A comparative study of fuzzy and neural network approaches to discriminant analysis with linguistic variables

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1 J. Idia Ist. Sci., Sep. Oct. 005, 85, Idia Istitute of Sciece. A comparative study of fuzzy ad eural etwork approaches to discrimiat aalysis with liguistic variables CHANDAN CHAKRABORTY AND DEBJANI CHAKRABORTY* Departmet of Mathematics, Idia Istitute of Techology, Kharagpur 7 30, Idia. debjai@maths.iitkgp.eret.i Received o Jue 5, 005; Revised o August, 005 Abstract This paper proposes a fuzzy discrimiat aalysis to solve the two-group classificatio problem where the measured variables are liguistic i ature. Especially uder imprecise framework, the liguistic variables capture more iformatio although vagueess is iheret. I aalogy to classical statistics, a fuzzy liear discrimiat fuctio is itroduced here, which directly deals with cotiuous fuzzy umbers as the represetative of liguistic values to obtai fuzzy scores for classificatio. To make a comparative study, the backpropagatio eural etwork approach has also bee studied i this paper. Fially admissio to maagemet programme is cosidered as a eample of the applicatio o two-level classificatio problem of the proposed method. Keywords: Liguistic variable, fuzzy umber, liear fuzzy discrimiat aalysis, eural etwork.. Itroductio Discrimiat aalysis [] for hard classificatio has bee widely applied for effective decisio-makig i may real-world problems for the last few decades. It becomes more effective for the followig advatages: firstly, separatio of classes as much as possible based o the measured variables ad secodly, classificatio of a ew etity ito a labeled class. As a multivariate data aalytic techique, the discrimiat aalysis is usually strogly recommeded for classificatio problems with precise data. But it becomes problematic if at least some of the variables are liguistic (i.e. qualitative) while desigig the classificatio problems uder imprecise eviromet. It must be logically accepted that the variables assessig the values as outcomes of huma factors, especially eperiece, perceptio, thikig, reasoig ad attitude, etc. become fuzzy. I practice, liguistic variables possess the liguistic or fuzzy values represeted i terms of atural laguages due to their fleibility ad simplicity although vagueess ad ambiguity are iheret. Therefore, there is a eed of developig fuzzy discrimiat techique for the classificatio problems o the basis of liguistic variables uder the paradigm of fuzzy statistics. The liguistic values are perfectly quatized by fuzzy sets ad subsequetly fuzzy umbers [, 3]. I doig so this paper itroduces a fuzzy approach to solve a two-group classificatio problem i discrimiat aalysis. Firstly, the fuzzy discrimiat scores for each of the cadidates computed ad the defuzzified to compare with the defuzzified threshold value for classificatio. I fact, *Author for correspodece.

2 66 CHANDAN CHAKRABORTY AND DEBJANI CHAKRABORTY we use here cetroid method [3] for defuzzificatio. Amog may statistical ad otraditioal techiques of classificatio, eural etwork has bee widely used for classificatio [4 6]. I may situatios, eural etwork approimates cotiuous fuzzy data ito discrete iputs. I this coectio, we have studied a backpropagatio eural etwork approach to make a comparative study where the liguistic data are iitially defuzzified. Here, we have cosidered a two-level classificatio problem to study whether the studets are eligible for admissio or ot i the maagemet programme. Some research developmets o fuzzy discrimiat aalysis have bee doe recetly. Li et al. [7] have show a fuzzy method for two-group discrimiat aalysis where the membership fuctios of the groups to be discrimiated is obtaied by miimizig the sum of squares of classificatio errors. A method for performig fuzzy multiple discrimiat aalysis [8] o groups of crisp data is proposed, which is able to detect membership fuctio of each group by miimizig the classificatio error usig geetic algorithm. A fuzzy mathematical programmig approach to develop fuzzy liear discrimiat fuctio has bee devised by Chiag et al. [9] for separable as well as oseparable data sets, which is ot at all based o fuzzy variables. Che et al. [0] have proposed a discrimiatio techique for chemical data sets with a few overlappig data poits that are cosidered equally importat for all classes i ordiary discrimiat aalysis. Watada et al. [] have proposed a fuzzy discrimiatio method oly for fuzzy data i fuzzy groups. A Hopfiled eural etwork approach [6] for classificatio is itroduced where the icomplete patter is first traslated ito fuzzy terms, but these terms have bee discretized. I fact, all the techiques for fuzzy classificatio are ot subjected to the liguistic data at all. Rather the methods cosider degrees of membership i discrete form that is othig but oversimplificatio. Sectio describes the defiitios ad otatios of fuzzy variable, -type fuzzy umber i Sub-sectios. ad., respectively. I Sub-sectio.3, the variace-covariace for fuzzy variables is itroduced. A fuzzy statistical approach to discrimiat aalysis with liguistic variables is described i Sub-sectio 3.. Sub-sectio 3. highlights the backpropagatio eural etwork framework for classificatio with liguistic iput. Fially, a twolevel classificatio problem whether the studets are eligible for admissio or ot i the maagemet programme is cosidered i Sectio 4. Coclusios are draw i Sectio 5.. Prelimiaries.. Fuzzy variable A fuzzy variable is a variable whose values are fuzzy. The cocept of fuzzy variable [3] is very useful i situatios where decisio problems are too comple or too ill-defied to be described properly usig covetioal quatitative epressios. For eample, the ability, performace ratigs, etc. could be well epressed usig fuzzy values such as very poor, poor, fair, good, very good, ecellet, etc... -type fuzzy umber Fuzzy set itroduced by Zadeh [] is a gradual trasitio from omembership to fullmembership. A fuzzy set = ( m, α, β) is said to be a -type fuzzy umber [3] where L ad R stad for left ad right refereces if

