Complex System Reliability Evaluation using Support Vector Machine for Incomplete Data-set

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1 Internatonal Journal of Performablty Engneerng, Vol. 7, No. 1, January 2010, pp RAMS Consultants Prnted n Inda Complex System Relablty Evaluaton usng Support Vector Machne for Incomplete Data-set YUAN FUQING *1, UDAY KUMAR 1 and K. B. MISRA 2 1 Dvson of Operaton and Mantenance Engneerng, Luleå Unversty of Technology, SE Luleå, Sweden 2 RAMS Consultants, 71 Vrndaban Vhar, Amer Road, Japur , Raasthan, Inda (Receved on March 16, 2010, revsed on July 6, 2010) Abstract: Support Vector Machne (SVM) s an artfcal ntellgence technque that has been successfully used n data classfcaton problems, takng advantage of ts learnng capacty. In systems modelled as networks, SVM has been used to classfy the state of a network as faled or operatng to approxmate the network relablty. Due to the lack of nformaton, or hgh computatonal complexty, the complete analytcal expresson of system states may be mpossble to obtan, that s to say, only ncomplete data-set can be obtaned. Usng these ncomplete data-sets, dependng on amount of mssed data-set, ths paper proposes two dfferent approaches named rough approxmaton method and smulaton based method to evaluate system relablty. SVM s used to make the ncomplete data-set complete. Smulaton technque s also employed n the so called smulaton based approxmaton method. Several examples are presented to llustrate the approaches. Keywords: Relablty Evaluaton, SVM, ncomplete data-set, Smulaton 1. Introducton Complex system relablty modelled as a network can facltate system relablty evaluaton. Usng a network, the relablty dependency between the system operatng (or faled) and that of the success (or falure) of ts subsystems and consttuent components or unts are observable. Furthermore, the system relablty evaluaton problem can be transformed nto a network problem. Network relablty evaluaton has been dealt wth extensvely n the lterature [1, 2, 3] usng varous technques. For a smple network, a smple method to evaluate network relablty s to enumerate all possble states of consttuent elements and the correspondng system states. Thereafter, sum up the probabltes of system beng n operatng state to obtan system relablty and the probabltes of system beng n faled to obtan system unrelablty. However, for more complex network, ths method s tme consumng or and sometmes nfeasble. Therefore, some other methods are requred to calculate system relablty and unrelablty. In ths paper, these subsystems, components or unts for the sake of ease can be desgnated as elements smply. The element s state vector and ts correspondng system state are named data sets. Also, t s assumed elements relabltes have been known. As mentoned before, enumeratng all data sets sometmes may not be possble or t may not be feasble to obtan the complete data set due to lack of nformaton or s hghly tme consumng, or structure of system may not be known at all. Ths paper nvestgates the problem of to evaluatng system relablty usng such an ncomplete data-set. * Correspondng author s emal: yuan.fuqng@ltu.se 32

2 Complex System Relablty Evaluaton usng Support Vector Machne for Incomplete Data-set 33 Support Vector Machne (SVM) s used n ths paper. As a artfcal ntellgence technque, support vector machne was ntally developed as a classfer for pattern recognton [4,5]. Currently, ts applcaton covers varous areas such as: data mnng, fnancal forecastng [6], fault dagnoss [7], and so on. The applcaton of SVM on system relablty evaluaton had been proposed by Rocco [8, 9]. In [8], Monte Carlo technque was employed to smulate states of each element and SVM s used as a bnary classfer to predct the system state. Ths synergy produces an estmaton of system relablty. Ths paper proposes two approaches, namely an approxmaton method and smulaton based approxmaton method respectvely. In the remanng secton of ths paper, the prncple of SVM classfer and dscusson on selecton of kernel functon are presented n Secton 2. Secton 3 descrbes the learnng capacty of SVM. Secton 4 descrbes an approxmaton approach, where a smple example s presented. Secton 5 descrbes the smulaton approxmaton approach. An example to llustrate the smulaton approxmaton approach s presented n Secton 5 as well. 2. Basc of Support Vector Machne Classfer 2.1 Support Vector Machne Classfer Regardless of support vectors regresson or classfer, support vector machne s desred to fnd a separator whch can partton data-set as far as possble. Let us take a bnary support vector machne as example, as shown n Fgure 1, the obectve of ths SVM s to separate the dark dots(e.g., represents falure state of system) from whte(e.g., represents operatng state of system). Evdently, any set of lnes (L1, L2,,) located between them dots can separate them. Among these lnes, the most reasonable separator should be the lne located n mddle of the two group dots. Fgure 1: Lnear SVM Ths optmal separator can obtaned from a constraned optmsaton formulaton [10,11]: 1 2 m n w (1) w h, b R 2 s. t. y ( w, x + b ) 1, = 1, 2,..., m. where w denotes the regularsed normal lne, b denotes the ntercept, x denotes the vector (representng a dot) and y denotes ts group label (for bnary classfer, represented by 1 and -1 respectvely). Each dot n Fgures 1 corresponds to a constrant n Formulaton (1). However, n practce, t may be mpossble to separate the mentoned dots lnearly. That means there s no lnear lne exstng to separate these dots. To deal wth ths problem, slack varables ξ are ntroduced to tolerate msclassfcaton. The above formulaton can be deduced as follows:

