th Word Congress on Structura and Mutdscpnary Optmzaton May 9-4, 3, Orando, Forda, USA Characterzng Probabty-based Unform Sampng for Surrogate Modeng Junqang Zhang, Souma Chowdhury, Ache Messac 3 Syracuse Unversty, Syracuse, NY, USA, jzhang6@syr.edu Syracuse Unversty, Syracuse, NY, USA, sochowdh@syr.edu 3 Syracuse Unversty, Syracuse, NY, USA, messac@syr.edu. Abstract Approprate sampng of tranng ponts s one of the prmary factors affectng the fdety of surrogate modes. Ths paper nvestgates the reatve advantage of probabty-based unform sampng over dstance-based unform sampng n tranng surrogate modes whose system nputs foow a dstrbuton. Usng the probabty of the nputs as the metrc for sampng, the probabty-based unform sampe ponts are obtaned by the nverse transform sampng. To study the sutabty of probabty-based unform sampng for surrogate modeng, the Mean Squared Error MSE) of a monoma form s formuated based on the reatonshp between the squared error of a surrogate mode and the voume or hypervoume per sampe pont. Two surrogate modes are deveoped respectvey usng the same number of probabty-based and dstance-based unform sampe ponts to approxmate the same system. Ther fdetes are compared usng the monoma MSE functon. When the exponent of the monoma functon s between and, the fdety of the surrogate mode traned usng probabty-based unform sampng s hgher than that of the other one traned usng dstance-based unform sampng. When the exponent s greater than or ess than, the fdety comparson s reversed. Ths theoretca concuson s successfuy verfed usng standard test functons and an engneerng appcaton.. Keywords: Surrogate modeng, Probabty-based sampng, Dstance-based sampng 3. Introducton Surrogate modeng s a statstca approach used to deveop approxmaton functons that adequatey represent the reatonshp between nputs and outputs based on known data[]. To aevate the burden of hgh expermenta or computatona costs resutng from compex engneerng desgn probems, surrogate modeng has been frequenty used as an effcent approach[, 3]. To deveop the surrogate mode of a system, sampe ponts of the nputs to the system need to be generated. These sampe ponts and ther correspondng system outputs are used to tran the surrogate mode. Dstance-based sampng uses coordnate-dstances as the metrcs between ponts n the sampe space. If the nputs to the system are physca parameters whose probabty of occurrence s known or predefned, probabty-based sampng can use the dfference of probabty as the metrcs between ponts. These sampng approaches can generate sampe ponts unformy n terms of dstance and probabty, respectvey. Rada Bass Functon RBF) and Krgng are expressed as combnatons of bass functons[]. Snce ther overa error s reated to the sampe densty, or equvaenty the voume or hypervoume per sampe pont, ths paper formuates the Mean Squared ErrorMSE) of a monoma form based on the reatonshp between the squared error of a surrogate mode and the voume or hypervoume per sampe pont. The overa MSEs of the two surrogate modes of the same system deveoped respectvey usng the probabtybased unform sampng and the dstance-based unform sampng are compared. Secton 4 ntroduces the probabty-based sampng approach. Secton 5 presents the estmaton of the MSE of a surrogate mode. Secton 6 ustrates the measure of a regon assocated wth a sampe pont. Secton 7 formuates the MSE of the monoma form used to anayze the fdety of surrogatemodes. The sutabty of probabty-based unform sampng s studed n Sec. 8. Secton 9 dscusses how to ft the MSE of the monoma form. Secton tests the concuson of the sutabty of probabty-based sampng. Secton appes probabty-based sampng to the deveopment of surrogate modes for wndow performance evauaton. The concudng remarks are presented n Sec.. 4. Probabty-based Sampng The popuar approaches to generate sampe ponts from a probabty dstrbuton ncude nverse transform sampng[4], rejecton sampng[5], mportance sampng[6], Markov Chan Monte Caro methods[7]. In ths paper, nverse transform sampng s used to evauate the vaues of random varabes correspondng