3 FUZZY AND NEURAL NETWORK APPROACHES TO DISCRIMINANT ANALYSIS 67 (i) µ is bouded ad upper semicotiuous; (ii) the membership fuctio µ is of the form µ m L for m α < m α ( ) = m R for m< m+ β β Now -type fuzzy umber reduces to a triagular fuzzy umber if L(y) = R(y) = ma {0, y }. Without loss of geerality we cosider -type fuzzy umbers as fuzzy realizatios of the fuzzy variables throughout the paper..3. Fuzzy variace covariace I classical sese, variace measures the dispersio aroud the cetral poit of a set of observatios, which is computed o the basis of crisp observatios. But whe we have fuzzy data, the computatio of crisp variace covariace is really oversimplificatio. Huma ituitio says that if the observatios are vague (i.e. ustable) the variace covariace will defiitely be imprecise i ature. I view of this, a cocept of computig fuzzy variace ad fuzzy covariace for fuzzy variables based o fuzzy arithmetic [3] is itroduced i this paper. Let us cosider two fuzzy variables, say, = ( m, α, β) ad Y = ( m, α, β ) defied o the uiverse say, U. A sample{(, y )} of size is draw y y y o ad Y, respectively. Now we defie fuzzy variace by as follows: S Y, Fuzzy meas: i S i ad fuzzy covariace by = ( m, α, β) where = m ; α i = αi i = ad β m = β () i Fuzzy variace: where ( i ) [(, α, β ) (, α, β)] S = = m m i i i = [( m m, α + α, β + β )] = ( s, α&, β & ) () s = ( m m ); i β & = ( m m )( β + β ). i i Therefore, i i i α& = ( m m )( α + α ); i i

4 68 CHANDAN CHAKRABORTY AND DEBJANI CHAKRABORTY Fuzzy covariace: S = ( )( ) Y, i Y i Y i = ( m m )( ), ( )( ) ( )( ), i my m i y m m i αy + α i y + my m i y α + α i = ( m m )( ) ( )( ) i βy β i y my m i y β β i = ( s, α&, β & ) (3) y y y where s = ( m m )( m m ); y i yi y α& = [( m m )( α + α ) + ( m m )( α + α )], ad y i yi y yi y i β& = [( m m )( β + β ) + ( m m )( β + β )]. y i yi y yi y i While makig a suitable decisio for certai purpose a decisio-maker should emphasize o precise ad cocrete decisio irrespective of fuzzy or crisp eviromet. Here we have used the well-kow cetroid method to defuzzify the computed fuzzy variace covariace for computig its iverse as follows: ad S = U U µ µ S = Y = ( d ) = ( d ) s s+ β& s s L d+ R d α& β& s α& s s s+ β& s s L d+ R d α& β& s α& s sy sy+ β& y y s t t sy tl dt + tr dt α& β& sy α& y y y sy sy sy+ β& y sy t t sy L dt + R dt α& β& sy α& y y y sy (4) (5) 3. Methodological developmet 3.. A fuzzy statistical approach to fuzzy liear discrimiat aalysis (FLDA) As a soft computig tool, fuzzy set theory [] has bee well established to deal with vagueess, which is especially reflected i all the atural laguages ad artificial itelligece-

