Clustering Algorithm Combining CPSO with K-Means Chunqin Gu 1, a, Qian Tao 2, b

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1 Internatonal Conference on Advances n Mechancal Engneerng and Industral Informatcs (AMEII 05) Clusterng Algorthm Combnng CPSO wth K-Means Chunqn Gu, a, Qan Tao, b Department of Informaton Scence, Zhongka Unversty of Agrculture and Engneerng, Guangzhou, 505, Chna Department of Computer Scence, Guangdong Unversty of Educaton, Guangzhou, Chna a emal: guchunqn@gmal.com, b emal:tao66@39.com(correspondng author) Keywords: K-Means; Clusterng; Partcle Swarm Optmzaton; Chaotc Abstract. A clusterng algorthm combnng partcle swarm optmzaton (CPSO) wth K-Means (KM-CPSO) s proposed, whch features better search effcency than K-Means, PSO and CPSO. The K-Means algorthms cannot guarantee convergence to global optma and suffer n local optmal cluster center because they are senstve to ntal cluster centers. Chaotc partcle swarm optmzaton (CPSO) can fnd global optmal soluton; meanwhle K-Means can acheve local optma. The CPSO-KM algorthm utlzes both global search capablty of CPSO and local search capablty of K-Means. CPSO-KM algorthm has been tested wth two synthetc datasets and three classcal data sets from UCI. Expermental results show better performance of the CPSO-KM as compared to K-Means, PSO and CPSO. Introducton Clusterng s a common unsupervsed learnng method, whch parttons a group of obects (nstances) nto groups (clusters) such that obects n the same cluster are smlar to each other and dssmlar to the obects n other clusters. K-Means [] algorthm parttons the groups of gven obects nto k clusters based on a dstance metrc. The K-Means algorthm s easy to mplement and very effcent. The man drawback of the K-Means algorthm s that the clusterng result s senstve to the ntal clusters centers and may converge to the local optma []. In recent years, swarm ntellgent algorthms has been combned wth K-Means and appled on many clusterng problems, because of the ablty of global search of the swarm ntellgent algorthms, such as GA[3], PSO[4], ACO[5]. The combnaton of swarm ntellgent algorthms and K-Means algorthm can take advantage of both global search ablty of swarm ntellgent algorthms and local search ablty of K-Means algorthm. CPSO[6] has the advantage of more precse global search ablty and more fast convergence speed n our prevous lterature. The CPSO s used to obtan better clusters centers for ntal clusters centers, then K-Means algorthm s used based on the ntal clusters centers found by CPSO. The rest of the paper s organzed as follows. Secton descrbes the model of clusterng problems. Secton 3 presents our clusterng algorthm combnng CPSO wth K-Means (CPSO-KM). Secton 4 llustrates expermental results. Fnally, Secton 5 makes concluson. Model of clusterng problem The clusterng problem s a collecton of data obects that are smlar to one another wthn the same cluster and are dssmlar to the obects n other clusters. Gven a data obects set L DS {X,X,...,X N}, where X ( x,,..., ) x x, L s the dmenson of a data obect, a clusterng problem tres to fnd a K-partton of DS, C { C, C,..., C K }, such that the smlarty of the data obects n the same cluster s maxmum and the dfference of the data obects between dfferent cluster centrod s maxmum. The obectve functon of clusterng problem s evaluated based on the sum of squared error (SSE), whch s defned as 05. The authors - Publshed by Atlants Press 749

