Fuzzy ID3 Decision Tree Approach for Network Reliability Estimation

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1 IJCSI Iteratoal Joural of Computer Scece Issues, Vol. 9, Issue 1, o 1, Jauary 2012 ISS (Ole): Fuzzy ID3 Decso Tree Approach for etwor Relablty Estmato A. Ashaumar Sgh 1, Momtaz Thgujam 2 1 Departmet of Computer Scece, Thoubal College, Mapur Uversty, Mapur , Ida. 2 Departmet of Computer Scece, Mapur Uversty, Imphal, Mapur Ida. Abstract The Computer commucato etwors have bee rapdly creasg recetly to share expesve hardware ad software resources, ad provde access to ma system from dstat locatos. The relablty ad the cost of these systems are mportat cosderatos that are largely determed by placemet of the odes ad the ls betwee odes. etwor Relablty Estmato s a process of predctg the efforts ad cost terms of moey, schedule ad staff for ay etwor system. I ths study, we vestgate the use of Fuzzy ID3 decso tree for etwor relablty estmato; t s desged by tegratg the prcples of ID3 decso tree ad the fuzzy settheoretc cocepts, eablg the model to hadle ucerta ad mprecse data whe descrbg the relabltes of the etwors, whch ca mprove greatly the accuracy of obtaed estmates. MMRE s used as measures of predcto accuracy for ths study. The results are compared wth those produced by the crsp verso of the ID3 decso tree. Keywords: etwor relablty, etwor desg, Decso Tree, Fuzzy ID3, Fuzzy Etropy. 1. Itroducto The desg of relable commucato etwors s a sgfcat problem the telecommucatos dustry. Improvg the accuracy of the relablty estmato models avalable to project maagers would facltate more effectve cotrol of tme ad budgets durg etwor desg. A mportat stage of etwor desg s to fd the best layout of compoets to mmze cost whle meetg a performace crtero such as trasmsso delay, throughput or relablty [1]. Geerally, a large scale etwor has a multlevel ad herarchcal structure cosstg of a bacboe etwor ad several local access etwors [2]. Ths paper s focused o large scale bacboe commucato etwor desg where the relevat relablty metrc s all termal etwor relablty [3]. I ths paper, we are cocered wth etwor cost estmato models based o fuzzy decso trees especally Fuzzy Iteractve Dchotomzer 3. There are three major advatages whe usg estmato by decso trees (DT). Frst, decso trees approach may be cosdered as whte boxes, t s smple to uderstad ad easy to expla ts process to the users, cotrary to other learg methods. Secod, t allows the learg from prevous stuatos ad outcomes. The learg crtero s very mportat for cost estmato models because etwor desg techology s supposed to be cotuously evolvg. Thrd, t may be used to feature subset selecto to avod the problem of cost drver selecto etwor cost estmato model. O the other had, fuzzy logc has bee used etwor cost estmato. It's based o fuzzy set theory, whch was troduced by Zadeh 1965 [4]. Attempts have bee made to rehabltate some of the exstg models order to hadle ucertates ad mprecso problems. The am of ths study s to evaluate ad to dscuss the use of fuzzy decso trees, especally the fuzzy ID3 algorthm desgg DT for etwor relablty estmato. Istead of crsp DT, fuzzy DT may allow to explot complemetary advatages of fuzzy logc theory whch s the ablty to deal wth exact ad ucerta formato whe descrbg the etwor desgg. The paper s orgazed as follows: I secto 2, we preset the fuzzy ID3 decso tree for etwor relablty estmato. The descrpto of datasets used to perform the emprcal studes ad the evaluato crtera adopted to measure the predctve accuracy of the desged models are gve secto 3. Secto 4 focuses o the expermetal desg. I Secto V, we preset ad dscuss the obtaed results whe the fuzzy ID3 s used to estmate the etwor desgg. A comparso of the estmato results produced by meas of the fuzzy ID3 Models ad the crsp ID3 model are also provded secto 5. A cocluso ad a overvew of future wor coclude the paper the last secto 6. 2.Fuzzy ID3 for etwor Relablty Estmato Based o the Cocept Learg System algorthm, Qula proposed a decso tree called the Iteractve Dchotomzer 3 (ID3). The ID3 techque s based o formato theory ad attempts to mmze the expected umber of comparsos. The fuzzy ID3 s based o a fuzzy mplemetato of the ID3 algorthm [5,6]. It s formed of oe root ode, whch s the tree top, or startg pot, ad a seres of other odes. Termal odes are leaves (effort). Each ode correspods to a splt o the values of oe put varable (cost drvers). Ths varable s chose order to reach a maxmum of homogeety amogst the examples that belog to the ode, relatvely to the output varable. Copyrght (c) 2012 Iteratoal Joural of Computer Scece Issues. All Rghts Reserved.

