J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Design and analysis of computer experiments. Statistical Science, 4:409--435, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....useful information from s. The problem of choosing a set of initial design sites, x 1 , x d #B, is a problem in the design of experiments. This problem has been studied extensively in the recent literature on the design and analysis of computer experiments (DACE) surveys of which include [31, 2, 22]. We seek designs that are space filling (for lack of a better term) i.e. that will allow us to sample the behavior of the objective function throughout the feasible region. We want to avoid designs that are tied to a narrow class of approximating functions, e.g. linear or quadratic ....

....(MLE) and thereby to specify a well defined procedure for selecting s. Although MLE has been criticized in the spatial statistics literature, e.g. 30] it has been defended by others as a crude form of cross validation [19, 10] Our experience to date has been similar to that reported in [31]: crude MLE s lead to useful prediction. Assuming that the covariances in question are a constant unknown variance times unknown correlations of a specified form, there exist closed form expressions for the MLEs of the mean and variance parameters. To obtain MLEs of the correlation ....

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn, Design and analysis of computer experiments, Statistical Science, 4 (1989), pp. 409--423.

....versions of one of the target problems for our collaboration, the design of a lower vibration helicopter rotor blade. The two versions are a 31 variable problem and an 11 variable problem. The smaller problem was obtained from the larger problem by an analysis of variance decomposition, or ANOVA, [9, 12, 13] performed on a kriging interpolatory model of the objective function [15, 61. The details of the ANOVA decomposition and the way the 11 variables were determined is covered elsewhere by Andrew Booker in this specially organized section. To us, ANOVA identifies the design parameters that have the ....

J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experi- ments. Statistical Science, 4(4):409-435, 1989.

....adjusted locally to some extent as new points are added. They are motivated by the supposition that the output being modeled is a realization of a Gaussian spatial process. Called kriging models, they are recommended in the Design and Analysis of Computer Experiments (DACE) literature, surveyed in [20]. These models are made to interpolate the observations, and they depend on a set of correlation parameters [20] that we estimated via maximum likelihood estimation (MLE) as in [9] In [13, 9] it is shown that MLE can be thought of as a form of cross validation. This Gaussian process ....

....modeled is a realization of a Gaussian spatial process. Called kriging models, they are recommended in the Design and Analysis of Computer Experiments (DACE) literature, surveyed in [20] These models are made to interpolate the observations, and they depend on a set of correlation parameters [20] that we estimated via maximum likelihood estimation (MLE) as in [9] In [13, 9] it is shown that MLE can be thought of as a form of cross validation. This Gaussian process supposition is a convenient fiction ( 23] that provides useful error estimates, termed mean squared errors (rose) ....

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):409-435, 1989.

....only on the covariance function and on the location of the design points y i , i = 1, n, but not on the responses Z(y i ) the question of optimal design can be discussed without knowledge of the actual response. The question remains which design criterion to choose. Natural choices are [23] . Minimization of the integrated mean square error SK (x)dx . Minimization of the maximum mean square error max x#R # SK (x) x 0.2 0.4 0.6 0.8 1 Figure 5: Weight # 1 (x) of the left design point in Fig. 4. Maximization of the prior entropy ## g(#) log g(#)d# where g(#) is ....

....for a weight function #(x) which, for the time being, may be set to 1 minimization of the integrated mean square error is equivalent to maximization of #(x)# (x)# 1 #(x)dx = # i (x)Cov(x, y i )dx (6) which is reminiscent of an unnormalized distortion in vector quantization. In [23], the maximization was e#ected with a quasi Newton optimizer. In analogy to the Linde Buzo Gray [24] and the Rose Gurewitz Fox [25] algorithm, we propose to minimize the contribution to the prediction error from each design point 13 2 0 2 4 4 2 0 2 4 0 0.5 1 4 2 0 2 4 ....

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Sacks, J.; Welch, W. J.; Mitchell, T. J.; Wynn, H. P. Design and analysis of computer experiments (with discussion). Stat. Sci. 1989. 4 , 409--435.

....is assumed to satisfy the normalization conditions [S : j ] 0, V S : j ,S : j =1,j=1, n , Y : j ] 0, V Y : j ,Y : j =1,j=1, q , 2.1) where X : j is the vector given by the jth column in matrix X, and [ and V , denote respectively the mean and the covariance. Following [9] we adopt a model y that expresses the deterministic response y(x) for an n dimensional input x#D#IR , as a realization of a regression model The user does not have to think of this: The first step in the model construction is to normalize the given S, Y so that (2.1) is satisfied, see ....

