Thailan Statistician July 0; 9() : 7-93 http://statassoc.or.th Contribute paper Exploratory Optimal Latin Hypercube Designs for Computer Simulate Experiments Rachaaporn Timun [a,b] Anamai Na-uom* [a,b] an Jaratsri Rungrattanaubol [c] [a] Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailan. [b] Centre of Excellence in Mathematics, CHE, Si Ayutthaya R., Bangkok 0400, Thailan. [c] Department of Computer Science an Information Technology, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailan. * Author for corresponence; e-mail: anamain@nu.ac.th Receive: April 0 Accepte: 9 June 0 Abstract The aim of this paper is to present the construction of the optimal esign for computer simulate experiments (CSE) base on three ifferent classes of Latin hypercube esign (LHD), ranom Latin hypercube esign (RLHD), symmetric Latin hypercube esign (SLHD), an orthogonal array-base Latin hypercube esign (OALHD), respectively. We first consier the property of esign through various optimality criteria such as φ p criterion, maximin istance criterion, an the mean of correlation coefficient between esign columns. After the esign properties of each class of esign are valiate, we compare the preiction accuracy of the surrogate moels namely Response surface methoology (RSM) an Kriging moel (KRG), conucte by using the optimal esign from those three classes of LHD. The results inicate that OALHD has the best esign property over all imensions of problem uner consieration. Moreover, OALHD is superior to SLHD an RLHD in terms of preiction accuracy when both of RSM an KRG moels are performe. Hence OALHD is recommene as the best esign choice for CSE.
7 Thailan Statistician, 0; 9():7-93 Keywors: computer simulate experiments, optimal Latin hypercube esign, statistical moeling metho.. Introuction Currently computer simulate experiments (CSE) have replace the physical experiments to investigate a physical complex phenomena, especially when physical experiments are not feasible. CSE is eterministic in nature; hence an ientical setting of input variables always prouces an ientical of output response. The CSEs are usually time consuming an computationally expensive to run. Moreover, the process of CSE is mainly relie on the imension of problems. Therefore the optimal Latin hypercube esigns (OLHD) that aim to sprea the esign points over the region of interest is more esirable. Normally, Kriging moels (KRG) along with the OLHD are practice in the context of CSE. The Latin hypercube esign (LHD) was originally propose by McKay et al. [] an has receive wie attention in various applications of CSE. LHD is a matrix which contains n rows columns, where n is the number of runs, an is the number of input variables. The total numbers of possible LHD are ( n!) an hence when the imension of problem is increase, the searching time for constructing OLHD is time consuming. The construction of optimal esign for CSE consists of two approaches calle non-search base an search base, respectively. The first approach (non-search base) aims to construct the optimal esign by using mathematical or statistical theory. These methos are complicate in the construction of esigns an the run size of esign is not flexible. For example, Tang [] presente the construction of orthogonal array-base Latin hypercube esign (OALHD) using the generation of the ranom orthogonal array (ranom OA). The OALHD has a goo property but the run size is not flexible as its construction requires the valiity of a ranom orthogonal array. Ye [3] use the algebraic metho calle Kronecker prouct to construct an orthogonal LHD. This metho performs well in the applications of CSE but there is one isavantage as the run size of esign is n = or + with + columns. Butler [4] propose the not flexible e.g. construction of optimal an orthogonal LHD generate by William transformation. Though this esign has a goo projection property, the number of run must be an o number or prime number only. Therefore the construction of this esign is quite
Rachaaporn Timun 73 complicate. The secon approach (search base) of constructing an optimal LHD is base on exchange of elements by using search algorithm. This metho first creates a esign with ifferent class of LHD e.g. ranom Latin hypercube esign (RLHD), symmetric Latin hypercube esign (SLHD), an OALHD, respectively. Then try to search for the optimal LHD using search algorithms uner pre-specifie optimality criteria. For instance, Ye et al. [5] presente the construction of symmetric Latin hypercube esign (SLHD), an compare the search algorithm which propose by Park [6], Morris an Mitchell [7] (SA), an Li an Wu [8] (Columwise-pairwise algorithm (CP)) by using the six ifferent imension problem. The authors recommene that SLHD is superior to ranom LHD in terms of esign property such as the maximin istance criterion. Further, the construction of optimal SLHD requires less searching time than ranom LHD with respect to entropy an maximin istance criteria. It has also been reporte that SA performe better than CP for the large imension of problem whereas CP performs much more efficient than SA for small imension of problem. Rungrattanaubol an Nauom [9] compare the efficiency of two popular evolutionary search algorithms for fining optimal LHD calle SA an Genetic algorithm (GA). The result inicate that SA performe better than GA for all cases uner consieration. There are various research papers that have been publishe in the area of construction of the optimal esign for CSE. Leary et al. [0] stuie the construction of LHD by using the iea of orthogonal array an compare the esign property obtaine from SA an