A METHOD FOR FACTOR SCREENING OF SIMULATION EXPERIMENTS BASED ON ASSOCIATION RULE MINING

Size: px

Start display at page:

Download "A METHOD FOR FACTOR SCREENING OF SIMULATION EXPERIMENTS BASED ON ASSOCIATION RULE MINING"

Sibyl Griselda Jacobs
5 years ago
Views:

1 A METHOD FOR FACTOR SCREENING OF SIMULATION EXPERIMENTS BASED ON ASSOCIATION RULE MINING Lngyun Lu (a), We L (b), Png Ma (c), Mng Yang (d) Control and Smulaton Center, Harbn Insttute of Technology, Harbn 158, P.R. Chna (a) jyzluht@gmal.com, (b) fran@ht.edu.cn, (c) pngma@ht.edu.cn, (d) myang@ht.edu.cn ABSTRACT Snce complex smulaton models contan a large number of nputs and parameters, t s meanngful to dentfy the most mportant ones for further studes. The tradtonal procedure of factor screenng s to desgn smulaton experment schemes and generate smulaton output data frst. Furthermore, due to rgorous hypothess and fewer levels for each factor, the exstng screenng methods are usually not approprate for complex smulaton. In ths paper, a new method for factor screenng of smulaton experments s proposed. It s based on data mnng technque and exstng smulaton data. The relatonshps between factors and the output are dscovered from the sets of smulaton data usng the assocaton rule mnng algorthm frstly. Then the mportant factors can be dentfed by syntheszng the assocaton rules. Fnally, the proposed method s verfed by the test functon. Keywords: smulaton experments, factor screenng, assocaton rules, quanttatve attrbutes 1. INTRODUCTION Complex smulaton models consst of a large number of nputs and parameters whch are generally referred to as factors n desgn of experments (DOE). DOE s essental for dong certan analyses nclude valdaton, uncertanty analyss, senstvty analyss, optmzaton, etc (Klejnen 28, L 216). However, t s usually prohbtve or mpractcal to do these analyses wth the large number of factors nvolved, especally for computer smulaton models wth hgh computatonal cost. An mportant queston s whch factors are really sgnfcant when there are potentally a large number factors nvolved (Alam 24)? The Pareto prncple or 2-8 rule mples that only a few factors are really mportant (Wrght 21). In order to mprove effcency, t s reasonable to screen a small number of factors before analyzed. Factor screenng s performed to elmnate unmportant factors so that the remanng mportant factors can be more thoroughly studed n later experments, regardless of the applcaton doman of area, ncludng natonal defense, logstcs, ndustry, healthcare, etc (L 216, Bruzzone 214, Padh 213, Rantanen 215). Several types of factor screenng methods have been developed to dentfy mportant factors wth a lmted set of smulaton experments. The most common ones are fractonal factoral, central composte, and Placett Burman desgns (Myers 22). However, these screenng desgns developed for physcal experments are not approprate for smulaton physcal experments. Procedures developed for stochastc smulaton experments nclude Morrs randomzed one-factor-ata-tme desgn (Morrs 1991), frequency doman method (Morrce 1995), edge desgns (Elster 1995), terated fractonal factoral desgns (Campolongo 2) and the Trocne screenng procedure (Trocne 21). Some methods developed for dscrete-event smulaton experments nclude sequental bfurcaton (SB), SB wth nteractons (SB-X), Cheng s method, controlled sequental bfurcaton (CSB), CSB wth nteractons (CSB-X), controlled sequental factoral desgn (CSFD), fractonal factoral controlled sequental bfurcaton (FFCSB), etc (Wan 27, Shen 29). However, the majorty of these methods have many assumptons or lmtatons, such as monotoncty of the functon, the smoothness of the nputs/outputs functon, etc. These methods are effcent and effectve when ther assumptons are satsfed. From the factor screenng methods descrbed above, t can be concluded that these methods easly neglect some mportant factors due to rgorous hypothess and few factor levels. It s possbly unsutable for the complex smulaton models. As s nown to all, the mportance of factors depends on the expermental doman. The effectve way s to select levels as many as possble durng factor screenng whch can represent smulaton model more precsely and mprove the accuracy of the classfcaton. So the data mnng algorthm s adopted to solve ths problem, whch can be based on the exstng smulaton data and regardless of mathematcs assumptons. In ths paper, the assocaton rule mnng algorthm s nvestgated as a potental factor screenng method. It should be based on smulaton data, ncludng smulaton nputs/parameters and the output. The remander of the paper s organzed as follows. In Secton 2, the procedure of the Apror algorthm and the algorthm of parttonng quanttatve attrbutes are descrbed. In Secton 3, a method for factor screenng of smulaton experments s proposed. Through ths method, mportant factors can be dentfed by syntheszng the assocaton rules. In Secton 4, the Proceedngs of the European Modelng and Smulaton Symposum, 217 ISBN ; Affenzeller, Bruzzone, Jménez, Longo and Pera Eds. 34