....denotes the size of the matrix and O is the matrix of all zeros) constant : J f = O n1 ] linear : J f = O n1 I nn ] quadratic : J f = O n1 I nn H] where we illustrate H 1) by n =2: H= 2x 1 x 2 0 0 x 1 2x 2 n=3: H= 2x 1 x 2 x 3 000 0x 102 x 2x 3 0 00x 10x 22 x 3 . As [9] we restrict our attention to correlations of the form R(#,w,x) j (#, w j x j ) i.e. to products of stationary, one dimensional correlations. More specific, the toolbox contains the following 7 choices Name j (#, d j ) exp exp( # j expg exp( # j #n 1 ) 0 # n 1 gauss ....

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J. Sacks, W.J. Welch, T.J. Mitchell, H.P. Wynn, Design and Analysis of Computer Experiments, Statistical Science, vol. 4, no. 4, pp. 409-435, 1989.

....for devising global optimization algorithms. We now describe our particular implementation of the P Algorithm. 1. Choose k points zi, i = 1, k uniformly from A using Latin Hypercube Sampling [9] and compute ( z) by simulation. Start iteration 1 = 1. 2. Using the BLUP and MSE expression in [15] (see Appendix 1, equations 17 and 19) find the mean m(zj) and variance s(zj) at N k uniformly distributed points in A. 3. Find the smallest value of m(x) i.e. m, c) minjcl. N m(zj) 9) Let Yo: m, z) e, At each z5 find the probability Pj(Yo) p 4. Choose mt points with largest ....

J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):pp. 409-435, 1989.

....predictors, Section 3 discusses generalized least squares, and in Section 4 we consider experimental design for the predictors. Section 5 is a reference manual for the toolbox, and finally examples of usage are given in Section 6. 2. Modelling and Prediction The following presentation is based on [4]. We adopt a model y that expresses the deterministic response y(x) IR, for an n dimensional input x#D#IR as a realization of a regression model and a random function (stochastic process) y(x) f(x)# z(x) 2.1) The regression model involves p chosen functions f j :IR IR , a n d ....

....in order to most e#ciently control or reduce the statistical uncertainty of the computed prediction. This section introduces three algorithms with space filling properties. Note that Latin Hypercube designs are based on random numbers, the two other algorithms produce deterministic designs. See [4] or [8] for further discussion and more advanced designs. 4.1. Rectangular Grid Assume that the region D#IR under interest is a box, defined by # j u j ,j=1, n. The simplest distribution of design sites is defined by all di#erent combinations of i = # j k u j q j ,k i =0, ....

J. Sacks, W.J. Welch, T.J. Mitchell, H.P. Wynn, Design and Analysis of Computer Experiments, Statistical Science, vol. 4, no. 4, pp. 409-435, 1989.

.... this has been done by using experimental design techniques for selecting points for simulation in the design space and then fitting polynomial models to the obtained responses through least square regression [1] 10] 15] The rationale of these methods has been brought to question in [13], for cases where the responses are obtained from circuit simulation. Regression techniques introduce systematic bias in the models. Also, classical experimental design techniques [4] are biased by the assumed form of the model. They [13] propose methods for optimizing the experimental design and ....

....rationale of these methods has been brought to question in [13] for cases where the responses are obtained from circuit simulation. Regression techniques introduce systematic bias in the models. Also, classical experimental design techniques [4] are biased by the assumed form of the model. They [13] propose methods for optimizing the experimental design and postulate novel prediction techniques, suitable for deterministic experiments. The main drawback of their method is the excessive time required to optimize the experimental design, though large increases in accuracy over classical ....

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):pp. 409-435, 1989.

....Characterization consists of repeatedly assigning values to the design variables from their given ranges, and simulating the interconnect structure with these values to obtain the responses. The choice of values to be assigned is determined by computer experimental design techniques [8] [9]. Basically, these techniques involve choosing a suitable data interpolating function, and several combinations of values assigned to variables (called samples ) so that the interpolating function accurately predicts the response at untried points in the design space. Sequential sampling is ....

J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):pp. 409-435, 1989.