CP. The results inicate that OALHD is superior to RLHD with respect to φ p criterion while SA performe better than CP in terms of rate of convergence an the optimum values of optimality criteria. Na-uom [] compare the efficiency of various types of LHD classes by consiering preiction accuracy. The results reveale that SLHD performe better than other classes of esign for small imension problem. For larger imension problem, the orthogonal Latin hypercube esigns generate by William transformation performe the best. Accoring to the results that have been publishe in the previous stuies, we observe that RLHD, SLHD an OALHD provie both of avantage an isavantage. Hence, it is crucial to explore the performance of RLHD, SLHD an OALHD when various types of problems are consiere. In this paper, we investigate the esign property of these three classes of LHD an compare the preiction accuracy of statistical moels when using the optimal esigns obtaine from these three classes. The esign property is valiate through the popular optimality criteria such as φ p criterion (Morris
74 Thailan Statistician, 0; 9():7-93 an Mitchell [7]), maximin istance criterion [], the mean of correlation coefficient between esign columns. The preiction accuracy is implemente by using Response Surface Methoology (RSM) [3] an KRG [4] with respect to the root mean square error (RMSE).. Methoology In this section, we present the etails of three classes of esigns, followe by the steps of SA for fining the optimal esign in the class. After the optimal esign from each imension is obtaine, the statistical moeling methos will be use to implement the preiction accuracy of statistical moels when three ifferent classes of esign are use. All simulation stuies iscusse here were implemente in R program version.0... Design Use.. Ranom Latin hypercube esign (RLHD) The RLHD was originally propose by McKay et al. [] an receive wie attention from the practitioners in the context of CSE. RLHD is a matrix which contains n rows columns, where n is the number of runs, an is the number of input variables, an be enote by RLHD( n, ). The total numbers of possible RLHD are ( n!). Generally the RLHD can be constructe base on the iea of stratifie sampling to ensure that all sub-regions in the ivie input variable space will be sample with equal probability. A LHD can be generate from π ij Uij ij = () n π, π,..., π are inepenent ranom permutation of {,,..., n } an U ij where i i i are n values of ii... uniform ( 0,) ( i =,,..., n; j =,,..., ). U ranom variables inepenent of the π ij In practice, RLHD can be easily generate by ranom permutation of each column which contains {,,..., n }. Thus the columns are combine together to form the esign matrix. For instance, in the case of RLHD (9, ) esign is
Rachaaporn Timun 75 consiere, this means that 9 n = an =, the total number of possible RLHD is ( ) 9!.368 0. Thus searching for the optimal esign in the class is time consuming. In practice the range of each input variable is scale into unit interval. Ye et al. [5] recommene the transformation as { 0,/ ( n ), / ( n ),...,} The example of RLHD (9, ) are visualize in Figure. () 0.000 0.50 0.500 0.750.000 0.000 0.50 0.500 0.750.000 0.000 0.5 0.50 0.375 0.500 0.65 0.750 0.875.000 (a) Figure. The example of RLHD (9, ). 0.000 0.5 0.50 0.375 0.500 0.65 0.750 0.875.000 (b) From the Figure, it shoul be note that RLHD represents a very ba esign space on two imensions. Hence the optimal RLHD coul be generate by using search algorithm uner pre-specifie optimality criteria... Symmetric Latin hypercube esign (SLHD) In this section we aapt the approaches of Ye et al. [5] an Na-uom [5] to stuy the construction of SLHD which is a special case of LHD an can be enote by SLHD( n, ). Any LHD is calle a SLHD if it has a reflection property. The SLHD comprises of n LHD with the levels {,,..., n }. Since SLHD has reflection of runs in the esign, thus if one of runs have been selecte as a esign point, then another esign point that reflects this esign point woul be selecte into the esign matrix as well. For instance, if ( ) a, a,..., a is one of the rows in esign matrix, then the vector
76 Thailan Statistician, 0; 9():7-93 ( n a n a n a ) + + + must be another row in esign matrix. The,,..., construction of SLHD can be ivie into two cases, where n is even number an n is o number, respectively. The steps of generating each case of esign runs are quite similar, except for the case of n ( ) is o number, the center point of the esign ( n + ) oes not play any role in the exchange of esign points in esign matrix, e.g. the construction of SLHD (9, ), when the first pair of element in the esign is ranomly generate, if the esign point (, ) (, ) ( n a, n a ) ( 9,9 ) ( 9,8) a a = is the first element of any row, then + + = + + = will be another row in esign matrix which it is the ninth rows in this case. The construction of other rows can be mae by using the same approach as mentione previously, in this case the fifth rows with the element ( 5,5 ) oes not have the reflection point. The example of SLHD (9, ) is shown in Figure (a) an the consequence esign which the range of input variables are scale into ( 0, ) is also visualize in Figure (b). The sprea of esign points for SLHD (9, ) is presente in Figure 3. It can be clearly seen from Figure 3 that the esign points sprea well over the esign space. Hence SLHD seems to have a goo space filling property. 0 0.5 7 0.5 0.75 3 4 0.5 0.375 4 0.375 0 5 5 center point 0.5 0.5 6 9 0.65 7 6 0.75 0.65 8 3 0.875 0.5 9 8 0.875 Figure. The esign matrix of SLHD (9, ) ; (a) real range (b) unit interval.