3 34 Yuan Fuqng, Uday Kumar and K. B. Msra mn s. t. 1 2 w y ( < 2 + C ξ w, x m = 1 > + b ) 1 ξ ζ 0, = 1,2,..., m. Formulaton (2) s called prmal problem n SVM. However, SVM usually uses ts dual problem to obtan optmal soluton. By ntroducng Lagrangan multplers, Formulaton (2) s rewrtten to: m m m 1 max α α y y < x, x > + α k 2 = 1 = 1 k = 1 (3) s. t. 0 α C, = 1,2,3..., m m = 0 α y = 0 where α represents Lagrangan multpler whch corresponds x. C s penalty parameter whch trade-offs classfcaton accuracy and computaton complexty, and x, x > can be < rewrtten to K < x, x >, whch s called the Kernel Functon. After the optmal soluton has been obtaned, the decson functon to determne the group for new x s as follows m (4) f ( x ) = sgn( α y K ( x, x ) + b ) = 1 The process to obtan optmal soluton to Formulaton (3) s called tranng. An nterestng property of the optmal soluton s the fact that (2) α 0 defnes the so-called Support Vector (SV), a sub set of the tranng data set. That means only the SVs can nfluence the decson functon. Hence, the problem can be smplfed snce only part of data sets takng effect. Notably, the set of SVs vares wth kernel functon. [12]. 2.2 Selecton of Kernel Functon The most successful part of SVM s ts Kernel functon. Kernel functon plays a key role n support vector machne. It defnes and measures the smlarty of two data-set, and Kernel functon determnes the SVM classfcaton capacty. By tunng kernel functon s parameter, SVM can control ts classfcaton accuracy. Currently, varous kernel functons, e.g., Gaussan functon, polynomal functon, wavelet functon, mult-layer perceptron functon, have been developed for specfed applcaton areas [10]. Among them, the commonest are Gaussan functon and polynomal functon. Gaussan and polynomal functons dffer n ther prncples to measure smlarty. Gaussan measures t by subtracton of the two vectors (x, x ). Polynomal functon does t by usng nner product. Both Gaussan functon and polynomal functon can perform equally for most applcatons. The form of Gaussan functon s: ' x x ' K( x, x ) = exp( ) (5) 2 2σ where σ s a varable parameter, t s a real contnuous value. The form of the polynomal functon s: ' ' d K ( x, x ) = ( < x, x > + 1) (6) where d s a varable parameter, d s usually a nteger. As mentoned above, parameterσ of Gaussan functon has a real contnuous value, whle d n polynomal functon s usually dscrete. In practce, contnuous parameter s