2 K mn D (x, C), C x x C C x C K () C DS, C C C, and, {,,..., K} where D(x, y) denotes the Eucldean dstance between x and y, the cluster C. C () (3) s the mean of data obects n Clusterng Algorthm Combnng CPSO wth K-Means In ths paper, we develop a hybrd algorthm for solvng clusterng problems, whose flowchart s shown n Fg.. It can be observed that CPSO s combned nto the clusterng algorthm. Detaled mplementatons of the algorthm are proposed and descrbed n ths secton. A. Partcle Swarm Optmzaton Partcle swarm optmzaton (PSO), whch was developed by Kennedy and Eberhart n 995[7,8], s a populaton-based swarm ntellgence algorthm. The PSO smulates the socal behavor of brds flockng and fsh schoolng. It has been successfully appled to many practcal optmzaton problems. In PSO, each partcle represents a canddate soluton n the soluton space of D D dmensons, whch has two vectors: a poston vector X [ x, x,..., x ] and a velocty D vectorv [ v, v,..., v ]. Durng the evolutonary process, the velocty vector and the poston vector of partcle on dmenson d at teraton t+ are updated as d d d d v ( t ) v () t c r ( pbest () t x ()) t (4) d d c r ( gbest ( t) x ( t)) d d d x ( t ) x ( t) v ( t ) (5) where d=,,,d represents each dmenson of the search space, s nerta weght, c and c are cogntve learnng coeffcent and socal learnng coeffcent, respectvely, r and r are two d unform random numbers n the range of [0, ], pbest () t s the poston on dmenson d wth the d best ftness found up to the tth teraton for the partcle, gbest () t s the best poston on dmenson d found by the whole partcle swarm. The nerta weght n (4) s usually updated as ( ) g g, max max mn max where and max are ntal and fnal weght, and set to 0.9 and 0.4, respectvely[9]. g s the mn current evolutonary generaton number, g s the maxmum number of generatons. max c and c are set.8, respectvely. Durng the evolutonary process, the velocty of each partcle on dmenson d d d s restrcted to the range of vmax, vmax, d vmax + d. The maxmum velocty vmax s usually set to 0% of the search range [0]. To fnd good ntal cluster center, the cluster centers should be encoded nto partcle s representaton. We employ a representaton that each partcle s characterzed by a D=Num_Attr*Num_Centrods dmensonal real number vector. Num_Attr s the number of attrbuton of nstances n the clusterng problems. Num_Centrods s the number of the centrods of nstances n the clusterng problems. Thus the partcle s encoded as a sequence of cluster centrods. The th partcle n the swarm s n the form of,,, Num _ Attr,,, Num _ Attr Num _ Centrods, Num _ Centrods, Num _ Centrods, Num _ Attr X [ x, x,..., x, x, x,..., x,..., x, x,..., x ] (6), k where x s the kth attrbuton value of the th centrod,,,..., Num_ Centrods, k,,..., Num_ Attr, Fg. shows the partcle representaton of the th partcle n a -D space. 750

3 Fg.. Partcle representaton n a -D space In the ntalzaton, the poston of all partcles are randomly generated, wth,,, Num_ Attr ( x, x,..., x ) s the attrbuton value of the th centrod(,,, K ). The velocty of each partcle s also randomly generated, whle the maxmum velocty of each dmenson s set to be 0% of the search range. B. K-Means The K-means algorthm s a straghtforward and wdely used clusterng algorthm. It s smple to mplement and run, relatvely fast. The K-Means algorthm parttons a gven dataset nto a user-specfed number of clusters. The standard K-Means algorthm s descrbed as follows: () Intalze the K cluster centrod M { m, m,..., m K } randomly. () Assgn each data pont x to the closest cluster based on Eucldean dstance. The dstance between data pont x to the closest cluster centrod mk s defned as Dx (, m) x m (7) k k (3) Recalculate the means of K cluster centers, and obtan new K centers. (4) Repeat step and 3 untl the average change n centrod vectors s less than a predefned value. C. Chaotc PSO algorthm(cpso) and ftness calculaton Otherwse, durng the evolutonary process, the PSO may trap nto a local optma soluton, then the exploraton performance wll not be mproved. CPSO ntroduces chaotc mappng wth certanty and stochastc property nto PSO n order to mprove the global convergence ablty []. The chaotc mappng method s the chaotc logstc sequence, whch s defned as follows: xt ( ) r x t x t, r N, x 0 [0,] (8) where r s the control parameter, y s a varable, r = 4, x 0,0.5,0.5,0.75, and t = 0,,,. Compute the radus of chaotc searchng regon to avod destroy the excellent soluton. The radus can be adaptvely adusted by the dstance between pbest and gbest and varance. The chaotc searchng tme s defned as ( pbest, gbest) ( pbest( partcle ) gbest ), PNum PNum d, (9) where s the threshold of the premature convergence. As descrbed n Secton II, the obectve of clusterng problem s the mnmzng of the SSE. 75

