Feature selection for classification using an ant colony system

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1 Bond Unversty Informaton Technology papers School of Informaton Technology Feature selecton for classfcaton usng an ant colony system Nada Abd-Alsabour Bond Unversty Marcus Randall Bond Unversty, Recommended Ctaton Nada Abd-Alsabour and Marcus Randall. (2010) "Feature selecton for classfcaton usng an ant colony system" e-scence 2010: Sxth IEEE nternatonal conference on e-scence. Brsbane, Australa.Dec Ths Conference Paper s brought to you by the School of Informaton Technology at epublcatons@bond. It has been accepted for ncluson n Informaton Technology papers by an authorzed admnstrator of epublcatons@bond. For more nformaton, please contact Bond Unversty's Repostory Coordnator.

2 2010 Sxth IEEE Internatonal Conference on e Scence Workshops Feature Selecton for Classfcaton Usng an Ant Colony System Nada Abd-Alsabour School of nformaton technology, Bond Unversty QLD, Australa nada.abdalsabour@eee.org Marcus Randall School of nformaton technology, Bond Unversty QLD, Australa mrandall@bond.edu.au Abstract Many applcatons such as pattern recognton requre selectng a subset of the nput n order to represent the whole set of. The am of feature selecton s to remove rrelevant or redundant whle keepng the most nformatve ones. In ths paper, an ant colony system approach for solvng feature selecton for classfcaton s presented. The proposed algorthm was tested usng artfcal and real-world datasets. The results are promsng n terms of the accuracy of the classfer and the number of n all the used datasets. The results of the proposed algorthm have been compared wth other results avalable n the lterature and found to be favorable. Keywords- Ant colony optmsaton, feature selecton. I. INTRODUCTION Pattern classfcaton s the assgnment of an nput pattern to one of several predefned categores/classes [1]. Pattern recognton can be vewed as a two-fold task: a) learnng (sometmes called the tranng phase) the nvarant and common propertes of a set of samples (tranng set) that characterse a class, and b) decdng (through the classfer) that a new sample s a possble member of the class by notng that t has propertes common to those of the set of samples [2]. The basc component of any pattern recognton system s the classfer whose task s to partton the feature space nto class-labelled decson regons, one regon for each category [1], [3]. The performance of classfers s senstve to the choce of the that are used for constructng those classfers. The choce of the that are presented to the classfers affects the followng mportant propertes: The accuracy of the classfers, the tme needed for learnng the classfers, and the number of examples needed for learnng the classfers [1]. Increasng the amount of tranng data may affect the tme needed for the learnng and may lead to fndng a less optmal soluton. Ths s because many n the tranng data may be rrelevant to the classfcaton task. For nstance, data that records the day of the week on whch a bank loan applcaton was completed s unlkely to be relevant to the success of the applcaton. Moreover, other may be redundant. Such may slow down and mslead the learnng step and do not contrbute to the classfcaton process. Hence, feature selecton (FS) from the orgnal set of s hghly desrable n many stuatons n order to remove any rrelevant or redundant [4]. Feature selecton s the problem of selectng a subset of wthout reducng the accuracy of representng the orgnal set of. Feature selecton s used n many applcatons to remove rrelevant and redundant where there are hgh dmensonal datasets. These datasets may contan a hgh degree of rrelevant and redundant that may decrease the performance of the classfers. The man approaches that are used for solvng feature selecton problems can be classfed nto flter or wrapper approaches, dependng on whether or not feature selecton s done ndependently of the classfer. Some researchers use hybrd methods to take advantage of these two approaches and to handle large datasets [5]. Feature selecton can be seen as an optmsaton problem that nvolves searchng the space of possble feature subsets to dentfy the optmal one. Many optmsaton technques such as genetc algorthms (GAs) [1], tabu search (TS), smulated annealng (SA) and ant colony optmsaton algorthms (ACO) have been used for solvng feature selecton. Ths paper ntroduces an mplementaton for ACO (usng Ant