4 Complex System Relablty Evaluaton usng Support Vector Machne for Incomplete Data-set 35 easer to be tuned n order to control the accuracy of classfcaton than the dscrete one. As a consequence, ths paper selects Gaussan functon for the kernel functon. 3. Proposed Approaches usng SVM The tranng process of SVM s essentally a learnng process, whch means absorbng knowledge from tranng data nto the SVM model. In order to llustrate the learnng process, we take a smple example of a 3-component system as shown n Fgure 2. Fgure 2: A 3-component system Table 1 tabulates all the 8 systems states, whch have been used as the tranng set. Durng tranng, the logcal structure of the system s beng learned nto SVM. After tranng, the learned knowledge s comprsed n ts decson functon. The last column n Table 1 lsted out all the system states generated from decson functon. It found they are all dentcal to the real system state. Hence, we can smply use ts decson functon to decde system states nstead of the data gven n Table 1. That shows the learnng capacty of SVM. Table 1: Tranng set and predcted result for Fgure 2 x 1 x 2 x 3 y Pred Our approach s usng the learnng capacty of SVM when full data sets are not avalable. For a network shown n Fgure 2, all ts system states can be enumerated. However, for some more complex system, the logc structure of the system may not been known, or the number of consttuent components n the system s too large to enumerate. The enumeraton of all system states s hence unavalable. We call t ncomplete data-set n ths paper. Snce the traned SVM buld the system s confguraton nformaton nto ts decson functon, straghtforwardly, we can use the decson functon to predct system state for mssed data-set. Thereafter, a complete data-set can be obtaned and ts system relablty can be evaluated. Dependng on the amount of mssed data-set, the paper proposes two approaches to evaluate system relablty: Approach I: For a case where a small amount of data mssed (e.g., only several data sets mssed out of decades data sets), we use an approach named rough approxmaton method to approxmate system relablty. Ths approach frst traned the SVM usng ncomplete data-set. Then use the traned SVM to approxmate the system state for mssed data sets. After that an approxmate full data-set s obtaned. Based on the approxmate full data-set, we evaluate the system relablty. Ths approach s sutable for a less complex system where number of system states s small. Approach II: For a complex system where all system states are not possble to be enumerated, a large amount of data-set are hence mssed. We use smulaton approach as an

5 36 Yuan Fuqng, Uday Kumar and K. B. Msra approxmaton method to evaluate system relablty. An example of ths approach s provded n Secton 5. In ths example, only part of the data set s actually used. 4. Approxmaton Method Ths secton frstly defnes an upper bound of relablty approxmate error so that relablty accuracy s controllable. Then a SVM based approach wll be dscussed to mprove the approxmate relablty. 4.1 Upper Bound of Relablty Approxmate Error Usng the ncomplete data-set, we can obtan a rough system relablty. It smply calculates the probablty of each system state (operatng or faled) based on probabltes of respectve element states. Later on sum up the probabltes of system beng n operatng state to obtan system relablty R, and sum up all the probabltes beng n ' faled state to obtan system unrelablty R unrel. It can also use mnmal cut set and mnmal cut sets approaches proposed by Msra and Rao [15] to compute relablty and ' unrelablty. Evdently, the real relablty and unrelablty s larger than R rel and R ' unrel. Hence an upper bound of evaluaton approxmate error s: ' ' E = R R (7) rel ' rel 1 rel unrel 4.2 Procedure of Rough Approxmaton Method We uses followng step to approxmate system relablty when a small amount data sets mssed. ) Intaton. Select a proper ntal parameter value σ for Gaussan kernel functon. ) Tran the SVM. Tran the SVM usng ncomplete data. Use those ncomplete data-set as nput of SVM, ts system states as correspondng output of SVM. ) Tune SVM parameter σ. Check f the tranng data set has been classfed successfully, where a tolerable msclassfcaton rato (Number of msclassfcaton dvded by number of tranng data sets) s defned as follows. N msclassfcaton r = (8) error NT When r error excesses to the predefned value, tune kernel functon parameter σ untl r error s less than the predefned value. v) Make up all mssed data set. Enumerate out all the mssed element state vectors. v) Approxmate system state. Approxmate system state for the mssed data usng the traned SVM decson functon (Formulaton 4). Therefore, the mssed data sets have been compensated and the data set has become complete. v) Compute system relablty and unrelablty. Usng the approxmate complete dataset, sum up all the probabltes of system beng operatng state to obtan relablty, sum up all the probabltes of system beng faled state to obtan unrelablty. Ths smple method can only be appled to small amount of data-set mssed case. When large data-set mssed, the approxmate error wll be too large to be accepted. 4.3 An Example of Smple Brdge System Let us take as an example, the relablty network as presented by Yoo and Deo [13]. The confguraton of the network s shown n Fgure 3.