to ther desgnated probabtes. It generates sampe ponts from a probabty dstrbuton gven the Cumuatve Dstrbuton Functon CDF). In an n-dmensona space R n, x k), k =,,...,n, s used to representthe k th randomvarabe. Suppose the n random varabesare ndependent. Functon F k x k) ) s the CDF of the varabe x k). A set of numbers c k) of x k). The probabty-based sampe ponts x k)p) evauated by x k)p) [,], =,,...,m, are the vaues of probabty correspondng to the set of probabtes c k) are ). ) = F k Suppose the ower and upper boundares of x k) are x k) mn sampng scaes the set of numbers c k) x k)d) c k) and xk) max, respectvey. The dstance-based to the coordnates x k)d) by ) = x k) mn +ck) x k) max x k) mn. ) Many sampng sequences have been deveoped to address dfferent desgn space exporaton demands. Fu factora sampng sequence[], ow-dsperson sequence[8], and ow-dscrepancy sequence[9] are popuar unform sequences used for optmzaton, surrogate modeng, and numerca ntegraton. Scaed or nversey transformed from the sequences, the dstance-based and probabty-based sampe ponts are unform n terms of dstance and probabty, respectvey. 5. Estmaton of MSE 5.. Integrated Estmaton of MSE The output y of a system s approxmated as a functon hx) of the system nput x. The CDF of x s Fx). The ntegrated MSE of the surrogate mode hx) s gven by[] MSE I = y hx)) dfx). 3) 5.. Emprca Estmaton of MSE In practca engneerng probems, the output y of a system s usuay ony known at a mted number of test ponts. The ntegrated estmaton of the MSE s not ready avaabe. Usng a set of test ponts x j,y j ), j =,,...,s, the emprca estmaton of the MSE s evauated by[] MSE E = s s y j hx j )). 4) j= If the probabty dstrbuton of the test ponts x j foows the dstrbuton of x, the emprca estmaton of the MSE shoud be cose to the ntegrated estmaton as the number of test ponts becomes suffcenty arge. 6. Measure of a Regon The measure of a one-dmensona, two-dmensona, three-dmensona, or hgher-dmensona regon s a ength, an area, a voume, or a hypervoume, respectvey. In ths paper, voume s used to represent the measure regardess of the number of dmensons. The measure of a regon and the probabty of a regon are two mportant concepts used n ths paper. Ther vaues are aways nonnegatve. The study n ths paper nvoves the reatonshp between the overa error of a surrogate mode and the sampe densty of ts nputs. In a sampe space, ts sampe densty can be equvaenty represented by the measures of regons assocated wth ndvdua sampe ponts. One approach to dvde a sampe space nto regons s the Vorono dagram[]. Before a sequence c n [,] n s scaed nto dstance-based sampe ponts, or nversey transformed nto probabty-based sampe ponts, the doman [,] n can be parttoned nto regons by the Vorono dagram of the sequence. For any pont n the Vorono regon of c, c s ts cosest sampe pont usng the Eucdean dstance. Fgures a) and b) show the Vorono dagrams for the Sukharev grda ow-dsperson sequence)[] and the Sobo sequence a ow-dscrepancy sequence)[3], respectvey. A sampe pont s approxmatey n the center of each Vorono regon. The ponts on an edge have equa dstances to the sampe ponts whose Vorono regons are bounded by
the edge. When the sequence s scaed nto dstance-based sampe ponts n the x space, or nversey transformed nto probabty-based sampe ponts n the x space, the edges of the ces n [,] n are aso converted nto edges n the x space. The whoe regon of the x sampe space s separated nto subregons by the edges. The measure of each regon assocated wth a sampe pont n the x space s used n the formuaton of the monoma MSE functon n Sec. 7..8.6.4...4.6.8 a) 64-pont Sukharev grd.8.6.4...4.6.8 b) 64-pont Sobo sequence Fgure : Vorono dagrams The sampe densty of the nputs to a system can aso be equvaenty represented by the measures of the regons confned by sampe ponts. The crces n Fg. are the 64 fu factora sampe ponts equay dstrbuted from edge to edge. In Fg. a), the squares that are confned by sampe ponts at ther vertces and have no nsde sampe ponts are the regons that represent sampe densty. If the ponts on the boundaresarevewedashafponts, andthe pontsat thefourvertces,),,),,), and,)) are vewed as quarter ponts, the area of each square s dvded by the equvaent number of sampe ponts. It s the same as the area of each Vorono regon n Fg. b). The Vorono regons borderng the boundares are vewed as haf regons, and the regons