The Black Hole Clustering Algorithm Based on Membrane Computing

Transcription

1 Internatonal Symposum on Computers & Informatcs (ISCI 2015) The Black Hole Clusterng Algorthm Based on Membrane Computng Qan L 1, a, Zheng Pe 2, b 1 Center for Rado Admnstraton & Technology Development, Xhua Unversty, Chengdu, Schuan, , Chna 2 Center for Rado Admnstraton & Technology Development, Xhua Unversty, Chengdu, Schuan, , Chna a l_qan111@163.com, b pqyz@263.net Abstract. Over the last few decades, nature has stmulated many successful heurstc optmzaton methods and computatonal tools for dealng wth optmzaton problems. In ths paper, we nvestgate the black hole clusterng algorthm based on membrane computng. Formally, we embed the black hole clusterng algorthm n the membrane structure, provde the three parts of the membrane system to perform black hole clusterng algorthm, whch are the membrane structure based on membrane computng, the obects n elementary membrane and the evoluton rules n the elementary membrane, and analyze the performance of the black hole clusterng algorthm based on membrane computng. The proposed algorthm n the paper s compared wth some other classcal algorthms, and the smulaton results ndcate the algorthm n the paper can acheve better performance n clusterng. Keywords: Heurstc algorthm; Black hole algorthm; Membrane computng Introducton In recent decades, varous heurstc algorthms have been developed [1] [2]. Most of the algorthms are nspred by the behavors of natural phenomena [3]. For example, genetc algorthms (GAs) are a knd of searchng optmal soluton whch was nspred by the process of the natural evoluton, usng natural genetc varaton and natural selecton operators [4]. The smulated annealng (SA) s a stochastc optmzaton algorthm developed by modelng the steel annealng process [5]. The ant colony optmzaton (ACO) was nspred from the behavor of the ant colony, whch s able to fnd the shortest path to the food source [6]. The partcle swarm optmzaton (PSO) was nspred from the swarm behavor, The authors - Publshed by Atlants Press 907

2 such as fsh and brd schoolng n nature [7]. The gravtatonal search algorthm (GSA) was developed based on the Newtonan gravty and the laws of moton [8]. The black hole algorthm () was proposed as a new heurstc optmzaton approach for data clusterng nspred from the black hole phenomena. In ths paper, the black hole algorthm based on membrane computng s proposed. Membrane computng was proposed by G. Păun nspred from the structure of the bologcal cells, amed at abstractng a calculaton model from the way of treatng compound n the herarchy of the cells [9]. Actually, there has been a growng nterest n computer model abstracted from the vewpont of bology [10] n the past few decades. For nstance, swarm ntellgence algorthms are evolutonary computaton technque abstracted from bologcal communtes [7]. The artfcal lfe nspred from the ndvdual lfe s a smulaton of the lfe system by artfcal [11]. As the optmzaton approach, the mmune algorthms were developed based on mmune system [12]. The Artfcal neural network s a knd of mathematcal model whch was nspred from the neural network of the human body [13]. Membrane computng has been a new branch of natural computaton and receved extensve attenton of many scholars snce t was put forward. So far, there have been many papers expound the membrane computng from varous subects, such as computer scence [14], bology [15], lngustcs [16], automatc control [17], etc, and appled n many felds ncludng computer graphcs [11], approxmate optmzaton [18], economcs [19], etc. The rest of the paper s organzed as follows. Secton 2 provdes a bref revew of the black hole algorthm and the membrane computng. In Secton 3, the black hole based on membrane computng s descrbed. The expermental results are demonstrated by comparng wth other algorthms n Secton 4. Fnally, the summary and the concluson are presented n Secton 5. Prelmnares In ths secton, we brefly revew the black hole algorthm and the membrane computng. Frstly, we brefly descrbe the orgn, process and the performance of the black hole algorthm, respectvely. The black hole algorthm () proposed n [20] s a new optmzaton method for data clusterng nspred by the black hole phenomenon n the unverse. Actually, John Mchell, the Brtsh natural phlosopher, has put forward that there s a knd of obect that runs faster than the lght n the eghteens-century. However, ths knd of obect was not known as a black hole durng that perod and t was only n 1967 that John Wheeler frst named the phenomenon of mass collapsng as the black hole. As we all known, black hole n unverse s what forms when a star of massve sze collapses. The gravtatonal power of the black hole s so hgh that there s no way for the nearby obects to escape from ts gravtatonal pull. Even for the lght, once t falls nto a black hole, t wll also be swallowed and gone forever. 908

