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In the | In [[computer science]], '''data stream clustering''' is defined as the [[cluster analysis|clustering]] of data that arrive continuously such as telephone records, multimedia data, financial transactions etc. Data stream clustering is usually studied as a [[streaming algorithm]] and the objective is, given a sequence of points, to construct a good clustering of the stream, using a small amount of memory and time. <!-- in contrary to the traditional clustering where data are static. --> | ||
== History == | |||
Data stream clustering has recently attracted attention for emerging applications that involve large amounts of streaming data. For clustering, [[k-means clustering|k-means]] is a widely used heuristic but alternate algorithms have also been developed such as [[k-medoids]], [[CURE data clustering algorithm|CURE]] and the popular [[BIRCH (data clustering)|BIRCH]]. For data streams, one of the first results appeared in 1980<ref>J.Munro and M. Paterson. [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4567985 Selection and Sorting with Limited Storage]. ''Theoretical Computer Science'', pages 315-323, 1980</ref> but the model was formalized in 1998.<ref>M. Henzinger, P. Raghavan, and S. Rajagopalan. [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.9554 Computing on Data Streams]. ''Digital Equipment Corporation, TR-1998-011'', August 1998.</ref> | |||
== Definition == | |||
The problem of data stream clustering is defined as: | |||
'''Input:''' a sequence of ''n'' points in metric space and an integer ''k''.<br /> | |||
'''Output:''' ''k'' centers in the set of the ''n'' points so as to minimize the sum of distances from data points to their closest cluster centers. | |||
This is the streaming version of the k-median problem. | |||
== Algorithms == | |||
<!-- Unlike online algorithms, algorithms for data stream clustering have only a bounded amount of memory available and they may be able to take action after a group of points arrives while online algorithms are required to take action after each point arrives. --> | |||
=== STREAM === | |||
STREAM is an algorithm for clustering data streams described by Guha, Mishra, Motwani and O'Callaghan<ref name=cds >S. Guha, N. Mishra, R. Motwani, L. O'Callaghan. [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.1927 Clustering Data Streams]. Proceedings of the Annual Symposium on Foundations of Computer Science, 2000</ref> which achieves a [[approximation algorithm|constant factor approximation]] for the k-Median problem in a single pass and using small space. | |||
'''''Theorem''': STREAM can solve the ''k''-Median problem on a data stream in a single pass, with time ''O(n<sup>1+e</sup>)'' and space ''θ(n<sup>ε</sup>)'' up to a factor ''2<sup>O(1/e)</sup>'', where ''n'' the number of points and ''e<1/2''. | |||
To understand STREAM, the first step is to show that clustering can take place in small space (not caring about the number of passes). Small-Space is a [[divide-and-conquer algorithm]] that divides the data, ''S'', into <math>\ell</math> pieces, clusters each one of them (using ''k''-means) and then clusters the centers obtained. | |||
[[File:Small-Space.jpg|thumb | 440x140px | right | Small-Space Algorithm representation]] | |||
'''Algorithm Small-Space(S)''' | |||
{{ordered list | |||
|1 = Divide ''S'' into <math>\ell</math> disjoint pieces ''X''<sub>1</sub>,...,''X''<sub><math>_{\ell}</math></sub>''. | |||
|2 = For each ''i'', find ''O(k)'' centers in ''X<sub>i</sub>''. Assign | |||
each point in ''X<sub>i</sub>'' to its closest center. | |||
|3 = Let ''X''' be the ''O(<math>\ell</math>k)'' centers obtained in (2), | |||
where each center ''c'' is weighted by the number | |||
of points assigned to it. | |||
|4 = Cluster ''X''' to find ''k'' centers. | |||
}} | |||
Where, if in Step 2 we run a bicriteria ''(a,b)''-[[approximation algorithm]] which outputs at most ''ak'' medians with cost at most ''b'' times the optimum k-Median solution and in Step 4 we run a ''c''-approximation algorithm then the approximation factor of Small-Space() algorithm is ''2c(1+2b)+2b''. We can also generalize Small-Space so that it recursively calls itself ''i'' times on a successively smaller set of weighted centers and achieves a constant factor approximation to the ''k''-median problem. | |||