4 Fg.. Flowchart of CPSO-KM Experments and comparsons In ths secton, seven data sets are employed to valdate the proposed CPSO-KM clusterng algorthm and to compare the CPSO-KM wth conventonal K-Means, PSO and CPSO. A. Data sets and algorthm confguraton The fve data sets nclude three synthetc data sets (SynSet, SynSet) and three real data sets. The three synthetc data sets are generated based on normal dstrbuton wth mean vector and covarance matrx. The parameters for the three synthetc data sets are shown n TableⅠ. The synthetc data sets are relatvely smple, and the real number and centers poston of ther clusters are known n advance, so experments on these data sets can reveal the vrtues and defects of the algorthms. Fg. 3-4 graphcally presents the two synthetc data sets. Table Ⅰ. The parameters for two synthetc data sets Data Sets No. of No. of Sze of Parameters of clusters clusters dmensons clusters [0,0], 0.4 SynSet 3 50 [7,7], [3,3], [30, 30, 30], 4 [0, 0, 0], 4 SynSet [4,4,4], 3 4 [5,5,5], B 6 B A Fg. 3. Clusters of the SynSet Fg. 4. Clusters of the SynSet The three real data sets that we consdered are Irs, Haberman, Hayes-roth, whch are avalable 75-0 A

5 onlne []. These data sets are very classcal and often used to examne and compare the performances of algorthms n the felds of classfcaton. The PSO, CPSO and CPSO-KM use the same populaton sze of 0 and the same number of 000 ftness evaluatons (FEs) for a far comparson. All the experments are carred out usng Vsual C++ on the same machne wth a Pentum(R) Dual-Core CPU 3.0 GHz,.0GB RAM, and Wndows 7 operaton system. All the data sets are ndependently smulated 30 tmes, and ther mean results are compared n order to reduce the statstcal errors. B. Performance and F-Measure comparson between CPSO-KM and other algorthms To valdate the proposed CPSO-KM, we compared the CPSO-KM wth K-Means, PSO and CPSO. Each data set was ndependently smulated 30 tmes for purpose of reducng statstcal errors. The statstcal results are shown n Table Ⅱ n terms of the mean and standard devaton of the ftness, and boldface n the table ndcates the best result obtaned. Because the real parttons of the data sets consdered here are already known, the performances of CPSO-KM algorthm can be evaluated by comparng the resultng cluster wth the real structures n terms of external crtera. A commonly used crtera s F-Measure. Assumng that S s a prespecfed partton of data sets DS, C s the resultng clusters by applyng the CPSO-KM algorthm. () TP s the numbers of pars of data ponts ( x, x ), where x, x Cm, x, x Sn,. () FP s the numbers of pars of data ponts ( x, x ), where x, x Cm, x, Sn, x Sn,, n n. (3) TN s the numbers of pars of data ponts ( x, x ), where x Cm, x Cm, x Sn, x Sn,, m m, n n. (4) P and R are the precson and recall of cluster. (.0)( P R) F Measure, where TP ( P) R P, TP R TP FP TP FN In experments, we set.0, whch weghts precson and recall equally. The performance comparson between CPSO-KM and other algorthms n terms of F-Measure s llustrated n Table Ⅱ. An nterestng result s that all algorthms have most relably acheved the mnmal ftness and F-Measure of SysSet. It s shown that CPSO-KM attans better ftness than K-Means, PSO and CPSO. The CPSO-KM s also observed to obtan the smallest SD of the ftness and F-Measure. TABLE Ⅱ: Performance (ftness) and F-Measure comparson of CPSO-KM wth other algorthms Datasets K-Means PSO CPSO CPSO-KM SynSet (N=50, C=3, D=) SynSet (N=400, C=4, D=3) Irs (N=50, C=3, D=4) Haberman (N=306, C=, D=3) Hayes-roth (N=3, C=3, D=4) Mean ftness Std. Dev F-Measure Std. Dev Mean ftness.8643e e e e+04 Std. Dev F-Measure Std. Dev Mean ftness Std. Dev F-Measure Std. Dev Mean ftness E E E E+04 Std. Dev F-Measure Std. Dev Mean ftness Std. Dev F-Measure Std. Dev