Colony System (ACS)) to solve feature selecton problems as explaned n Secton IV. The rest of ths paper s organsed as follows. Secton II ntroduces the bascs of ACO. Secton III addresses the fundamentals of ACS on whch the proposed algorthm was developed. The fourth secton brefly descrbes some of ACO algorhtms used to solve feature selecton problem. The ffth secton explans the proposed algorthm. Secton VI detals the experments carred out and presents the results. The dscusson of the results s gven n Secton VII. Secton VIII concludes ths paper and hghlghts future work n ths area. II. ANT COLONY OPTIMISATION Real ants are able to fnd the shortest path between ther nest and food sources because of the chemcal substance (pheromone) that they depost on ther way. The pheromone evaporates over tme so the shortest paths wll contan more pheromone and wll subsequently attract a greater number of ants /10 $ IEEE DOI /eScenceW

3 ACO algorthms smulate the foragng behavor of some ant speces [6]. ACO algorthms use two factors for gudng the search process. These are: 1) the pheromone values (numercal values as a smulaton for the pheromone that real ants depost on ther way to and from ther nest), and 2) heurstc nformaton for the selecton of soluton component values. There are two types of heurstc nformaton used by ACO algorthms; statc heurstc nformaton (that s computed at ntalsaton tme and then remans unchanged throughout the whole algorthm s run, such as the dstances between ctes n the travellng salesman problem (TSP)) and dynamc heurstc nformaton (that depends on the partal soluton constructed so far and therefore s computed at each step of an ant s walk) [7]. One of the recent trends n ACO s to solve ndustral problems provng that t s useful for real-world applcatons [8]-[9]. Recently, researchers have adopted ACO to solve feature selecton problems ([10]-[14]). We wll hghlght the man dfferences between these algorhtms and the proposed algorthm n the followng sectons. In tradtonal ACO algorthms, the pheromone values are assocated wth the nodes or the edges of the constructon graph representng the problem (dependng on the chosen problem representaton), whch may also contan heurstc nformaton representng pror nformaton about the gven problem [7]-[8]. In the proposed algorthm, a constructon graph s not used. Pheromone s assocated wth each feature. Although there s no heurstc nformaton known n advance for ths type of optmsaton problems, t s used n the proposed algorthm n computng the move probablty that s used by each ant to select a partcular feature n each constructon step. The proporton of ants that choose a partcular feature s used as the heurstc nformaton. Ths s explaned n detal n Secton IV. III. ANT COLONY SYSTEM In ths secton, a varant of ACO s explaned, namely ant colony system (ACS) for solvng the travellng salesman problem. Ths s because many aspects of the applcaton have commonalty wth feature selecton problem. Ant colony system s consdered one of the most successful ACO algorthms [15]. Snce ts appearance, t has been used to solve a varety of optmsaton problems. ACS s a constructve algorthm where for each teraton, each ant chooses the next cty to be vsted (j) usng the pseudorandom proportonal rule that s computed accordng to the followng equaton: β arg max p { τ. η } ( ) l C N s l (1) Ths equaton s used by each ant at each constructon step to choose the next cty dependng on a random varable q unformly dstrbuted n [0, 1] and a parameter q 0 (0 q 0 1). The equaton s used f q q 0. After each constructon step, the local pheromone update s performed by all ants to the last edge traversed accordng to the followng equaton: τ + = ( 1 ϕ). τ ϕ. τ 0 (2) where ϕ (0,1] and τ 0 s the ntal pheromone. The local pheromone update leads to decreasng the pheromone values on the edges and hence encourages subsequent ants to choose other edges and to therefore produce dfferent solutons. Ths s essental to prevent pre-mature convergence. At the end of each teraton, the pheromone values are updated by the best ant only accordng to the followng equaton: τ (1 ρ). τ + ρ. Δτ f (, j) best tour, = τ otherwse where Δτ = 1/ Lbest and L best s the length of the tour constructed by the best ant. IV. RELATED WORK In ths secton, we explan brefly some of the ACO algorthms that have been used to solve feature selecton problem. An ACO approach called antselect for varable selecton n quanttatve structure actvty relatonshp (QSAR) has