2 IJCSI Iteratoal Joural of Computer Scece Issues, Vol. 9, Issue 1, o 1, Jauary 2012 ISS (Ole): Fg. 1 A example of fuzzy ID3 decso tree for etwor relablty estmato. The major characterstc of fuzzy ID3 s that a example belogs to P of examples a ode to a certa degree. The proporto of wth classfcato a t ode s calculated usg the membershp degrees as follows: Where H s the etropy the ode. H l s the etropy of the ode that belogs to the fuzzy set L of the j varable. W l s the fuzzy relatve weght. The ode s splt to as may sub-odes as there are attrbutes. The algorthm termated whe all attrbutes are used for splts, or whe all examples at a ode have the same classfcato. The fuzzfcato of the etwor cost drvers coverts crsp cost drvers to membershp degrees to the dfferet fuzzy sets of the partto. May algorthms ca be foud the specalzed lterature for geeratg parttos from data, we choose the herarchcal fuzzy parttog [7]. It correspods to a ascedg procedure. At each step, for each gve varable, two fuzzy sets are merged. Ths method combes two dfferet clusterg techques, herarchcal clusterg ad fuzzy clusterg techques. The tragular membershp fuctos are used to represet the fuzzy sets because of ts smplcty, easy compreheso ad computatoal effcecy. Fgure 2 llustrates the membershp fucto assocated to the fuzzy sets of the SeTode attrbute. P Where K represets the classes ad s the umber of examples the subset. u ( y ) s the membershp degree o the attrbute that belogs to the class ad u ( x ) s the membershp degree of the attrbute at ode. Λ represets the cojucto operator. T-orm, whch geeralzes tersecto the doma of fuzzy sets, s usually used for fuzzy cojucto. The most popular T- orms are mmum ad product. The fuzzy etropy uses the membershp degree of examples at a partcular ode ad cotrbutes to ehace the dscrmatve power of a attrbute, s computed as: H p * log( p ) (2) 1 c1 1 u ( y ) u c u ( y ) u ( x ) ( x ) The growth of the fuzzy ID3 s realzed by expadg a ode of tree characterzed by the hghest formato ga. The formato ga s calculated as follows: (1) (a) Membershp fucto of 3 fuzzy sets defed for the SeTode cost drver (b) Membershp fucto of 5 fuzzy sets defed for the SeTode cost drver G j H M l1 w H l l (3) Copyrght (c) 2012 Iteratoal Joural of Computer Scece Issues. All Rghts Reserved.

3 IJCSI Iteratoal Joural of Computer Scece Issues, Vol. 9, Issue 1, o 1, Jauary 2012 ISS (Ole): (c) Membershp fucto of 7 fuzzy sets defed for the SeTode cost The MRE value are calculated for each attrbute the dataset, whle drver mea magtude of relatve error (MMRE) computes the average Fg. 2 Membershp fucto assocated to the fuzzy of the SeTode over tems attrbute. The fuzzy decso tree s terpreted by rules, each path of the braches from root to leaf ca be coverted to a rule wth codto part represets the attrbutes o the passg braches from the root to the leaf ad codto part represets the class at the leaf of the form: IF (codto 1 ad codto 2.. ad codto ) THE C, where the codtos are extracted from the odes ad C s the leaf. 3. Data Descrpto ad Evaluato Crtera 3.1 Data Descrpto Ths secto descrbes the dataset used to perform ths emprcal study ad the evaluato crtera adopted to measure the estmates accuracy of the desged etwor cost estmato model based o fuzzy ID3 method. The dataset s descrbed usg 9 umercal attrbutes as gve table 1. Table 1: Attrbutes for etwor cost estmato Attrbutes SeTode SeTL Lodej LRela DV LTopo TerRela RelaReq CostL Descrpto Set of odes (termal) Set of ls (edge, arcs) L betwee odes ad j L Relablty Decso varables L Topology All termal Relablty etwor Relablty Requremet Cost of L 1 MMRE Relabltyactual, Relablty Relablty I 1 actual, estmated, j 100 The acceptable target values for MMRE are MMRE 25. Ths dcates that o the average, the accuracy of the establshed estmato model would be less tha 25%. 