5 FUZZY AND NEURAL NETWORK APPROACHES TO DISCRIMINANT ANALYSIS 69 related problems. I such situatios, while huma cogitive aspects like eperiece, kowledge, ad reasoig drive the classificatio problems, the data become imprecise or fuzzy i ature. Practically, there are may real-world problems where some of the decisio variables are purely fuzzy i ature. I those cases, cosiderig the crisp values, istead of fuzzy values, may be crude oversimplificatio. Fuzzy logic has the power of cosiderig the whole cotet of fuzzy values, which are represeted by fuzzy umbers. Here a paradigm of classificatio problem cosistig of all fuzzy variables is cosidered. Firstly, a umber of groups is cosidered ad hece collect the sample fuzzy data of differet sizes for each of the groups. Now, the classical discrimiat aalysis is eteded to fuzzy discrimiat aalysis for fuzzy data. Fuzzy discrimiat aalysis is a fuzzy statistical approach to model with fuzzy variables. Let us desig the problem iterface as follows: (a) a umber of fuzzy variables is cosidered; ad (b) a ew etity is to be classified ito oe of the labeled groups. For the sake of simplicity we assume here oly two classes, say C ad C. All the variables, say,, K, m are cosidered here as fuzzy where a crisp value ca be treated as fuzzy umber with zero vagueess. The fuzzy data of fuzzy variables ca be sematically represeted as follows: Let us cosider a fuzzy vector as = [,, K, ] T m such that ( k) ( k) j = ( m, α, ) j β j ad ij = ( m,, ) j α ij β ij, where ( k) ij deote the fuzzy ij opiio of the i th (i =,,..., ) perso for the j th (j =,,, m) criterio i the k th (here k =, ) group. Two groups are to be discrimiated based o m triagular fuzzy variables. The key idea here is to trasform the multifuzzy variables (,, K, m) ito uivariate fuzzy variable Z such that Z s derived from two groups are separated as much as possible. I doig so, the fuzzy meas usig eq () are computed as follows: Table I Fuzzy values of fuzzy variables ( k) ( ) ( ) (, k, k ) j j j = m α β j Sample Class L m () () () L m () () () L m M C () () () M i i L im i M () () () L m () () () L m () () () L m M C M i M () () () i i L im () () () L m ad t = ( m, α, β ) t t t (6)

6 70 CHANDAN CHAKRABORTY AND DEBJANI CHAKRABORTY where m ( k) ( k) ( k) ( k) ( k) ( k) =,, m j α = ij α j β = ij β j ij m ( k) ( k) ( k) ( k) ( k) ( k) = m,, t α = α it t β = β it t it. (7) Therefore, the fuzzy mea vector ad covariace matri for the k th group based o observed fuzzy observatios ca be theoretically calculated usig equatios () (5) as follows: ( m, α, β ) ( m, α, β ) ( k) ad S M M ( k) m ( m, α, β ) m m m ( k) ( k) = = = = L L L L sm sm s L mm s s L s m ( k) ( k) s s L sm (( sjt )) m m Hece, the objective is to select the liear combiatio of fuzzy variables to achieve maimum separatio of fuzzy sample meas Z ad Z for C ad C, respectively. Let us cosider the fuzzy discrimiat fuctio as a liear combiatio of,, K, m i.e. Z = b b + L b = b, (8) m m where ad deote the eteded sum ad multiplicatio operator, respectively. Suppose ad umbers of fuzzy resposes are sampled from the two groups say, ( Z, Z, L, Z ) ad ( Z, Z, L, Z ). Now the fuzzy discrimiat aes for two groups are calculated as Z = b () ; Z = b () ad ( Z j Z ) + ( Z j Z ) j= j= Var ( Z ) =. + The pooled covariace matri for all the groups is computed by S p = k ( ) S The separatio betwee two groups is defied by k ( ) k ( k). squared distace betwee sample meas of Z sample variace of Z