....corresponding to goals, Eqns. 19 23) Satisfy: SFC 1.030 lbm h lbf normalized [8] Thrust 990 lbf normalized [9] Mixing pressure ratio (Phot Pcold) 0.94 [10] SFC 0.96 d 1 = 1 [12] STDSFC 0. 01 d 2 = 1 [13] THRUST 1052 d 3 = 1 [14] STDTHR 29 d 4 = 1 [15] Weight 1425 lbs [16] Length 7 ft [17] Fan diameter 38.4 in [18] Thegoals: WEIGHT 1350 d 5 = 1 [19] STDWGT 0.005 d 6 = 1 [20] LENGTH 6 d 7 = 1 [21] STDLNG 0.02 d 8 = 1 [22] FANDIA 37 d 9 = 1 [23] Bounds on the control factors (Table 3) d i . d i = 0, with d i , d i 0 ....

Sacks, J., Welch, W.J., Mitchell, T.J., and Wynn, H.P., "Design and Analysis of Computer Experiments," Statistical Science, Vol. 4, No. 4, 1989, pp. 409-435.

....provide an approximation to another physical model, where a set of independent input variables lead to a dependent output variable in a function like manner. Many approximation techniques may be used as metamodels. These include response surfaces, lookup and interpolation tables, kriging [1, 2], neural networks [3, 4] and Gaussian processes[5] While a complete taxonomy of metamodels is beyond the scope of this paper, a discussion of the techniques used in this study is appropriate, refer to Simpson et al. for a more complete discussion[6] All the metamodels used herein are based on ....

J. Sacks et. al. Designs and Analysis of Computer Experiments. Statistical Science, vol. 4(no. 4):pp. 409--435, 1989. With discussion.

....to the data. One would like to be able to predict the output of the computer code for various combinations of inputs without going through the expense of running the code at all input combinations of interest. Recent work in the area of design and analysis of computer experiments has been reviewed [24]. Computer experiments do not share the concepts of experimental unit, blocking, replication, and randomization with physical experiments, and the lack of random error combined with the complex systems being modeled by the computer make it unclear that traditional designs are ideal or even ....

....replication, and randomization with physical experiments, and the lack of random error combined with the complex systems being modeled by the computer make it unclear that traditional designs are ideal or even reasonable for computer experiments. In fact, some evidence to the contrary exists [24]. This has led to the search for what are generally referred to as space filling designs. Ideally, a space filling design should allow for flexibility in the number of factor levels, the number of factors, and the number of sites while not being tied to a particular model for the data. Just as ....

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J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Design and analysis of computer experiments. Statistical Science, 4:409--435, 1989.

.... of difficulty that arise are choosing inputs to tune the model using possibly noisy data, optimising certain criteria for outputs, and gaining information useful for general inferential and predictive purposes; for a discussion of statistical issues arising in the treatment of such models, see [12]. In this paper, we describe a general approach for specifying prior beliefs which is applicable to a wide class of such computer models. Our intention is to produce a methodology for simple, semi automatic belief construction for routine analyses which are not overly sensitive to prior ....

Sacks, J. Welch, W., Mitchell, T. and Wynn, H. (1989) Design and analysis of computer experiments. Statistical Science, 4, 409-435.

....models to alleviate this difficulty. One widely used method in design engineering is the Response Surface Methodology, which uses low order polynomials and the least square estimations [6] The Kriging model, which is also called the Design and Analysis of Computer Experiments (DACE) model [7], is another very useful tool. In this method, a global polynomial approximation is combined with a local Gaussian process and the maximum likelihood method is used for parameter estimation. In the last few years, artificial neural networks, including Multi layer Perceptrons and Radial Basis ....

J. Sacks, W. J. Welch, T. J. Michell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4:409--435, 1989.

....of geostatistics (also known as kriging ) although the latter is concerned primarily with functions of two or three variables while DACE will often demand much higher dimensional input spaces. The problem of the choice of x i s is one of experimental design. An important review of DACE work is Sacks et al. (1989), and for more recent work on computer experiment design see Bates et al. (1996) A good reference for geostatistics is Stein (1999) 1.3 SACCO The main focus in DACE was to predict the output (x) of the computer code at some untried input con. guration. The result is a statistical approximation ....