Rachaaporn Timun 77 0.000 0.50 0.500 0.750.000 0.000 0.5 0.50 0.375 0.500 0.65 0.750 0.875.000 Figure 3. The scatter iagram for SLHD (9, )...3 Orthogonal Array-base Latin hypercube esign (OALHD) In this section, we present the construction of OALHD which aapte from the approach of Tang [], Leary et al. [0] an Fang et al. [6]. An n matrix A is an orthogonal array of strength r, where r, comprises of n runs, input variables, an q levels. If n r is a sub-matrix of A, which contains all possible r rows with the same frequency λ, where λ represents the inex of an array which r λ = n/ q. The OALHD is enote by OA( n,, q, r ). The construction of an OALHD can be performe by generating the matrix A. For each column of A, the permutation of r λq positions with entry k will be replace by the {( ) r, ( ) r,..., ( ) r r k λq k λq k λq λq kλq r } + + + = (3) for all k,,..., q =. After every columns of A is fully replace by using equation (3), a new esign matrix A will be the LHD class. The OALHD can be constructe if an only if the ranom orthogonal arrays are available; hence in this stuy we limit the imension of problem by consiering the availability of a ranom orthogonal array propose by Tang []. For the sake of completeness, we present the steps of constructing an OALHD as follows.
78 Thailan Statistician, 0; 9():7-93 Step : Specify imension of problem, e.g. OA ( 8,,,). Step : Construct the ranom orthogonal arrays (matrix A ), an fin the value ( 8/ ) Figure 4(a). λ = =. The esign matrix of ranom orthogonal arrays is shown in Step 3: Create the OALHD. Since r λ =, then the ( )( ) λq λ = = 4 position in each value of k ( k =, ) must be ranomly replace by the elements compute from equation (3). If k =, the position with the element equal to must be,,3,4. If k = replace by the permutation of { }, the position with element equal to must be replace by the permutation of { 5,6,7,8 }. Thus, the new esign matrix is an OA ( 8,,,) an presente in Figure 4(b). The esign matrix in the scale format is also shown in Figure 4(c). The sprea of esign points is visualize in Figure 5. 3 6 4 5 8 4 6 3 5 8 7 7 0 0.43 0.86 0 0.43 0.74 0.49 0.57 0.49 0.74 0.86 0.57 0.857 0.857 (a) (b) (c) Figure 4. Design matrix: (a) ranom orthogonal arrays (b) OA ( 8,,,) with real range (c) OA( 8,,,) with unit range.
Rachaaporn Timun 79 0.0000000 0.85743 0.57486 0.85749 0.0000000 0.85743 0.57486 0.85749 Figure 5. The sprea of esign points for OA ( 8,,,). Clearly, the sprea of esign points presente in Figure 3 an Figure 5 show better space filling property than the esign presente in Figure. This inicates that SLHD an OALHD classes are superior over RLHD class of esign in terms of space filling property.. Search Algorithms As we alreay mentione that the optimal LHD can be constructe by using the search algorithm uner a pre-specifie optimality criterion. In this paper, we aopt a version of SA propose by Morris an Mitchell [7] to construct the optimal esigns with respect to φ p criteria. For each esign class, SA was repeate for 0 times to vary the starting point. The preiction accuracy values are implemente base on these 0 optimal esigns. The steps an parameter setting of SA can be foun in Morris an Mitchell [7], Leary et al. [0], an Rungrattanaubol an Na-uom [9]..3 Optimality Criteria For a given imension of problem, the optimality criteria are use to consier the gooness of esigns conucte from three ifferent classes of LHD. In this stuy we consier φ p criterion, maximin istance criterion, an the mean of correlation coefficient between esign columns, respectively. The etails of these criteria are presente as follows.