6 Complex System Relablty Evaluaton usng Support Vector Machne for Incomplete Data-set 37 Fgure 3: A Smple Brdge System All path relablty are r = 0.9. The exact network relablty s The complete data set, part of whch wll be used as tranng data sets for SVM, s shown n Table 2. Select Gaussan functon as kernel functon. Suppose the data sets (4 data sets) shown n talc (n Table 2) are mssed, the remanng 28 data sets are as tranng data sets. Gaussan functon parameter σ =0.3. After tranng, the data sets can be classfed successfully. Therefore the Step 3 (Tune SVM parameter) mentoned n prevous secton can be omtted. Thereafter we use the traned SVM to approxmate system state for the 4 mssed data sets. The approxmate result s lsted out on last column n Table 2(n talc). Table 2: Tranng set for Fgure 1 x 1 x 2 x 3 x 4 x 5 y y app x 1 x 2 x 3 x 4 x 5 y y app After approxmatng, t s found 1 out of 4 s msclassfed (No.22). Usng the approxmate data sets, we calculated out ther correspondng relablty and unrelablty are and respectvely. To facltate comparson, we provde n Table 3, the relablty and unrelablty, whch are calculated from ncomplete data set, real complete data set and approxmate complete data set, respectvely. Table 3: Comparson of Exact and Approxmate relablty Relablty Unrelablty Complete data set Incomplete data set Approxmate complete data set

7 38 Yuan Fuqng, Uday Kumar and K. B. Msra After approxmatng, the relablty has been mproved from to Ths approach can compensate the ncomplete data for complete data set. However, t s only sutable for small amount of mssed data set. The next secton presents another approach to evaluate system relablty, when large amount of data set s mssed. 5. Smulaton Based Approxmate Method When large amount of data sets have been mssed, or the complete data sets are mpossble to be obtaned due to hgh computaton complexty, the approxmaton approach presented n the prevous secton s not sutable. In graph theory, the problems such as fndng shortest path, longest path are NP hard. That s, computaton complexty s ncreasng exponentally wth the number of nodes or adacent edges [14]. The approach n ths secton s tryng to evaluate system relablty by usng part of those data-sets. 5.1 Relablty Approxmaton Procedure Element state vector (comprsng 1 or -1) s obtaned va smulaton, where each element state s generated randomly by ts relablty one by one. Thereafter, we use an algorthm descrbed n Secton 5.2 to fnd ts system state (1 or -1). After a certan number of smulaton teratons, t s found there are some vectors are most frequent, whle some others seldom occur. We select the most frequent vectors as tranng data-set, snce the most frequent vectors s also the most sgnfcant (mportant) contrbutng to system relablty. The tranng data-set generated va such procedure s better than those selected arbtrarly. The detaled procedure to evaluate system relablty s as follows: ) Generate tranng data sets. After certan number of smulaton teratons, we select the most frequent vector as tranng data-set. Then use the algorthm developed by Secton 5.2 to fnd ts correspondng system state. ) Tran SVM. Select Gaussan functon as kernel functon. Set parameter value σ arbtrarly. Tran the SVM. After tranng, check f all tranng data-set can be separated successfully. Because the classfcaton performance of SVM s hgh, normally, for a data-set sze wthn thousands, SVM can separate them successfully. Nevertheless, a tolerable msclassfcaton rate s defned, see Formulaton (8). When the msclassfcaton rate exceeded the predefned value, tune parameter σ untl msclassfcaton rate satsfed. Actually, σ s tuned by grd search method. ) Compute relablty and unrelablty. Dependng on the number of nodes or adacent edges, two approaches can be used to evaluate relablty and unrelablty. For a smaller scale network, we employ followng steps to evaluate system relablty: a. Enumerate all element states; Fnd ther correspondng system states usng decson functon of the traned SVM to obtan an approxmate complete data-set. b. Compute the relablty by summng up all probabltes beng operatng system states and compute unrelablty by summng up all probabltes beng faled system states. For a larger scale network where enumeraton s mpossble, we employ followng approach to compute system relablty and unrelablty: a. Generate a large number of element states, regardless of replcaton. Fnd ther correspondng system states usng decson functon of the traned SVM. b. Compute system relablty usng followng formulaton. The unrelablty s 1- R rel.