been developed by Izralev and Agrafots [10]. In antselect, a weght s assocated wth each feature and the selecton of each varable depends on t,.e., the varable wth a larger weght has a hgher probablty of beng than the ones wth smaller weghts. In Guntur et al. [11], antselect was used wth two extra : Only varables wth an nter-correlaton coeffcents less than 0.75 are, and The number of varables to be s fxed at the begnnng of the selecton process. Shen et al. [12] express the feature selecton problem n bnary notaton. The move probablty for any feature s zero or one. A one means that ths feature wll be and zero means t wll not. Sh et al. [13] used ths algorthm wth a dfferent dataset. The work of Shamspur et al. [14] s smlar to Shen et al. [12] but the move probablty depends on the weghts assocated wth the varables and the number of the varables by an ant s randomly chosen. V. THE PROPOSED ALGORITHM The proposed algorthm s a wrapper-based system that deals wth the problem of feature selecton as a bnary problem where a set of bnary bts (of a length equvalent to the number of the n the gven dataset) s assocated wth each ant to represent ts feature selecton. If the n th bt s a 1 ths means that the feature number n n the gven dataset s, otherwse ths feature s not. Thus, the concept of path n the tradtonal ACO algorthms s not meanngful here. At (3) 87

4 the start of the algorthm, the bts are randomly ntalsed to zeros and ones. The pheromone values are assocated wth the. At each constructon step, each ant selects a feature out of all the wth the probablty computed accordng to the followng equaton: Ρ = τ. Δτ (4) where τ s the pheromone value assocated wth feature and Δτ s the proporton of ants that have ths feature. It acts as heurstc nformaton that represents the desrablty of feature. In ACO algorthms, the desgn of the move probablty s crtcal. Here both the pheromone values and heurstc nformaton are used to compute t. After each constructon step, the local pheromone update s performed by all ants accordng to the followng equaton: τ + = ( 1 ϕ). τ ϕ. τ 0 (5) At the end of each teraton, the global pheromone update s performed accordng to the followng equaton: τ = ( 1 ρ). τ + ( ρ.1/ L ) β where L best s the number of by the best ant and ρ s the evaporaton rate. Ths equaton s used by the best ant at the end of each teraton and s appled to all that t has chosen. The updatng of the pheromone values s mportant n order to renforce those that lead to hgh qualty feature subsets. Features that belong to good solutons wll contan larger pheromone values. Consequently, these tend to be more often. The man steps of the proposed algorthm are as follows: Intalsaton: n ths phase, the parameters of the proposed algorthm are set. Construct a soluton by each ant: at each constructon step, each ant selects the next feature usng the move probablty that s calculated based on the pheromone values and the heurstc nformaton accordng to Equaton 4. Each feature can be at most once. An ant s soluton represents ts own feature subset. The length of each ant s feature subset does not need to be equal to other ants solutons. Buld the support vector machne (SVM) model: each ant passes ts feature subset to the classfer and receves ts accuracy. Update the best soluton and the best ant based on the accuracy of the classfer. best (6) Update the pheromone values usng Equaton 6 by the best ant to ts subset. If a predefned maxmum number of teratons s reached, the algorthm halts and outputs the best soluton encountered. VI. COMPUTATIONAL EXPERIENCE In order to test the proposed algorthm, a SVM learnng algorthm s used. Although not all machne learnng algorthms requre the phase of feature selecton, feature selecton s mportant n buldng a SVM-based classfcaton [16]. We used the accuracy of the classfer as a ftness functon where each ant evaluates ts soluton based on ts rato of correct classfcatons. Classfer accuracy ndcates how accurately a gven classfer wll label future data on whch the classfer has not been traned. One of the ways used to estmate classfer accuracy s k-fold cross valdaton n whch the ntal data are randomly parttoned nto k mutually exclusve folds s 1, s 2,, s k each of whch s of approxmately equal sze. Tranng and testng s performed k tmes. The classfer of the frst teraton s traned on folds s 2, s 3,., s k and the frst fold s kept for testng. In teraton, the fold s s the test fold and the remanng folds are collectvely