2 proposed method s llustrated and verfed by the test functon. Fnally, the paper s concluded n Secton RELATED WORK The objectve of factor screenng s to dentfy the mportant factors among many factors nvolved. In ths secton, the procedure of the Apror algorthm wll be descrbed. However, ths algorthm can only cope wth boolean attrbutes nstead of quanttatve attrbutes. But quanttatve attrbutes usually exst n smulaton models. So the algorthm of parttonng quanttatve attrbutes s adopted Apror Algorthm The Apror algorthm s proposed by Agrawal (1994) whch s used to fnd potental relatonshps between the tems (Data_Attrbutes). The result of the algorthm wll be expressed n the form of assocaton rules. In order to clarfy the Apror algorthm clearly, some notatons s llustrated n the followng. -temset: An temset havng tems. L : Set of large -temsets wth mnmum support. C : Set of canddate -temsets whch has potentally large temsets. D : Set of transactons where each transacton T s a set of tems,.e., T D. Gven a set of transactons D, the use of the Apror algorthm s to generate all assocaton rules whch the support and confdence are larger than the gven mnmum values. The mnmum support and mnmum confdence are referred to as mn_sup and mn_conf respectvely. The procedure of the Apror algorthm s descrbed as follows: Apror algorthm 1: L 1 ={large 1-temsets}; 2: For ( 2 ; L ; ) do begn 3: C apror-gen( L ); 4: For all transactons t D do begn 5: Ct subset( C, t); 6: For all canddates c Ct do 7: c.count ; 8: end 9: end 1: L { c C c.count mn_sup }; 11: end 12: return L L. At the begnnng, the large 1-temsets s determned by smply counts tem occurrences. Then subsequent pass conssts of two phases. One s the canddate temsets C whch s generated by the apror-gen functon; and the other s the support of canddates n C whch s counted by scannng the database. The apror-gen functon descrbed above returns a superset of the set of all large -temsets. And the subset functon returns all the canddates contaned n a transacton t wth some rules. The more detals of these two functons can be referred to the lterature Agrawal (1994) The Algorthm of Parttonng Quanttatve Attrbutes However, quanttatve attrbutes need to be dscretzed when usng the Apror algorthm. Two problems may be occurred after dscretzaton (Srant 1996). mn_sup. If the number of ntervals for a quanttatve attrbute (or values, f the attrbute s not parttoned) s large, the support for any sngle nterval can be low. Hence, wthout usng larger ntervals, some rules nvolvng ths attrbute may not be found because they lac mnmum support. mn_conf. There s some nformaton lost whenever we partton values nto ntervals. Some rules may have mnmum confdence only when an tem n the antecedent conssts of a sngle value (or a small nterval). Ths nformaton loss ncreases as the nterval szes become larger. In order to solve problems descrbed above, the algorthm of the equ-depth parttonng (Fuuda 1999) s adopted, whch can dvde N data nto M bucets almost evenly. The procedure of the algorthm s descrbed as follows. Step 1: Mae an S-szed random sample from N data. Step 2: Sort the sample n OS ( log S ) tme. Step 3: Scan the sorted sample and set SM ( )-th the smallest sample to p for each 1,, M. Let p be and p M be. Step 4: For each tuple x n the orgnal N data, fnd such that p x p and assgn x to the -th bucet. Ths chec can be done n O(log M ) tme by usng the bnary search tree for the bucets. Thus, for all, the sze u of B can be computed n ON ( log M) tme. 3. THE METHOD FOR FACTOR SCREENING OF SIMULATION EXPERIMENTS To dentfy mportant factors from many factors nvolved n smulaton models, the Apror algorthm descrbed n Secton 2 s adopted. However, n many cases, factors used n smulaton are quanttatve attrbutes, whch the Apror algorthm cannot cope wth them. So data preprocessng s used to partton the doman of quanttatve attrbutes nto several ntervals and mapped these ntervals nto dscrete attrbutes. And the algorthm of the equ-depth parttonng s adopted to prunng the search space. What need to be ponted out that calculaton of strong rules s not nterestng whle usng support-confdent framewor, lft for mnng postve non-redundant assocaton rules s adopted here. It s assume that temsets A and B are denote the premse and consequent of assocaton rules, respectvely, where A, B, A B. Here A B s used to represent the relatonshp between Proceedngs of the European Modelng and Smulaton Symposum, 217 ISBN ; Affenzeller, Bruzzone, Jménez, Longo and Pera Eds. 341

