# Ensemble of surrogates with hybrid method using global and local measures for engineering design

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## Abstract

Surrogate models are usually used as a time-saving approach to reduce the computational burden of expensive computer simulations for engineering design. However, it is difficult to choose an appropriate model for an unknown design space. To tackle this problem, an effective method is forming an ensemble model that combines several surrogate models. Many efforts were made to determine the weight factors of ensemble, which include global and local measures. This article investigates the characteristics of global and local measures, and presents a new ensemble model which combines the advantages of these two measures. In the proposed method, the design space is divided into two parts, and different strategies are introduced to evaluate the weight factors in these two parts respectively. The results from numerical and engineering design cases show that the proposed ensemble model has satisfactory robustness and accuracy (it performs best for most cases tested in this article), while spending almost the equivalent modeling time (the additional cost is not more than 6.7% for any case tested in this article) compared with the combined global and local ensemble models.

## Keywords

Ensemble model Global measure Local measure Surrogate models## Nomenclature

*d*Number of design variables.

*E*_{i}Root generalized mean square cross-validation error of the

*i*^{ th }surrogate.*e*_{ik}Cross-validation error of the

*i*^{ th }surrogate at the*k*^{ th }sample point.- \( {\widehat{f}}^{ens} \)
Predictor of the ensemble.

- \( {\widehat{f}}_i \)
Predictor of the

*i*^{ th }surrogate.*N*Number of test points.

*N*_{s}Number of surrogates used in the ensemble.

*n*Number of sample points.

*P*_{k}Ratio of the global cross-validation error to the local cross-validation error at the

*k*^{ th }sample point.*R*^{o}Outer region.

*R*^{i}Inner region.

*r*_{k}Radius of the

*k*^{ th }point’s inner region.- \( {r}_k^{\mathrm{max}} \)
Euclidean distance between the

*k*^{ th }sample point and the closest sample point.*S*Sample points set.

*WCVE*Weighted cross-validation error.

*w*_{i}Normalized weight of the

*i*^{ th }surrogate.- \( {w}_i^{\ast } \)
Unnormalized weight of the

*i*^{ th }surrogate.*w*_{ik}Pointwise weight of the

*i*^{ th }surrogate at the*k*^{ th }sample point.*x*^{nearest}Sample point which is nearest to the prediction point.

- \( {\widehat{y}}_{ik} \)
Response predicted by the

*i*^{ th }surrogate at the*k*^{ th }point, the surrogate is constructed by using leave-one-out cross-validation.*y*_{k}True response at the

*k*^{ th }sample/test point.- \( {\widehat{y}}_k \)
Prediction response at the

*k*^{ th }sample/test point.*ρ*Impact metric of local measure.

## Notes

### Acknowledgments

Financial support from the National Natural Science Foundation of China under Grant No. 51675198, 973 National Basic Research Program of China under Grant No. 2014CB046705 and National Natural Science Foundation of China under Grant No. 51421062 are gratefully acknowledged.

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