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== 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{{citation needed|date=July 2017}} [[BIRCH (data clustering)|BIRCH]]. For data streams, one of the first results appeared in 1980<ref>{{cite journal | first1 = J. | last1 = Munro | first2 = M. | last2 = Paterson | url = http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4567985 | title = Selection and Sorting with Limited Storage | journal = Theoretical Computer Science | volume = 12 | issue = 3 | pages = 315–323 | date = 1980 | doi = 10.1016/0304-3975(80)90061-4 | doi-access = free }}</ref> but the model was formalized in 1998.<ref>{{cite journal | first1 = M. | last1 = Henzinger | first2 = P. | last2 = Raghavan | first3 = S. | last3 = Rajagopalan | citeseerx = 10.1.1.19.9554 | title = Computing on Data Streams | journal = Digital Equipment Corporation | volume = TR-1998-011 | date = August 1998 }}</ref>
== Definition ==
=== STREAM ===
STREAM is an algorithm for clustering data streams described by Guha, Mishra, Motwani and O'Callaghan<ref name=cds >{{cite paperbook | first1 = S. | last1 = Guha | first2 = N. | last2 = Mishra | first3 = R. | last3 = Motwani | first4 = L. | last4 = O'Callaghan | title = Proceedings 41st Annual Symposium on Foundations of Computer Science | chapter = Clustering data streams | citeseerx = 10.1.1.32.1927 | titlepages = Clustering Data Streams359–366 | journaldate = Proceedings2000 of| thedoi Annual= Symposium10.1109/SFCS.2000.892124 on| Foundationsisbn of= Computer Science0-7695-0850-2 | dates2cid = 20002767180 }}</ref> which achieves a [[approximation algorithm|constant factor approximation]] for the k-Median problem in a single pass and using small space.
'''''Theorem''':{{math 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 ''{{tmath|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</submath>X_1,... \ldots,''X''<sub><math>_ X_{\ell}</math></sub>''.
|2 = For each ''i'', find ''{{tmath|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>O(\ell k)</math>k)'' centers obtained in (2),
where each center ''c'' is weighted by the number
of points assigned to it.
}}
Where, if in Step 2 we run a bicriteria ''{{tmath|(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 ''{{tmath|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>n / \ell</math>) and so that the weighted <math>\ell k</math>''k'' centers also fit in memory, <math>\ell k < M</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 ''{{tmath|O(k)''}} (say 2''2kk'') points.
|2 = Repeat the above till we have seen ''m''<sup>2</sup>/(2k)2''k'') 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 2''2kk'' second-level medians and proceed.
|4 = In general, maintain at most ''m'' level-''i'' medians, and, on seeing ''m'', generate 2''2kk'' 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>{{cite journalbook | first1 = K. | last1 = Jain | first2 = V. | last2 = Vazirani | url = http://portal.acm.org/citation.cfm?id=796509 | title = Primal-dual approximation algorithms for metric facility ___location and k-median problems | journal = Proc. FOCS | pages = 2– | date = 1999 | isbn = 9780769504094 | series = Focs '99 }}</ref>
}}
=== Other Algorithmsalgorithms ===
Other well-known algorithms used for data stream clustering are:
* [[BIRCH (data clustering)|BIRCH]]:<ref name="birch">{{cite journal | first1 = T. | last1 = Zhang | first2 = R. | last2 = Ramakrishnan | first3 = M. | last3 = Linvy | url = http://doi.acm.org/10.1145/235968.233324 |doi=10.1145/235968.233324 | title = BIRCH: An Efficientefficient Datadata Clusteringclustering Methodmethod for Veryvery Largelarge Databasesdatabases | journal = Proceedings of the ACM SIGMOD ConferenceRecord on Management of Data|doi=10.1145/235968.233324 | date = 1996 | volume=25 | issue = 2 | pages=103–114| doi-access = free }}</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 ''{{tmath|O(N)''}} since one pass suffices to get a good clustering (though, results can be improved by allowing several passes).
* [[Cobweb (clustering)|COBWEB]]:<ref>{{cite journal | first = D. H. | last = Fisher | url = http://link.springer.com/article/10.1023%2FA%3A1022852608280 | title = Knowledge Acquisition Via Incremental Conceptual Clustering | journal = Machine Learning | date = 1987 | doi=10.1023/A:1022852608280 | volume=2 | issue = 2 | pages=139–172| doi-access = free }}</ref><ref>{{cite journal | first = D. H. | last = Fisher | citeseerx = 10.1.1.6.9914 | title = Iterative Optimization and Simplification of Hierarchical Clusterings | journal = Journal of AI Research | volume = 4 | date = 1996 | arxiv = cs/9604103 | bibcode = 1996cs........4103F }}</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>{{cite journal | first = F. | last = Can | url = http://dl.acm.org/citation.cfm?doid=130226.134466 | title = Incremental Clustering for Dynamic Information Processing | journal = ACM Transactions on Information Systems | volume = 11 | issue = 2 | date = 1993 | pages = 143-164143–164 | doi=10.1145/130226.134466| s2cid = 1691726 | doi-access = free }}</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.
* [[CluStream (data clustering)|CluStream]]:<ref>{{cite journal |last1=Aggarwal |first1=Charu C. |last2=Yu |first2=Philip S. |last3=Han |first3=Jiawei |last4=Wang |first4=Jianyong |title=A Framework for Clustering Evolving Data Streams |journal=Proceedings 2003 VLDB Conference |date=2003 |pages=81–92 |doi=10.1016/B978-012722442-8/50016-1 |isbn=9780127224428 |s2cid=2354576 |url=http://www.vldb.org/conf/2003/papers/S04P02.pdf |ref=CluStream}}</ref> uses micro-clusters that are temporal extensions of [[BIRCH]]<ref name="birch" /> cluster feature vector, so that it can decide if a micro-cluster can be newly created, merged or forgotten based in the analysis of the squared and linear sum of the current micro-clusters data-points and timestamps, and then at any point in time one can generate macro-clusters by clustering these micro-clustering using an offline clustering algorithm like [[K-Means]], thus producing a final clustering result.
== References ==
{{reflist}}
[[Category:DataCluster clusteringanalysis algorithms]]
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