3 C C apror-gen( ) L Fgure 1: Flow of Factor Screenng Based on the Improved Apror Algorthm factors and the output. Then the confdence of A B can be calculated as support( A B) confdence( A B) P( B A) (1) support( A) where support( A ) s the support of A. The flow of factor screenng based on the mproved Apror algorthm s shown n Fgure 1. The procedure of the proposed method manly conssts of four steps: Step 1: Data preprocessng conssts of partton quanttatve attrbutes nto several ntervals and dscretzaton. Step 2: Obtan the set of all frequent temsets by fndng all temsets where the support of the correspondng temsets satsfes mn_sup. Step 3: Generate assocaton rules usng frequent emsets where the confdence of the correspondng frequent temsets satsfes mn_conf. Step 4: Calculate the senstvty ndces of factors by Equaton (2), and eventually obtan the mportant factors by comparng the values of senstvty ndces. The equaton of calculatng senstvty ndces s gven as r max(confdence(x y)) mn(confdence(x y)) (2) r S r where x s factor and y s the output. 4. CASE STUDY The method proposed n ths paper can be used for factor screenng of smulaton experment. In ths secton, a test functon s used to demonstrate the effectveness of the proposed method. Consder the test functon y 1x 9x 2x x 15x 3x 2x 16x x (3) where x1, x2,, x 7 s an..d. sample from U (,1). The set of data { x1, x2,, x7, y } s obtaned frst, where the sample sze s 5. Then each factor x s parttoned nto seven ntervals through the algorthm of the equ-depth parttonng. Each nterval of x and y are dscretzed n the form of { x _1, x _2,, x _7} and { y_1, y_2,, y _7}, respectvely. The assocaton Proceedngs of the European Modelng and Smulaton Symposum, 217 ISBN ; Affenzeller, Bruzzone, Jménez, Longo and Pera Eds. 342