8 Complex System Relablty Evaluaton usng Support Vector Machne for Incomplete Data-set 39 N succ R rel = (9) NTotal where N succ denotes number of operatng system states, whch s determned from decson functon, N total denotes number of total smulaton teratons. The latter approach (smulaton) s an approxmate approach to the former (enumeraton), whle the computaton complexty of smulaton approach s much less than the enumeraton approach. 5.2 An Algorthm to fnd System State After element states vectors have been generated by smulaton, an algorthm s developed to fnd system state. As s known, n network relablty, when there exsts a path from the start termnal to the end termnal, the correspondng system state s then beng operatng state(normally denoted by 1); otherwse, system state s beng faled state (-1). Some network has a addtonal mnmal flow lmt, that s, unless the flow cost from start termnal to end termnal s greater than a predefned mnmal flow, the system state s beng operatng state; otherwse, t s beng faled state. Ths s a so called constrant network problem. The paper uses followng procedure to fnd feasble path (system state) for such a constrant problem. The correspondng flow dagram s shown n Fgure 4. ) Preparaton. Suppose start termnal s s, end termnal s e. Set ntal total flow cost 0. Specfy one stack to store vsted paths. ) Intaton. Check f there exst path from s to all other nodes. If paths exst, push those paths nto stack. ) Search path further. If stack s empty, go to step (v); otherwse, pop up one path (p ) from stack f stack s nonempty. Take out the last node (n l ) from p. Fnd all the adacent nodes (whch have not been n p ) wth n l. v) Check the termnaton crtera. If the adacent node of n l s e, calculate the path cost. If t exceeds predefned mnmal flow, stop the search. The system state has been found 1; otherwse, push the new path nto stack and go to step (). v) Check Result. Check f the stack s empty. If t s empty and a feasble path has not been found, the system state s then -1, that s, no path exstng or flow s less than the predefned mnmal flow. 5.3 An Example of Complex Network Let us now take a classcal brdge network adopted by some papers to demonstrate ther algorthms as s gven n Yoo and Deo [14]. It s a confguraton of 21 adacent edges as shown n Fgure 5. In ths case, enumeratng all system states s computatonally tedous.

9 40 Yuan Fuqng, Uday Kumar and K. B. Msra Preparaton Fnd all adacent nodes of s and put the path nto Stack Yes Stack s empty? No Pop up one path p Take out the last node nl from path p Does nl have adacent node? Yes No Delete p Fnd all adacent nodes of nl Put new path {p, the new adacent node} Select one node adacent to n l Is ths adacent node e? No Yes t exceeds predefned mnmal flow? Delete path {p, the new adacent node} Yes System state -1 System state 1 End Fgure 4: Flow Dagram The network sown n Fgure 5 has 2 21 elements states (around 2 mllons). Here we use the smulaton based approach to approxmate system relablty. Further, we know that elements have the followng relabltes: r 7 = 0.81, r 4 =r 12 =r 13 =r 19 = 0.981, and all other r = 0.9. s Fgure 5: A complex network After smulaton, sze of 200, 500, 1000 sgnfcant non-replcated tranng data-set are generated respectvely. The SVM parameter value s set asσ = 2.3, C= After tranng, t s found no msclassfcaton. Therefore, the tunng parameter σ can be omtted. Then we use the approach descrbed n Secton 5.1 to approxmate system relablty (The number of smulaton s 200,000 n (9)). The number of smulaton to fnd non-replcated data sets s far less than the full number of combnatons of element states (full data sets). Therefore, the computatonal complexty of ths proposed approach s e