....issues. The most detailed description of these is in Kennedy and O Hagan (2001) The earliest references to Bayesian analysis of computer code outputs are within the DACE literature, where the Bayesian approach is exploited in the context of interpolation. The Bayesian formulation is mentioned in Sacks et al. (1989), and more recent references are Currin et al. (1991) and Morris et al. (1993) Problems of sensitivity and uncertainty analysis, where a distribution is placed on x, have typically been addressed in the past by Monte Carlo methods. That is, the x i s are random draws from that distribution, so ....

Sacks, J., Welch, W.J., Mitchell, T.J. and Wynn, H.P. (1989) Design and Analysis of Computer Experiments, Statist. Sci., 4, 409--435.

....is constrained by the local nature of SAL computations. For example, global, least squares type approximations are inappropriate since measurements at all locations are equally considered to uncover trends and patterns in a particular region. We advocate the use of kriging type interpolators [Sacks et al. 1989] , which are local modeling methods with roots in Bayesian statistics. Kriging can handle situations with multiple local extrema (for example, in weather data, remote sensing data, etc. and can easily exploit anisotropies and trends. Given k observations, the interpolated model gives exact ....

....and the true values at the chosen k sites, and that all variability in the model arises from the design of Z (the derivation is beyond the scope of this paper) The multi dimensional optimization is often performed by gradient descent or pattern search methods. More details are available in [Sacks et al. 1989] , which demonstrates this methodology in the context of the design and analysis of computer experiments. 3.2 Bottom Up Detection of Ambiguity The SAL equivalence class clustering mechanism (operating on the sparse input data or the dense surrogate model) exploits continuity, grouping ....

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J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Design and Analysis of Computer Experiments. Statistical Science, 4(4):409--435, 1989.

....space where data can be collected) A cheap surrogate model can then be constructed that can serve as an alternative starting point for data mining. This methodology, known quite simply as Design and Analysis of Computer Experiments (DACE) is prevalent in the statistical design literature [Sacks et al. 1989] but hasn t received much attention in the data mining community. The goal is to extract useful patterns from the surrogate models, rather than the original, costly codes (or field data, which cannot be controlled) Considerable attention has been devoted to the use of sampling for mining very ....

Sacks, J., Welch, W., Mitchell, T., and Wynn, H. (1989). Design and Analysis of Computer Experiments. Statistical Science, Vol. 4(4):pp. 409--435.

....involving high dimensional searches in complex computer experiments. Such computer experiments have the special feature that repeated runs at the same input settings always produce the same output, so that the classical concept of random error is not present; an overview of the eld is given in [SWMW89]. We shall describe a quite general approach to pressure matching, which may be similarly applied to many related types of computer search problems. Our approach to prior speci cation is based around combining expert judgements of reservoir engineers with information gained from the analysis of ....

J. Sacks, W. Welch, T. Mitchell, and H. Wynn. Design and analysis of computer experiments. Statistical Science, 4:409{ 435, 1989. 50 Peter S. Craig, Michael Goldstein, Allan H. Seheult, James A. Smith

....codes. Other issues include choosing design points to run the model at, to gain the maximum possible amount of information from a small number of runs of the model, and how to calibrate the model to observations of reality. The use of statistical methods in computer experiments is the subject of Sacks et al. 1989), and a Bayesian approach to various inference problems is given in O Hagan et al. 1999) For previous work in the Bayesian approach to uncertainty analysis see Haylock and O Hagan (1996) and Haylock (1997) This includes estimates of the mean and variance of the uncertainty distribution. It has ....

Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments, Statist. Sci., 4: 409--435.

....accurate model (Gangadharan, et al. 1995) When applying the design of experiment approach, there is always a tradeoff between the number of experiments conducted and the accuracy of the response model. Since computer experiments differ from physical experiments in that there is no random error (Sacks, et al. 1989; Welch, et al. 1990) classical methods for the design and analysis of physical experiments (e.g, Box and Draper, 1987; Box, et al. 1978) are no long ideal for complex, deterministic computer models. In this paper, we present a sequential experimentation strategy as well as the heuristic rules ....

Sacks, J., Welch, W. J., Mitchell, T.J., and Wynn, H.P., 1989, "Design and Analysis of Computer Experiments", Statistical Science, Vol. 4, No. 4, pp. 409-435.

....Neural Network (ANN) methods (Smith, 1993; Cheng and Titterington, 1994) are two well known approaches for constructing simple and fast approximations of complex computer codes. An interpolation method known as Kriging is becoming widely used for the design and analysis of computer experiments (Sacks, et al. 1989; Booker, et al. 1999) Finally, other statistical techniques that hold a lot of promise, such as Multivariate Adaptive Regression Splines (Friedman, 1991) and radial basis function approximations (Hardy, 1971; Dyn, et al. 1986) are beginning to draw the attention of many researchers. An ....

....with mean zero and spatial correlation function given by Cov[Z(x ) Z(x ) s 2 R(x i , x j ) 3) where s 2 is the process variance and R is the correlation. A variety of correlation functions can be chosen (cf. Simpson, et al. 1998) however, the Gaussian correlation function proposed in (Sacks, et al. 1989) is the most frequently used. Furthermore, f (x) in Eqn. 2 is typically taken as a constant term. In our study, we use a constant term for f (x) and a Gaussian correlation function with p=2 and k q parameters, one q for each of the k dimensions in the design space. In addition to being extremely ....

Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P., 1989, "Design and Analysis of Computer Experiments," Statistical Science, 4(4), pp. 409-435.

....models. One widely used method in design engineering is the Response Surface Methodology, which uses low order polynomials and the least square estimation [9] A more statistically sound method is the Kriging model, which is also called the Design and Analysis of Computer Experiments (DACE) model [10]. In this method, a global polynomial approximation is combined with a local Gaussian process and the Maximum Likelihood measure is used for parameter estimation. Arti cial neural networks, including Multi layer Perceptrons (MLP) and Radial Basis Function Networks (RBFN) have also been employed to ....

J. Sacks, W. J. Welch, T. J. Michell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4:409-435, 1989.

....for modeling h(x 1 ; xm ) When the function g is periodic, a Fourier regression model is recommended by Riccmagno, Schwabe and Wynn (1997) The spatial modeling technique of Kriging (Koehler and Owen 1996) is based on a stationary Gaussian stochastic process and a Bayesian approach. Sacks, Welch, Mitchell and Wynn (1989) and MMY use the prior information. Finding an appropriate prior distribution, however, may not be obvious. Fang and Wang (1994) on the other hand, prefer polynomial regression models. The E(f NOD ) optimal mixed level supersaturated design can be viewed as a type of space lling design for ....

Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989), \Design and Analysis of Computer Experiments," Statistical Science, 4, 409-435.

....offer are based on random error (e) estimation and do not reflect the accuracy of the metamodel, since random error does not exist in deterministic computer simulation. A discussion on accuracy and behavior of statistical regression metamodels in computer simulations can be found in [12] 24] [25], and [26] In general, the best validation of the accuracy of the metamodel is an extensive test at randomly distributed points over the design space to compare predicted values with the exact computer simulation values. Unfortunately, this test increases the computational effort put into the ....

Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P., Design and Analysis of Computer Experiments, Statistical Science, Vol. 4, No. 4, February, 1989

....of an arbitrary number of variables with various numbers of local minimizers. 2.1 A Probability Model Our approach to generating objective functions for global optimization was inspired by the literature on the design and analysis of computer experiments. Surveys of this literature include [13] and [5] A computer experiment is a set of planned evaluations of a deterministic function f : p , typically an expensive computer simulation of some physical phenomenon. Usually the goal of a computer experiment is to construct an inexpensive interpolation of f , f : p . 2 Many ....

J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4:409-435, 1989. Includes discussion.

....Unfortunately, even with the use of numerical computation only a limited number of parameter combinations can be investigated and, like the experimental approach, the theoretical approach loses its generality. In fact, computational studies are often referred to as numerical experiments [81]. Though it is known in the engineering community that successful analyses rest upon the proper balance of all three approaches, it is noted that few attempts have been made in uniting experimental, theoretical and numerical methods in the literature. It is the overall objective of this report to ....

....function assigned to the response function F (x) Obviously this stochastic approach for selecting the smoothness requirements on f a (x) may be fully justified only in the case when the response function can be realistically treated as a realization of a random field. Nevertheless, Sacks et.al. [81] and Welch et.al. 88] insisted on the application of statistical procedures, even when the response function is deterministic, as in the case of computer experiments. The statistical framework introduced for analysis of deterministic data was proven to be useful in yielding an accurate ....

J. Sacks, W.J. Welch, T.J. Mitchell, and J.P. Wynn. "Design and analysis of computer experiments". Statistical Science, 4(4):409--435, 1989.

....may have a form, such as an anova decomposition, which yields insight into f . We suppose that the function f can be evaluated at any point x 2 [0; 1] s , and that f is to be based on n function values f(x 1 ) f(xn ) We are motivated here by problems arising in computer experiments [4, 8, 16]. In such applications, a function f describes the performance of a product such as an aircraft or semiconductor as a function of s variables x = x 1 ; x 2 ; x s ) chosen to describe how it is manufactured. In semiconductor applications f may describe how fast and how stably a ....

....present state of the art consists primarily of kriging methods. They originated in geostatistics; see for example, Journel and Huijbregts [7] The value and elegance of kriging for computer experiments was shown by Currin, Mitchell, Morris and Ylvisaker [4] and by Sacks, Welch, Mitchell and Wynn [16]. Kriging allows one to incorporate derivative information on the function, and the mathematical framework supports a notion of optimal designs. Section 2 also presents regression and quasi regression methods. Kriging becomes awkward numerically when n increases, eventually becoming infeasible, ....

Jerome Sacks, William J. Welch, Toby J. Mitchell, and Henry P. Wynn. Design and analysis of computer experiments (c/r: P423-435). Statistical Science, 4:409--423, 1989.

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J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Design and analysis of computer experiments. Statistical Science, 4:409--435, 1989.

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J. Sacks, W. J. Welch, T. J. Mitchell and H. P. Wynn, "Design and Analysis of Computer Experiments ". Statistical Science, 4(4):409--435, 1989.

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):409--435, 1989.

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn, "Design and analysis of computer experiments," Statistical Science, vol. 4, no. 4, pp. 409--435, 1989.

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Sacks, J., Welch, W.J., Mitchell, W.J., Wynn, H.P.: Design and analysis of computer experiments. Statistical Science 4 (1989) 409--435

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J. Sacks, W. J. Welch, W. J. Mitchell and H.-P. Wynn. Design and analysis of computer experiments, Statistical Science (4) (1989) pp. 409-435 10

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J. Sacks, W.J. Welch, W.J. Mitchell, and H.-P. Wynn. Design and analysis of computer experiments. Statistical Science, (4):409--435, 2000.

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):409--435, 1989.

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J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Design and Analysis of Computer Experiments. Statistical Science, Vol. 4(4):pages 409--435, 1989.

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J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Design and Analysis of Computer Experiments. Statistical Science, Vol. 4(4):pages 409--435, 1989.

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J. Sacks, W.J. Welch, W.J. Mitchell, and H.P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):409--435, 1989. (with discussion) .

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J. Sacks, W. J. Welch, T. J Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):409--423, 1989.

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4):409--435, 1989.

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Sacks J, Welch W J, Mitchell T J, Wynn H P (1989) Design and Analysis of Computer Experiments. Statistical Science, Vol. 4, No. 4, pp. 409-435

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J. Sacks, W. J. Welch, T. J. Mitchell and H. P. Wynn, 1989, "Designs and analysis of computer experiments", Statistical Science, Vol. 4, No. 4, pp. 409--423.

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J. Sacks, W. J. Welch, T. J. Mitchell, H. P. Wynn, Design and analysis of computer experiments (c/r: P423-435), Statistical Science 4 (1989) 409-423.

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Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P., 1989, "Design and analysis of computer experiments," Statistical Science, 4(4), 409-435.

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- Sacks, J. Welch, W. J., Mitchell T. J., Wynn, H. M., 1989: Design and Analysis of Computer Experiments. Statistical Science, Volume 4, Issue 4. November 1989. pp 409-423.

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Sacks, J.; Welch, W. J.; Mitchell, T. J.; Wynn, H. P. Design and Analysis of Computer Experiments. Statistical Science 1989, 4 , 409.

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J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Designs and analysis of computer experiments. Statistical Science, 4(4):409--423, 1989.

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Sacks, J., Welch, W.J., Mitchell, T.J., and H.P. Wynn (1989). Design and Analysis for Computer Experiments. Statistical Science. 4, 409-423.

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Sacks, J., Welch, W.J., Mitchell, T.J. and Wynn, H.P. (1989). Design and analysis of computer experiments. Statistical Science 4, 409-435.

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J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn, "Design and Analysis of Computer Experiments, Statistical Science, 4, pp. 409-435 (1989).

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