented in Table 16.1 in the context of their Lance-Williams parameters and cluster centers. 16.4.2 Reallocation Methods 1 . 1. HARTIGAN, J. London: Butterworths. The best-known algorithm for implementing the complete link method is the CLINK algorithm developed by Defays (1977). J. Comparative studies suggest that the bottom-up search gives the best results (apart from the complete link method), particularly when the search is limited to the bottom-level clusters (Willett 1988). The thoughtful user of a clustering method or algorithm must answer two questions: (i) Which clustering method is appropriate for a particular data set? The thoughtful user of a clustering method or algorithm must answer two questions: (i) Which clustering method is appropriate for a particular data set? Finding Groups in Data: An Introduction to Cluster Analysis. New York: Wiley. Voorhees algorithm Because of the heavy demands of processing and storage requirements, much of the early work on cluster analysis for information retrieval was limited to small data sets, often only a few hundred items. Subsequent study has concentrated on the effectiveness of retrieval from hierarchically clustered document collections, based on the cluster hypothesis, which states that associations between documents convey information about the relevance of documents to requests (van Rijsbergen 1979). /* add this point to the MST; store this point and their */ Because the similarity between two clusters is determined by the average value of all the pairwise links between points for which each is in one of the two clusters, no general O (N2) time, O (N) space algorithm is known. Initially the emphasis was on efficiency: document collections were partitioned, using nonhierarchical methods, and queries were matched against cluster centroids, which reduced the number of query-document comparisons that were necessary in a serial search. 1983. MURTAGH, F. 1985. P. R. Krishnaiah and J. N. Kanal, pp. Automatic Text Processing. This requires a sorted list of document-document similarities, and a means of counting the number of similarities seen between any two active clusters. Advances in Computers, 19, 113-227. SLINK algorithm Clustering Algorithms. 1971. WARD, J. H., JR. and M. E. HOOK. The time requirement is O (N2), rising to O (N3) if a simple serial scan of the similarity matrix is used; the storage requirement is O (N2). 1973. Lecture Notes in Computer Science, 111, 328-42. Finding Groups in Data: An Introduction to Cluster Analysis. London: Wiley. 16.4 ALGORITHMS FOR NONHIERARCHICAL METHODS 1988. This is true of both the van Rijsbergen and SLINK algorithms for the single link method, and the CLINK algorithm for the complete link method, all of which operate by iteratively inserting a document into an existing hierarchy. DIJKSTRA, E. W. 1976. London: Pitman (Research Monographs in Parallel and Distributed Computing). The computational requirements range from O (NlogN) to O (N5). 16.8.1 Approaches to Retrieval
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