The problem with the Small-Space is that the number of subsets <math>\ell</math> that we partition ''S'' into is limited, since it has to store in memory the intermediate medians in ''X'''. So, if ''M'' is the size of memory, we need to partition ''S'' into <math>\ell</math> subsets such that each subset fits in memory, (n/<math>\ell</math>) and so that the weighted <math>\ell</math>''k'' centers also fit in memory, <math>\ell</math>''k<M''. But such an <math>\ell</math> may not always exist. | |||
The STREAM algorithm solves the problem of storing intermediate medians and achieves better running time and space requirements. The algorithm works as follows:<ref name=cds /> | |||
{{ordered list | |||
|1 = Input the first ''m'' points; using the randomized algorithm presented in<ref name=cds /> reduce these to ''O(k)'' (say ''2k'') points. | |||
|2 = Repeat the above till we have seen ''m<sup>2</sup>/(2k)'' of the original data points. We now have ''m'' intermediate medians. | |||
|3 = Using a [[Local search (optimization)|local search]] algorithm, cluster these ''m'' first-level medians into ''2k'' second-level medians and proceed. | |||
|4 = In general, maintain at most ''m'' level-''i'' medians, and, on seeing ''m'', generate ''2k'' level-''i''+ 1 medians, with the weight of a new median as the sum of the weights of the intermediate medians assigned to it. | |||
|5 = When we have seen all the original data points, we cluster all the intermediate medians into ''k'' final medians, using the primal dual algorithm.<ref>K. Jain and V. Vazirani. [http://portal.acm.org/citation.cfm?id=796509 Primal-dual approximation algorithms for metric facility location and k-median problems.] Proc. FOCS, 1999</ref> | |||
}} | |||
=== Other Algorithms === | |||
Other well-known algorithms used for data stream clustering are: | |||
* [[BIRCH (data clustering)|BIRCH]]:<ref>T. Zhang, R. Ramakrishnan, M. Linvy. [http://doi.acm.org/10.1145/235968.233324 BIRCH: An Efficient Data Clustering Method for Very Large Databases], Proceedings of the ACM SIGMOD Conference on Management of Data, 1996</ref> builds a hierarchical data structure to incrementally cluster the incoming points using the available memory and minimizing the amount of I/O required. The complexity of the algorithm is ''O(N)'' since one pass suffices to get a good clustering (though, results can be improved by allowing several passes). | |||
* [[Cobweb (clustering)|COBWEB]]:<ref>D.H. Fisher [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.6.9914 Iterative Optimization and Simplification of Hierarchical Clusterings]. Journal of AI Research, Vol 4, 1996</ref> is an incremental clustering technique that keeps a hierarchical clustering model in the form of a [[Decision tree learning|classification tree]]. For each new point. COBWEB descends the tree, updates the nodes along the way and looks for the best node to put the point on (using a [[Category utility| category utility function]]). | |||
* [[C2ICM(incremental clustering)|C2ICM]]:<ref>F. Can. [http://dl.acm.org/citation.cfm?doid=130226.134466 Incremental Clustering for Dynamic Information Processing], ACM Transactions on Information Systems, Vol. 11, No. 2 1993, pages 143-164</ref> builds a flat partitioning clustering structure by selecting some objects as cluster seeds/initiators and a non-seed is assigned to the seed that provides the highest coverage, addition of new objects can introduce new seeds and falsify some existing old seeds, during incremental clustering new objects and the members of the falsified clusters are assigned to one of the existing new/old seeds. | |||
== References == | |||
{{reflist}} | |||
[[Category:Data clustering algorithms]] | |||
Revision as of 21:01, 7 March 2013
In computer science, data stream clustering is defined as the clustering of data that arrive continuously such as telephone records, multimedia data, financial transactions etc. Data stream clustering is usually studied as a streaming algorithm and the objective is, given a sequence of points, to construct a good clustering of the stream, using a small amount of memory and time.
