Sequential Projection Maximin Distance Sampling Method

Transcription

1 APCOM & ISCM th December, 2013, Sngapore Sequental Projecton Maxmn Dstance Samplng Method J. Jang 1, W. Lm 1, S. Cho 1, M. Lee 2, J. Na 3 and * T.H. Lee 1 1 Department of automotve engneerng, Hanyang Unversty, 222 Wangsmn-ro, Seongdong-gu, Seoul, Korea 2 Korea Insttute of Ocean Scence and Technology, beon-gl, Yuseong-daero, Yuseong-gu, Daejeon, Korea 3 GMKorea, Cheongcheon 2-dong, Bupyeong-gu, Incheon , Korea *Correspondng author: thlee@hanyang.ac.kr Abstract Desgn optmzaton for engneerng problems often requres severe computer smulatons. Thus, to perform a desgn optmzaton effcently, surrogate models replacng the tme-consumng smulator by usng the adequate number of computer experments,.e., desgn and analyss of computer experments (DACE) have been developed. Our goal n ths paper s to propose a sequental desgn of experments to construct a global surrogate model. The proposed method employs the prorty of varables defned from non-lnearty, contrbuton rato or global senstvty. The prorty gves a chance to have better projectve property to more mportant varable, because relatvely more mportant varable sgnfcantly nfluences on the accuracy of surrogate model. Consequently ths causes a decrease n the error of surrogate model and a reducton of the total number of sample ponts. The proposed method s compared wth sequental maxmn dstance desgn and optmal Latn hypercube desgn wth two examples. Keywords: Desgn of experment (DOE), Sequental desgn, Maxmn dstance desgn, Space fllng desgn, Projectve property, Surrogate model Introducton In engneerng problems, desgn often requres computer smulatons to evaluate desgn objectves and constrants. If a sngle smulaton s severe tme-consumng, desgn optmzaton becomes mpossble because t often requres the consderable number of smulatons. One way of allevatng ths burden s to employ surrogate models, for nstance, response surface model (RSM), radal bass functon (RBF) and krgng model. The basc concept of surrogate model s to approxmate relaton between nput and output for predctng responses at untred nput wthn the adequate number of computer experments. Ths can reduce the computatonal cost by replacng the hgh-fdelty smulator. However, snce naccurate surrogate model can gve ncorrect responses, approprate desgn of experment s necessary to generate accurate surrogate model. Thus, many studes have been performed to suggest crtera of superor desgn of experment and to mplement the algorthm to enhance the effcency. Among many crtera, we focus on two crtera such as space fllng and projectve property. And as a method for enhancng effcency, sequental desgn method s adopted. In the followng paragraphs, we brefly revew earler researches for two crtera and sequental desgn methods. One of representatve crtera for computatonal experment s the space fllng that samplng ponts fll desgn space unformly. The space fllng crteron has been developed to obtan nformaton effectvely on the overall desgn doman because 1

2 computatonal experment s determnstc. Many researchers have proposed dfferent crtera to defne the space fllng desgn. Thus, the crtera provde optmal sample set accompaned wth optmzaton algorthm and a sample set can be dfferent accordng to the crtera n spte of the same number of sample ponts. One of these crtera s maxmn dstance that tres to maxmze the smallest Eucldean dstance between any two sets of ponts over desgn doman (Johnson, 1990). It s smple and easy to mplement, so t s wdely used n practce. Maxmum entropy desgn was ntroduced (Shewry and Wynn, 1987). Entropy s defned as the matrx that conssts of entropy functon value of each sample pont such as Gaussan form. By maxmzng determnant of entropy, evenly dstrbuted sample set can be acheved. As a further study, to reduce ts computaton cost and resolve sngularty problem, maxmm egenvalue desgn was suggested (Lee and Jung, 2004). In general, however, above space fllng crtera cannot smultaneously consder projectve property that s a space fllng n terms of each axs. Fg. 1-(a) shows the best space fllng, but projectve property s not consdered at all,.e., collapsng arose. In addton, the crtera cannot reflect the behavor of output but consder only relatons of nput. Thus, to obtan more nformaton of an mportant varable, the scaled maxmn dstance desgn was proposed (Jn Chen, 2002) that gves dmensonal weghtng to more mportant varable but t stll could not solve the problem of overlappng of sample ponts as shown n Fg. 1-(b). Another crteron s projectve property. It s also called non-collapsng or nonoverlappng property. It s mportant to consder dstances that are projected to axs of each varable. If a certan varable x has no nfluence on the output, two desgn ponts that are only dfferent a coordnate of the varable x are consdered as the same pont. Therefore, two desgn ponts should not share any coordnate values. In the early days, desgn method consderng projectve property were used n the feld of safety dagnoss, relablty analyss or uncertanty propagaton. Latn hypercube desgn (LHD) s representatve method (Mckay, 1979). Even now, varous desgn method based on LHD have been steadly proposed. However, there are some problems n LHD whose sample ponts are based, dstorted or clustered as shown n Fg. 1-(c). In order to resolve ths problem, optmal Latn hypercube desgn (OLHD) was developed by employng optmzaton concept n prevous work (Morrs and Mtchell, 1995; Park, 1994). OLHD compromses between optmal crteron such as entropy and Latn hypercube wth the good projecton propertes. However, dependng on ncrease of the number of sample ponts and varables, t takes too much tme to optmze and optmal sample set can be unstable. And desgn based on LHD s dffcult to employ sequental desgn snce area of one sample pont s determned n advance accordng to the number of sample ponts and varables. Meanwhle, most of the authors are concerned these crtera as one shot approach that sample ponts are selected over the desgn space n advance. However, snce a smulator s nonlnear and complex, a desgner s hard to predct how many sample ponts are necessary to acheve the suffcent accuracy of surrogate model. Thus, n order to solve ths problem, sequental desgn methods have been proposed. It allows the samplng process to be stopped as soon as there s suffcent nformaton as data accumulate. Also, t takes nformaton such as predcted response, contrbuton, nonlnearty of varables and mean squared error (MSE) gathered from exstng surrogate model updated sequentally wth new sample ponts and the assocated 2

3 response evaluatons. These are sgnfcantly advantageous compared to one shot approach. Sequental desgn can be used for both global surrogate model and surrogate modelbased desgn optmzaton. Sequental desgn for global surrogate model focuses on sequentally mprovng the accuracy of a surrogate model over the entre desgn space, but sequental desgn for surrogate model-based desgn optmzaton fnds promsng area where optmum pont can be exst. The latter s also called nfllng samplng method that gves up space fllng. As one of the latter method, mean squared error gathered from krgng model based desgn of experment (Sacks and Welch, 1989). And expected mprovement (EI) was suggested n work by Mockus, Tess, and Zlnskas (1978), and has been popularzed n work by Jones, Schonlau, and Welch (1998) as an effcent global optmzaton (EGO) algorthm. EI s the functon whereby ponts that have ether low objectve functon value or hgh uncertanty are preferred. Ths paper focuses on a global surrogate model by usng sequental desgn method. To enhance the effcency, the prorty of varables s defned, that derved from output nformaton,.e., nonlnearty of varable, contrbuton of varable, global senstvty or even ntuton of a desgner. In addton, both space fllng and projectve property s smultaneously consdered to mprove the accuracy quckly. At last, the proposed method, sequental projecton maxmn dstance desgn, overcome drawbacks of earler space fllng desgn and projectve property based desgn. The proposed method s compared wth sequental maxmn dstance desgn and optmal Latn hypercube desgn wth two examples. (a) space fllng desgn, (b) scaled space fllng desgn, (c) LHD, (d) OLHD Fgure 1. Examples of exstng desgn methods: sample ponts on 2-D (blue) and those projected to axs (red) Sequental projecton maxmn dstance desgn Formulaton of the proposed method Sequental projecton maxmn dstance desgn s proposed based on maxmn dstance desgn. Orgnal maxmn dstance desgn doesn t use output nformaton gathered from exstng surrogate model but use only nput nformaton,.e., dstance between pre-sampled ponts. Thus, to select a new sample pont(s), we ntroduce sequental projecton maxmn dstance desgn. The proposed method employs prorty of varables. If one varable s more mportant than the other varables, prorty should be assgned to that varable. The prorty gves chance to have better projectve property to more mportant varable. In other word, relatvely less mportant varable s projectve property does not sgnfcantly nfluences on accuracy of surrogate model. Thus, accordng to prorty, each varable s sequentally optmzed n teraton. And n order to satsfy space fllng crteron, frst optmzed 3

4 varables contnually nfluence on an objectve functon,.e. modfed dstance. The steps of new method are: Step 1 Defne prorty measure and gather nformaton of prorty from exstng surrogate model. Step 2 Maxmze the proposed crteron made up of Mn. l 1 norm of 1 st prorty varable Step 3 Maxmze the proposed crteron made up of Mn. l 2 norm and Mn. l 1 norm of 2 nd prorty varable wth optmzed 1 st prorty varable. Step 4 Repeat step 3 untl the last varable. Above steps can be formulated as Eq. (1) Fnd x new a set of prorty order, whle n f 1, Maxmze mn x else, whle Maxmze end d E, x 1 n 1,2,... n new 1 d d mn x, 1 j E, x new 2 mn x E, x new 1, j 1,...,, 1 (1) where x E are exstng sample ponts and n d s the number of varables. In ths method, t s mportant to defne the prorty snce t decsvely determnes accuracy of surrogate model. The prorty of varable can be defned from nonlnearty of varable, contrbuton of varable, global senstvty or even ntuton of desgners. Among them, we employ the nonlnearty of varable that can be alternated by correlaton parameters, θ k, n krgng model. The correlaton parameters ndcate smoothness of x k coordnate. The smaller θ k lnear effect on the response of the varable, mpact on the response s non-lnear as the θ k ncreases. Proposed method comparson wth sequental maxmn dstance desgn and OLHD The proposed method s compared wth sequental maxmn dstance desgn and OLHD n order to show or not t meets above crtera, space fllng and projectve property. Experments are carred out sequentally one by one on 2-dmensonal doman from ntal 6 sample ponts, 4 on vertex and 2 on center. And snce OLHD cannot provde sequental desgn, we perform experments at the same number of sample ponts n order to compare the surface of dstrbuton of sample ponts. We use genetc algorthm as an optmzer provded by matlab R2011 to select a new pont, and OLHD s also desgned by matlab toolbox. Fg. 2-(a) show results of sequental maxmn dstance desgn. We can check a frst optmzed sample pont s located n bottom lne. It s the best poston as an aspect of space fllng, but t can be the worst poston n terms of vertcal axs. A second optmzed sample pont also smlar. Also after addng 18 ponts, the trend of result s same. 4

5 The results of OLHD n Fg. 2-(b) are the opposte of the results of sequental maxmn dstance desgn. Whle projectve property s suffcently conscous, space fllng s poor. Even f one sample pont s selected, the results of the proposed method n Fg 2-(c) show ts characterstc well. The frst optmzed pont satsfes projectve property both horzontal and vertcal axs. And after addng 18 ponts lkewse above method, whle not lose space fllng, projectve property s mantaned very well. (a) Sequental maxmn dstance desgn (sequentally sampled from 6 to 24) (b) OLHD (one shot approach) (c) Sequental projecton maxmn dstance desgn (sequentally sampled) Fgure 2. Surface of dstrbuton of samples usng three methods: pre-sampled ponts (blue), projected ponts (black) and a new pont (red) Examples The two examples are utlzed n order to show the performance of proposed method. Snce we focus on buld up accurate global surrogate modelng, the accuracy,.e. error s used as a performance measure. Mathematcal example The frst example s a mathematcal example n 2-D that can easly obtan responses and know real response. The equaton of example s as followng; f x ) 2cos(8 x ) 0.25cos( x ) 12, 0 x, x 1 (2) ( The experment s carred out n the followng procedure. Step 1 Select ntal sample ponts. Step 2 Buld up surrogate model,.e. krgng model wth ntal sample ponts. Step 3 Predct responses at valdaton ponts, and measure the error. Step 4 Select a new sample pont accordng to each method. 5

6 Step 5 Repeat steps 2~4 untl pre-defned maxmum teraton and skp step 4 n last teraton. Valdaton ponts are 9 2 from full factoral desgn (FFD) and we employ a mean relatve error (MRE) as follows; Y Yˆ MRE % E 100 (3) Y Fgure 3. Hstory of error for mathematcal example as addng sample ponts; 21, 29 and 30 mean the frst number of sample pont satsfyng 1% error Errors of krgng model made by the three methods are decrease as addng sample ponts. It means that krgng model becomes more accurate as added pre-sampled ponts. However, error usng the proposed method (SP-maxmn) consderng projectve property accordng to prorty of varables decreases faster than two other methods. Thus, the proposed method can reduce 8 or 9 experments. Errors of OLHD as the one shot approach fluctuate snce ts dstrbuton of sample ponts s change. Engneerng example The second example s an engneerng example wth 10 desgn varables that takes more tme and cost. Target model s a front cradle n a passenger vehcle released by GM Korea. 10 thckness values are consdered as desgn varables. Analyss purpose s to make plastc deformaton of front cradle under target value. Abaqus s used as analyzer. The experment s carred out n the followng procedure. Step 1 Select ntal sample ponts. Step 2 Buld up surrogate model,.e. krgng model wth ntal sample ponts. Step 3 Predct responses at valdaton ponts. Step 4 Measure the error. Step 5 Stop f error s under 1% fve tmes n a row. Step 6 Select a new sample pont accordng to each method. And go step. 2 6

7 Fgure 4. Fnte element model of a front cradle Cross valdaton (one-leave-out) s employed snce addng ponts s also tmeconsumng. Intal sample ponts are obtaned from OLHD and they are used for valdaton ponts. Lastly OLHD s not consdered n ths example. Fgure 5.. Hstory of error for an engneerng example as addng sample ponts: 53 and 89 mean the last number of sample pont satsfyng 1% error 5 tmes n a row Lkewse a mathematcal example, the error of the propose method decreases faster whle the proposed method reduce the total number of sample ponts. It means the proposed method save about 6 hours snce 1 smulaton takes about 10 mnute. Conclusons The sequental desgn method to create global surrogate model accurately s proposed n ths paper. The proposed method employs the prorty of varables defned from non-lnearty, contrbuton rato or global senstvty. The prorty gves a chance to have better projectve property to more mportant varable. Consequently more 7

8 nformaton of varables n the prorty can be obtaned by usng the proposed method. Ths decreases the error of surrogate model and reduces the total number of sample ponts. In order to show the performance of the proposed method, krgng model s ntroduced and correlaton coeffcents of krgng model are consdered as a crteron defnng the prorty. And sequental maxmn dstance desgn and optmal Latn hypercube desgn are used for comparson. The mathematcal example that conssts of a hghly nonlnear varable and a moderately lnear varable shows an advantage of the proposed method well. And there s a remarkable dfference between the convergence hstores. Ths s because a curve of the hghly nonlnear varable s well ftted when projectve property of t s fully represented. Even f the response s unpredctable, ths mert stll exsts. In engneerng example, the fnte element model of front cradle n the vehcle, a samplng wth the proposed method s stopped after 53th teraton under defned stop crteron. In other word, the proposed method uses 119 sample ponts to create a suffcently accurate surrogate model and t uses less 36 sample ponts than usng sequental maxmn dstance desgn. In terms of tme, an engneer can save 6 hours snce 1 smulaton takes about 10 mnute. As a result, the sequental projecton maxmn dstance desgn helps engneers to solve the problems n that only a part of varables are mportant. References Johnson, M., Moore, L. and Ylvsaker, D., (1990), Mnmax and maxmn dstance desgns. Journal of Statstcal Plannng and Inference. 26, pp Shewry, M.C, and Wynn, H.P (1987), Maxmum entropy samplng. Journal of Appled Statstcs. 40(4), pp Lee, T.H., and Jung, J.J. (2004), Maxmum egenvalue samplng of krgng model. 10th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference. Albany, New York, August, AIAA Morrs, M. D. and Mtchell, T. J. (1995), Exploratory desgns for computatonal experments. Journal of Statstcal Plannng and Inference. 43(3), pp McKay, M.D., Beckman, R.J. and Conover, W.J. (1979), A comparson of three methods for selectng values of nput varables n the analyss of output from a computer code. Technometrcs. 21(2), pp Park, J.S. (1994), Optmal Latn-hypercube desgns for computer experments. Journal of statstcal plannng and Inference. 39(1), pp Sacks, J., Welch, W.J., Mtchell, T.J. and Wynn, H.P. (1989), Desgn and analyss of computer experments. Statstcal Scence. 4, pp Mockus, J., Tess, V. and Zlnskas, A. (1978), The applcaton of Bayesan methods for seekng the extremum. Towards Global Optmsaton. North Holland, Amsterdam, 2, pp Jones, D.R., Schonlau. M., and Welch, W.J (1998), Effcent global optmzaton of expensve blackbox functons. Journal of Global Optmzaton. 13, pp Jn, R., Chen, W., and Sudjanto, A. (2002), On sequental samplng for global metamodelng for n engneerng desgn. Proceedngs ASME 2002 Desgn Engneerng Techncal Conferences and Computer and Informaton n Engneerng Conference, Montreal, Canada, September, DETC2002/DAC Crombecq, K., Laermans, E. and Dhaene, T. (2011), Effcent space-fllng and non-collapsng sequental desgn strateges for smulaton-based modelng. European Journal of Operaton Research. 214(3), pp Xong, F., Xong, Y., Chen, W. and Yang, S. (2009), Optmzng Latn Hypercube desgn for sequental samplng of computer experments. Engneerng Optmzaton. 41(8), pp Tmothy W. S., Denns, K. J. Ln, Chen, W. (2001), Samplng strateges for computer experments: desgn and analyss. Internatonal Journal of Relablty and Applcatons. 2(3), pp Jn, R., Chen, W., Sudjanto, A. (2005). An effcent algorthm for constructng optmal desgn of computer experments. Journal of Statstcal Plannng and Inference.134, pp van Dam, E.R., Husslage, B., den Hertog, D., Melssen, H. (2007). Maxmn latn hypercube desgn n two dmensons. Operatons Research. 55, pp