3 Smlar to other populaton-based algorthms, the black hole algorthm starts wth an ntal populaton of canddate (called star n [20]) solutons and an obectve functon that s calculated for them. In every teraton of, the best canddate s selected to be the black hole accordng to the obectve functon. Then the black hole starts pullng other stars around t. In other words, all the stars start movng towards the black hole. If a star gets too close to the black hole, t wll be swallowed by the black hole and gone forever and a new star wll be randomly generated and placed n the search space and starts a new search. Brefly, the algorthm can be summarzed as the followng steps: a) Intalze a populaton of stars randomly n the search space. b) For each star, evaluate the obectve functon and select the best star whch has the best ftness value as the black hole. c) Change the locaton of each star. d) If a star reaches a locaton wth lower cost than the black hole, exchange ther locatons. e) If a star crosses the event horzon of the black hole, replace t wth a new star n a random locaton n the search space. f) Repeat steps c) to e) untl the stop crtera s reached. g) End From the process of the algorthm, we can see that t has a smple structure and t s easy to mplement. The results n [20] show that the algorthm has the greater advantages than other algorthms n solvng clusterng problems. Secondly, the development and the man structure of the membrane system are dscussed below. Membrane computng s also known as P system. It s ndcated from [21] that P system conssts of three parts: membrane structure, multsets of obects and evoluton rules. The membrane structure s a herarchcal rooted tree of compartments that delmt regons, where the root s called skn. Multsets of obects are chemcal substances present nsde the compartments (membranes) of a cell and the evoluton rules are chemcal reactons that can take place nsde the cell. In the P system, the multsets of obects are placed n compartments surrounded by membranes, and evolved by some gven rules. In ths way, every compartment can exchange nformaton wth other compartments accordng to these evoluton rules. P system s a model wth actve membrane, and the model s formed as follows: = ( O, µω, 1, ω2,, ωq, R) (1) where q 1 s the ntal degree of the P system. O s the alphabet of obects. µ represents the membrane structure (a rooted tree). ω1, ω2,, ω are strngs q over O, descrbng the multsets of obects placed n the q regons of µ, and R s a fnte set of rules. 909

4 Generally, there are three man types of P system: Cell-lke P systems, Tssue-lke P systems and Neural-lke P systems [22]. Cell-lke P systems are developed manly accordng to the lfe cycle of bologcal cells and ntercellular substance exchange [23]. Tssue-lke P systems are mportant expanson of cell membrane system whch can exchange nformaton between varous cells [24]. Neural-lke P systems are another type of P systems whch were nspred by the bologcal neural system and are also the current hot topcs n the study of membrane computng [25]. To sum up, P systems are a class of dstrbuted parallel computng models whch have synchronous and non-determnstc propertes and maxmally parallel computng ablty [13]. Because of the parallel character, not only NP complete problems can be solved n lnear tme n an easy way by P systems wth actve membranes [26], even the lmtatons of Turng machne can be exceeded by the smple models of the P systems [15]. The black hole algorthm based on MC (-MC) Although the black hole algorthm mentoned above outperforms other tradtonal heurstc algorthms descrbed n [20], the rate of convergence n t s so slow that we should use more teratons to acheve the effect. In order to mprove the algorthm performance, we ntroduce the membrane computng nto the black hole algorthm (-MC, for short). As mentoned above, the membrane system conssts of three parts: membrane structure, multsets of obects and the evoluton rules. Combned wth the model descrbed by Eq.(1), we gve the model of membrane system whch has been concretzed n -MC as follows: 1 2 q = ( O, µ, ω, ω,, ω, ω, R ) where q 1 s the ntal degree of the P system n -MC. (2) O s the alphabet of obects and µ represents the membrane structure n -MC. ω s the fnte set of O and R s the rule n -MC algorthm. The three parts of the membrane system combned wth the algorthm wll be ntroduced respectvely as follows. The frst s the membrane structure of the -MC. We adopt the Cell-lke P system wth a two-layer membrane structure whch conssts of a skn membrane and q elementary membrane n -MC. Fg.1 shows the membrane structure of the -MC brefly. As shown n Fg.1, 1, 2,, q are elementary membranes and each of them contans one or more obects and some evoluton rules. In the desgned cell-lke P system, each elementary membrane contans n obects and evoluton rules whle the skn membrane wthout any evoluton rules contans only one obect whch represents the global best obect. The second s the obects n elementary membrane. Lke other populaton-based methods, we generate the obects randomly n every 910