borderng the four vertces are vewed as quarter regons. The measure of the confned regon or the Vorono regon s aso the same as that n Fg. a), snce the number of sampe ponts s the same. Therefore, the Vorono regons and the regons confned by sampe ponts can be vewed equvaenty for the purpose of anayzng the reatonshp between the overa error of a surrogate mode and ts sampe densty..8.6.4...4.6.8 a) Confned regons.8.6.4...4.6.8 b) Vorono dagram Fgure : 64-pont fu factora sampng equay from edge to edge The Sobo sequence has a desrabe property that the sequence of m ponts n an n-dmensona space s a subset of m+ or more ponts n an n+ or hgher-dmensona space. The number of regons n an x sampe space can ncrease contnuousy. The fu factora sampng and the Sukharev grd do not have ths property. The Sobo sequence s recommended for the study of the monoma MSE of surrogate modes. The consdered regon n an x sampe space s A. It conssts of subregons A, =,,...,m, assocated wth sampe ponts x, =,,...,m. Any two subregons A and A j n A satsfy A A j = for j. Therefore, the regon A = A. The measure of the regon A s V. The measure of the entre regon A s V = V. The probabty of the regon A s F. The probabty of the regon A s F = F. If the entre x sampe space s consdered, ts probabty F s. Dstance-based unform sampng dvdes a sampe space nto m equa regons. The measure of each 3
regon, V d), s gven by V d) = V m. 5) Probabty-based unform sampng dvdes a sampe space nto m regons wth equa probabty. The probabty of each regon, F p), s gven by F p) = F m. 6) 7. MSE of a Monoma Form RBF and Krgng are expressed as combnatons of bass functons[]. A bass functon s a functon of the dstance between a sampe pont and the nput to RBF or Krgng. If the number of sampe ponts ncreases, the number of bass functons w aso ncrease, and the dstance between the nput and ts cosest sampe pont w decrease. Meanwhe, the overa error of the surrogate mode s expected to decrease. The error of a surrogate mode s expected to be reated to the dstances between sampe ponts, or equvaenty the voume per sampe pont. Ths paper formuates the Mean Squared Error MSE) of a monoma form based on the reatonshp between the squared error of a surrogate mode and the voume per sampe pont. Snce the change of the MSE s reated to the change of the voume per sampe pont, the MSE of the subregon A assocated wth x s statstcay approxmated as a monoma functon of ts voume V, whch s gven by MSEA ) = av. 7) Equaton 7 statstcay refects the reatonshp between the MSE and the measure of a regon. It s not necessary accurate for each ndvdua subregon. When V changes, a and can aso change. Snce V >, then V >. SnceMSEs nonnegatve,the parameterasasononnegatve. Theparameter a and the exponent determne how the MSE changesas the measure V changes. Generay, the exponent >. It ndcates that, as V becomes smaer, the error aso becomes smaer. It s not common that the exponent s <. For dfferent vaues of the exponent, the shapes of the MSE are dfferent. For the same vaue of a, Fg. 3 shows the shapes for fve dfferent scenaros: ) =, ) =, 3) < <, 4) >, and 5) <. MSE = = < < > < a V Fgure 3: Monoma MSE for dfferent As descrbed n Sec. 6, n a sampe space A wth a measure of V and probabty of F, V = V and F = F. The Mean Square Error MSE) of the surrogate mode n regon A s evauated by A MSEA ) = y hx)) dfx) A = y hx)) dfx). 8) A dfx) F Therefore, MSEA )F = y hx)) dfx). 9) A The MSE over the entre space A s gven by MSE = y hx)) dfx) = y hx)) dfx) = MSEA )F. ) A A = 4 =
Substtutng MSEA ) by the monoma MSE expressed by Eq. 7, the overa MSE s gven by MSE = = av F. ) 8. Fdety Comparson usng the MSE of a Monoma Form 8. Power Mean Inequaty Power mean[4]: If q s a nonzero rea number, the weghted power mean wth exponent q of postve rea numbers b, =,,...,m s defned as /q M q b,...,b m ) = b q m ). ) Power mean nequaty[4]: For power means, f q < q, then M q b,...,b m ) M q b,...,b m ), and the two means are equa f and ony f b = = b = = b m. 8.. Comparson of Overa MSE Usng the MSE of a monoma form, the fdetes of the two surrogate modes of the same system deveoped respectvey usng probabty-based and dstance-based unform sampng are compared. For dstance-based unform sampng, substtutng V n Eq. by V d) n Eq. 5, the overa MSE s ) ) V V m ) V MSE d) = a F = a F =. 