元模型為解釋模型的模型，透過執行隨機模擬，模擬模型其輸出值提供了元模型所需的估計值。在隨機模擬實驗中，多項式迴歸與隨機克利金法皆為常見元模型建模方法。其中迴歸是找一個函數，使此函數盡量符合已知點的資料，此函數稱作迴歸函數；而隨機克利金法是為隨機模擬實驗開發的一種元建模方法，在克利金法的基礎上開發新的模型設計。隨機克利金法將模型輸出性能的不確定性與隨機模擬中固有的取樣不確定性區分開來，因此隨機克利金模型既要描述隨機模擬中原有的固有不確定性，又要考慮未知輸出的外部不確定性。
本篇論文比較了隨機克利金元模型與迴歸兩種元模型，在限制其計算成本的條件下，具異質性變異數輸出的模擬模型，不需經過複雜的轉換運算使其變異數趨於相似，便可直接用以建立元模型。在這項研究中，通過模擬實驗的設計和分析，經過共計100次的實驗，透過數據分析的結果，證明所提出的隨機克利金元模型相對於競爭方法，在限制條件下其模型估計性能更優於迴歸元模型。

Ｍetamodel is a model for explaining the model. By running a stochastic simulation, we can know the number specified by the random model but cannot be analyzed and calculated. The output value of the simulation model provides the estimated value required by the metamodel. Polynomial Regression and stochastic kriging are both common metamodeling methods in stochastic simulation experiments. Regression method is to find a function that could match the known data as much as possible, this function called regression function. Another method is a metamodeling method developed for random simulation experiments named stochastic kriging. The design of the model is based on the kriging method, this method characterized both the intrinsic uncertainty inherent in a stochastic simulation and the extrinsic uncertainty about the unknown response surface.
In this study, we compared two different metamodels, stochastic kriging metamodel and regression metamodel. Under the condition of limiting calculation budget, the simulation model with heterogeneous variable output does not need to undergo complex conversion operations to make the variation tend to be similar. It can be used directly to build a metamodel. Through the design and analysis of simulation experiments, after a total of one hundred experiments, and through the results of data analysis, it is proved that stochastic kriging metamodel has the better performance of estimated.

[1] Ankenman, B., Nelson, B. L., Staum, J. "Stochastic Kriging for Simulation Metamodeling. " Operations Research, 58(2), pp. 371–382, 2010.
[2] Banks, J., J. S. Carson II, B. L. Nelson, D. M. Nicol. "Discrete-Event System Simulation (5th ed.). " Pearson Prentice Hall, 2009.
[3] Barton, R. R., Nelson, B. L., Xie, W. "Quantifying Input Uncertainty via Simulation Confidence Intervals. " INFORMS Journal on Computing, 26(1), pp. 74–87, 2014.
[4] Barton, Russell R. "Simulation metamodels." 1998 Winter Simulation Conference. Proceedings (Cat. No. 98CH36274), 1998.
[5] Barton, Russell R., Meckesheimer M. "Metamodel-based simulation optimization." Handbooks in operations research and management science, 13, pp. 535-574, 2006.
[6] Bartz-Beielstein, T., Zaefferer, M. "Model-based methods for continuous and discrete global optimization." Applied Soft Computing, 55, pp. 154-167, 2017.
[7] Chen, X., Ankenman, B. E., Nelson, B. L. "The effects of common random numbers on stochastic kriging metamodels." ACM Transactions on Modeling and Computer Simulation (TOMACS) , 22(2) , pp. 1-20, 2012.
[8] Chen, X., Ankenman, B. E., Nelson, B. L. "Enhancing stochastic kriging metamodels with gradient estimators." Operations Research , 61(2) , pp. 512-528, 2013.
[9] Chen, X., Ankenman, B., Nelson, B. L. "Common random numbers and stochastic kriging." Proceedings on the 2010 Winter Simulation Conference, 2010.
[10] Chen, X., Kim, K.-K. "Stochastic kriging with biased sample estimates." ACM Transactions on Modeling and Computer Simulation (TOMACS) , 24(2) , pp. 1-23, 2014.
[11] Cressie, N. "Aggregation in geostatistical problems." Geostatistics Troia’92. Springer, pp. 25-36, 1993.
[12] Fang, K. T., R. Li, Sudjianto, A. "Design and Modeling for Computer Experiments Chapman & Hall." CRC, 304, 2006.
