On Applications of a Generalized Hyperbolic Measure of Entropy

Transcription

1 I.J. Itellget Systems ad pplcatos, 205, 07, Publshed Ole Jue 205 MECS ( DOI: 0.585/sa O pplcatos of a Geeralzed Hyperbolc Measure of Etropy P.K Bhata Departmet of Mathematcs, DCR Uversty of Scece ad Techology, Murthal-3039 (Haryaa), Ida Emal: bhatapk@redffmal.com Sureder Sgh School of Mathematcs, Shr Mata Vasho Dev Uversty, Sub post offce, Katra (J & K) Ida Emal: sureder976@gmal.com Vod Kumar Departmet of Mathematcs, Govt. Wome College, Madlauda -323 (Haryaa) Ida Emal: vkmara@gmal.com bstract fter geeralzato of Shao s etropy measure by Rey 96, may geeralzed versos of Shao measure were proposed by dfferet authors. Shao measure ca be obtaed from these geeralzed measures asymptotcally. atural questo arses the parametrc geeralzato of Shao s etropy measure. What s the role of the parameter(s) from applcato pot of vew? I the preset commucato, super addtvty ad fast scalablty of geeralzed hyperbolc measure [Bhata ad Sgh, 203] of probablstc etropy as compared to some classcal measures of etropy has bee show. pplcato of a geeralzed hyperbolc measure of probablstc etropy certa stuatos has bee dscussed. lso, applcato of geeralzed hyperbolc measure of fuzzy etropy mult attrbute decso makg have bee preseted where the parameter affects the preferece order. Idex Terms Probablstc Etropy, Fuzzy Etropy, Super ddtve Etropy, Mult ttrbute Decso I. INTRODUCTION The ma use of formato s to remove ucertaty ad ma obectves of formato theoretc studes are: To develop ew measures of formato ad ther applcatos To develop etropy optmzato prcples To develop coectos ad terrelatos of formato theory wth other dscples such as scece, egeerg, maagemet, operato research etc. Explorato of role of addtoal parameters geeralzed formato/dvergece measures. Etropy s cetral cocept the feld of formato theory ad was orgally troduced by Shao hs semal paper [], the cotext of commucato theory. The etropy of a expermet has dual terpretatos. It ca be cosdered both as a measure of the ucertaty that prevaled before the expermet was accomplshed ad as a measure of the formato expected from a expermet. expermet mght be a formato source emttg a sequece of symbols (.e., a message) M { s, s2,..., s}, where successve symbols are selected accordg to some fxed probablty law. For the smplest kd of source, we assume that successve symbols emtted from the source are statstcally depedet. Such a formato source s termed a zero-memory source ad s completely descrbed by the source alphabet ad the probabltes wth whch the symbols occur P { p, p2,..., p}. We may calculate the average formato provded by a zero-memory formato source usg several etropes. The Shao Etropy [] s a well-kow ad hghly used measure of formato. Cosder a set E of mutually exclusve evets E ( =,..., ) each of whch has the probablty of occurrece p, so that the p s add up to uty. The formato cotet of the occurrece of evet E, s defed []: If ( E ) log p log. p The expected formato cotet of a evet from our set of evets, the etropy of the set E, s defed []: H( P) p If ( E ) p log p. () H (P) s always o egatve. Its maxmum value depeds o. It s equal to log whe all p are equal. (P) H s kow as Shao Etropy or Shao s measure of Iformato. I 96, to add flexblty to Shao s measure, Rey [2] proposed a oe parametrc geeralzato of Shao s measure. fter Rey, may oe, two, three ad four parametrc geeralzatos have bee proposed by the scholars the feld of formato theory. Bhata ad Sgh[3] proposed a oe parametrc hyperbolc measure of etropy as follows: Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43