5 elementary membrane. The multset of the obects n each elementary membrane of the form ω = { O, O,, O } 1 2 where = 1, 2,, n q. ω s the set of the obects n the th elementary membrane and n s the number of the obects n elementary membrane. Actually, each obect of elementary membrane represents the cluster centers of the canddate. Assume that D= { x d R = 1, 2,, m} s the dataset to be clustered, whch s dvded nto k clusters. z1, z2,, z k are the k clusters centers respectvely, where z d R ( = 1, 2,, k ). Thus, the obect s an ( k d ) -dmensonal vector of the form O = {( z, z,, z ),,( z, z,, z ),,( z, z,, z )} d 1 2 d k1 k 2 kd (4) where O s the th obect n elementary membrane and ( z 1, z2,, z d ) s the th cluster center z. By the way, the obects n dfferent elementary membrane may be exactly the same n theoretcally because of the randomness n the process of obects generaton. Therefore, we adopt the followng way n order to solve ths problem: ω ω ω ω δ 1 2 q (5) where ω s the set of obects n the th elementary membrane. δ s the threshold n the nterval [0, 1]. In ths way, we can say that the obects n each elementary membrane are not exactly the same and the dversty of the obects s ncreased by t to a certan extent. The thrd s the evoluton rules n the elementary membrane. There are three evoluton rules n -MC, whch conssts of selecton, move and absorb rules. The three evoluton rules are descrbed as follows:. Selecton rule. For each of the n obects (called stars) n elementary membrane, evaluate the ftness values and select the best star whch has the best ftness value as the black hole. In the -MC, the ftness values of the obects are assessed by the followng equaton: k F( x, z) = w x z m = 1 = 1 (3) 911

6 where cluster center (6) x z s the Eucldean dstance between data pont x and the number of the clusters. z. m s the number of the data pont n D and k s the w s the assocaton weght of data pont x wth cluster, whch wll be ether 1 or 0 (f data pont x s assgned to cluster : w s 1, otherwse 0).. Move rule. After a star was selected as the black hole by the selecton rule, the rest stars start movng towards the black hole accordng to the followng equaton: y ( t + 1) = y () t + rand ( y y ()) t (7) where = 1,2,, n. y () t and y ( t + 1) are the locatons of the th star at teratons t and t + 1. y s the locaton of the black hole n the search space. rand s a random number n the nterval [0, 1]. Whle movng towards the black hole, a star may reach a locaton wth better effect on the data clusterng than the black hole. In such a case, the star wll replace the black hole and become the new black hole for the next teraton.. Absorb rule. Durng the process of movng towards the black hole, some stars may cross the event horzon of the black hole. In ths stuaton, these stars wll be sucked by the black hole and the same number of the stars wll be generated randomly at the same tme to keep the number of canddate solutons constant. The radus of the event horzon s formulated as follows: R = f n = 1 f (8) where f s the ftness value of the black hole and f s the ftness value of the th star. n s the number of stars. If the dstance between a star and the black hole s less than R, that canddate wll be sucked by the black hole and a new canddate s created randomly n the search space. 912

7 0 1 2 q Fg.1.The membrane structure of the -MC algorthm Based on the above descrpton, the man steps of the proposed algorthm are shown as follows: a) Determne the number of the elementary membrane. b) For each elementary membrane, ntalze the populaton of stars and select the best star whch has the best ftness value as the black hole. c) Change the locaton of each star accordng to Eq.(7) n every elementary membrane. d) For each elementary membrane, evaluate the ftness of all stars agan. If a star has the better ftness than the black hole, exchange ther locatons. e) In every elementary membrane, f a star crosses the event horzon of the black hole, t wll be swallowed by the black hole and replaced by a new star n the search space. f) Repeat steps c) to e) untl the stop crtera s reached. g) End. Test results In ths paper, we use two datasets whch are Irs and Wne to evaluate the performance of the proposed approach. The evaluaton crteron of the algorthm s the sum of ntra-cluster dstance called M value n ths paper, whch was defned n Eq. (6). Clearly, the smaller the M value, the hgher the qualty of the clusterng. On one hand, the performance of the -MC algorthm s compared aganst some classcal algorthms, ncludng K-means, PSO, GSA and. On the other hand, 2, 4 and 8 elementary membranes are used respectvely n the membrane structure of the -MC to compare wth algorthm. Theoretcally, the bgger the number of the elementary membrane whch means the more the dversty of the obects n our algorthm, the better the effect wll be. We descrbe the process of the -MC algorthm wth 2 elementary membranes n detals agan, takng the Irs for example. The process of the -MC algorthm s descrbed as follows: Frstly, Irs has 150 data ponts whch composed of 3 clusters and each data pont has 4 features. Therefore, m = 150 andk = 3 n Eq. (6) for Irs. Assume that we set the number of the obects (stars) n each elementary membrane s 100. Accordng to Eq. (3), the multsets of obects n elementary membrane can be descrbed as follows: ω = { O, O, O }