[13] Gupta, A., Ding, Y., Xu, L., Reinikainen, T. "Optimal parameter selection for electronic packaging using sequential computer simulations." Journal of Manufacturing Science and Engineering, 128(3), pp. 705-715, 2006.
[14] Huang, D., Allen, T. T., Notz, W. I., Zeng, N. "Global optimization of stochastic black-box systems via sequential kriging meta-models." Journal of global optimization, 34(3), pp. 441-466, 2006.
[15] Kleijnen, J. P. C. "Design and analysis of simulation experiments, second edition." Springer, 2015.
[16] Kleijnen, J. P. C. "Kriging metamodeling in simulation: A review." European journal of operational research , 192(3) , pp. 707-716, 2009.
[17] Kleijnen, J. P. C. "Simulation Optimization through Regression or Kriging Metamodels." High-Performance Simulation-Based Optimization. Springer, pp.115-135, 2020.
[18] Li, Y. F., Ng, S. H., Xie, M., Goh, T. N. "A systematic comparison of metamodeling techniques for simulation optimization in decision support systems." Applied Soft Computing, 10(4), pp.1257-1273, 2010.
[19] Pham, T., Wagner, M. "Filtering noisy images using kriging." ISSPA'99. Proceedings of the Fifth International Symposium on Signal Processing and its Applications (IEEE Cat. No. 99EX359), 1999.
[20] Qu, H., Fu, M. C. "Gradient extrapolated stochastic kriging." ACM Transactions on Modeling and Computer Simulation (TOMACS), 24(4) , pp.1-25, 2014.
[21] Quan, N., Yin, J., Ng, S. H., Lee, L. H. "Simulation optimization via kriging: a sequential search using expected improvement with computing budget constraints." IIE Transactions, 45(7), pp.763-780, 2013.
[22] Roshan, U., Livesay, D. R. "Probalign: multiple sequence alignment using partition function posterior probabilities." Bioinformatics , 22(22), pp. 2715-2721, 2006.
[23] Sacks, J., Welch, W. J., Mitchell, T. J., Wynn, H. P. "Design and analysis of computer experiments." Statistical science, pp. 409-423, 1989.
[24] Sakata, S., F. Ashida, M. Zako. "On applying Kriging-based approximate optimization to inaccurate data." Computer methods in applied mechanics and engineering, pp.2055-2069, 2007.
[25] Sargent, R. "Verification and validation of simulation models." Journal of simulation ,7(1), pp.12-24, 2013.
[26] Shen, H., Hong, L. J., Zhang, X. "Enhancing stochastic kriging for queueing simulation with stylized models." IISE Transactions, 50(11), pp.943-958, 2018.
[27] Shi, X., Tong, C., Wang, L. "Evolutionary optimization with adaptive surrogates and its application in crude oil distillation." 2016 IEEE Symposium Series on Computational Intelligence, pp.1-8, 2016.
[28] Simpson, T. W., Poplinski, J. D., Koch, P. N.,Allen, J. K. "Metamodels for computer-based engineering design: survey and recommendations." Engineering with computers, 17(2), pp.129-150, 2001.
[29] Staum, J. "Better simulation metamodeling: The why, what, and how of stochastic kriging." Proceedings of the 2009 Winter Simulation Conference (WSC), pp.119-113, 2009.
[30] Stein, M. L. "Interpolation of spatial data: some theory for kriging." Springer Science & Business Media, 2012.
[31] Tew, J.D., Wilson, J.R. "Estimating simulation metamodels using combined correlationbased variance reduction techniques." IIE Transactions, 26(3), pp.2-16, 1994.
[32] Xie, W., Nelson, B. L., Barton, R. R. "A Bayesian framework for quantifying uncertainty in stochastic simulation." Operations Research, 62(6), pp.1439-1452, 2014.
[33] Yin, J., Ng, S. H., Ng, K. M. "A study on the effects of parameter estimation on Kriging model's prediction error in stochastic simulations." Proceedings of the 2009 Winter Simulation Conference (WSC), 2009.
[34] Yin, J., Ng, S. H., Ng, K. M. "Kriging metamodel with modified nugget-effect: The heteroscedastic variance case." Computers & Industrial Engineering, 61(3), pp.760-777, 2011.
[35] Zou, L., Zhang, X. "Stochastic kriging for inadequate simulation models." arXiv preprint arXiv:1802.00677, 2018.