船舶设计多维问题的元建模技术英文文献和中文翻译(3)
MARS adaptively partitions the design space into sub-regions that have their own regression equations, which enables it to model nonlinear and multimodal responses in high dimensional space.MARS is shown to predict as well as other methods, but only when a large set of training data is available (Wang et al., 1999, Jin et al., 2001).Using MARS is particularly challenging compared with other methods due to the number of user defined parameters that must be selected. When a large number of basis functions is required to build an accurate model (i.e. nonlinear, multimodal), the build time for MARS can be prohibitively large.Based on the literature reviewed in the previous section, several conclusions can be made about the various metamodeling methods. In general, polynomial regression models can be built and executedvery quickly, even in high dimensions. However, polynomial response surfaces are unable to predict highly non-linear and multimodal functions in multiple dimensions. On the other hand, kriging and radial basis functions are both capable of modeling nonlinear and multimodal functions with higher computation time than polynomial regression. Multivariate adaptive regression splines are capable of modeling multimodal functions in high dimensional space, (7)(11)(12)(8)(9)(10)Methods Selected for this Study but often require large training data sets and are computationally expensive to build. Support vector regression, which appears to be the most promising method reviewed here, is shown to be capable of modeling high dimensional multimodal functions accurately with minimal computational expense.The functions that are modeled in this study range from one to ten dimensions and have modality ranging from one to twenty modes. Polynomial response surfaces are tedious and difficult to construct for multimodal functions due to the exploding number of possible interaction terms that are available for inclusion in the final model. MARS has been shown to be able to model the general behavior of multimodal functions (Jin et al., 1999), but it fails to predict new points accurately at the local maxima and minima. For these reasons, kriging, radial basis functions, and SVR are the primary methods considered in this study.To test the ability of polynomial regression, kriging, RBF, and SVR to model functions in high dimensional space, four test functions are used. The first three are generated using a kernel density estimation (KDE) method. The KDE method generates functions in any number of dimensions, containing any number of kernels, where the number of kernels dictates the modality of the resulting surface. The fourth test function is an analytical model of a common engineering system: a two stream counter-flow heat exchanger.The need to create test problems of arbitrary dimensionality, D, and arbitrary modality, N, (the number of local maxima or minima) motivates the kernel density estimation (KDE) method. The KDE, also known as a Parzen window (Parzen, 1992), is formulated according to (13) as an average of N kernel functions, K, in product form (Scott, 1992).The shape of the KDE is controlled by the kernel function, the kernel centers, xj, and the smoothing parameters, hi. For this research, the triweight kernel function (Scott, 1992), shown in Equation (14), was used for its smoothness and for the fact that it is not a Gaussian function which was used as a basis function for some of the metamodeling techniques studied.Creating a kernel function of arbitrary dimensionality is straightforward. However, controlling the modality is a bigger challenge for which careful consideration of the kernel centers and smoothing parameters is necessary. For the choice of kernel centers, a certain amount of randomness is desired in the resulting function such that it is unique and its structure is unknown in advance of modeling it. However, a certain amount of control over placement of the kernel centers is needed for creating the requested number of distinct peaks and distributing them throughout the design space. This challenge was met by choosing the kernel centers sequentially such that the next center, xN+1, is the minimum of the KDE based on the previous N center points (Equation 15).