3) m m m = = For probabty-based unform sampng, substtutng F n Eq. by F p) MSE p) = a = V p) = ) F m = m = V p) n Eq. 6, the overa MSE s ). 4) The comparson between MSE d) and MSE p) are performed for fve dfferent scenaros: ) =, ) =, 3) < <, 4) >, and 5) <. Scenaro : =. ) V MSE d) = =. 5) m MSE p) = m = V =. 6) When =, the overa MSE s for both dstance-based and probabty-based unform sampng. If the entre x space s consdered, ts probabty s F =, and the overa MSE s a. Scenaro : =. MSE d) = V m. 7) MSE p) = m = V = V m. 8) When =, the overa MSE s V/m for both dstance-based and probabty-based unform sampng. If the entre x space s consdered, the probabty s F =, and the overa MSE s av/m. To compare MSE d) and MSE p) under scenaros 3, 4, and 5, rewrte the two equatons 3 and 4 as foows. MSE d) MSE p) ) ) = m = = m 5 = V p) V p) ) ). 9) ) ). )
Scenaro 3: < <. Usng the power mean nequaty, when < <, Snce >, MSE p) ) MSE p) MSE d) ). ) MSEd). ) Snce a > and F >, MSE p) MSE d). MSE p) = MSE d) ony and ony f V p) = = V p) = = V m p) = V/m. Scenaro 4: >. Usng the power mean nequaty, when >, Snce >, MSE p) ) MSE p) MSE d) ). 3) MSEd). 4) Snce a > and F >, MSE p) MSE d). MSE p) = MSE d) ony and ony f V p) = = V p) = = V m p) = V/m. Scenaro 5: <. Usng the power mean nequaty, when < <, Snce <, MSE p) ) MSE p) MSE d) ). 5) MSEd). 6) Snce a > and F >, MSE p) MSE d). MSE p) = MSE d) ony and ony f V p) = = V p) = = V m p) = V/m. Concuson to fdety comparson: Two surrogate modes of the same system are deveoped respectvey usng probabty-based and dstance-based unform sampng wth the same number of sampe ponts. Suppose the MSE of the voume per sampe pont has the form of av. The MSEs of the two surrogate modes, MSE p) and MSE d), have the foowng reatons for dfferent vaues of exponent.. =. MSE p) = MSE d) =.. =. MSE p) = MSE d) = V/m. 3. < <. MSE p) MSE d). MSE p) = MSE d) ony and ony f V p) = = V p) = = V m p) = V/m. 4. >. MSE p) MSE d). MSE p) = MSE d) ony and ony f V p) = = V p) = = V m p) = V/m. 5. <. MSE p) MSE d). MSE p) = MSE d) ony and ony f V p) = = V p) = = V m p) = V/m. 9 Fttng Monoma MSE The expresson of the MSE Eq. 3) for dstance-based unform sampng provdes an approach to ft the parameter a and the exponent. The probabty F and the entre voume V of a sampe space are known for a specfc probem. If pars of MSE d) and m are avaabe, a and can be obtaned by regresson. Dstance-based unform sampng can generate a seres of sets of sampe ponts. The numbers of ponts n these sets arem,m,...,m t. The correspondngvoumes per pont arev/m,v/m,...,v/m t. The vauesofmse d) ofthesurrogatemodesdeveopedusngthesesetsofpontsaremse d),msed),...,msed) t. 6
The parameter a and the exponent n Eq. 3 are ftted usng the pars of V/m,V/m,...,V/m t and MSE d) t.,msed),...,msed) Snce the parameter a and the exponent can change for dfferent vaues of V, the seres of the numbers of sampe ponts shoud be appropratey seected. If many numbers are used to ft one set of a and, the ftted vaues cannot accuratey show how the vaue of changes when the number of sampe ponts s sghty changed. The detas of the change of exponent are ost. However, f the seres s too sma, the ftted resut w have consderabe noses. If the voumes of subregons for probabty-based unform sampng are not very dfferent from V/m,V/m,...,V/m t, the fdety comparson concuson s expected to be accurate.. Testng the Fdety Comparson Concuson In ths secton, RBF and Krgng are deveoped for test functons. The probabty dstrbutons of a the varabes are assumed as ndependent Gaussan dstrbutons. The fu factora sequence s scaed to dstance-based unform sampe ponts, and aso nversey transformed to probabty-based sampe ponts. For each test functon, two seres of RBF and Krgng modes are deveoped respectvey usng these two sampng approaches. The parameter a and the exponent are ftted usng the surrogate modes deveoped usng the dstance-based unform sampng. The Root Mean Squared Error RMSE) s the root of the MSE. The overa RMSE of a surrogate mode s evauated usng test ponts. Test functon : -Varabe functon fx) = 6x ) sn6x )), 7) where x [,] The probabty dstrbuton of x s a Gaussan dstrbuton wth a mean of.5 and standard devaton of.5. Fgure 4a) shows the ftted vaues of exponent for dfferent numbers of sampe ponts. Fgures 4b) and 4c) show the RMSE of the surrogate modes deveoped usng Krgng and RBF, respectvey. The vaue of exponent s consstenty greater than. The RMSE of the surrogate modes deveoped usng probabty-based unform sampng s consstenty arger than that deveoped usng dstance-based unform sampng. 