Comparison of the COG Defuzzification Technique and Its Variations to the GPA Index

Transcription

1 Compariso of the COG Defuzzificatio Techique ad Its Variatios to the GPA Idex Michael Gr. Voskoglou Departmet of Mathematical Scieces, School of Techological Applicatios Graduate Techological Educatioal Istitute (T. E. I.) of Wester Greece Patras, Greece Abstract The Ceter of Gravity (COG) method is oe of the most popular defuzzificatio techiques of fuzzy mathematics. I earlier works the COG techique was properly adapted to be used as a assessmet model (RFAM) ad several variatios of it (GRFAM, TFAM ad TpFAM) were also costructed for the same purpose. I this paper the outcomes of all these models are compared to the correspodig outcomes of a traditioal assessmet method of the bi-valued logic, the Grade Poit Average (GPA) Idex. Examples are also preseted illustratig our results. Keywords: Grade Poit Average (GPA) Idex, Ceter of Gravity (COG) Defuzzificatio Techique. Rectagular Fuzzy Assessmet Model (RFAM), Geeralized RFAM (GRFAM), Triagular (TFAM) ad Trapezoidal (TpFAM) Fuzzy Assessmet Models.. Itroductio Fuzzy Logic (FL), due to its ature of characterizig the ambiguous situatios of our day to day life by multiple values, offers rich resources for the assessmet of such kid of situatios. A characteristic example is the process of learig a subject matter, where the ew kowledge is frequetly coected to a degree of vagueess ad/or ucertaity from the learer s, as well as the teacher s poit of view. I 999 Voskoglou [0] developed a fuzzy model for the descriptio of the process of learig a subject matter i the classroom i terms of the possibilities of the studet profiles ad later he assessed the studet learig skills by calculatig the correspodig system s total possibilistic ucertaity []. Meawhile, Subboti et al. [], based o Voskoglou s model [0], adapted properly the frequetly used i fuzzy mathematics Ceter of Gravity (COG) defuzzificatio techique ad used it as a alterative assessmet method of studet learig skills. Sice the, Voskoglou ad Subboti, workig either joitly or idepedetly, applied the COG techique ad a umber of variatios of it for assessig several huma or machie (Decisio Makig, Case-Based Reasoig, etc.) skills, e.g. see [-7, -6],, etc. I the preset paper the outcomes of the COG techique ad its variatios are compared to the correspodig outcomes of a traditioal assessmet method of the bivalued logic, the Grade Poit Average (GPA) idex. The rest of the paper is formulated as follows: I Sectio we describe the classical GPA assessmet method. I Sectio 3 we sketch the use of the COG techique as a assessmet method, while i Sectio 4 we briefly describe the variatios of the COG techique costructed i earlier papers ad the reasos who led to the developmet of these variatios. I Sectio the outcomes of the COG techique ad its variatios

2 are compared to the outcomes of the GPA idex ad examples are preseted to illustrate our results. The last Sectio 6 is devoted to our coclusio ad a discussio o the perspectives for future research o the subject.. Traditioal Assessmet Methods The assessmet methods which are commoly used i practice are based o priciples of the bi-valued logic. The calculatio of the mea value of the scores achieved by each oe of its members is the classical method for assessig the mea performace of a group of objects (e.g. studets, players, machies, etc.) with respect to a actio. O the other had, a very popular i the USA ad other Wester coutries assessmet method is the calculatio of the Grade Poit Average (GPA) idex. This idex is a weighted average i which greater coefficiets (weights) are assiged to the higher scores. GPA, which is coected to the quality group s performace, is calculated by 0F D C 3B 4A the formula GPA = (), where is the total umber of the group s members ad A, B, C, D ad F deote the umbers of the group s members that demostrated excellet (A), very good (B), good (C), fair (D) ad usatisfactory (F) performace respectively [8]. I case of the worst performace ( F = ) formula () gives that GPA = 0, while i case of the ideal performace ( A = ) it gives GPA = 4. Therefore we have i geeral that 0 GPA 4. Cosequetly, values of GPA greater tha idicate a more tha satisfactory performace. Fially ote that formula () ca be also writte i the form GPA = y + y 3 +3y 4 + 4y (), where y = F, y = D, y 3 = C, y 4 = B ad y = A deote the frequecies of the group s members which demostrated usatisfactory, fair, good, very good ad excellet performace respectively. 3. The COG Defuzzificatio Techique as a Assessmet Method (RFAM) The solutio of a problem i terms of FL ivolves i geeral the followig steps: Choice of the uiversal set U of the discourse. Fuzzificatio of the problem s data by defiig the proper membership fuctios. Evaluatio of the fuzzy data by applyig rules ad priciples of FL to obtai a uique fuzzy set, which determies the required solutio. Defuzzificatio of the fial outcomes i order to apply the solutio foud i terms of FL to the origial, real world problem. Oe of the most popular i fuzzy mathematics defuzzificatio methods is the Cetre of Gravity (COG) techique. For applyig this method, let us assume that A = {(x, m(x)): x U} is the fial fuzzy set determiig the problem s solutio. We correspod to each x U a iterval of values from a prefixed umerical distributio, which actually meas that we replace U with a set of real itervals. The, we costruct the graph of the membership fuctio y=m(x) ad we cosider the level s area F cotaied betwee this graph ad the OX axis. There is a commoly used i FL approach (e.g. see [9]) to represet the system s fuzzy data by the coordiates (x c,