80 Thailan Statistician, 0; 9():7-93.3. Maximin Distance Criterion Maximin istance criterion was evelope by Johnson et al. []. Any esign is calle a maximin esign if it maximizes the minimum intersite istance: where ( i., j. ), ( i. j. ) maximin = min, ; i j (4) i j n is the Eucliean istance between ( i., j. ) = ( ik jk ) th i an th j esign points: / (5) k = This criterion guarantees that the esign points are not close to each other. In this stuy, each class of esign is constructe for 0 times to vary the starting esigns point. Once the optimal esign from each class is constructe, the Eucliean istance matrix is calculate using equation (5) an the maximin value as state in equation (4) is obtaine. After this process is repeate for 0 times, the average of maximin value is calculate. Any class of esign which maximizes the average of maximum value is consiere as the best class of esign for a specific imension of problem..3. φ p Criterion The φ p criterion, an extension of maximin istance criterion, was propose by Morris an Mitchell [7]. This criterion can be calculate as m p φp = J j j j= where p is a positive integer, / p (6) J is inex list ( ) number of pairs of runs in the esign separate by istance ( ),,..., m with... m this stuy, we use the aaptive form of equation can be expresse as j J, J,..., J m which j j, J is the j is istance list n < < <, an m is the value between an. In φ p [7], which is simpler than equation (4). This
Rachaaporn Timun 8 where element in n n φp = p i= j= i+ ij / p (7) ( ) ( l) ( l) ij is Eucliean istance, ( xi, xj ) = ij = xi xj th i an l= /, of the th j runs, an l is the number of input variables ( l,,..., ) =..3.3 Mean of correlation coefficient between esign columns The mean of correlation between esign columns is the mean of Pearson correlation coefficient [8] which use to measure relationship between any pairs of input variables, can be calculate from r ij = n ( xui xi )( xuj xj ) u= n n ( xui xi ) ( xuj xj ) u= u= where r ij is Pearson correlation coefficient between Once we obtain all th i an th j esign columns. (8) values of the correlation coefficient between the esign columns, we then calculate the mean of these values. The class of esign which maximizes the mean of correlation coefficient is consiere as the best class of esign..4 Statistical Moels.4. Response Surface Methoology (RSM) RSM has receive wie attention in the statistical moeling metho. The secon-orer polynomial moel has been extensively use. All unknown parameters can be estimate by using the metho of least squares [9]. RSM is base on assumption of ranom error that arising from a large number of insignificant input variables that are iscare from statistical moeling metho. For a given output variables y, an input variables x= ( x, x,..., x ), the relationship between y an x can be written as y = f ( x) + ε (9)
8 Thailan Statistician, 0; 9():7-93 where ε is ranom error that is assume to be normal istribution with mean zero an constant variance ( ) σ. Since the true response surface function ( ) the response surface g( x ) is constructe to approximate f ( ) f x is unknown, x. Thus, the preiction values are obtaine by using ŷ = g( x) + ε, which is the polynomial function of ( ),,...,. Simpson et al. [3] an Fang an Horstemeyer [0] rewrite the function g( x ) as ( ) 0 i i ii i ij i j i= i= i= i< j (0) yˆ x = β + β x + β x + β xx where β, β ( =,,..., ) an (,,..., ) i ii i β ij i< j = are unknown parameters. The function in equation (0) can be written in terms of the observe value as i = 0 + ij ij + jj ij + jk ij ik + i j= j= j= j< k () y β β x β x β xx ε Equation () can be expresse in matrix form, as where y y y = yn 0, y = β + ε 0 () x x x x x x x x x = n n n, β β0 ε β ε, ε = β ε n = The approximation of regression coefficients, β, in equation () can be obtaine by using the metho of least squares. This metho is base on the minimization of The equation (3) can be simplifie to n εi 0 β 0 β i= T ( ) ( ) L = = y y (3) T ˆ T = y Hence, the least squares estimator of β is β (4) 0