7 FUZZY AND NEURAL NETWORK APPROACHES TO DISCRIMINANT ANALYSIS 7 ( Z Z ) = = Var ( Z ) / () / () ( b b ). / bs pb ˆ () () b = S from the liear The maimum is achieved for the choice of b where ( ) / programmig: Ma / () / () ( b b ). / b bs pb The deviatios betwee two fuzzy vectors are calculated as follows: () () () () () () ( m m, α + α, β + β ) () () () () () () () () ( m m, α α, β β ) + + =. M () () () () () () ( m m, α + α, β + β ) m m m m m m Therefore, the estimated fuzzy discrimiat fuctio from eq (8) is formulated as follows: Z = ( ) S. (9) () () T p Ad the fuzzy threshold o the basis of which a fuzzy classificatio rule is set up is computed m = ( Z + Z ), () () T () where Z = ( ) Spooled ad () () T () Z = ( ) Spooled. Therefore, the fuzzy discrimiat scores for two groups are obtaied based o the fuzzy discrimiat aes ad the etities are categorized based o the defuzzified value of m, deoted by d( m ). Here, we actually have applied the cetroid method for defuzzificatio. The obtaied fuzzy scores are also defuzzifed to make a better compariso with d( m ). Hece, a fuzzy classificatio rule ca be formulated as follows: IF ( is ) & ( is ) & & ( m is m) THEN decisio class is G if d( Z ) d( m ). G if d( Z ) < d( m ) 3.. Backpropagatio eural etwork framework The most commoly used ANN is the feedforward etwork traied usig the backpropagatio algorithm [], which is adopted i the preset study. The eural etwork model is desiged with the liguistic variables (first layer) where the liguistic values are defied i the secod layer, called fuzzificatio layer. The the third layer iputs defuzzified values that costitute a represetative of fuzzy rule. p

8 7 CHANDAN CHAKRABORTY AND DEBJANI CHAKRABORTY FIG.. Backpropagatio eural etwork topology. Now the backpropagatio algorithm ca be described i three equatios for classificatio. First, weight coectios are chaged i each learig step (k) with Secod, for output odes it holds that ad third, for the remaiig odes it hold that [ s] [ s] [ s ] [ s] ij( k) ηδ pj j ij( k ). w = + m w (0) [ o] / [ s] pj dj oj f j I j δ = ( ) ( ) () [ s ] / [ ] [ ] [ ] ( s ) s s pj f j I j pj wjk k δ = δ + + () where [ s] [ s] j is the actual output of ode j i layer s; w ij, the weight of the coectio betwee ode i at layer (s ) ad ode j at layer (s); δ pj, the measure for the actual error of [ s] [s] ode j; I j, the weighted sum of the iputs of ode j i layer s; η, the time-depedet learig rate; f( ), the trasfer fuctio; m, the mometum factor (betwee 0 ad ); ad d j, o j are the desired ad actual activity of ode j (for output odes oly). Parameter values (i.e. the learig rate η, mometum factor m, ad the umber of hidde odes h j ) are selected eperimetally. The iput ad output odes are selected accordig to the liguistic variables ad class of the objects to be classified. 4. Eample: Admissio to maagemet programme A practical eample o admissio to maagemet programme i a busiess school has bee cosidered here to employ the proposed methodology. There are several factors o the basis of which studets are to be evaluated by the evaluators. Especially i maagemet programme, it becomes importat to focus o some cogitive factors [3]. Perceptio, attitude reasoig ad thikig, etc. are the cogitive factors. Here we are cosiderig oly three major factors: JMAT score, Work eperiece ad Overall performace (commuicatio

9 FUZZY AND NEURAL NETWORK APPROACHES TO DISCRIMINANT ANALYSIS 73 FIG.. Liguistic scale for. FIG. 3. Liguistic scale for ad 3. skill ad other abilities) of a cadidate. I coectio with eperiece, kowledge, perceptio, attitude, reasoig ad thikig, the factors i fact lead to three fuzzy variables. The variables cosider the eperts fuzzy resposes for evaluatio of a cadidate whether s/he will be either admitted or ot admitted ito the programme. Therefore, we have two groups: Admitted (G ) ad Not-admitted (G ) ad three fuzzy variables: : JMAT score; : Work eperiece ad 3: Overall performace. Also the fuzzy scales for the variables are defied o the domai [0, 00] accordig to epert. Here, a data set of size 60 (see Appedi A) has bee surveyed durig the admissio of the maagemet programme where the first 45 data are set as traiig data ad the last 5 are to be tested usig fuzzy liear discrimiat aalysis ad eural etwork. 4.. Results: fuzzy liear discrimiat aalysis (FLDA) Here the fuzzy meas ad variace covariace are computed usig eqs () (5) for two classes. (a) Class (C ): Admitted (A) = (79., 5.,.9) = (73.6, 33.3,.) = (6., 33.3, 7.8) 3 (b) Class (C ): Not-Admitted (NA) ad S A = = (4.7, 3.9, 5) = (57., 30., 5.4) = (4.3, 3., 33.3) ad S NA = Therefore, the pooled estimated covariace matri is computed by S ( S S ) Hece, the liear fuzzy discrimiat fuctio usig eq (9) is obtaied as = +. p A NA Z = (0.7, 0.6, 0.0) (0.53, 0.8, 0.6) (0.04, 0.7, 0.5) (0) 3

10 CHANDAN CHAKRABORTY RTY Table II Classificatio usig FLDA for 5 testig samples (Appedi) Sample Class 46 (8.69, 07.48, 63.49) Admitted (3.08,.89, 35.87) Admitted (0.7, 3.4, 53.49) Not Admitted (8.83, 7.74, ) Not Admitted (4.4, 00.98, 58.49) Admitted 5 53 (.75, 6.54, 47.0) Admitted (3.08, 68.53, 53.53) Admitted **55 -Admitted **56 4.4, 7.57, 40.84) Admitted (8.67, 34.07, 45.84) Admitted 58 -Admitted 59 -Admitted 60 (.75, 7.5, 47.4) Not-Admitted Therefore, the fuzzy threshold is automatically computed by the methodology as m = ( 47, 39.6, 5.7). After buildig the fuzzy discrimiat fuctio based o the traiig fuzzy data set, the traiig samples are tested i the followig. I doig so, the fuzzy discrimiat scores ad threshold value are defuzzified usig MATLAB 7.0 to classify the objects either i Admitted class or i Not-Admitted class. Now the tested results are give i Table II: It ca be observed from the above table that oly three samples (marked by * ) have bee misclassified. 4.. Results: eural etwork approach The etwork is iitially traied with 45 samples where the error history is depicted i Fig. 4. Here, we have cosidered four odes i the hidde layer with η = 0.90 ad m = FIG. 4. History of error o traiig data.

11 FUZZY AND NEURAL NETWORK APPROACHES TO DISCRIMINANT ANALYSIS 75 Table III Classificatio by backpropagatio eural etwork for the testig data The eural etwork approach to fuzzy variables (Table III) leads to five samples (from 53 to 57) to be misclassified (marked by ) by meas of comparig the predicted values with the actual values. 5. Coclusios The results i Tables II ad III obtaied by the proposed FLDA ad eural etwork depict almost the same classificatio ecept sample os 53 ad 54. Though sample 53 is correctly classified by eural etwork, sample o. 54 is ot. The reverse case ca be ivestigated i the case of FLDA. But it is recommedable here that both the methods are able to detect the misclassified etities. I FLDA, the fuzzy threshold is automatically computed o the basis of traiig liguistic data. The computatioal compleities of eural etwork are relatively higher tha FLDA while usig the cotiuous fuzzy umbers. Rather the proposed tool is a simplistic approach to obtai similar classificatio results i compariso with backpropagatio eural etwork. This method directly deals with the fuzzy umbers ad ehaces the fuzzy scores, those of which ca also be compared with fuzzy threshold by approimate reasoig [3]. This method ca also be applicable i desigig a fuzzy decisio-makig iterface for uderstadig group membership based o fuzzy perceptio. Ackowledgemet The authors are grateful to the aoymous referees for their valuable commets ad suggestios. The authors also ackowledge the support provided by the Coucil of Scietific ad Idustrial Research, Govt of Idia (scheme o. 9/8(5)/05-EMR-II), ad the Departmet of Sciece ad Techology, Govermet of Idia (DST/MS/57/0). Refereces. J. F. Hair, R. E. Aderso, R. Tatham, ad W. C. Black, Multivariate data aalysis, fifth editio, Aad Sos (003).. L. A. Zadeh, Fuzzy Sets, Ifo. Cotrol, 8, (965). 3. D. Dubois, ad H. Prade. Fuzzy sets ad systems (Theory ad Applicatios), Academic Press (980).

12 76 CHANDAN CHAKRABORTY AND DEBJANI CHAKRABORTY 4. K. Y. Tam ad N. Y. Kiag, Maagerial applicatios of eural etworks: the case of bak failure predictios, Mgmt Sci., 38, (99). 5. B. Baeses, R. Setioo, C. Mues, ad J. Vathiee, Usig eural etwork rule etractio ad decisio tables for credit-risk evaluatio, Mgmt Sci., 49, 3 39 (003). 6. Shouhog Wag, Classificatio with icomplete survey data: A Hopfield eural etwork approach, Computers Oper., Res., 3, (005). 7. C. C. Li, ad A. P. Che, A method for two group fuzzy discrimiat aalysis, It. J. Fuzzy Systems, 3, (00). 8. C. C. Li, ad A. P. Che, Fuzzy discrimiat aalysis with outlier detectio by geetic algorithm, Computers Oper. Res., 3, (004). 9. C. Chiag, Q. Liu, ad Y. Liu, Fuzzy liear programmig ad discrimiat aalysis, It. J. Fuzzy Systems, 3, (00). 0. Z. P. Che, J. H. Jiag, Y. Li, Y. Z. Liag, ad R. Q. Yu, Fuzzy liear discrimiat aalysis for chemical data sets, Chemometrics Itelliget Laboratory Systems, 45, (999).. J. Watada, H. Taaka, ad K. Asai, Fuzzy discrimiat aalysis i fuzzy groups, Fuzzy Sets Systems, 9, 6 7 (986).. D. E. Rumelhart, ad J. McClellad, Parallel distributed processig, Vol., MIT Press (986). 3. Chada Chakraborty, ad Debjai Chakraborty, A decisio scheme based o OWA operator for a evaluatio programme: A approimate reasoig approach, It. J. Appl. Soft Computig, 5, (004). Appedi A Fuzzy data set of size 60 (Traiig data: 5 ad Test data: 46 60) ID Class JMAT Work Overall score eperiece performace Admitted G G E Admitted VG E M 3 Admitted G E G 4 Admitted E G M 5 Admitted V G G 6 Admitted E G G 7 Admitted VG G M 8 Not-Admitted G M G 9 Not-Admitted S E G 0 Not-Admitted VG G M Not-Admitted VG M G Not-Admitted S E G 3 Not-Admitted G E M 4 Not-Admitted S E M 5 Not-Admitted G G M 6 Admitted G E M 7 Not-Admitted G M M 8 Admitted VG G G 9 Admitted E M G 0 Admitted E G M Not-Admitted S E G Not-Admitted S G M 3 Admitted VG G M 4 Admitted VG G G 5 Admitted VG E M 6 Not-Admitted G M P 7 Not-Admitted G P G 8 Admitted G E G 9 Not-Admitted S G P 30 Admitted VG M G ID Class JMAT Work Overall score eperiece performace 3 Admitted E E E 3 Admitted VG G M 33 Admitted E G E 34 Admitted VG M G 35 Admitted VG E E 36 Not-Admitted S G M 37 Not-Admitted G VG M 38 Not-Admitted S M M 39 Not-Admitted B G M 40 Not-Admitted VG M M 4 Not-Admitted G P G 4 Not-Admitted S M M 43 Admitted E G G 44 Admitted E G E 45 Admitted VG E G 46 Admitted VG G E 47 Admitted G E M 48 Admitted VG G G 49 Not-Admitted S P G 50 Not-Admitted G P M 5 Admitted VG G G 5 Admitted G E M 53 Admitted G M P 54 Admitted G M M 55 Admitted S P P 56 Not-Admitted G E E 57 Not-Admitted VG E G 58 Not-Admitted S P M 59 Not-Admitted B M P 60 Not-Admitted G P P E = Ecellet; VG = Very Good; G = Good; S = Satisfactory; M = Medium; P = Poor; B = Bad.

13 FUZZY AND NEURAL NETWORK APPROACHES TO DISCRIMINANT ANALYSIS 77 Appedi A Defuzzificatio usig Matlab 7.0 ID Class JMAT Work Overall score eperiece performace Admitted Admitted Admitted Admitted Admitted Admitted Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Admitted Not-Admitted Admitted Admitted Admitted Not-Admitted Not-Admitted Admitted Admitted Admitted Not-Admitted Not-Admitted Admitted Not-Admitted Admitted ID Class JMAT Work Overall score eperiece performace 3 Admitted Admitted Admitted Admitted Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Admitted Admitted Admitted Admitted Admitted Admitted Not-Admitted Not-Admitted Admitted Admitted Admitted Admitted Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted Not-Admitted