8 ω = { O, O,, O } ' ' ' (9) (10) where ω 1 s the set of the obects n the frst elementary membrane and ω 2 s the set of the obects n the second elementary membrane. O represents the obect n the set accordng to Eq. (4) of the form O = { z, z, z, z, z, z, z, z, z, z, z, z } (11) where = 1,2,,100. O s a (3 4) -dmensonal vector and ( z11, z12, z13, z 14) s the frst cluster center z 1. Secondly, calculate the M value of these obects n elementary membrane respectvely accordng to Eq. (6). In ths step, we need to select the best star as the black hole whch has the mnmum value of the100 M values n each elementary membrane. We mght assume that O s selected as the black hole n ' the frst elementary membrane and O s selected as the black hole n the second elementary membrane. Thrdly, change the locaton of stars n each elementary membrane accordng to Eq. (7). The rest of stars n the frst elementary membrane wll move towards O and stars n the second elementary membrane wll move towardso. Fourthly, calculate the M value agan for all the stars n each elementary membrane and select the best star whch has the best ftness as the new black hole. In ths step, we suppose O and O are selected n the 2 elementary p q membranes respectvely and then we save the better one between of the two obects as the global best obect. Then, the global best obect wll replace the old black hole n each elementary membrane as the new black hole n the next teraton. Fnally, calculate the radus of the event horzon n each elementary membrane accordng to Eq. (8). Smlarly, the process of the experment on Wne s the same as Irs. We wll not repeat here. A summary of the ntra-cluster dstances ( M values) obtaned by the classcal algorthm s shown n Table 1. As seen from the results n Table 1, the -MC algorthm acheves the best results among all the algorthms. For the Irs dataset, the best, average and worst M values obtaned by -MC are , , and respectvely. The worst M value of -MC s much better than the best values of other algorthms. For the Wne ' 914

9 dataset, the -MC algorthm acheves the optmum value of whch s much better than other algorthms. Table 1.The summary of the ntra-cluster dstances obtaned by dfferent algorthms Datase Crter K-means PSO GSA -MC t a Best Average Irs Worst Std wne Best Wne Average Worst Std x 10 4 wne -2 membranes -4 membranes -8 membranes M nteraton x 10 4 Fg.2. The comparson on Wne dataset between and -MC Actually, the algorthm has acheved the same effect as our algorthm presented n Table 1. However, the rate of convergence s so slow that the algorthm needs to sacrfce more teraton to acheve the best results. Fg.2 above shows the comparson on Wne dataset between algorthm and the -MC algorthm whch used 2, 4 and 8 membranes respectvely. As seen from Fg.2 clearly, the -MC algorthm acheves the better effect of the rate of convergence compared wth the algorthm. Fg.3 below shows the 915

10 experment results for Irs dataset. It can be ndcted by the Fg.3 that the results become better and better as the ncrease of the number of the membrane. rs membranes -4 membranes -8 membranes 97.4 M nteraton x 10 4 Fg.3. The comparson on Irs dataset between and -MC Concluson In recent years, modellng and smulatng the natural phenomena for solvng complex problems has been a hot research area. In ths paper, the -MC algorthm s proposed whch combned wth the advantage of the black hole algorthm and the membrane computng. The results of the experment show that the -MC algorthm outperforms other classcal algorthms. In future research, the -MC algorthm wll fnd more and more dfferent areas of applcatons. Acknowledgement The authors would lke to thank the fund of Schuan key laboratory of ntellgent network nformaton processng (SGXZD ) and key laboratory of the rado sgnals ntellgent processng (Xhua Unversty) (XZD ). References [1]O. Castllo, R. Martnez-Marroqun, P. Meln, et al, Comparatve study of bo-nspred algorthms appled to the optmzaton of type-1 and type-2 fuzzy controllers for an autonomous moble robot, Informaton Scences192(2012)

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