used to tran the classfer [4]. In ths case, the classfer accuracy s computed accordng to the followng equaton: classfer accuracy = Overall number of correct classfcatons from k teratons The total number of samples n the ntal data A. Datasets In order to test the proposed algorthm, ten experments usng ten artfcal and real world datasets were carred out. The datasets comprse two categores: datasets from statstcal datasets [17] and datasets from the UCI (Unversty of Calforna, Irvne) machne learnng repostory. The latter s a collecton of databases, doman theores, and data generators that are used all over the world as a prmary source of machne learnng datasets for the emprcal analyss of machne learnng algorthms [18]. The detals of the datasets are shown n Table I. (7) 88

5 TABLE I. Dataset Name THE DETAILS OF THE USED DATASETS. classes Class label type nstances Statstcal datasets Backache 32 2 Numerc 180 Prnn_vrus Numerc 38 Prnn_vruses 17 6 Numerc 61 Analcatdata_authorshp 70 4 Numerc 841 Analcatdata_marketng 32 5 Nomnal 364 UCI datasets Kdd_synthetc_control 61 6 Nomnal 600 Sonar 60 2 Numerc 208 Vehcle 18 4 Numerc 846 Dermatology 34 6 Nomnal 366 Wne 13 3 Nomnal 178 B. Method In our experments, the followng two systems were developed: SVM: that uses the entre set of (wthout the phase of feature selecton) [19], and SVM-FS: that uses a subset of by the proposed algorthm (wth the phase of feature selecton). In ths paper, we focus on testng the effect of feature selecton on the performance of the classfer. The performance of SVM tself was not optmsed, although further nvestgaton s requred snce t affects the performance of the whole system. We used the default values to ts parameters n both cases, wth and wthout the use of feature selecton. The C- classfcaton SVM of package e1071 of the R language wth the default values to ts parameters was used. In these two systems, 5-fold cross valdaton was used. The number of ants was set to the number of the n the gven dataset. The ntal pheromone was set to 1. The number of teratons s 10 teratons. ρ was set to 0.4 for all experments except for datasets that contan more than 30 for whch t was 0.3. β was set to 0.2. φ was set to 0.2 and α to 1. After many experments, these values of the parameters were used snce they gve the best performance of the proposed algorthm. C. Results Table II shows the results of these two systems usng the above mentoned datasets. The results for the algorthm represent the average of ten ndependent runs. These systems are mplemented usng the R language [20]-[21] and the WEKA machne learnng tool [22]-[23]. All the experments were run on a PC wth a 2 GHz CPU and 2 GB RAM. TABLE II. Dataset Name THE ACCURACY OF SVM WITH AND WITHOUT THE USE OF FEATURE SELECTION. orgnal Avg. no. of usng ACS_SVM SVM (wthout FS) SVM-FS (wth FS) Statstcal datasets Backache Prnn_vrus Prnn_vruses Analcatdata_authorshp Analcatdata_marketng UCI datasets Kdd_synthetc_control Sonar Vehcle Dermatology Wne The prevous results show that SVM-FS wth the proposed algorthm for performng feature selecton outperforms SVM that uses all the n all the datasets (the accuracy of SVM (the last column) s larger than that of SVM wth all (the fourth column)). The number of by the proposed algorthm (the thrd column) s sgnfcantly smaller than the total number of the n the orgnal datasets (the second column) n all of the datasets. It should be noted that the proposed algorthm can gve better results f we set the parameters ndvdually for each dataset. Table III compares some of the results of the proposed algorthm wth the results of other feature selecton algorthms from the lterature as follows: 1. Genetc algorthm (GA) for feature selecton wth the use of DstAI classfer (GADstAI). These results are taken from Yang and Honavar [1]. GAs are one of the most common approaches for feature selecton. 2. Partcle swarm optmsaton (PSO). These results are taken from Tu et al. [24]. 3. Chaotc bnary partcle swarm optmsaton (CBPSO). These results are taken from Chuang et al. [25]. 89