6 Concluson In ths paper, a CPSO-KM algorthm s proposed to solve the clusterng problem, whch s based on combnaton of the partcle swarm optmzaton (PSO) and the K-Means algorthm. CPSO and K-Means are run sequentally and the cluster centers obtaned by CPSO are the ntal centers of the K-Means. The combnaton method solves the problem of senstvty to the choce of ntal cluster centers. The ftness experments on three synthetc and four real data sets show that the CPSO-KM algorthm can obtan better performance than K-Means, PSO and CPSO. The expermental results also show that CPSO-KM can attan the better clusters than other algorthms n terms of F-Measure. Future work ncludes research nto the applcaton of the CPSO-KM algorthm to solve more demandng data sets wth more complex structure, such as document clusterng. Acknowledgment Ths work s supported by the Postdoctoral Scence Foundaton of Chna (No. 03M549), Research Fund for the Doctoral Program of Hgher Educaton of Chna (No ), Guangdong Provnce Natural Scence Fund Proects (No. S ), Doctoral Foundaton of Guangdong Unversty of Educaton(No. 0ARF05), and Guangdong Provnce Colleage Specalty Constructon Proects (No. 03LYM_006), Guangdong Provncal Key Platform and Maor Scentfc Research Proects of Educaton(04KQNCX69, 04KTSCX94). References [] E. Forgy, Cluster analyss of multvarate data: Effcency vs. nter-pretablty of classfcatons, Bometrcs, ( 3), 965, [] S. Z. Selm, M. A. Ismal, K-means type algorthms: A generalzed convergence theorem and characterzaton of local optmalty, IEEE Transacton Pattern Analyss Machne Intellgence, 6, 984, 6, 8-87 [3] P. Sunantasaengtong, S. Chvapreecha, Mxed K-means and GA-based weghted dstance fngerprnt algorthm for ndoor localzaton system, n: Proceedngs of the 04 IEEE Regon 0 Conference TENCON, 04, -5 [4] M. Davod, H. R. Modares, E. Rehan, M. Davod, A. Sarkhan, Coherency approach by hybrd PSO, K-Means clusterng method n power system, n: Proceedngs of the IEEE nd Internatonal Power and Energy Conference, 008, [5] X. Sun, B. Zhang, L. Yang, S. L, L. Gao, Hyperspectal mage clusterng usng ant colony optmzaton(aco) mproved by K-means algorthm, n: Proceedngs of the 00 3rd Internatonal Conference on Advanced Computer Theory and Engneerng (ICACTE),, 00, V-474-V-478 [6] Q. Tao, Z. Huang, C. Gu et al., Optmzaton of green agr-food supply chan network usng chaotc PSO algorthm, n: Proceedngs of the 03 IEEE Internatonal Conference on Servce Operatons and Logstcs, and Informatcs (SOLI), 03, [7] J. Kennedy, R. C. Eberhart, Partcle swarm optmzaton, n: Proceedngs of the IEEE Internatonal Conference Neural Network, 995, 4, [8] R. C. Eberhart, J. Kennedy, A new optmzer usng partcle swarm theory, n: Proceedngs of the 6th Internatonal symposum on Mcromachne Human Scence, 995, [9] Y. Sh, R. C. Eberhart, A modfed partcle swarm optmzer, n Proceedngs IEEE World Congress Computer Intellgence, 998, pp [0] R. C. Eberhart, Y. H. Sh, Partcle swarm optmzaton: Developments, applcatons and resources, n: Proceedngs IEEE Congress Evolutonary Computaton, 00,

7 [] L. Zhang, Y. Zhang, Y. Yang. A Decorrelaton algorthm based on chaos adaptve. Journal of Electroncs & Informaton Technology, 009, 3(8), [] ftp://ftp.cs.uc.edu/pub/machne-learnng-databases/ 755

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