4. Expermet Desg Ths secto descrbes the expermet desg of the fuzzy ID3 decso tree o the gve datasets. The use of fuzzy ID3 to estmate etwor desg effort requres the determato of the parameters, amely the umber of put varables, the maxmum umber of fuzzy sets for each put varable, the sgfcat level value ad the cojucto operator. The last two parameters play a essetal role the geerato of Fuzzy Decso trees. It greatly affects the calculato of fuzzy etropy ad classfcato results of Fuzzy Decso trees. The umber of put varables s the umber of the attrbutes descrbg the used dataset. Therefore, whe applyg fuzzy ID3 to the gve dataset, the umber of put varables s equal to 9. Cocerg the maxmum umber of fuzzy sets s the maxmum partto sze for each varable, s fxed to 7 for all expermets. I the preset paper we are terested studyg the mpact of the fuzzy cojucto operators (t-orms) ad the sgfcat level parameter (β) o the accuracy of fuzzy ID3. The sgfcat level s the membershp degree for a example to be cosdered as belogg to the ode. For each dataset, two models of fuzzy ID3 were geerated. The frst Fuzzy ID3 effort estmato model uses the product etropy cojucto operator to measure the fuzzy etropy (t-orm=product), ad the secod model uses the mmum etropy cojucto operator to calculate the fuzzy etropy t-orm=m). These cojucto operators are the two commoly used t-orm operators because of ther well behavor ad ther computatoal smplcty [8] The mmum etropy cojucto operator s defed as: u y ) u ( x ) m[ u ( y ), u ( x )] (6) ( Cocerg, the product etropy cojucto operator s gve as: (5) 3.2 Evaluato crtera We employ the followg crtera to measure the accuracy of the estmates geerated by the fuzzy ID3. A commo crtero for the evaluato of relablty estmato models s the magtude of relatve error (MRE), whch s defed as MRE Relablty Relablty actual estmated (4) Relabltyactual Where Relablty actual s the actual relablty of the etwor the dataset ad Relablty estmated s the estmated Relablty that was obtaed usg a model or a techque. u ( y ) u ( x ) u ( y ) * u ( x ) (7) For each model, a seres of expermets s coducted wth the fuzzy ID3 algorthm each tme usg a dfferet value of the sgfcat level parameter (β). The sgfcat level s vared wth the terval [0, 1]. 5. Results Ths secto presets ad dscusses the results obtaed whe applyg the fuzzy ID3 to the gve datasets. The calculatos were made usg MatLab software. We coducted several expermets Copyrght (c) 2012 Iteratoal Joural of Computer Scece Issues. All Rghts Reserved.

4 IJCSI Iteratoal Joural of Computer Scece Issues, Vol. 9, Issue 1, o 1, Jauary 2012 ISS (Ole): usg dfferet cofguratos of fuzzy ID3. For these expermets, a holdout valdato o the etre datasets was performed. Datasets were radomly splt to two groups: trag set ad test set. Two models of fuzzy ID3 were desged. The frst Fuzzy ID3 effort estmato model (Model 1) uses the formula of the cojucto operator gve Eq. (7) to compute fuzzy etropy, ad the secod model (Model 2) uses the formula of the cojucto operator gve Eq. (6). For each model, dfferet cofguratos have bee obtaed by varyg the sgfcat level (β). The am s to determe whch cofgurato mproves the estmates. We have traed ad tested the two models usg the gve dataset. The results for the dfferet cofguratos have bee compared. Fgure 3 show the accuracy of the two fuzzy ID3 models, measured terms of MMRE Table 3: Result of the dfferet models used o the gve dataset Performace crtera Crsp ID3 Model 1 Model 2 MMRE Fg. 3: Relatoshp betwee the accuracy of fuzzy ID3(MMRE), the used cojucto operator ad the SL value. Fgure 3 compares the accuracy of the two models, terms of MMRE, whe varyg the sgfcat level. We ote that the fuzzy ID3 model usg the product cojucto operator geerates a lower MMRE that the other model usg the mmum cojucto operator for sgfcat level value less tha 0.2. For example, for β=0.1 the model 1 geerates a lower predcto error (MMRE=2.45) tha the model 2 (MMRE=5.31). By agast, model 2 geerates a lower MMRE tha the model 1 for sgfcat level value greater tha or equal to 0.2. For example, for β=0.5 the model 2 geerate a lower predcto error (MMRE=9.09) tha the model 1 (MMRE=34.48). Table 2 summarzes the results obtaed usg dfferet cofguratos of fuzzy ID3 for the gve dataset. It shows the varato of the accuracy accordg to the sgfcat level value ad to the used cojucto operator. Table 2: MMRE results of dfferet fuzzy ID3 cofguratos for the gve dataset Accuracy of Fuzzy ID3 Sgfcat level (β) t-orm=product MMRE t-orm=mmum MMRE The expermetal results show that the fuzzy ID3 models show better estmato accuracy tha the crsp ID3 model terms of MMRE. 6. Coclusos I the paper, we have emprcally studed two fuzzy ID3 models for etwor relablty estmato. Each oe used a dfferet formula to compute the fuzzy etropy. These fuzzy ID3 models were traed ad tested usg the gve datasets. The results show that the use of a optmal sgfcat level value ad a adequate cojucto operator for computg the fuzzy etropy mproves greatly the estmates geerated by fuzzy ID3 model. The comparso wth the crsp verso of ID3 decso tree shows ecouragg results. Refereces [1] R.H.Ja, F.J.Hwag, ad S.T.Che, Topologcal optmzato of a commucato etwor subject to a relablty costrat, IEEE Trasactos o Relablty, Vol.42(1993), [2] R.R.Boorsty ad H.Fra, Large scale etwor topologcal optmzato, IEEE Trasactos o commucato, Com- 25(1977), 29-7 [3] C.J.Colbour, The Combatores of etwor Relablty, Oxford Uversty Press(1987). [4] L.A.Zadeh, Fuzzy sets, Iformato ad Cotrol, Vol. 8, 1965, [5] M.Umao, H.Oamoto, I.Hatoo, H.Tamura, F.Kawach, S.Umedzu, J.Koshta, Fuzzy Decso Trees by Fuzzy ID3 algorthm ad ts Applcatos to Dagoss System, I Copyrght (c) 2012 Iteratoal Joural of Computer Scece Issues. All Rghts Reserved.

5 IJCSI Iteratoal Joural of Computer Scece Issues, Vol. 9, Issue 1, o 1, Jauary 2012 ISS (Ole): proceedgs of the thrd IEEE Coferece o Fuzzy Systems, vol. 3, Orlado, 1994, [6] R.Weber, Fuzzy ID3: a class of methods for automatc owledge acqusto, proceedgs of the 2 d Iteratoal Coferece o Fuzzy Logc ad eural etwors, Izua, Japa, July, 17-22, 1992, [7] S.Gullaume ad B.Charomordc, Geeratg a terpretable famly of Fuzzy parttos, IEEE Trasactos o Fuzzy Systems, 12(3), Jue [8] E.H.Mamda, Applcatos of Fuzzy Set Theory to Cotrol Systems: A survey, Fuzzy Automata ad Decso Processes, M.M.Gupta, G..Sards ad B.R.Gaes, eds., orth-hollad, ew Yor, 1977, Bography: A. Ashaumar Sgh graduated Mathematcs from Mapur Uversty, Imphal ad passed MCA the year 2000 from the same varsty. He was awarded Ph.D. the area of Computer Scece from the Dept. of Mathematcs of the same varsty the year The produced eght M.Phl scholars Computer Scece ad ow supervsg three scholars leadg to Ph.D. Computer Scece. The area of research s o Soft computg ad related applcatos of computer scece. Ms. Momtaz Thgujam had receved her B.Tech degree (C.Sc.) from IET, Lucow U.P(IDIA) ad M.Tech degree from AAIDU, Allahabad U.P (IDIA) respectvely. She s curretly a Assstat Professor (Seor Scale) Computer Scece Departmet, Mapur Uversty. Prevously she was a Programmer Departmet of Scece ad Techology, Mapur ad Computer Ceter, Mapur Uversty. Her prmary research terests are atural Laguage Processg ad Computer etwor. She also wored as a Co- Ivestgator for a project (Developmet of Mapur laguage Techology) sposored by Departmet of Iformato Techology, ew Delh. Copyrght (c) 2012 Iteratoal Joural of Computer Scece Issues. All Rghts Reserved.