3 y c ) of the COG, say F c, of the area F, which we calculate usig the followig wellkow [9] from Mechaics formulas: x c F F xdxdy, yc dxdy F F ydxdy dxdy Cosider ow the special case where oe deals with the assessmet of a group s performace The, we choose as set of the discourse the set U = {A, B, C, D, F} of the fuzzy liguistic labels (characterizatios) of excellet (A), very good (B), good (C), fair (D) ad usatisfactory (F) performace respectively of the group s members. Whe a score, say y, is assiged to a group s member (e.g. a mark i case of a studet), the its performace is characterized by F, if y [0, ), by D, if y [, ), by C, if y [, 3), by B if y [3, 4) ad by A if y [4, ] respectively. Cosequetly, we have that y = m(x) = m(f) for all x i [0,), y = m(x) = m(d) for all x i [,), y 3 = m(x) = m(c) for all x i [, 3), y 4 = m(x) = m(b) for all x i [3, 4) ad y = m(x) = m(a) for all x i [4,]. Therefore, the graph of the membership fuctio y = m(x), takes the form of Figure, where the area of the level s sectio F cotaied betwee the graph ad the OX axis is equal to the sum of the areas of the rectagles S i, i=,, 3, 4,. (3). y m(b) s 4 m(d) m(f) m(c) m(a) O s s s 3 s F D C B A 3 4 x Figure : The graph of the COG method It is straightforward the to check (e.g. see Sectio 3 of []) that i this case formulas (3) take the form: x c = (y +3y +y 3 +7y 4 +9y ), y c = (y +y +y 3 +y 4 +y ) (4), mx ( i ) with x =F, x =D, x 3 =C, x 4 =B, x =A ad y i =, i =,, 3, 4,. Note that the mx ( ) membership fuctio y = m(x), as it usually happes with fuzzy sets, ca be defied, accordig to the user s choice, i ay compatible to the commo logic way. However, i order to obtai assessmet results compatible to the correspodig results of the GPA idex, we defie here y = m(x) i terms of the frequecies, as i formula () of Sectio. The m ( x i ) = (00%). j j 3

4 Usig elemetary algebraic iequalities ad performig elemetary geometric observatios (e.g. Sectio 3 of []) oe obtais the followig assessmet criterio: Amog two or more groups the group with the biggest x c performs better. If two or more groups have the same x c., the the group with the higher y c performs better. If two or more groups have the same x c <., the the group with the lower y c performs better. As it becomes evidet from the above statemet, a group s performace depeds maily o the value of the x-coordiate of the COG of the correspodig level s area, which is calculated by the first of formulas (4). I this formula, greater coefficiets (weights) are assiged to the higher grades. Therefore, the COG method focuses, similarly to the GPA idex, o the group s quality performace. I case of the ideal performace (y = ad y i = 0 for i ) the first of formulas (4) gives that x c = 9. Therefore, values of x c greater tha 4 9 =. demostrate a more tha satisfactory performace. Due to the shape of the correspodig graph (Figure ) the above method was amed as the Rectagular Fuzzy Assessmet Model (RFAM). 4. Variatios of the COG Techique (GRFAM, TFAM ad TpFAM) A group s performace is frequetly represeted by umerical scores i a climax from These scores ca be coected to the liguistic labels of U as follows: A (8-00), B(7-84), C (60-74), D(0-9) ad F (0-49) *. Ambiguous cases appear i practice, beig at the boudaries betwee two successive assessmet grades; e.g. somethig like 84-8%, beig at the boudaries betwee A ad B. I a effort to treat better such kid of cases, Subboti [4] moved the rectagles of Figure to the left, so that to share commo parts (see Figure ). I this way, the ambiguous cases, beig at the commo rectagle parts, belog to both of the successive grades, which meas that these parts must be cosidered twice i the correspodig calculatios. The graph of the resultig fuzzy set is ow the bold lie of Figure. However, the method used i Sectio 3 for calculatig the coordiates of the COG of the area cotaied betwee the graph ad the OX-axis is ot the proper oe here, because i this way the commo rectagle parts are calculated oly oce. The right method for calculatig the coordiates of the COG i this case was fully developed by Subboti & Voskoglou [7] ad the resultig framework was called the Geeralized Rectagular Fuzzy Assessmet Model (GRFAM). The developmet of GRFAM ivolves the followig steps:. Let y, y, y 3, y 4, y 3 be the frequecies a group s members who obtaied the grades F, D, C, B, A respectively. The yi = (00%).. We take the heights of the rectagles i Figure to have legths equal to the correspodig frequecies. Also, without loss of geerality we allow the sides of the * This way of coectio, although it satisfies the commo sese, it is ot uique; i a more strict assessmet, for example, oe could take A(90-00), B(80-89), C(70-79), D(60-69) ad F (0-9), etc. 4

5 adjacet rectagles lyig o the OX axis to share commo parts with legth equal to the 30% of their legths, i.e. 0.3 uits. Figure : Graphical represetatio of the GRFAM 3. We calculate the coordiates ( x, y ) of the COG, say F i, of each rectagle, i =, ci ci, 3, 4, as follows: Sice the COG of a rectagle is the poit of the itersectio of its diagoals, we have that yc y i i. Also, sice the x-coordiate of each COG F i is equal to the x- coordiate of the middle of the side of the correspodig rectagle lyig o the OX axis, from Figure it is easy to observe that x = 0.7i We cosider the system of the COGs F i ad we calculate the coordiates (X c, Y c ) of the COG F of the whole area cosidered i Figure as the resultat of the system of the GOCs F i of the five rectagles from the followig well kow [0] formulas X c = Sx i c S i, Y c = i Sy i c i (). S i I the above formulas Si, i=,, 3, 4, deote the areas of the correspodig rectagles, which are equal to y i. Therefore S = Si = yi = ad formulas () give that X c = yi (0.7i 0.), Y c = yi( yi) or X c = (0.7 iyi ) 0., Y c = yi (6).. We determie the area i which the COG F lies as follows: For i, j =,, 3, 4,, we have that 0 (y i - y j ) = y i + y j - y i y j, therefore y i + y j y i y j, with the equality holdig if, ad oly if, y i = y j. Therefore = ( yi ) = i = yi + yy i j yi i, j, i j + ( yi yj ) = yi or i, j, i j ci yi (7), with the equality holdig if, ad oly if, y = y = y 3 = y 4 = y =. I case of the equality the first of formulas (6) gives that X c = 0.7( ) =.9. Further, Sice the ambiguous assessmet cases are situated at the boudaries betwee the adjacet grades, it is logical to accept a percetage for the commo legths of less tha 0%.

6 combiig the iequality (7) with the secod of formulas (6), oe fids that Y c 0 Therefore the uique miimum for Y c correspods to the COG F m (.9, 0.). The ideal case is whe y = y = y 3 = y 4 = 0 ad y =. The formulas () give that X c = 3.3 ad Y c =. Therefore the COG i this case is the poit F i (3.3, 0.). O the other had, the worst case is whe y = ad y = y 3 = y 4 = y = 0. The from formulas () we fid that the COG is the poit F w (0., 0.). Therefore, the area i which the COG F lies is the area of the triagle F w F m F i (Figure 3). Figure 3: The triagle where the COG lies 6. From elemetary geometric observatios o Figure 3 oe obtais the followig assessmet criterio: Betwee two groups, the group with the greater X c performs better. If two groups have the same X c.9, the the group with the greater Y c performs better. If two groups have the same X c <.9, the the group with the lower Y c performs better From the first of formulas (6) it becomes evidet that the GRFAM measures the quality group s performace. Also, sice the ideal performace correspods to the 3.3 value X c = 3.3, values of X c greater tha =.6 idicate a more tha satisfactory performace. At this poit oe could raise the followig questio: Does the shape of the membership fuctio s graph of the assessmet model affect the assessmet s coclusios? For example, what will happe if the rectagles of the GRFAM will be replaced by isosceles triagles? The effort to aswer this questio led to the costructio of the Triagular Fuzzy Assessmet Model (TFAM), created by Subboti & Bilotskii [] ad fully developed by Subboti & Voskoglou [3]. 6

7 Figure 4: Graphical Represetatio of the TFAM The graphical represetatio of TFAM is show i Figure 4 ad the steps followed for its developmet are the same with the correspodig steps of GRFAM preseted above. The oly differece is that oe works with isosceles triagles istead of rectagles. The fial formulas calculatig the coordiates of the COG of TFAM are: X c = (0.7 iy ) 0. y (8) i, Y c = ad the correspodig assessmet criterio is the same with the criterio obtaied for GRFAM. A alterative to the TFAM approach is to cosider isosceles trapezoids istead of triagles [4, ]. I this case we called the resultig framework Trapezoidal Fuzzy Assessmet Model (TpFAM). The correspodig scheme is that show i Figure. I this case the y - coordiate of the COG F i, i=,, 3, 4,, of each trapezoid is calculated i terms of the fact that the COG of a trapezoid lies o the lie segmet joiig the midpoits of its parallel sides a ad b at a distace d from the loger side b h give by d= ( a b ), where h is its height [8]. Also, sice the x-coordiate of the 3( a b) COG of each trapezoid is equal to the x-coordiate of the midpoit of its base, it is easy to observe from Figure that x = 0.7i i y y B C y 3 B 3 C 3 y 4 H B 4 C 4 y y H H3 H 3 B C 0=A A D A 3 D A D 3 A D 4 D Figure : The TpFAM s scheme x 7

8 Oe fially obtais from formulas () that X c = ad the assessmet criterio is the same agai. i, Y c = 7 (0.7 iy ) 0. 3 i y (9). Compariso of the Assessmet Methods Oe ca write formulas (6), (8) ad (9) of Sectio 4 i the sigle form: X c = (0.7 iyi ) 0., Y c = a yi (0), where a = for the GRFAM, a = for the TFAM ad a = 3 for the TpFAM. 7 Combiig formulas (0) with the commo assessmet criterio stated i Sectio 4 oe obtais the followig result:. THEOREM: The three variatios of the COG techique, i.e. the GRFAM, the TFAM ad the TpFAM are equivalet assessmet models. Further, the first of formulas (0) ca be writte as X c = 0.7(y + y + 3y 3 + 4y 4 + y ) 0. = 0.7 [(y + y 3 + 3y 4 + 4y ) + y i ] 0.. Therefore, by formula () of Sectio 3, oe fially gets that X c = 0.7(GPA + ) 0., or X c = 0.7GPA + 0. (). I the same way, the first of formulas (4) of Sectio 3 for RFAM ca be writte as x c = (y + 3y + y 3 +7y 4 + 9y ) = (GPA + ), or x c = GPA + 0. (). We are ready ow to prove:. THEOREM: If the values of the GPA idex are differet for two groups, the the GPA idex, the RFAM ad its variatios (GRFAM, TFAM ad TpFAM) provide the same assessmet outcomes o comparig the performace of these groups. Proof: Let G ad G be the values of the GPA idex for the two groups ad let x c, x c be the correspodig values of the x-coordiate of the COG for the RFAM. Assume without loss of geerality that G>G, i.e. that the first group performs better accordig to the GPA idex. The, equatio () gives that x c > x c, which, accordig to the first case of the assessmet criterio of Sectio 3, shows that the first group performs also better accordig to the RFAM. I the same way, from equatio () ad the first case of the assessmet criterio of Sectio 4, oe fids that the first group performs better too accordig to the equivalet assessmet models GRFAM, TFAM ad TpFAM.- I case of the same GPA idex we shall show the followig result:.3 THEOREM: If the GPA idex is the same for two groups the the RFAM ad its variatios (GRFAM, TFAM ad TpFAM) provide the same assessmet outcomes o comparig the performace of these groups. Proof: Sice the two groups possess the same value of the GPA idex, equatios () ad () show that the values of X c ad x c are also the same. Therefore, oe of the last two cases of the assessmet criteria of Sectios 3 ad 4 could happe. The possible values of x i these criteria lie i the itervals [0, 9 ] ad [0, 3.3] respectively, while the critical poits correspod to the values x c =. ad X c =.9 respectively. Obviously, if both values of x are i [0,.9), or i [., 9 ], the the two criteria 8

9 provide the same assessmet outcomes o comparig the performace of the two groups. Assume therefore that.9 < X c ad x c <.. The, due to equatio (),.9 < X c.9< 0.7GPA <0.7GPA GPA >. Also, due to equatio (), x c <. GPA + 0. <. GPA >. Therefore, the iequalities.9 < X c ad x c <. caot hold simultaeously ad the result follows.- Combiig Theorems. ad.3 oe obtais the followig corollary:.4 COROLLARY: The RFAM ad its variatios GRFAM, TFAM ad TpFAM provide always the same assessmet results o comparig the performace of two groups. The followig example shows that i case of the same GPA values the applicatio of the GPA idex could ot lead to logically based coclusios (see also paragraph (vii) of Sectio 4 of [7]). Therefore, i such situatios, our criteria of Sectios 3 ad 4 become useful due to their logical ature.. EXAMPLE: The studet grades of two Classes with 60 studets i each Class are preseted i Table Table : Studet Grades Grades Class I Class II C 0 0 B 0 0 A 0 40 *0 4*0 3*0 4*40 The GPA idex for the two classes is equal to 3. 67, which meas that the two Classes demostrate the same performace i terms of the GPA idex. Therefore equatio () gives that X c = 0.7* , while 6 equatio () gives that x c = 4.7 for both Classes. But yi = ( ) ( ) for the first ad yi = ( ) ( ) = for the secod Class. Therefore, accordig to the assessmet criteria of Sectios 3 ad 4 the first Class demostrates a better performace i terms of the RFAM ad its variatios. Now which oe of the above two coclusios is closer to the reality? For aswerig this questio, let us cosider the quality of kowledge, i.e. the ratio of the studets received B or better to the total umber of studets, which is equal to for the first 6 ad for the secod Class. Therefore, from the commo poit of view, the situatio i Class II is better. However, may educators could prefer the situatio i Class I havig a greater umber of excellet studets. Coclusively, i o case it is logical to accept that the two Classes demostrated the same performace, as the calculatio of the GPA idex suggests. The ext example shows that although the RFAM, GRFAM, TFAM ad TpFAM provide always the same assessmet results o comparig the performace of two groups (Corollary.4), they are ot equivalet assessmet models..6 EXAMPLE: Table depicts the results of the fial exams of the first term mathematical courses of two differet Departmets, say D ad D, of the School of 9

10 Techological Applicatios (future egieers) of the Graduate T. E. I. of Wester Greece. Note that the cotets of the two courses ad the istructor were the same for the two Departmets. Table : Results of the two Departmets Grade D D A B 3 6 C 3 D 9 0 F 6 Total No. of studets 30 3 *9 * 3*3 4* The GPA idex is equal to. 47 for D ad 30 *0 *3 3*6 4*.66 for D. Therefore, the two Departmets 3 demostrated a less tha satisfactory performace (sice GPA < ), with the performace of D beig better. Further, equatio () gives that X c.3 for D ad X c.66 for D. Therefore, accordig to the first case of the assessmet criterio of Sectio 4, D demostrated (with respect to GRFAM, TFAM ad TpFAM) a better performace tha D. 3.3 Moreover, sice.3 < =.6 <.66, D demostrated a less tha satisfactory performace, while D demostrated a more tha satisfactory performace. I the same way equatio () gives that x c.97 for D ad x c.6 for D. Therefore, accordig to the first case of the assessmet criterio of Sectio 3, D demostrated (with respect to RFAM) a better performace tha D. But i this case, 4. sice for both Departmets X c < =., the two Departmets demostrated a less tha satisfactory performace. REMARK: Note that, if GPA > (more tha satisfactory performace), the X c = 0.7GPA + 0. > 0.7 * + 0. =.9 >.6 ad x c = GPA + 0. > =.>.. Therefore the correspodig group s performace is also more tha satisfactory with respect to GRFAM, TFAM, TpFAM ad RFAM. However, if GPA < (less tha satisfactory performace), the X c <.9 ad x c <., which do ot guaratee that X c <.6 ad x c <.. Therefore the assessmet characterizatios of RFAM ad the equivalet GRFAM, TFAM, TpFAM ca be differet oly whe GPA <. 6. Coclusio ad Perspectives for Future Research From the discussio performed i this paper it becomes evidet that the RFAM ad its variatios GRFAM, TFAM ad TpFAM, although they provide always the same assessmet outcomes o comparig the performace of two groups, they are ot 0

11 equivalet assessmet methods. The above assessmet outcomes are also the same with those of the GPA idex, uless if the groups uder assessmet possess the same values. I the last case the GPA idex could ot lead to logically based coclusios. Therefore, i this case either the use of RFAM or of its variatios must be preferred. Other fuzzy assessmet methods have bee also used i earlier author s works like the measuremet of a system s ucertaity [] ad the applicatio of the fuzzy umbers [7]. These methods, i cotrast to the previous oes which focus o the correspodig group s quality performace, they measure its mea performace. The plas for our future research iclude the effort to compare all these methods i order to obtai the aalogous coclusios. Refereces. Subboti, I. Ya., Badkoobehi, H., Bilotckii, N. N. (004), Applicatio of fuzzy logic to learig assessmet, Didactics of Mathematics: Problems ad Ivestigatios,, Subboti, I. Ya., Bilotskii, N. N. (04), Triagular fuzzy logic model for learig assessmet, Didactics of Mathematics: Problems ad Ivestigatios, 4, Subboti, I. Ya, Voskoglou, M. Gr. (04), A Triagular Fuzzy Model for Assessig Critical Thikig Skills, Iteratioal Joural of Applicatios of Fuzzy Sets ad Artificial Itelligece, 4, Subboti, I. Ya. (04), Trapezoidal Fuzzy Logic Model for Learig Assessmet, arxiv [math..GM]. Subboti, I. Ya, Voskoglou, M. Gr. (04), Fuzzy Assessmet Methods, Uiversal Joural of Applied Mathematics, (9), Subboti, I. Ya. (0), O Geeralized Rectagular Fuzzy Model for Assessmet, Global Joural of Mathematics, (), Subboti, I.Ya. & Voskoglou, M.Gr. (06), A Applicatio of the Geeralized Rectagular Fuzzy Model to Critical Thikig Assessmet, America Joural of Educatioal Research, 4(), Swibure.edu.au (04), Grade Poit Average Assessmet, retrieved o October, 04 from: 9. va Broekhove, E. & De Baets, B. (006). Fast ad accurate cetre of gravity defuzzificatio of fuzzy system outputs defied o trapezoidal fuzzy partitios, Fuzzy Sets ad Systems, 7(7), Voskoglou, M. Gr. (999), A Applicatio of Fuzzy Sets to the Process of Learig, Heuristics ad Didactics of Exact Scieces, 0, Voskoglou, M. Gr. (009), Trasitio across levels i the process of learig: A fuzzy Model, Iteratioal Joural of Modellig ad Applicatio, Uiv. Blumeau,, Voskoglou, M. Gr. (0), A Study o Fuzzy Systems, America Joural of Computatioal ad Applied Mathematics, (), Voskoglou, M. Gr., Subboti, I. Ya. (03) Dealig with the Fuzziess of Huma Reasoig, Iteratioal Joural of Applicatios of Fuzzy Sets ad Artificial Itelligece, 3, Voskoglou, M. Gr. (03), Case-Based Reasoig i Computers ad Huma Cogitio: A Mathematical Framework, Iteratioal Joural of Machie Itelligece ad Sesory Sigal Processig, Idersciece Publishers,, 3-.

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