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Accuracy: Capability of predicting new points that closely match those generated by the base model.• Training Speed:
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Time to build the metamodel with training data from the base model.• Prediction Speed: Time to predict new points using the constructed metamodel.• Scalability: Capability of accommodating additional independent variables.• Multimodality: Capability of modeling highly nonlinear functions with multiple regions of local optimality (modes).No single metamodeling method has emerged as universally dominant. Rather, inpidual techniques have strengths and weaknesses.Selection of the method is dependent on several factors such as the nature of the response function, and the availability of training data. Methods that most frequently appear in the literature are response surface methodology (RSM) (Box and Wilson, 1951), multivariate adaptive regression splines (MARS) (Friedmanl, 1991), support vector regression (SVR) (Vapniket al., 1997), kriging (Sacks et al., 1989), radial basis functions (RBF) (Hardy, 1971), and neural networks (Haykin, 1999).In this research study, kriging, radial basis functions (RBF), and support vector regression (SVR) are used to model a set of functions with varying degrees of scale and multimodality. In Section 2, related research is reviewed, and the rationale for selecting kriging, RBF, and SVR for this study is discussed. In Section 3, the test functions and experimental design for this study are explained. Results are presented in Section 4, and closing remarks are made in Section 5.Polynomial RegressionPolynomial regression (PR) (Box and Wilson, 1951) IntroductionReview of Metamodeling MethodsDetailed Descriptions of Methods Under Consideration models the response as an explicit function of the independent variables and their interactions.
The second order version of this method is given in (1):where the xi are the independent variables and the βi are coefficients that are obtained with least squares regression. Polynomial regression is a global approximation method that presumes a specific form of the response (linear, quadratic, etc.). Therefore, polynomial regression models are best when the base model is known to have the same behavior as the metamodel. Studies have shown that polynomial regression models perform comparable to kriging models, provided that the base function resembles a linear or quadratic function (Giunta and Watson, 1998; Simpson et al., 1998).KrigingKriging (Sacks et al., 1989) consists of a combination of a known global function plus departures from the base model, as shown in (2):where βi are unknown coefficients and the fi(x)’s are pre-specified functions (usually polynomials). Z(x) provides departures from the underlying function so as to interpolate the training points and is the realization of a stochastic process with a mean of zero, variance of σ2, and nonzero covariance of the formwhere R is the correlation function which is specified by the user. In this study, a constant term is used for f(x) and a Gaussian curve of the form in (4) is used for the correlation function:where the θi terms are unknown correlation parameters that are determined as part of the model fitting process. The automatic determination of the θi terms makes kriging a particularly easy method to use. Also, contrary to polynomial regression, a krigingmetamodelof this form will always pass through all of the training points and therefore should only be used with deterministic data sets.Kim et al. (Kim et al., 2009) show that kriging is a superior method when applied to nonlinear, multimodal problems. In particular, kriging outperforms its competitors when the number of independent variables is large. However, it is well documented that kriging is the slowest with regard to build time and prediction time compared with other methods (Jin et al., 2001; Ely and Seepersad, 2009).Radial Basis FunctionsRadial basis functions (Hardy, 1971) use a linear combination of weights and basis functions whose values depend only on their distance from a given center, xi. Typically, a radial basis function metamodel takes the form (5):where the wi is the weight of the ith basis function, ϕi. In this study, a Gaussian basis function of the form in (6) is used to develop the metamodels for testing:where k is a user specified tuning parameter.Radial basis function metamodels are shown to besuperior in terms of average accuracy and ability to accommodate a large variety of problem types (Jin et al., 1999). However, Kim et al. (Kim et al., 2009) show that prediction error increases significantly for RBF as the number of dimensions increases. This pattern suggests caution must be used when applying RBF to high dimensional problems to ensure that proper tuning parameters are used. RBF is also shown to be moderately more computationally expensive than other methods such as polynomial regression (Fang et al., 2005).Support Vector RegressionIn support vector regression (Vapniket al., 1997), the metamodel takes the form given in (7):where the a terms are Lagrange multipliers, k(xi,x) is a user specified kernel function, and b is the intercept. The a terms are obtained by solving the following dual form optimization problem:In (8) and (9), l is the number of training points, ɛ is a user defined error tolerance, and C is a cost parameter that determines the trade-off between the flatness of the ŷ and the tolerance to deviations larger than ɛ. In this study, a Gaussian kernel function of the form in (10) is used to construct the metamodels for testing:where g is a user specified tuning parameter.Clarke et al. (Clarke et al., 2005) indicate that SVR has the lowest level of average error when applied to a set of 26 linear and non-linear problems when compared to polynomial regression, kriging, RBF, and MARS. SVR has also been shown to be the fastest method in terms of both build time and prediction time (Ely and Seepersad, 2009). An unfortunate drawback of SVR is that accurate models depend heavily on the careful selection of the user defined tuning parameters (Lee and Choi, 2008).Multivariate Adaptive Regression SplinesMultivariate adaptive regression splines (MARS) (Friedmanl, 1991) involves partitioning the response into separate regions, each represented by their own basis function. In general, the response has the form given in (11):where am are constant coefficients and Bm(x) are basis functions of the formIn the Equation (12), Km is the number of splits in the Mth basis function, sk,m take values ±1, xv(k,m) are the test point variables, and tk,m represent the knot locations of each of the basis functions. The subscript “+” indicates that the term in brackets is zero if the argument is negative.