Rachaaporn Timun 83 ( ) 0 ˆ T T β (5) = y.4. Kriging Moel (KRG) Kriging moel (KRG), in the context of CSE, was propose by Sacks et al. [4]. The functional form of this moel can be written as k β j j( x) ( x ) (6) y = f + Z j= where (,,..., ) f ( x ) is a polynomial function of input variables ( j =,,..., ), an ( ) j β j j = is the parameter for polynomial function of input variables, Z x is a realization of stochastic process with zero mean an some forms of correlation function. In practice, the polynomial function in equation (6) is consiere as a constant [4,], the subsequent equation can be expresse by y = β + Z x (7) ( ) The secon part on the right of equation (6) an (7), Z( x ) can be consiere as a Gaussian correlation function. The covariance between Z( x i ) an Z( x j ) can be written as Cov Z( xi), Z( xj) = σ R( xi, xj) where σ is the process variance an ( i, j) frequently use form can be written as where 0 p an θ > 0. (8) R x x is the correlation function. The most p ( i., j. ) = exp ( θ j i., j. ) (9) R j j= KRG is normally fitte using the iea of generalize least squares metho. The problem of estimating all unknown parameters is reuce to the estimation of the parameter of the correlation function. The metho of maximum likelihoo estimation (MLE) propose by Welch et al. [] are wiely use. The maximum likelihoo estimators can be obtaine by maximizing the log likelihoo function
84 Thailan Statistician, 0; 9():7-93 l Ry R y R (0) ( β, σ, 0) = nlnσ + ln + ( 0 β ) T where is a column vector of length which all elements are one. Given the correlation parameters θ an p in equation (9), the generalize least squares estimate of β is an the MLE of maximize σ is Substituting ( ) ˆ T T β () ˆ σ = R R y T ( 0 β) ( 0 β) 0 = y ˆ R y ˆ () n ˆ β an ˆ σ into equation (0), the problem is to numerically ( ln ˆ ln ) n σ + R (3) After all unknown parameter are estimate, the next step is to construct a preictor, ŷ( x ), at preiction point x. This preictor can be written as T where r ( ) ŷ x is the n ( ) ˆ T x = + r ( x) R ( y ˆ 0 ) β β (4) vector of correlation function between error Z ( x) T preiction point x. The correlation vector r ( ) ŷ x can be compute from ( ) ˆ T x = + r ( x) R ( y ˆ 0 ) an β β (5) KRG has receive wie attention in many applications of CSE ue to its interpolation property [4]..5 Test Problems In orer to investigate the relation of esign property an preiction accuracy, we fit RSM an KRG by using the obtaine optimal esign for a specifie imension of problem as escribe in Section.. Five test problems are employe an the preiction accuracy is valiate through root mean square error (RMSE). A range of test problems are inclue in this stuy, the complex test problem are Welch function [], Cyclone
Rachaaporn Timun 85 moel [5], an Borehole function [], while the 3D function [3] an 0D function [4] are non-complex test problems. In orer to valiate the preiction accuracy of RSM an KRG moels through RMSE values, the aitional test points are use in this stuy. For two-imensional test problem, the 8 test points are constructe using the gri points over the esign space. For larger imension of problems the 500 ranom test points are selecte to valiate the preiction accuracy. All test problems along with their imensions inclue in this stuy are summarize in Table. Table. The etails of test problems. n Designs Functions 9 RLHD SLHD OALHD 3 6 RLHD SLHD OALHD 7 49 RLHD SLHD OALHD x ( ) ( ) ( ) f x, x = 30 + xsin x 4 + e ;0 x, x 5 ( ) ( ) ( ) f x, x, x = x + x + x + x ; 0 x, x, x 0 3 3 3 x x f x, x,..., x 74.4 ( ) 7 0.85 3 = x5 x x 3/ { ( x4 x) } ( x4 x) 6 7 0.56.6.6 0.36 / / xx ;0.09 x 0., 0.7 x 0.33, 0.09 x 0., 3 0.09 x 0.,.35 x.65,4.4 x 7.6, 4 5 6 0.675 x 0.85 7
86 Thailan Statistician, 0; 9():7-93 n Designs Functions 8 49 RLHD SLHD OALHD 0 8 RLHD SLHD OALHD ( ) π x3 x4 x6 f ( x, x,..., x8) = x xx 7 3 x3 ln + + x x x5 ln xx 8 x ;0.05 x 0.5,00 x 50000, 63070 x 5600, 3 990 x 0, 63. x 6, 700 x 80, 4 5 6 0 x 680,9855 x 045 7 8 0 3 6 6 f ( x, x,..., x0 ) = + sin xj + sin xj j= 0 5 5 ; x, x,..., x 0 The RMSE values is calculate from RMSE = k i= ( y yˆ ) i k i (6) where k is the number of test points (specifie in Table ), from the test point. y i is the actual response th i test point an y ˆi is the preicte value by using RSM or KRG for the i th 3. Result In this section we present the esign property of the optimal esigns obtaine from three classes of esign. The esign property consiere here are φ p criterion, maximin istance criterion, an the mean of correlation between columns in the esign matrix. All results are presente in Table.
Rachaaporn Timun 87 Table. Design property of three classes of LHD. n Design φ p Maximin Correlation 9 RLHD 4.30 (0.884) 0.3786 (5.69) 0.067 (0.885) SLHD 4.777 (6.0697) 0.333 (4.445) 0.733 (53.347) OALHD 4.30 (0.884) 0.3786 (5.69) 0 (-) 3 6 RLHD 4.3875 (0.6897) 0.49 (3.5648) 0.0079 (46.7345) SLHD 4.4948 (.406) 0.405 (4.7570) 0.06 (374.460) OALHD 4.3749 (0.58) 0.49 (4.48) -0.009 (-405.7984) 7 49 RLHD 4.50 (0.87) 0.738 (.0447) 0.007 (57.805) SLHD 4.6 (0.559) 0.74 (.486) -0.009 (-403.840) OALHD 4.36 (0.76) 0.743 (.4894) 0.003 (339.777) 8 49 RLHD 3.77 (0.037) 0.864 (.7346) 0.000 (339.098) SLHD 3.843 (0.0600) 0.888 (0.7805) -0.0038 (-00.96) OALHD 3.774 (0.375) 0.830 (.36) -0.000 (-70.578) 0 8 RLHD 4.368 (0.060) 0.909 (.033) 0.000 (883.9840) SLHD 4.755 (0.066) 0.93 (0.3999) 0.005 (46.437) OALHD 4.363 (0.0530) 0.96 (.9795) 0.000 (579.4033) The results in Table inicate that OALHD shows a better esign property than RLHD an SLHD for all imensions of problem since OALHD has a minimum mean of φ p criterion, while the mean value of maximin istance criterion for OALHD is larger than those obtaine from RLHD an SLHD. OALHD also has a minimum average value of the mean of correlation between esign columns, which inicates the orthogonal between columns in the esign matrix. Hence it coul be conclue that OALHD is the best esign class when comparing with RLHD an SLHD for 3-0 imensional problems. In the case of two imensional problems, it shoul be note that RLHD an OALHD have the same esign property when φ p criterion an maximin istance criterion are consiere. Further, OALHD is the most consistence class of esign since the coefficient of variation (C.V.) obtaine from ten ifferent starting points is very small. After consiering the esign properties of three ifferent esign classes, we use them to construct the statistical moels. For each test problem, 0 LHDs from each
88 Thailan Statistician, 0; 9():7-93 class of esign are use to get RMSE values an the escriptive statistics are presente in Table 3. The box plot of RMSE generate from each statistical moeling metho is presente in Figure 6 to Figure 0. Table 3. RMSE values from RSM an KRG for all test problem. Test RSM KRG Designs problems Mean S.D. C.V. (%) Mean S.D. C.V. (%) RLHD 3.309 0.4 3.366 3.8744 0.497.830 Welch SLHD 3.6703 0.34 8.53 4.5583 0.6699 4.6973 Function OALHD 3.308 0.0898.740 3.733 0.5079 3.606 RLHD 5.655 5.8368 5.0467 8.6686.984 6.93 3D SLHD 09.345 6.64 6.0566 8.8497 8.704 30.6 Function OALHD 97.094 4.6363 4.7743 4.980.4909 5.9807 Cyclone RLHD 0.0494 0.070 34.4645 0.0534 0.050 8.537 moel SLHD 0.057 0.065 3.989 0.0569 0.057 7.663 OALHD 0.0487 0.0 3.0076 0.0537 0.0097 8.85 RLHD 0.6870 4.853.4853 5.90 0.538 3.3465 Borehole SLHD 46.4 5.50 3.48 6.070 0.445.6486 function OALHD 34.47 58.8035 70.8773 4.976 0.6337 4.37 0D RLHD 0.307 0.44 44.979 0.645 0.0058.078 function SLHD 0.3057 0.008.6869 0.830 0.00 3.569 OALHD 0.3030 0.507 49.7409 0.496 0.0065.593 It can be clearly seen from the Table 3 that the statistical moels constructe from OALHD performs best in terms of preiction accuracy, as it provies lower RMSE values on average, when both of RSM an KRG are use. It is, however, when Cyclone moel ( = 7) is consiere, the KRG constructe from RLHD performs slightly better than OALHD, but the RMSE values obtaine from these two esign classes are still close to each other. It shoul be note that RSM an KRG create from SLHD perform worst among three esign classes when Welch function an 0D test problems are consiere. SLHD, however, is superior to RLHD when 3D an 0D with RSM test problems are consiere. In the case of preiction capability between RSM an Kriging is compare, we observe that Kriging moel performs much better than RSM for Borehole an 0D functions. This inicates that Kriging moel along with optimal LHD is the best choice to use for moeling response from CSE. RSM moel, however, performs not far from Kriging moel when less complex test functions are employe (Cyclone
Rachaaporn Timun 89 function an 0D). Accoring to the C.V. of RMSE values, it coul be conclue that the preiction accuracy of RSM an KRG moels conucte from OALHD are quite stable as the C.V. values obtaine from this esign are small. Boxplot of Welch function Boxplot of Welch function RMSE value 3. 3.4 3.6 3.8 4.0 RMSE value 3.0 3.5 4.0 4.5 5.0 5.5 6.0 RSM_RLHD RSM_SLHD RSM_OALHD RSM conucte from the optimal three KRG_RLHD KRG_SLHD KRG_OALHD KRG conucte from the optimal three c (a) (b) Figure 6. Box plot of RMSE values for Welch function; (a) RSM (b) KRG. Boxplot of 3D function Boxplot of 3D function RMSE value 0 40 60 80 00 0 RMSE value 0 40 60 80 00 0 RSM_RLHD3 RSM_SLHD3 RSM_OALHD3 RSM conucte from the optimal three KEG_RLHD3 KEG_SLHD3 KEG_OALHD3 KRG conucte from the optimal three c (a) Figure 7. Box plot of RMSE values for 3D function; (a) RSM (b) KRG. (b)
90 Thailan Statistician, 0; 9():7-93 RMSE value 0.04 0.05 0.06 0.07 0.08 0.09 Boxplot of Cyclone moel RMSE value 0.04 0.05 0.06 0.07 0.08 0.09 Boxplot of Cyclone moel RSM_RLHD7 RSM_SLHD7 RSM_OALHD7 KRG_RLHD7 KRG_SLHD7 KRG_OALHD7 RSM conucte from the optimal three KRG conucte from the optimal three c (a) Figure 8. Box plot of RMSE values for Cyclone moel; (a) RSM (b) KRG. (b) Boxplot of Borehole function Boxplot of Borehole function RMSE value 0 00 00 300 400 500 600 RMSE value 4.0 4.5 5.0 5.5 6.0 6.5 RSM_RLHD8 RSM_SLHD8 RSM_OALHD8 RSM conucte from the optimal three KRG_RLHD8 KRG_SLHD8 KRG_OALHD8 KRG conucte from the optimal three c (a) Figure 9. Box plot of RMSE values for Borehole function; (a) RSM (b) KRG. (b)
Rachaaporn Timun 9 Boxplot of 0D function Boxplot of 0D function RMSE value 0.3 0.4 0.5 0.6 0.7 RMSE value 0.4 0.5 0.6 0.7 0.8 0.9 0.30 RSM_RLHD0 RSM_SLHD0 RSM_OALHD0 RSM conucte from the optimal three KRG_RLHD0 KRG_SLHD0 KRG_OALHD0 KRG conucte from the optimal three c (a) Figure 0. Box plot of RMSE values for 0D function; (a) RSM (b) KRG. (b) 4. Conclusions The construction of optimal esign base on three ifferent classes of Latin hypercube esign has been stuie. Accoring to the esign property results presente in the previous section, it coul be conclue that OALHD is the best esign for most of the imensional problems with respect to the optimality criteria uner consieration. It shoul be note that for two imensional problems, both of RLHD an OALHD perform best with respect to φ p an maximin istance criterion. When the preiction accuracy base on RMSE values is consiere, OALHD with RSM moel performs best for all imension problems, while OALHD with KRG moel is the most accurate approach for Welch function, 3D function, Borehole function, an 0D function. It is, however, RLHD with Kriging moels can perform well when Cyclone moel is valiate. Further, SLHD with RSM can perform as well as OALHD for small imensional problem. As alreay presente in the previous section, OALHD shoul be recommene as the best esign choice for esigning an moeling the response in the context of CSE. Since the imension of problem is assigne with respect to the valiity of the ranom orthogonal arrays, hence SLHD can be use as an alternative of OALHD whenever the ranom orthogonal array oes not exist. As we restrict our approach to three classes of Latin hypercube esign only, hence other methos shoul be further investigate.
9 Thailan Statistician, 0; 9():7-93 Acknowlegements This research is supporte by the Centre of Excellence in Mathematics, the Commission on Higher Eucation, Thailan. References [] McKay, M.D., Beckman, R.J., an Conover, W.J., A Comparison of three Methos for Selecting Values of Input Variables in the Analysis of Output from a Computer Coe, Technometrics, 979; : 39-45. [] Tang, B., Orthogonal Array-Base Latin Hypercubes, Journal of American Statistical Association, 993; 88: 39-397. [3] Ye, K.Q., Orthogonal Column Latin Hypercubes an Their Application in Computer Experiments, Journal of American Statistical Association, 998; 93: 430-439. [4] Butler, N.A., Optimal an Orthogonal Latin Hypercube Designs for Computer Experiments, Biometrika, 00; 88: 847-857. [5] Ye, K.Q., Li, W., an Sujianto, A., Algorithmic construction of optimal symmetric Latin hypercube esigns, Journal of Statistical Planning an Inference, 000; 90: 45-59. [6] Park, J.S., Optimal latin hypercube esigns for computer experiments, Journal of Statistical Planning an Inference, 994; 39: 95-. [7] Morris, M.D., an Mitchell, T.J., Exploratory esigns for computational experiments, Journal of Statistical Planning an Inference, 995; 43: 38-40. [8] Li, W., an Wu, C.F.J., Columnwise-pairwise algorithms with applications to the construction of supersaturate esigns, Technometrics, 997; 39: 7-79. [9] Rungrattanaubol, J., an Na-uom, A., Comparison of Evolutionary Search Algorithms in Computer Simulate Experiments, Proceeings of the National Computer Science an Engineering Conference, 007; 0-06. [0] Leary, S., Bhaskar, A., an Keane, A., Optimal Orthogonal Array-Base Latin Hypercubes, Journal of Applie Statistics, 003; 30: 585-598. [] Na-uom, A., Comparison of Design for Computer Simulate Experiments, Proceeings of the Annual meeting on Statistics, 007; 48-5.
Rachaaporn Timun 93 [] Johnson, M.E., Moore, I.M., an Ylvisaker, D., Minimax an Maximin istance esigns, Journal of Statistical Planning an Inference, 990; 6: 3-48. [3] Simpson, T.W., Lin, D.K.J., an Chen, W., Sampling Strategies for Computer Experiments: Design an Analysis, International Journal of Reliability an Applications, 00; : 09-40. [4] Sacks, J., Welch, W.J., Mitchell, T.J., an Wynn, H.P.. Design an analysis of computer experiments, Statistical Science, 989; 4: 409-435. [5] Na-uom, A., Review of Design an Analysis of Computer Simulate Experiments, KKU Science Journal, 008; 36: 06-0. [6] Fang, K.T., Li, R., an Sujianto, A., Design an Moeling for computer experiment, America, Chapman & Hall/CRC. [7] Jin, R., Chen, W., an Sujianto, A., An efficient algorithm for constructing optimal esign of computer experiments, Journal of Statistical Planning an Inference, 005; 34: 68-87. [8] Montgomery, D.C., Peck, E.A., an Vinning, G.G., Introuction to Linear Regression Analysis, 4 th e, New Jersey, Wiley, 006. [9] Myers, R.H., Montgomery, D.C., an Anerson-Cook, C.M., Response Surface Methoology: Process an Prouct Optimization Using Designe Experiments, 3 r e, Unite States of America, Wiley, 009. [0] Fang, H., an Horstemeyer, M.F., Global Response Approximation with Raial Basis Functions, Engineering Optimization, 006; 38: 407-44. [] Welch, W.J., Buck, R.J., Sacks, J., Wynn, H.P., Mitchell, T.J., an Morris, M.D., Screening, Preicting, an Computer Experiments, Technometrics, 99; 34: 5-5. [] Palmer, K., an Tsui, K.L., A minimum bias latin hypercube esign, IIE Tarnsactions, 00; 33: 793-808. [3] Hock, W.F., an Schittkowski, K., Test examples for nonlinear programming coes, New York, Springer, 98. [4] Muller, A.A., an Messac, A., Metamoeling using Extene Raial Basis Functions: A Comparative Approach, Engineering with Computers, 006; : 03-7.