6 TABLE III. COMPARISON BETWEEN THE PROPOSED ALGORITHM AND OTHER FEATURE SELECTION ALGORITHMS Dataset Name PSO_SVM CBPSO_KNN GADstAI ACS_SVM (proposed) Avg. no. of accuracy Avg. no. of accuracy Avg. no. of accuracy Avg. no. of accuracy Sonar Vehcle Wne Ths comparson shows that the proposed algorthm outperforms the other approaches n solvng feature selecton problems n terms of the accuracy of the classfer. The accuracy of SVM compared wth the proposed algorthm (the last column) s hgher than that of the other approaches (the thrd, the ffth, and the seventh columns) n all of the datasets used n ths comparson. VII. DISCUSSION The proposed algorthm deals wth the feature selecton problem as a bnary problem and ntegrates ths nto an ACS framework. The proposed algorthm s general as t can be used wth any dataset as shown n the prevous sectons (snce these datasets are from dfferent domans). Ths s motvated by the need for feature selecton n dfferent applcaton areas. One example, the dermatology dataset, s llustrated here. It s known that the dfferental dagnoss of erythemato-squamous dseases s a real problem n dermatology. They all share the clncal of erythema and scalng, wth very few dfferences. The dseases n ths group are psorass, seborec dermatts, lchen planus, ptyrass rosea, cronc dermatts, and ptyrass rubra plars. Usually a bopsy s necessary for the dagnoss but unfortunately these dseases share many hstopathologcal as well. Another dffculty for the dfferental dagnoss s that a dsease may show the of another dsease at the begnnng stage and may have the characterstc at the later stages. Patents were frst evaluated clncally wth 12. Afterwards, skn samples were taken for the evaluaton of 22 hstopathologcal. The values of the hstopathologcal are determned by an analyss of the samples under a mcroscope. All of that makes selectng the most nformatve s hghly desrable n many applcatons related to ths doman [26]. The mert of the proposed algorthm s that t s a combnaton between the tradtonal ACO and the ACO algorthms that deals wth feature selecton as a bnary problem. The man characterstcs of the proposed algorthm are as follows: The move probablty s not zero or one as n Shen et al. [12] or uses only the weght (pheromone) assocated wth each feature as n Izralev and Agrafots [10]. Yet, t uses a smlar equaton to ACS where two factors are used to compute t. These two factors are the old pheromone value assocated wth a feature and the heurstc nformaton. Ths heurstc nformaton ndcates how often a partcular feature has been chosen by dfferent ants. Therefore the value of the move probablty depends on the result of the equaton. It uses the local pheromone update as n ACS. Usng the local pheromone update leads to decreasng the pheromone values on the that encourages subsequent ants to choose other and subsequently produce dverse solutons. A dfferent global pheromone update equaton rather than the usual one s used for feature selecton. Ths equaton s used by the best ant as n ACS. The global pheromone update n ths algorthm uses the old pheromone value assocated wth the feature multpled by the nverse of the length of the best ant s soluton. The length of the soluton of each ant s not fxed as n Guntur et al. [11] and n Shamspur et al. [14] allowng each ant to construct ts soluton wthout any pror restrcton. The only crteron used n evaluatng the constructed soluton by each ant s the accuracy of the classfer. VIII. CONCLUSIONS AND FUTURE WORK In ths paper, the feature selecton problem for classfcaton was solved usng an ant colony system approach on an SVM classfer wth ten artfcal and real world datasets. The proposed algorthm deals wth the feature selecton problem as a bnary one. We used heurstc nformaton n order to gude the search process besdes the pheromone values as n most of the conventonal ACO algorthms. The results are promsng n terms of the soluton qualty and the number of n all the datasets. Although the obtaned results are encouragng, further nvestgaton nto the adjustment of the values of the large number of parameters that the proposed algorthm has, s requred. These are ts parameters besdes the parameters of SVM. All of these parameters need to be adjusted so that the performance of the proposed algorthm s enhanced. In order to sensbly attempt ths parameter exploraton, as well as to be able to solve large real-world feature selecton problems n reasonable amounts of computatonal tme, the use of parallel and hgh performance computaton wll be necessary. It should be noted that by usng ensemble feature selecton the performance of the proposed algorthm could be enhanced, especally when dealng wth large datasets. Recently, the ensemble feature selecton technque has been used to select dfferent feature subsets for each base learner to construct an ensemble classfer. Ths s exemplfed by L et al. [27] who they used dynamc Adaboost learnng wth feature 90