History
Data stream clustering has recently attracted attention for emerging applications that involve large amounts of streaming data. For clustering, k-means is a widely used heuristic but alternate algorithms have also been developed such as k-medoids, CURE and the popular BIRCH. For data streams, one of the first results appeared in 1980[1] but the model was formalized in 1998.[2]
Definition
The problem of data stream clustering is defined as:
Input: a sequence of n points in metric space and an integer k.
Output: k centers in the set of the n points so as to minimize the sum of distances from data points to their closest cluster centers.
This is the streaming version of the k-median problem.
Algorithms
STREAM
STREAM is an algorithm for clustering data streams described by Guha, Mishra, Motwani and O'Callaghan[3] which achieves a constant factor approximation for the k-Median problem in a single pass and using small space.
Theorem: STREAM can solve the k-Median problem on a data stream in a single pass, with time O(n1+e) and space θ(nε) up to a factor 2O(1/e), where n the number of points and e<1/2.
To understand STREAM, the first step is to show that clustering can take place in small space (not caring about the number of passes). Small-Space is a divide-and-conquer algorithm that divides the data, S, into pieces, clusters each one of them (using k-means) and then clusters the centers obtained.
Algorithm Small-Space(S)
Where, if in Step 2 we run a bicriteria (a,b)-approximation algorithm which outputs at most ak medians with cost at most b times the optimum k-Median solution and in Step 4 we run a c-approximation algorithm then the approximation factor of Small-Space() algorithm is 2c(1+2b)+2b. We can also generalize Small-Space so that it recursively calls itself i times on a successively smaller set of weighted centers and achieves a constant factor approximation to the k-median problem.
The problem with the Small-Space is that the number of subsets that we partition S into is limited, since it has to store in memory the intermediate medians in X'. So, if M is the size of memory, we need to partition S into subsets such that each subset fits in memory, (n/) and so that the weighted k centers also fit in memory, k<M. But such an may not always exist.
The STREAM algorithm solves the problem of storing intermediate medians and achieves better running time and space requirements. The algorithm works as follows:[3] Template:Ordered list
Other Algorithms
Other well-known algorithms used for data stream clustering are:
- BIRCH:[4] builds a hierarchical data structure to incrementally cluster the incoming points using the available memory and minimizing the amount of I/O required. The complexity of the algorithm is O(N) since one pass suffices to get a good clustering (though, results can be improved by allowing several passes).
- COBWEB:[5] is an incremental clustering technique that keeps a hierarchical clustering model in the form of a classification tree. For each new point. COBWEB descends the tree, updates the nodes along the way and looks for the best node to put the point on (using a category utility function).
- C2ICM:[6] builds a flat partitioning clustering structure by selecting some objects as cluster seeds/initiators and a non-seed is assigned to the seed that provides the highest coverage, addition of new objects can introduce new seeds and falsify some existing old seeds, during incremental clustering new objects and the members of the falsified clusters are assigned to one of the existing new/old seeds.
References
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- ↑ J.Munro and M. Paterson. Selection and Sorting with Limited Storage. Theoretical Computer Science, pages 315-323, 1980
- ↑ M. Henzinger, P. Raghavan, and S. Rajagopalan. Computing on Data Streams. Digital Equipment Corporation, TR-1998-011, August 1998.
- ↑ 3.0 3.1 S. Guha, N. Mishra, R. Motwani, L. O'Callaghan. Clustering Data Streams. Proceedings of the Annual Symposium on Foundations of Computer Science, 2000
- ↑ T. Zhang, R. Ramakrishnan, M. Linvy. BIRCH: An Efficient Data Clustering Method for Very Large Databases, Proceedings of the ACM SIGMOD Conference on Management of Data, 1996
- ↑ D.H. Fisher Iterative Optimization and Simplification of Hierarchical Clusterings. Journal of AI Research, Vol 4, 1996
- ↑ F. Can. Incremental Clustering for Dynamic Information Processing, ACM Transactions on Information Systems, Vol. 11, No. 2 1993, pages 143-164