4 1 Emprcal CDF x 1 x 2 x 3 x 4 x 5 F( x ).5 x 6 x x Fgure 2: An Ensemble of 7 CDFs of the Factors rules are generated by the Apror algorthm, as shown n Table 1. Only 2-temsets s taen to be consdered,.e., A only contans one factor n each rule. Accordng to the fourth column of Table 1, the cumulatve dstrbuton functon (CDF) for the confdence of each factor x s drawn n Fgure 2. The relatve mportance of the factors s qualtatvely expressed by the gradents of the curves, whch the smaller one s more mportant. In other words, the range of confdence for each factor reflects the mportance of factors. Furthermore, the senstvty ndces for each factor are calculated and shown n Fgure 3. It can be quanttatvely conclude that the order of factors s x6 x3 x1 x2 x4 x5 x7 accordng to the mportance. The mportant factors are x 6, x 3, x 1 and x 2. The expermental results obtaned above are n agreement wth the theoretcal results whch are easly concluded from Equaton (3). Table 1: Lst of the Assocaton Rules ID A B confdence( A B) 1 x _ 2 1 y _ x _ 2 1 y _ x _ 4 1 y _ x _1 y _ x _ 7 y _ x _ 3 y _ Above all, t s demonstrated that the proposed method n ths paper s an effectve method whch can be used for factor screenng. Compared wth the tradtonal screenng methods, the proposed method can dentfy mportant factors based on the exstng smulaton data and regardless of mathematcs assumptons. S x1 x2 x3 x4 x5 x6 x7 Fgure 3: Hstogram of Senstvty Indces for Test Functon Usng the Proposed Method 5. CONCLUSIONS Factor screenng s very mportant whle smulaton models to be analyzed wth many factors nvolved. In ths paper, the use of assocaton rule mnng algorthm as a nd of factor screenng method has been studed. The relatonshps between factors and the output are consdered to be a measure of ndex to dentfy mportant factors from many factors nvolved. As a classc algorthm of assocaton rule mnng, Apror algorthm can only be used to solve the problem of mnng boolean assocaton rules. So the algorthm of equ-depth parttonng s adopted to transform quanttatve attrbutes nto boolean attrbutes. After the assocaton rules are generated, the results of confdence are used to calculate the senstvty ndces of factors. And the mportant factors are dentfed accordng to the senstvty ndces. It s demonstrated from the case study that the proposed method appears to perform well n factor screenng. To mprove the effcency of factor screenng and nsure ts accuracy, two aspects are deserved to be further Proceedngs of the European Modelng and Smulaton Symposum, 217 ISBN ; Affenzeller, Bruzzone, Jménez, Longo and Pera Eds. 343