4 3.5 3.5.5 Krgng RBF RMSE of Krgng.5.5.5 Dstance Probabty RMSE of RBF.5.5.5 Dstance Probabty 5 5 5 3 Number of sampe ponts a) Ftted exponent 5 5 5 3 Number of sampe ponts b) RMSE of Krgng 5 5 5 3 Number of sampe ponts c) RMSE of RBF Fgure 4: Test functon Test functon : Booth functon n two-dmensona space fx) = x +x 7) +x +x 5), 8) where x [,], x [,] The probabty dstrbuton of x and x s a bvarate Gaussan dstrbuton wth both means of and standard devatons of 3.5. Fgure 5a) shows the ftted vaues of exponent for dfferent numbers of sampe ponts. These ftted vaues for Krgng and RBF are sgnfcanty dfferent. Fgures 5b) and 5c) show the RMSE of the surrogate modes deveoped usng Krgng and RBF, respectvey. For the surrogate modes constructed usng Krgng, the exponent s very cose to. The RMSE of Krgng for probabtybased unform sampng s generay ower than that for dstance-based unform sampng. The RMSE vaues of Krgng for both sampng approaches are cose to. For the surrogate modes constructed usng RBF, the exponent s arger than. The RMSE of RBF for probabty-based unform sampng s hgher than that for dstance-based unform sampng. 7
.5.5.5 Krgng RBF 4 6 8 Number of sampe ponts a) Ftted exponent RMSE of Krgng.6 x 3.4. Dstance Probabty.8 4 6 8 Number of sampe ponts b) RMSE of Krgng RMSE of RBF 8 6 4 Dstance Probabty 4 6 8 Number of sampe ponts c) RMSE of RBF Fgure 5: Test functon Test functon 3: Hartmann functon n three-dmensona space 4 n fx) = c exp A j x j P j ), 9) = j= where x = x,x,...,x n ) x [,] The parameter vector, c, s gven by c = [,.,3,3.] T. The parameters, A and P, are gven by 3. 3. 35 A = 3. 3, and. 35.3689.7.673.4699.4387.747 P =.9.873.5547..385.5743.888 The probabty dstrbuton of x, x, and x 3 s a trvarate Gaussan dstrbuton wth a three means of.5 and standard devatons of.5. Fgure 6a) shows the ftted vaues of exponent for dfferent numbers of sampe ponts. Fgures 6b) and 6c) show the RMSE of the surrogate modes deveoped usng Krgng and RBF, respectvey. When the number of sampe ponts s sma, s smaer than. The RMSE of both Krgng and RBF deveoped usng probabty-based unform sampng are smaer than those deveoped usng dstance-based unform sampng. When the number of sampe ponts becomes arge, becomes arger than. The RMSE of the surrogate modes deveoped usng probabty-based unform sampng are arger than those deveoped usng dstance-based unform sampng..5 Krgng RBF RMSE of Krgng.8.6.4. DIstance Probabty RMSE of RBF.5.4.3.. Dstance Probabty.5 4 6 8 Number of sampe ponts a) Ftted exponent 4 6 8 Number of sampe ponts b) RMSE of Krgng 4 6 8 Number of sampe ponts c) RMSE of RBF Fgure 6: Test functon 3. Surrogate Modes Used for Wndow Performance Evauaton Dstance-based and probabty-based unform sampng approaches are used to deveop surrogate modes 8
representng the heat transfer rates of a trpe pane wndow under varyng cmatc condtons[5]. Three cmatc condtons, namey, the ar temperature, the wnd speed, and the soar radaton, are the nputs to the surrogate modes. The heat transfer rate of the wndow s the output of each of the surrogate modes. January and August are chosen as the typca months n wnter and summer, respectvey. The heat transfer rates under samped cmatc condtons are evauated by Fuent[6] smuatons. The target ocaton for wndow performance evauaton s Mchgan, ND. Its cmatc data are obtaned from the North Dakota Agrcutura Weather Network[7]. The dstrbutons of ar temperature, wnd speed, and soar rradance are assumed to be a Gaussan dstrbuton, a Webu dstrbuton, and a Gamma dstrbuton, respectvey. A the dstrbutons are ftted usng the maxmum kehood estmaton method[8]. In ether January or August from 6 to, there are 37 observatons of cmatc condtons. They are used as test ponts to evauate the MSE and RMSE of the surrogate modes. The ftted vaue of s constraned to be grater than n ths secton. The Krgng modes are deveoped usng the DACE Matab Krgng toobox[9]... Sampe Ponts Transformed from the Sobo Sequence Probabty-based and dstance-based sampe ponts are generated from Sobo sequences for both January and August. Four seres of Krgng surrogate modes are deveoped respectvey usng probabty-based sampe ponts for January, dstance-based sampe ponts for January, probabty-based sampe ponts for August, and dstance-based sampe ponts for August. The numbers of tranng ponts for Sobo sequences used for these four surrogate modes are 7 to. The exponent of the monoma MSE s ftted usng the MSE d) of the surrogate modes deveoped usng consecutvey ncreasng numbers of dstance-based sampe ponts. To ft the exponent for a specfc number of sampe ponts, 9 surrogate modes traned by consecutvey-ncreasng numbers of sampe ponts are used. The ftted vaue of the exponent s for the mdde number of each group of 9 numbers. The RMSEs of the two surrogate modes for January are shown n Fg. 7a). Fgure 7b) shows the vaues of the exponent of the monoma MSE functon for January. The RMSEs of the two surrogate modes for August are shown n Fg. 8a). Fgure 8b) shows the vaues of the exponent of the monoma MSE functon for August. Generay, when the exponent s between and, the RMSE of the surrogate modes deveoped usng probabty-based sampng are ess that of the surrogate modes deveoped usng dstance-based sampng. Generay, when the exponent s greater than, the comparson resut s reversed...5 Dstance Probabty 4 3 RMSE..5 4 6 8 Number of ponts a) RMSE 4 6 8 Number of ponts b) Fgure 7: Krgng modes for January usng Sobo sequences.. Sampe Ponts Transformed from Fu Factora Sampng Sequence Probabty-based and dstance-based sampe ponts are generated from fu factora sampng sequences equay dstrbuted from edge to edge for both January and August. The numbers of ponts n each dmenson are 3, 4, 5, 6, 7, 8, 9, and. Correspondngy, the tota numbers of tranng ponts n the three-dmensona sampe space are 3 3, 4 3, 5 3, 6 3, 7 3, 8 3, 9 3, and 3 ; and the tota numbers of confned cubes are 3, 3 3, 4 3, 5 3, 6 3, 7 3, 8 3, and 9 3. Four seres of Krgng surrogate modes are deveoped respectvey usng probabty-based sampe ponts for January, dstance-based sampe ponts for January, probabty-based sampe ponts for August, and dstance-based sampe ponts for August. The exponent of the monoma MSE s ftted usng the MSE d) of surrogate modes traned by consecutvey ncreasng numbers of dstance-based sampe ponts. To ft the exponent for a specfc number of sampe ponts, 3 consecutvey-ncreasng numbers of tranng ponts are used. The ftted vaue of the exponent s used for the mdde number of the 3 numbers of tranng ponts. 9
RMSE.8.6.4. Dstance Probabty 4 3 4 6 8 Number of ponts a) RMSE 4 6 8 Number of ponts b) Fgure 8: Krgng modes for August usng Sobo sequences The RMSEs of the two surrogate modes for January are shown n Fg. 9a). Fgure 9b) shows the vaues of the exponent of monoma MSE functon for January. The RMSEs of the two surrogate modes for August are shown n Fg. a). Fgure b) shows the vaues of the exponent of monoma MSE functon for August. The RMSE vaues of the two on Fg. 9a) are very cose. Fgure b) shows that the vaues of exponent for the surrogate modes of August are between and. Fgure a) shows that the RMSE vaues of surrogate modes deveoped usng nverse transform sampng are ess than those usng drect fu factora sampng. RMSE.4.3.. Dstance Probabty.5.5 3 4 3 6 3 Number of cubes 8 3 a) RMSE 3 4 3 6 3 Number of cubes 8 3 b) Fgure 9: Krgng modes for January usng fu factora sampng RMSE.5.5 Dstance Probabty.8.6.4. 3 4 3 6 3 Number of cubes 8 3 a) RMSE 3 4 3 6 3 Number of cubes 8 3 b) Fgure : Krgng modes for August usng fu factora sampng. Concudng Remarks The mean squared error of a monoma form s formuated n ths paper based on the reatonshp between the mean squared error of a surrogate mode and the voume or hypervoume per sampe pont. Probabty-based and dstance-based unform sampng approaches generate ponts unformy dstrbuted n sampe space n terms of probabty and dstance, respectvey. Usng these two sampng approaches wth the same number of ponts, two surrogate modes are deveoped to approxmate the same system.