2 O pplcatos of a Geeralzed Hyperbolc Measure of Etropy 37 H ( P) sh( log ), 0 sh( ) p p. (2) lm 0 H ( P) H( P). (3) Thus the Shao measure s the lmtg case of the measure proposed Eq.(2). fter the tato of fuzzy theory by Zadeh[4], the cocept of fuzzess has flueced almost each ad every brach of research. The fluece of fuzzy theory the feld of formato theory gave brth to oclasscal formato theory. De Luca ad Term[5] proposed a measure of fuzzy etropy correspodg to probablstc etropy of Shao gve by H( ) ( x )log ( x ). (4) ( ( x )) log( ( x )) where, s a fuzzy set ad (x ) s membershp value of x. fter De Luca ad Term may geeralzed versos of ths fuzzy etropy have bee proposed. Bhata et al. [6] proposed a oe parametrc geeralzed hyperbolc measure of etropy as follows: H ( ) sh( ) ( x )sh( log ( x )) ( ( x ))sh( log( ( x ))) (5) lm 0 H ( ) H( ). (6) The real umber α s assocated wth the o extesveess of the system. The cocepts of etropy ad fuzzy etropy have bee extesvely utlzed umerous applcatos scece, egeerg ad maagemet[7,8]. I the preset paper, some applcatos of geeralzed hyperbolc measure of probablstc etropy (Eq.(2)) ad geeralzed hyperbolc measure of fuzzy etropy (Eq.(5)) have bee preseted. Ths paper s orgazed as follows: I Secto II, super addtvty ad scalablty of geeralzed hyperbolc formato measure (2) are vestgated ad ts applcato certa stuatos s dscussed. Secto III presets a ew model for mutple attrbute decso makg usg geeralzed fuzzy etropy. Secto IV cotas cocludg remarks. II. PPLICTION OF GENERLIZED HYPERBOLIC INFORMTION MESURE For otatoal coveece, let us call the etropy measure proposed Eq.(2) as hyperbolc etropy (hyp etropy) ad deote t as H hyp( P) sh( log ), 0 sh( ) p p. (7) May oe parametrc measures are suggested lterature. But from applcato pot of vew t has bee observed that most of the applcatos revolves aroud Shao etropy[], Rey etropy[2], Havrda-Charvat etropy(hc)[9] ad Tsalls etropy[0]. Shao etropy[], Rey etropy[2] are addtve ad ther applcato s sutable for extesve systems. O the other had, Havrda-Charvat etropy [9] ad Tsalls etropy[0] are sub addtve ad ther applcato s sutable for oextesve systems. The Hyperbolc etropy proposed Eq.(7) s compared wth Rey etropy[2] ad Havrda- Charvat etropy[9] for arbtrarly chose eght complete probablty dstrbutos ad dfferet values of parameter α. For comparso, all of three etropes have bee ormalzed. Table. Values of ormalzed Rey, Hyperbolc ad Havrda-Charvat Etropes at hypothetcally choose eght probablty dstrbutos at α=0. α=0. Normalzed Etropes α=0. P Rey Hyp. Havrda-Charvat Rey Hyp. Havrda-Charvat P P P P P P P P Let P 0.,0.2, 0.05,0.3,0.35, P 2 0., 0.22, 0.23,0.3,0.4, P 3 P , 0.32, 0.09, 0.22, 0.34, 0., 0.2, 0.3, 0.03, 0.34, P , 0.37, 0.05, 0.2, 0.3, , 0.23, 0.2, 0.05, 0.27 P 7 0.7, 0.09, 0., 0.33, 0.3 ad P ,0.,0.,0.2,0.37 P, Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43

3 38 O pplcatos of a Geeralzed Hyperbolc Measure of Etropy be hypothetcally chose eght complete probablty dstrbutos. For these probablty dstrbutos we calculate three ormalzed etropes amely, Rey[2], Hyp. etropy gve by Eq.(7) ad Havrda-Charvat etropy[9] for dfferet values of α Tables -5. Here we compare the three etropes uder cosderato at same scale for gve α=0.. I Table., ormalzed values of etropes uder cosderato have bee calculated for hypothetcally chose eght probablty dstrbutos because ormalzed value makes comparso smple. Fg. 3. Fg.. I Fgure., the graph of the ormalzed values of three etropes uder cosderato for α=0. wth respect to hypothetcally chose eght probablty dstrbutos s gve. Table 2-5 ad Fgure 2-5 have bee costructed for smlar purpose for α = , 0.7, Fg. 4. Fg. 2. Fg. 5. Table 2. Values of ormalzed Rey, Hyperbolc ad Havrda-Charvat Etropes at hypothetcally choose eght probablty dstrbutos at α=0.4 α=0.4 Normalzed Etropes α=0.4 P Rey Hyp. Havrda-Charvat Rey Hyp. Havrda-Charvat P P P P P P P P Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43

4 O pplcatos of a Geeralzed Hyperbolc Measure of Etropy 39 Table 3. Values of ormalzed Rey, Hyperbolc ad Havrda-Charvat Etropes at hypothetcally choose eght probablty dstrbutos at α=0.5 α=0.5 Normalzed Etropes α=0.5 P Rey Hyp. Havrda-Charvat Rey Hyp. Havrda-Charvat P P P P P P P P Table 4. Values of ormalzed Rey, Hyperbolc ad Havrda-Charvat Etropes at hypothetcally choose eght probablty dstrbutos at α=0.7 α=0.7 Normalzed Etropes α=0.7 P Rey Hyp. Havrda-Charvat Rey Hyp. Havrda-Charvat P P P P P P P P Table 5. Values of ormalzed Rey, Hyperbolc ad Havrda-Charvat Etropes at hypothetcally choose eght probablty dstrbutos at α=0.75 α=0.75 Normalzed Etropes α=0.75 P Rey Hyp. Havrda-Charvat Rey Hyp. Havrda-Charvat P P P P P P P P Table 6. Values of ormalzed Hyperbolc Etropes H hyp(p P ), for hypothetcally chose eght probablty dstrbutos H hyp ( P P 2 ) H hyp P P ) H hyp P P ) H hyp P P ) H hyp P P ) H hyp P P ) P P ) ( 3 ( 4 ( 5 ( 6 ( 7 H hyp ( Table 7. Values of ormalzed Hyperbolc Etropes H hyp(p ) H hyp(p ), for hypothetcally choose eght probablty dstrbutos H hyp ( P ) H hyp ( P 2 ) H hyp ( P ) H hyp ( P 3 ) H hyp ( P ) H hyp ( P 4 ) H hyp ( P ) H hyp ( P 5 ) H hyp ( P ) H hyp ( P 6 ) H hyp ( P ) H hyp ( P 7 ) H hyp ( P ) H hyp ( P 8 ) Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43

5 40 O pplcatos of a Geeralzed Hyperbolc Measure of Etropy Based o the precedg comparsos the followg observatos ca be made:. The geeralzed etropes, wth the Shao Etropy as a specal case, are almost cosstet for dfferet values of α, creasg α makes the measure s values spa a smaller terval. Ths meas that as α creases, the measures become coarser ad ther dscrmatg power decreases. By chagg value of α, Hyperbolc etropy ca be made to oscllate o large terval or small terval. Further from the graphs Fgure -4, t ca be see that scalablty of Hyperbolc etropy s much faster tha Rey ad Havrda-Charvat Etropy. 2. I Table -4, t s observed that maxmum etropy correspods to the probablty dstrbuto P 2 for all the three etropes uder cosderato. But, from Table 5 t ca be see that Hyperbolc etropy s ot agreemet wth Rey ad Havrda-Charvat etropes cotext of maxmum etropy prescrptos of Jayes[]. That s, for 0. 75, Hyperbolc etropy behaves dfferetly. I other words, we ca say that whe extraeous factor α assumes value greater tha equal to 0.75, we observe that the most ubased probablty dstrbuto s ot that what s expected from classcal measures of etropy. Further, usg the probablty dstrbutos P, P2,..., P8, we costruct Table 6 ad Table 7 to show the superaddtvty of Hyperbolc etropy. From Table 6 ad Table 7 t s cocluded that: H ( P) H ( P ) H ( PP ), hyp hyp hyp Smlarly, we ca prove H hyp 2,3,,8. ( P ) H hyp ( P ) H,,2,3,,8 ; hyp ( P P ), Therefore, hyperbolc etropy s super addtve. So, t ca be appled to measure the formato cotet of those systems whch have several subsystems ad total formato provded by dvdual subsystems s less tha that of formato provded by whole system. I bologcal studes t s observed that amog the great amout of gees preseted mcroarray gee expresso data, oly a small fracto are effectve for performg a certa dagostc test. I ths regard, mutual formato has bee show to be successful for selectg a set of relevat ad o redudat gees from mcroarray data. However, formato theory offers may more measures such as the f-formato measures whch may be sutable for selecto of gees from mcroarray gee expresso data. Ma [2] tested performace of some f-formato measures ad compared wth that of the mutual formato based o the predctve accuracy of ave bayes classfer, K-earest eghbour rule, ad support vector mache ad foud that some f-formato measures are show to be effectve for selectg relevat ad o redudat gees from mcroarray data. I ths type of study, the geeralzed formato measure gve Eq.(7) proposed here may perform better, as t s more. dscrmatve comparso of some classcal measures. Further, ths measure seems to perform better bary mage segmetato. III. PPLICTIONS OF GENERLIZED HYPERBOLIC FUZZY INFORMTION MESURE IN MULTIPLE TTRIBUTE DECISION MKING model to fd the best alteratve o the bass of multple attrbutes s proposed here. The model requres maly the followg:. valable alteratves. 2. ttrbutes. 3. Weght of each attrbute. 4. Parameter α wth whch preferece order of alteratves may vary. Let X x, x,..., x } be a fte set of alteratves { 2 { 2 m ad a, a,..., a } be a fte set of attrbutes, ad ( 2 w m for 0, W w, w,..., ), m w 0, w be the weght vector of the attrbutes a (,2,..., m) whch s ot predefed. Now, the model ca be gve followg three steps: STEP : Let R ( ( x, a )) be fuzzy decso m matrx, where x, a ) dcates the degree rage that ( the alteratve x satsfes the attrbute a, the fuzzy etropy proposed by DeLuca ad Term[5] gve Eq. (4) takes the form H( a ) ( x, a )log ( x, a ) ( x, a )) log( ( x, a )) (8) (,,2,.., m. ad the geeralzed hyperbolc measure of etropy gve Eq.(5) takes the form H ( a ) ( x, a )sh( log ( x, a )) sh( ) ( ( x, a ))sh( log( ( x, a,2,.., m. STEP 2: The weght assocated wth attrbute defed as follows: w m H H ( a ) ( a ), (,2,..., m.) ))) (9) a s STEP 3: Fally, we costruct a score fucto as follows: Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43

6 O pplcatos of a Geeralzed Hyperbolc Measure of Etropy 4 m S ( x ) ( x, a ) w,,2,..., lteratve wth hghest score s the best choce. Now, the above model s llustrated wth the help of followg example: Example 3. Let us cosder a customer who teds to buy a car. Fve types of cars (alteratves) x (,2,3,4,5) are avalable. The customer takes to accout four attrbutes to decde whch car to buy: () a : Prce; (2) a 2 : Comfort; (3) a3 : Desg; ad (4) a 4 : Safety ssume that the characterstcs of the alteratves x (,2,3,4,5) are represeted by fuzzy decso matrx Let R R ( ( x, a )) We dscuss the result for varous values of. For 0 ; From fuzzy decso matrx we have Table 8. Values of formula Eq.(9) at α=0 H 0 ( a ) H 0 ( a 2 ) H 0 ( a 3 ) H 0 ( a 4 ) ad H ( ) a w ( w, w, w, w ) (0.2346, 0.269, , ) By calculatg we have S ( ) , S ( 2 ) , 0 x 0 x S 0 ( x 3 ) , S 0 ( x 4 ) , S 0 ( x 5 ) Based o the calculated values of S ( x ),as above, we get the followg ordergs of raks of the alteratves x ( =,2,3,4,5) ; x x4 x2 x3 x5. Therefore the optmal alteratve s x. I the same maer we ca fd the orderg of raks of the alteratves x ( =,2,3,4,5) for dfferet values of α Table 9. Table 9. Values of S α(x ) for α = 0.3, 0.5, 0.7, 0.9, 0.99 α =0.3 α =0.5 α =0.7 α =0.8 α =0.9 α =0.99 S α(x ) S α(x 2) S α(x 3) S α(x 4) S α(x 5) ga, based o results Table 9, we get the followg ordergs of raks of the alteratves x ( =,2,3,4,5) ; For 0. 3, x x4 x2 x3 x5 For 0. 5, x x4 x2 x3 x5 For 0. 7, x x4 x2 x3 x5 For 0. 8, x x4 x2 x3 x5 For 0. 9, x x4 x2 x3 x5 For 0. 99, From ths dscusso t s clear that x s most preferable alteratve. But at 0. 99, the raks of preferece of alteratves chages. Thus extraeous factor α plays ts role order of preferece ad does ot affect the choce of best alteratve. The proposed algorthm s also helpful surace sector (example 3.2), medcal dagoss (example 3.3), educato (example 3.4) etc. Example 3.2 Let us cosder a customer who teds to get sured at a surace compay. Fve surace compaes (alteratves) x (,2,3,4,5 ) are avalable. The compay takes to accout four attrbutes to decde the sutablty of customer for surace: () a : Smokg habt; (2) a 2 : Cholesterol level; (3) a 3 : Blood pressure; ad (4) a 4 : dequate weght ssume that the characterstcs of the alteratves x (,2,3,4,5) are represeted by fuzzy decso matrx R ( ( x, a )) 5 4 where x, a ) dcates the degree rage that the ( alteratve x satsfes the attrbute a. Proceedg wth the data of example 3., t s cocluded that compay x s best choce to get sured. Example 3.3 Let us cosder a doctor teds to dagose a patet based o some symptoms of a dsease. Let fve possble dseases (alteratves) say (x :Vral fever, x 2: Malara, x 3: Typhod, x 4 : Stomach problem, x 5 : Chest problem) x (,2,3,4,5 ) have closely related symptoms or characterstcs. The doctor takes to accout four symptoms to decde the possblty of a partcular dsease: Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43

7 42 O pplcatos of a Geeralzed Hyperbolc Measure of Etropy () a : Temprature ; (2) a 2 : Headache ; (3) a 3 : Stomach pa ; ad (4) a 4 : Cough. ssume that the characterstcs of the alteratves x (,2,3,4,5) are represeted by fuzzy decso matrx R ( ( x, a )) 5 4 where x, a ) dcates the degree rage that the ( dsease x satsfes the symptom a. Proceedg wth the data of example 3., doctor cocluded that patet s sufferg from vral fever. Example 3.4 Let us cosder a perso who teds to select a school for hs ward s admsso. Fve schools (alteratves) x (,2,3,4,5 ) are avalable. Ma takes to accout four attrbutes to decde the sutablty of school for hs ward: () a : Trasportato faclty; (2) a cademc profle of teachers; (3) a : Past results of 2 : school; ad (4) a 4 : Dscple. ssume that the characterstcs of the alteratves x (,2,3,4,5 ) are represeted by fuzzy decso matrx R ( ( x, a )) 5 4 where x, a ) dcates the degree rage that the alteratve ( x satsfes the attrbute a. Proceedg wth the data of example 3., father cocludes that school x s best choce to secure admsso for hs ward. It has bee observed that the extraeous factor α plays a mportat role order of rakg of alteratves. So, two parametrc geeralzed verso of formula gve Eq.(9) may gve better sght ad flexblty certa cases of multple attrbute decso makg. IV. CONCLUDING REMRKS probablstc etropy measure ca be addtve, sub addtve ad super addtve. I ths paper, super addtvty of geeralzed hyperbolc etropy measure (2) s tested wth the help of hypothetcal data. Therefore, oe ope problem, the applcato of super addtve formato measures s atural ths cotext. Secodly, we observed the fast scalablty of geeralzed hyperbolc etropy measure (2) as compared to some classcal geeralzed etropy measures [2,9] wth respect to parameter. Ths proposes aother ope problem: How ths scalablty s useful varous applcatos? Sahoo et al.[3], Sahoo ad rora [4,5] appled oe parameter etropy measures mage thresholdg ad aalyzed the mages o the bass of fact that how much formato s lost due to thresholdg. They observed that correspodg to the certa value of the loss of formato s least ad produces best optmal threshold value. Thus, the parameter the geeralzed etropy measures s very mportat from applcato pot of 3 vew. The etropy measure (2) may serve well mage thresholdg problems case of certa mages.two, three or four parameter etropy measures provde more flexblty of applcato. model for multple attrbute decso makg(mdm) usg geeralzed hyperbolc fuzzy etropy (5) s proposed here. The advatage of ths method s that here we calculate the weght of a attrbute from etropy formula tself whereas the avalable methods of MDM weght of attrbutes s determed by experts. Moreover, fuzzy etropy has vtal applcato mage processg problems [6,7]. Therefore, geeralzed hyperbolc fuzzy etropy (5) seems to be useful mage processg ad patter recogto problems. CKNOWLEDGMENTS We thaks the referees for ther helpful commets for the mprovemet of the paper. REFERENCES [] C.E. Shao, The mathematcal theory of commucatos, Bell Syst. Tech. Joural, Vol 27, pp , 948. [2]. Réy, O measures of etropy ad formato, Proc. 4th Berk. Symp. Math. Statst. ad Probl., Uversty of Calfora Press, pp , 96. [3] P.K. Bhata ad S. Sgh, O a New Csszar s f- Dvergece Measure, Cyberetcs ad Iformato Techologes, Vol.3, No. 2, pp , 203. [4] L.. Zadeh, Fuzzy Sets, Iformato ad Cotrol, Vol.8, pp , 965. [5]. De Luca ad S. Term, defto of oprobablstc etropy the settgs of fuzzy set theory, Iformato ad Cotrol, Vol. 20, pp , 97. [6] P.K. Bhata, S. Sgh ad V. Kumar, O a Geeralzed Hyperbolc Measure Of Fuzzy Etropy, Iteratoal Joural of Mathematcal rchves, Vol.4, No. 2,pp.36-42, 203. [7] W. LL ad C. Yufag, Dversty Based o Etropy: Novel Evaluato Crtero Mult-obectve Optmzato lgorthm, Iteratoal Joural of Itellget Systems ad pplcatos, Vol. 4, No. 0, pp. 3-24, 202. [8] L. bdullah ad. Othema, New Etropy Weght for Sub-Crtera Iterval Type-2 Fuzzy TOPSIS ad Its pplcato, Iteratoal Joural of Itellget Systems ad pplcatos, Vol.5, No. 2, 203, [9] J. Havrda ad F. Charvát, Quatfcato method of classfcato processes: Cocept of structrual α-etropy, Kyberetka, Vol.3, pp.30-35, 967. [0] C. Tsalls, Possble geeralzato of Boltzma-Gbbs statstcs, J. Statst. Phys., Vol.52, pp , 988. [] E. T. Jayes, Iformato theory ad statstcal mechacs, Phys. Rev., Vol.06, pp , 957. [2] P. Ma, f-iformato measures for effcet selecto of dscrmatve gees from mcroarray data, IEEE Trasactos O Bomedcal Egeerg, Vol.56, No.4, pp , Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43

8 O pplcatos of a Geeralzed Hyperbolc Measure of Etropy 43 [3] P.K. Sahoo, C.Wlks ad R. Yager, Threshold selecto usg Rey s etropy, Patter Recogto,Vol.30: pp.7-84, 997. [4] P. K. Sahoo ad G. rora, thresholdg method based o two dmesoal Rey s etropy, Patterm Recogto, Vol. 37, pp.49-6, [5] P. K. Sahoo ad G. rora, Image thresholdg method usg two dmesoal Tsalls-Havrda-Charvat etropy, Patterm Recogto Letters, Vol. 27, pp , [6] L.K. Huag ad M. J. Wag, Image thresholdg by mmzg the measure of fuzzess, Patter Recogto, Vol. 28, No., pp.4-5, 995. [7] Z.-W. Ye, et al., Fuzzy etropy based optmal thresholdg usg bat algorthm, ppl. Soft Comput. J. (205), /.asoc uthors Profles P.K Bhata, Professor of Mathematcs at the DCR Uversty of Scece & Techology, Murthal (Soepat), Ida. He has taught udergraduate ad post graduate studets for over twety fve years. He has bee supervsg Ph.D. studets. He has publshed a umber of research papers ourals of atoal ad teratoal repute. Hs areas of terest are Mathematcal Iformato Theory & Relablty Modelg. Sureder Sgh, ssstat Professor at School of Mathematcs, Shr Mata Vasho Dev Uversty, Katra (J & K), Ida. He has taught udergraduate ad post graduate studets for over te years. He s also workg towards hs Ph.D uder supervso of Prof. P.K Bhata ad Dr. Vod Kumar. Hs research drectos clude Iformato, dvergece measures ad ther applcatos. VodKumar, ssstat Professor at Deptt. of Mathematcs, Govt. Wome College Madlauda (Haryaa), Ida. He has taught udergraduate ad post graduate studets for over fftee years. Hs research terests clude Gbbs- Shao type equaltes, Iformato, dvergece measures ad ther applcatos. How to cte ths paper: P.K Bhata, Sureder Sgh, Vod Kumar,"O pplcatos of a Geeralzed Hyperbolc Measure of Etropy", Iteratoal Joural of Itellget Systems ad pplcatos (IJIS), vol.7, o.7, pp.36-43, 205. DOI: 0.585/sa Copyrght 205 MECS I.J. Itellget Systems ad pplcatos, 205, 07, 36-43