5 studes. One s to study how to synthesze the assocaton rules n other ways, and the other s how to process the unbalanced data. ACKNOWLEDGMENTS Ths research s supported by the Natonal Natural Scence Foundaton of Chna (Grant No ). REFERENCES Klejnen J.P.C., 28. Desgn and analyss of smulaton experments. New Yor:Sprnger. L W., Lu L.Y., Lu Z.Z. et al., 216. HIT-SEDAES: An ntegrated software envronment of smulaton experment desgn, analyss and evaluaton. Internatonal Journal of Modelng, Smulaton, and Scentfc Computng, 7 (3), Alam F.M., McNaught K.R., Rngrose T.J., 24. Usng Morrs' randomzed OAT desgn as a factor screenng method for developng smulaton metamodels. Proceedngs of the 24 Wnter Smulaton Conference, pp December 5-8, Washngton DC, Unted States. Wrght A., Bates D.W., 21. Dstrbuton of Problems, Medcatons and Lab Results n Electronc Health Records: The Pareto Prncple at wor. Appled Clncal Informatcs, 1 (1), Sanchez S.M., Wan H., 215. Wor smarter, not harder: a tutoral on desgnng and conductng smulaton experments. Proceedngs of the 215 Wnter Smulaton Conference, pp December 6-9, Huntngton Beach, Unted States. Bruzzone, A., Longo, F., 214. An applcaton methodology for logstcs and transportaton scenaros analyss and comparson wthn the retal supply chan. European Journal of Industral Engneerng, 8 (1), Padh S.S., Wagner S.M., Nranjan T.T. et al., 213. A smulaton-based methodology to analyse producton lne dsruptons. Internatonal Journal of Producton Research, 51 (6), Rantanen J., Khnast J., 215. The future of pharmaceutcal manufacturng scences. Journal of Pharmaceutcal Scences, 14 (11), Myers, R.H., Montgomery, D.C., 22. Response surface methodology: process and product optmzaton usng desgned experments, second ed. John Wley & Sons, New Yor. Morrs M.D., Factoral samplng plans for prelmnary computatonal experments. Technometrcs, 33: Morrce, D.J., Bardhan, I.R., A weghted least squares approach to computer smulaton factor screenng. Operatons Research, 43 (5), Elster, C., Neumaer, A., Screenng by conference desgns. Bometra 82 (3), Campolongo, F., Klejnen, J.P.C., Andres, T., 2. Screenng methods. In: Saltell, A., Chan, K., Scott, E.M. (Eds.), Senstvty Analyss. John Wley & Sons, New Yor, pp Trocne L., Malone L.C., 21. An overvew of newer advanced screenng methods for the ntal phase n an expermental desgn. Proceedngs of the 21 Wnter Smulaton Conference, pp December 9-12, Arlngton, Unted States. Wan H., Anenman B., 27. Two-stage Controlled Fractonal Factoral Screenng for Smulaton Experments. Journal of Qualty Technology, 39 (2), Shen H., Wan H., 29. Controlled sequental factoral desgn for smulaton factor screenng. European Journal of Operatonal Research, 198, Agrawal R., Srant R., Fast algorthms for mnng assocaton rules. Proceedngs of the 2th Internatonal Conference on Very Large Data Bases, pp September 12-15, Santago, Chle. Srant R., Agrawal R., Mnng quanttatve assocaton rules n large relatonal tables. Proceedngs of the 1996 ACM SIGMOD Internatonal Conference on Management of Data pp June 4-6, Montreal, Canada. Fuuda T., Mormoto Y., et al., Mnng optmzed assocaton rules for numerc attrbutes. Journal of Computer and System Scences, 58 (1), Rastog R., Shm K., 22. Mnng optmzed assocaton rules wth categorcal and numerc attrbutes. IEEE Transactons on Knowledge and Data Engneerng, 14 (1), Ke Y.P., Cheng J., Ng W., 28. An nformatontheoretc approach to quanttatve assocaton rule mnng. Knowledge and Informaton Systems, 16 (2), AUTHORS BIOGRAPHY LINGYUN LU s a Ph.D. student at Harbn Insttute of Technology (HIT), and receved the M.E. from HIT n 213. Hs research nterests nclude smulaton experment desgn and smulaton data analyss. Hs emal s jyzluht@gmal.com. WEI LI s an assocate professor at HIT, and receved the B.S., M.E. and Ph.D. from HIT n 23, 26 and 29 respectvely. Hs research nterests nclude smulaton evaluaton, smulaton data analyss, and dstrbuted smulaton. Hs emal s fran@ht.edu.cn. PING MA s a professor at HIT, and receved Ph.D. from HIT n 23. Her research nterests nclude dstrbuted smulaton technque and VV&A. Her emal s pngma@ht.edu.cn. MING YANG s the correspondng author. He s a professor and the drector of control and smulaton center at HIT. Also he s the vce edtor-n-chef for Journal of System Smulaton and edtor for Internatonal Journal of Modelng Smulaton and Scence Computng. Hs research nterests nclude system smulaton theory and VV&A. Hs emal s myang@ht.edu.cn. Proceedngs of the European Modelng and Smulaton Symposum, 217 ISBN ; Affenzeller, Bruzzone, Jménez, Longo and Pera Eds. 344

Parallelism for Nested Loops with Non-uniform and Flow Dependences

Parallelism for Nested Loops with Non-uniform and Flow Dependences Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr