Metamodeling Based On SVR And Its Application In Robust Parameter Design

In order to deal with the increasing global competition, industrial and academic circles focus on how to have an advantage over their competitors by high quality, low cost, and short development cycle. On the view of modern quality engineering, quality originates from designing and manufacturing. Variation is the basic factor influencing the quality of products. Quality design, which is an useful tool to reduce variation, is widely employed in the design phase of product/process. Classic design of experiment is an important mean for quality design. Nevertheless, there are some shortcomings, such as, high experimental cost, time-consuming, being difficult to implement, and so on, in its application in industrial areas. Considering these inefficiencies above, some foreign scholars proposed the design of computer experiment, the typical literature of which are those two papers published by Sacks. These two papers laid the foundation for design of computer experiment, and have deep influence on it.Based on several metamodeling techniques, regarding the robust parameter design based on metamodel as its research object, and using systematic modeling, simulation techniques, and empirical research, this paper systematically studied support vector regression (SVR) based on non-positive semi-definite (non-PSD) kernels, SVR based on gradient information, ensemble of surrogates, and the robust parameter design (RPD) based on metamodeling.Firstly, the stand-alone surrogates are studied. SVR based on non-PSD kernels and SVR based on gradient information are studied in this part of the paper. On the one hand, considering the traditional sequential minimal optimization (SMO) algorithms, which is often used to solve the optimization problem in SVR, can not deal with the SVR metamodel with non-PSD, the original quadratic programming is spreaded and the Karush-Kuhn-Tucker (KKT) condition is solved in this part. After employing this strategy, the number of the Lagrange multipliers which is needed to consider is diminished, and the tedious judgments are avoided, therefore, the implementation of solving the optimization problem is simplified. The classic test functions and the abalone data are used to test the efficiency of this algorithm. On the other hand, considering the bad performance of the traditional SVR with small samples, the gradient information around the samples are added into the construction of the SVR metamodel after changing the objective function and constrained functions, then the decision function is reconstructed. Three benchmark functions are employed to test the improved SVR metamodel, and the results of the experiment show that the proposed method has better prediction accuracy than the traditional ones.Secondly, the ensemble of surrogates is studied. Considering the choice of metamodel is highly depended on the set of the samples which is used to construct the metamodels, the simple arithmetic average method and the philosophy of recursion are employed, and the root mean square error (RMSE) of prediction is adopted as the stop criterion of the algorithm. After continually replacing the worst candidate stand-alone metamodel with the arithmetic average model, the weight of the best stand-alone metamodel is raised while that of the worst one is reduced.2-dim,3-dim, and 6-dim test functions, and 8-dim abalone data are used to verify the performance of ensemble technique. The results show that the ensemble of surrogates efficiently kicks out the negative effects of the improper stand-alone metamodels, and the performance of the ensemble of surrogates does not vary apparently with samples. Therefore, ensemble of surrogates to a certain extent is a robust model.Thirdly, the RPD based on the stand-alone metamodel and the ensemble of surrogates is studied. Considering the dual response surface model in RPD are highly depended on the metamodeling, the SVR metamodel, Kriging metamodel, and radius basis function(RBF) metamodel, and especially the ensemble of surrogates made from these three metamodels are applied into the dual response surface in RPD. Above all, the SVR metamodel, Kriging metamodel, and RBF metamodel are constructed respectively, and then the ensemble of surrogates is also constructed using the above-mentioned metamodels. The next, the mean response and the variation response are built using these stand-alone metamodels and the ensemble of surrogates respectively. In addition, the random optimization process is taken, and the best recommended setting is obtained. The printing ink experiment is employed to test the performances of these stand-alone metamodels and the ensemble of surrogates. These metamodels proposed in this paper have similar recommended settings to the previous polynomial models, which indicates the effectiveness of these metamodels, and the mean square errors (MSEs) with SVR and ensemble of surrogates are lower than the traditional polynomial models, which indicates the superiority of our proposed metamodels.The stand-alone metamodels, the ensemble of surrogates, and the RPD based on the metamodels are studied systematically, which extends the scientific connotation of RPD. Finally, this paper points out the topics for further study.