An Implementation of Integer Programming Techniques in Clustering Algorithm

Transcription

1 S. Shebaga Ezhl et al / Ia Joural of oputer Scece a Egeerg (IJSE) A Ipleetato of Iteger rograg echques lusterg Algorth S. Shebaga Ezhl a Dr.. Vayalaksh 2 Departet of Matheatcs Sathyabaa Uversty, hea 9 e-al: shebaga_ezhl@reff.co, 2 vusesha22@yahoo.co. Abstract hs paper aly eals wth the aalyss of I clusterg. lusterg s eeplfe by the usupervse learg of patters a clusters that ay est a gve atabase a s a useful tool for owlege Dscovery Database (DD). A atheatcal prograg forulato of ths proble s propose that s theoretcally ustfable a coputatoally pleetable a fte uber of steps. he clusterg algorth apples the herarchcal clusterg ethoology [] where pots or clusters wth the shortest stace are erge to a cluster utl the esre uber of clusters s acheve. Nuercal eaples are gve for the above algorthc approach.. Itroucto he Matheatcal prograg oel subect to soe costrats s a broa scple that has bee apple for varous theoretcal a apple probles. I ths paper the escrbe varous teger prograg oel for clusterg s pleete. he fuaetal No Lear prograg proble, cossts of zg the obectve fucto subect to equalty a equalty costrats a t s wrtte as g() subect to f(), h() = where s a -esoal vector of real varables, f s a real-value fuctos of, f a h are fte esoal vector fuctos of. If all the fuctos f, g a h are lear the the proble s splfe to a lear progra [8] whch s the classcal proble of atheatcal prograg. he clusterg proble cosere ths paper s that of assgg pots the -esoal real space R to k clusters. If a polyheral stace (such as the -or stace) s use, the oel ca be forulate as that of zg a pece-wse lear cocave fucto. Although a blear progra s a o cove optzato proble (.e., zg a fucto that s ot valley-lke) a fast fte -ea ISSN : Vol. 3 No. Feb-Mar 22 73

2 S. Shebaga Ezhl et al / Ia Joural of oputer Scece a Egeerg (IJSE) algorth cosstg of solvg few lear prograg close fro leas to a statoary pot. O four other publcally avalable atabase of the ea a ea algorth best o two of the atabase. 2. he Me Iteger Optzato Moel he trag ata cossts of observatos (, y ), =, 2,, wth R a y (, ). Let M = {, 2,, }, M = {, 2,, }, = {, 2,, }. Let a be the uber of class a class pots respectvely. Note that class a class pots by, =, 2,, a, =, 2,, respectvely. Let G be the set ces of class pots that are Group, where G M a G. hus the followg syste s feasble, for all M a λ λ G, λ, G, G. Fro Farka s lea, syste () s ufeasble f a oly f the followg proble s feasble. We coser the optzato proble δ p p subect to q q q q, z aze δ k, k, δ,, M G M ; k, k, (3) ;, G I orer to etere f we ca assg class pots to groups such that z >, we efe ecso varable for a M. a k,, y s assge to Group otherwse. We clue the costrats k, q k, probles (3) f a oly f a k, =..e., q M(a ) where M s the large postve costat. Now we ca assue M = k, k, k,-.e., k, q k, a k,-.. () (2) ISSN : Vol. 3 No. Feb-Mar 22 74

3 S. Shebaga Ezhl et al / Ia Joural of oputer Scece a Egeerg (IJSE) hus we ca check whether we ca partto class pots to sot groups such that o class pots s assge by solvg by the e-teger prograg. 3. Matheatcal Forulato z * = aze subect to k δ a k, k, k, a k, q q k, k,, {,} δ, a k,, M, M ; M ;, M If z * >, the partto to groups s feasble, whle f z * =, t s ot requrg us to crease the value of. (4) 4. he lusterg Algorth A herarchcal clusterg algorth s evelope base o preprocesses the ata to create clusters of class a class pots. ollecto of class (class ) pots are cosere a cluster f there are o class (class ) pots ther cove hull. Equato (4) becoes epesve to solve. Alteratvely, we ca rastcally ecrease the eso of proble (4) by solvg hyperplae for clusters of pots at a te stea of pot by pot. he followg lear optzato proble s solve to check whether class clusters r a s ca be erge. * = aze (5) subect to q δ, M q δ, s. r where r a s are set of ces of class pots. 5. lusterg va Matheatcal rograg he usupervse assget of eleets of a gve set to groups or clusters of lke pots, s the obectve of cluster aalyss. A prcpal otvato beh the atheatcal prograg approach s a precse a cocse stateet of the clusterg probles as a cocave zg probles. ISSN : Vol. 3 No. Feb-Mar 22 75

4 S. Shebaga Ezhl et al / Ia Joural of oputer Scece a Egeerg (IJSE) For a gve set of pots R represete by the atr A R a a uber of esre clusters, we forulate the clusterg probles as follows. F cluster ceters l, l =,, k such that the a over l {, 2,, k} of the -or stace betwee each pot, A, =, 2,,, a the cluster ceters l, l =,, k s ze so that the followg teger prograg proble ca be solve. subect to ze,d {e,...k D, D,, } A D,,2,...,,,2,...,k (6) hs s ot the case of 2-or or p-or, p. We state the blear prograg forulato a ea algorth for solvg the clusterg proble. 6. lusterg as a Blear rogra he clusterg proble (6) s equvalet to the followg blear progra subect to the costrats Mze e D,, D,, R,D, R,, R, A D,,2,...,,,2,...,k,,,2,...,,,2,...,k (7) Here because of the sple structure of the blear progra (7), the two lear progras ca be solve eplctly close for. hs leas to the followg algorthc pleetato. 6. Algorth of Mea steps. (a) (b) Gve the clusters luster Assget: For each oe or. A at terato, copute,a 2,..., A k luster eter Upate: For l =, 2,, k choose A,2,..., A k, =, 2,, etere l() such that A as a ea of all by the followg two A () s close to assge to A. the Stop whe A A. Assg each pot to a cluster where ceter s close the -or to the pot. 7. Matheatcal Forulato of Meas lusterg usg Iteger rograg roble he obectve fucto s to aze the total uber of correctly classfe ata pots as equato (). here are two sets of costrats use to esure that the trag saples are classfe base o the otg ISSN : Vol. 3 No. Feb-Mar 22 76

5 S. Shebaga Ezhl et al / Ia Joural of oputer Scece a Egeerg (IJSE) earest eghbor rules as equato (2) a (3). he equato (4) s a logcal costrat use to esure that at least oe feature s use the votg earest eghbor rule. he e teger prograg s gve by he obectve fucto s to aze the total correct classfcato equato (6). here are two sets of costrats use to esure that the trag saple are classfe base o the stace averagg earest eghbor rules as equato (6) a (8). here s a set of logcal costrat (9) use to esure that at least oe feature s use the stace averagg earest eghbor rule. he teger prograg proble for average SFM s gve by a (6) y subect to the costrats M y,2,..., (7) M 2 ( y ),2,..., (8) (9) {, }, y {, } where s the average statstcal stace betwee saple a all other saples fro the sae class at feature (tra-class stace) s the average statstcal stace betwee saple a all other saples fro fferet class at feature (ter class stace), M a M. 2 Data Aalyss usg eas lusterg. able. Statstcs for cotuous varable: south west Nuber of clusters: 2 otal uber of trag cases: 2 luster luster 2 Overall Mu Mau Mea Staar evato ISSN : Vol. 3 No. Feb-Mar 22 77

6 S. Shebaga Ezhl et al / Ia Joural of oputer Scece a Egeerg (IJSE) able 2. Statstcs for cotuous varable: orth east Nuber of clusters: 2 otal uber of trag cases: 2 luster luster 2 Overall Mu Mau Mea Staar evato able 3. Statstcs for cotuous varable: wter se Nuber of clusters: 2 otal uber of trag cases: 2 luster luster 2 Overall Mu Mau Mea Staar evato able 4. Statstcs for cotuous varable: hot seaso Nuber of clusters: 2 otal uber of trag cases: 2 luster luster 2 Overall Mu Mau Mea Staar evato ocluso Accorg to the great eghteeth cetury Matheatca Leohar Euler, Nothg happes the uverse that oes ot have a sese of ether cota au or u fro hs pot of vew applcato of large scale ata g probles ca be solve easly usg quaratc prograg wth lear progras cotas llo of varables. But f soe of the features are screte a ca be represete usg tegers, the the techques of teger prograg ca be aapte. Hece teger prograg approaches ISSN : Vol. 3 No. Feb-Mar 22 78

7 S. Shebaga Ezhl et al / Ia Joural of oputer Scece a Egeerg (IJSE) have bee apple for eaples of aual rafall ata for ayakuar strct fro 997 to 2 obtae fro eteorogcal epartet. By usg Algorthc approach, the bary progra for screte ata s pleete by usg clusterg techque. Refereces [] Johso, R.A., Wcher, D.W., (998), Apple ultvarate statstcal aalyss, 4 th eto, rectce Hall, N.J. [2] Robert, J. Veerte, Lear prograg: Fouatos a etesos, luwer Acaec ublsheres, Hgha, M.A., 997. [3] W..G. Icootz,.M. Narera a. Fukuga, A brach a bou clusterg algorth, IEEE rasactos o oputers -24 (975), [4] J.F. Macrotorcho a. Mchau, Optzato e aalyss oale es oees (Masso, ars 979). [5] S. Reger, Sur quelques aspects atheatques at Sceces huates 82 (983), 85. [6] G. Dehr, Evaluato of a brach a bou for clusterg, SIAM Joural of Scetfc a Statstcal oputg, 6 (985), [7] L. aufa a.j. Rousseeuw, Fg groups ata: A troucto to clusterg aalyss (New York; Wley, 99). [8] S. hopra a M.R. Rao, O the ultway cut polyhero, Networks, 2 (99), 589. [9] G. le a J.E. Aroso, Optal clusterg: A oel a etho, Novel Research Logstcs, 38 (99), [] U. Dororf a E. esch, Fast lusterg Algorths, ORSA Joural o oputg, 6 (994), 453. [] S. hopra a J.H. Owe, Etee forulatos for the A-cut proble, Matheatcal rograg, 73 (996) 73. ISSN : Vol. 3 No. Feb-Mar 22 79