7 selecton based on a parallel genetc algorthm. Another research drecton s to use hgh performance computng technques such as parallel multcategory support vector machnes [28]. Ths s because usng support vector machnes for large datasets s very computatonally ntensve and usng hgh performance computng technques can mprove ts performance. For example, the experments of Zhang et al. [28] showed that usng parallel multcategory support vector machnes enhanced the accuracy of the classfcaton. ACKNOWLEDGMENT The authors would lke to thank Dr. Andrew Lews for hs valuable comments on ths paper. REFERENCES [1] Yang, J., Honavar, V.: Feature subset selecton usng a genetc algorthm, IEEE Intellgent Systems, vol. 13, no. 2, pp , (1998) [2] Bandyopadhyay, S., and Pal, S.: Classfcaton and Learnng Usng Genetc Algorthms-Applcatons n Bonformatcs and Web Intellgence, Sprnger-Verlag Berln Hedelberg, (2007) [3] Duda, R.O., Hart, P.E.: Pattern classfcaton and scene analyss, John Wley & Sons, Inc. (1973) [4] Han, J., and Kamber, M.: Data mnng concepts and technques, Academc Press, ( 2001) [5] Lu, H., Motoda, H.: Feature Selecton for Knowledge Dscovery and Data Mnng, Kluwer Academc, Norwell, MA, (1998) [6] Dorgo, M., Bonabeou, E., and Theraulaz, G.: Inspraton for optmzaton from socal nsect behavor, Nature, vol. 406, pp , (2000) [7] Dorgo, M., and Stutzle, T.: Ant Colony Optmzaton, MIT Press, (2004) [8] Dorgo, M., Brattar, M., and Stutzle, T.: Ant colony optmzatonartfcal ants as a computatonal ntellgence technque, IEEE computatonal ntellgence magazne, (2006) [9] Dorgo, M., Caro, G.D., and Sampels, M.: Ant Algorthms, Sprnger- Verlag (2002) [10] Izralev, S., Agrafots, D.: Varable selecton for QSAR by artfcal ant colony systems, SAR QSAR n envronmental research, vol. 13, pp , (2002) [11] Guntur, S., Narayanan, R., Khandelwal, A.: In slco ADME modelng 2: Computatonal models to predct human serum albumn bndng affnty usng ant colony systems, Bo-organcs & Medcnal Chemstry, vol. 14, pp , (2006) [12] Shen, Q., Jang, J.H., Tao, J.C., Shen, G.L., and Yu, R.Q.: Modfed Ant Colony Optmzaton Algorthm for Varable Selecton n QSAR Modelng: QSAR Studes of Cyclooxygenase Inhbtors. J Chem. Inf. Model, vol. 45, pp , (2005) [13] Sh, W., Shen, Q., Kong, W., and Ye, B.: QSAR analyss of tyrosne knase nhbtor usng modfed ant colony optmzaton and multple lnear regresson, European Journal of Medcnal Chemstry. vol. 42, pp , (2007) [14] Shamspur, M., Zare-Shahabad, V., Hemmateenejad, B., and Akhond, M.: Ant colony optmzaton: a powerful tool for wavelength selecton, Journal of Chemometrcs, vol. 20, pp , (2006) [15] Dorgo, M., and Gambardella, L.M.: Ant colony system: A cooperatng learnng approach to the travelng salesman problem, IEEE Transactons on evolutonary computaton, vol.1, no. 1, pp , (1997) [16] Haste, T., Tbshran, R., Fredman, J.: The Elements of Statstcal Learnng-Data Mnng, Inference, and Predcton, Second Edton, Sprnger, (2008) [17] vewed 25th June [18] Murphy, P.M., Aha, D.W.: UCI Repostory of machne learnng databases, Department of nformaton and computer scence, Unversty of Calforna, Irvne, CA, vewed 25th June [19] Abd-Alsabour, N.: Feature Selecton for Classfcaton Usng an Ant System Approach, n Hnchey, M., et al. (Eds.): IFIP Advances n Informaton and Communcaton Technology, vol. 329, pp , (2010) [20] R: A Language and Envronment for Statstcal Computng 2006 [ R Foundaton for Statstcal Computng, Venna, Austra. [21] Dalgaard, P.: Introductory statstcs wth R, Sprnger, (2008) [22] vewed 25th June [23] Wtten, I., and Frank, E.: Data Mnng: Practcal machne learnng tools and technques, 2nd Edton, Morgan Kaufmann, San Francsco, (2005) [24] Tu, C.J., Chuang, L.Y., Chang, J.Y, and Yang, C.H.: Feature Selecton usng PSO-SVM, Internatonal Journal of Computer Scence, vol. 18, pp. 33-1, (2007) [25] Chuang, L.Y., L, J.C., and Yang, C.H.: Chaotc Bnary Partcle Swarm Optmzaton for Feature Selecton usng Logstc Map, Proceedngs of the Internatonal Conference of Engneers and Computer Scentsts, vol. I, Hong Kong (2008) [26] vewed 10th October [27] L, R., Lu, J., Zhang, Y., Zhao, T.: Dynamc Adaboost learnng wth feature selecton based on parallel genetc algorthm for mage annotaton, Knowledge-Based Systems, vol. 23, pp , (2010) [28] Zhang, C., L, P., Rajendran, A., Deng Y., and Chen D.: Parallelzaton of multcategory support vector machnes (PMC-SVM) for classfyng mcroarray data, BMC Bonformatcs, vol. 7, Suppl. 4, (2006) 91