Ther fdetes are compared usng the monoma MSE functon. When the exponent of the monoma functon s between and, the fdety of the surrogate mode traned usng probabty-based unform sampng s hgher than that of the other one traned usng dstance-based unform sampng. When the vaue of the exponent s greater than or ess than, the fdety comparson s reversed. Ths theoretca concuson s successfuy verfed usng standard test functons and an engneerng appcaton. 3. References [] A. I. J. Forrester, A. Sobester, and A. J. Keane, Engneerng Desgn va Surrogate Modeng: A Practca Gude, Wey, Chchester, West Sussex, UK, st edton, 8. [] J. I. Madsen, W. Shyy, and R. T. Haftka, Response surface technques for dffuser shape optmzaton, AIAA Journa, 389):5-58,. [3] J Zhang, A Messac, J Zhang, and S Chowdhury, Adaptve Optma Desgn of Actve Thermoeectrc Wndows Usng Surrogate Modeng, Optmzaton and Engneerng Accepted). [4] F. P. Mer, A. F. Vandome, and M. B. John, Inverse Transform Sampng, VDM Pubshng, Saarbrcken, Germany,. [5] J. von Neumann, Varous technques used n connecton wth random dgts, Nat. Bureau Stand. App. Math. Ser., :36-8, 95. [6] A. W. Marsha, The use of mut-stage sampng schemes n Monte Caro computatons, H. A. Meyer ed.), Symposum on Monte Caro Methods, Wey, Hoboken, NJ, 956. [7] W. R. Gks, S. Rchardson, and D. J. Spegehater, Markov Chan Monte Caro n Practce, Interdscpnary Statstcs. Chapman & Ha, London, 996. [8] S. M. LaVae. Pannng Agorthms, Cambrdge Unversty Press, Cambrdge, UK, 6. [9] H. Nederreter, Pont sets and sequences wth sma dscrepancy, Monatshefte fr Mathematk, 4:73-337, December 987. [] E. L. Lehmann and G. Casea, Theory of Pont Estmaton, Sprnger, New York, nd edton, 998. [] F. Aurenhammer, Vorono dagrams - a survey of a fundamenta geometrc data structure, ACM Computng Surveys, 33):345-45, 99. [] A. G. Sukharev, Optma strateges of the search for an extremum, Computatona Mathematcs and Mathematca Physcs, 4):9-94, 97. [3] I. M. Sobo, Unformy dstrbuted sequences wth an addtona unform property, USSR Computatona Mathematcs and Mathematca Physcs, 6:36-4, 976. [4] P. S. Buen, Handbook of Means and Ther Inequates, Mathematcs and Its Appcatons, Kuwer Academc Pubshers, Dordrecht, The Netherands, 3. [5] J. Zhang, A. Messac, J. Zhang, and S. Chowdhury, Improvng the accuracy of surrogate modes usng nverse transform sampng, 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structura Dynamcs and Materas Conference, number AIAA -49, Honouu, Hawa, Apr. [6] ANSYS Fuent gettng started gude,, http://www.ansys.com. [7] NDAWN, The North Dakota agrcutura weather network, http://ndawn.ndsu.nodak.edu. [8] A. Had, On the hstory of maxmum kehood n reaton to nverse probabty and east squares, Statstca Scence, 4):4-, 999 [9] S. N. Lophaven, H. B. Nesen, and J. Sondergaard, DACE: A Matab Krgng toobox, Technca Unversty of Denmark, http://www.mm.dtu.dk.