10 Complex System Relablty Evaluaton usng Support Vector Machne for Incomplete Data-set 41 reduced greatly compared wth some approaches based on full data sets. The results are tabulated n Table 4. The last column n Table 4 lsts the (relablty + unrelablty) evaluated from the ncomplete tranng data sets. The result of the last column (Raw Cut/Path Set) s calculated usng the mnmal path sets and cut sets approach orgnally proposed by Msra and Rao [15]. Table 4: Comparson of Several Approaches Sample Sze Enumeraton Smulaton Raw Cut/Path Set Approach Approach (Relablty+Unrelablty) When data sets sze s 200, the relablty calculated by path set and cut set s very low. It means the evaluaton error wll be hgh and therefore the sample sze s too small. We mprove the sample sze from 200 to Then the evaluatng error s reduced consderably. By usng the approach descrbed n Secton 5.1, the approxmate relablty s then mproved from to further (The actual relablty R actual = ). Another concluson s that the enumeraton approach and smulaton approach descrbed n Secton 5.2 are almost equvalent when number of smulaton teratons s large. However, the tme consumed by smulaton aapproach s much less. Therefore, obvously, the smulaton approach s preferable over the enumeraton approach. 6. Concluson For system modelled as network relablty, when complete data-set s mpossble to be obtaned, ths paper proposes two approaches to evaluate ts relablty dependng amount of mssed data-set. When a small amount of data-set mssed, the paper uses Support Vector Machne (SVM) to make up the mssed data set. Thereafter, compute the system relablty approxmately usng the approxmate complete data-set; when large data set mssed, for example, a complex network where all system states can not possbly be enumerated, the paper employs SVM, combnng smulaton to evaluate relablty. From the llustrated examples, t s found the evaluatng error can be reduced by usng these approaches. References [1] Msra, K.B. Relablty Analyss and Predcton. Elsever: Amsterdam; [2] Msra, K.B. (Edtor). New Trends n System Relablty Evaluaton. Elsever: Amsterdam; [3] Msra, K.B. (Edtor). Handbook of Performablty Engneerng. Sprnger Verlag:London; [4] Vapnk, V.N. The Nature of Statstcal Learnng Theory. Sprnger: New York; [5] Vapnk, V.N. Statstcal Learnng Theory. John Wley and Sons, Inc: New York; [6] Francs, E.H.T. and L. Cao Applcaton of Support Vector Machnes n Fnancal Tme Seres Forecastng. Internatonal Journal of Management Scence 2001; 29(4): [7] Haozhong,C., Z. Habao and D. Lxn. Fault Dagnoss of Power Transformer based on Mult- Layer SVM Classfer. Electrc Power Systems Research 2005; 75: [8] Rocco, C.M. and J.A.Moreno. Fast Monte Carlo Relablty Evaluaton usng Support Vector Machne, Relablty Engneerng and System Safety 2002; 76: [9] Rocco,C.M., and M. Mussel. Emprcal Models based on Machne Learnng Technques for Determnng Approxmate Relablty Expressons. Relablty Engneerng and System Safety 2004; 83:

11 42 Yuan Fuqng, Uday Kumar and K. B. Msra [10] Scholkopf, B. and A.J. Smola. Learnng wth Kernels. London: The MIT Press; [11]. Gunn,S.R. Support Vector Machnes for Classfcaton and Regresson. Techncal report, School of Electroncs and Computer Scence, Unversty of Southampton, [12] Lng, Z. and Bo,Z. Relatonshp between Support Vector Set and Kernel Functon n SVM. J. Comput.Sc.and Technol 2002; 17(5): [13] Yoo, Y.B. and N. Deo. A Comparson of Algorthms for Termnal-par Relablty. IEEE Transactons on Relablty 1988; 37(2): [14] Ahmad,R.S. and R.M.Omd. All-Termnal Network Relablty Usng Recursve Truncaton Algorthm. IEEE Transacton on Relablty 2009; 58(2): [15] Msra, K.B. and T.S.M.Rao. Relablty Analyss of Redundant Networks usng Flow Graphs. IEEE Transactons on Relablty 1970; 19(1): Yuan Fuqng obtaned hs master s degree n System Engneerng at Beng Unversty of Aeronautcs and Astronautcs, Chna, n the year He oned the Dvson of Operaton and Mantenance engneerng, Luleå Unversty of Technology, Sweden n September 2007 for Ph.D. degree programme. Hs area of research deals wth relablty data analyss and statstcal learnng theory. Uday Kumar obtaned hs B. Tech from Inda n the year After workng for 6 years n Indan mnng ndustres, he oned the postgraduate program of Luleå Unversty of Technology, Luleå, Sweden and obtaned a Ph.D. degree n feld of Relablty and Mantenance n Afterwards, he worked as a Senor Lecturer, Assocate Professor at Luleå Unversty and n 1997, he was apponted as a Professor of Mechancal Engneerng (Mantenance) at Unversty of Stavanger, Stavanger, Norway. Presently, he s Professor of Operaton and Mantenance Engneerng at Luleå Unversty of Technology. Hs research nterests are equpment mantenance, equpment selecton, relablty and mantanablty analyss, system analyss, etc. He has publshed more than 170 papers n Internatonal Journals and Conference Proceedngs. For K.B.Msra s bography, refer to: