{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T21:27:09Z","timestamp":1760909229945,"version":"3.41.0"},"reference-count":40,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T00:00:00Z","timestamp":1604102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Model. Comput. Simul."],"published-print":{"date-parts":[[2020,10,31]]},"abstract":"<jats:p>\n            Simulations are often used for the design of complex systems as they allow one to explore the design space without the need to build several prototypes. Over the years, the simulation accuracy, as well as the associated computational cost, has increased significantly, limiting the overall number of simulations during the design process. Therefore, metamodeling aims to approximate the simulation response with a cheap to evaluate mathematical approximation, learned from a limited set of simulator evaluations. Kernel-based methods using stationary kernels are nowadays widely used. In many problems, the smoothness of the function varies in space, which we call\n            <jats:italic>nonstationary<\/jats:italic>\n            behavior [20]. However, using stationary kernels for nonstationary responses can be inappropriate and result in poor models when combined with sequential design. We present the application of two recent techniques: Deep Gaussian Processes and Gaussian Processes with nonstationary kernel, which are better able to cope with these difficulties. We evaluate the method for nonstationary regression on a series of real-world problems, showing that these recent approaches outperform the standard Gaussian Processes with stationary kernels. Results show that these techniques are suitable for the simulation community, and we outline the variational inference method for the Gaussian Process with nonstationary kernel.\n          <\/jats:p>","DOI":"10.1145\/3384470","type":"journal-article","created":{"date-parts":[[2020,7,7]],"date-time":"2020-07-07T12:34:43Z","timestamp":1594125283000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":["Hierarchical Gaussian Process Models for Improved Metamodeling"],"prefix":"10.1145","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5322-5930","authenticated-orcid":false,"given":"Nicolas","family":"Knudde","sequence":"first","affiliation":[{"name":"Ghent University\u2013imec, Belgium"}]},{"given":"Vincent","family":"Dutordoir","sequence":"additional","affiliation":[{"name":"PROWLER.io, UK"}]},{"given":"Joachim Van Der","family":"Herten","sequence":"additional","affiliation":[{"name":"Ghent University\u2013imec, Belgium"}]},{"given":"Ivo","family":"Couckuyt","sequence":"additional","affiliation":[{"name":"Ghent University\u2013imec, Belgium"}]},{"given":"Tom","family":"Dhaene","sequence":"additional","affiliation":[{"name":"Ghent University\u2013imec, Belgium"}]}],"member":"320","published-online":{"date-parts":[[2020,11,25]]},"reference":[{"volume-title":"Challenor","year":"2012","author":"Andrianakis Ioannis","key":"e_1_2_1_1_1"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1090.0754"},{"volume-title":"Proceedings of the 33rd International Conference on Machine Learning (ICML\u201916)","year":"2016","author":"Bui Thang D.","key":"e_1_2_1_3_1"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1120.1143"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/090761811"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1162\/089976602317250933"},{"volume-title":"Proceedings of the 16th International Conference on Artificial Intelligence and Statistics (AISTATS\u201913)","author":"Andreas","key":"e_1_2_1_7_1"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1002\/cpe.3097"},{"volume-title":"International Conference on Learning Representations.","author":"Kingma Diederik","key":"e_1_2_1_9_1"},{"key":"e_1_2_1_10_1","first-page":"I","article-title":"Variational inference via x upper bound minimization","volume":"30","author":"Dieng Adji Bousso","year":"2017","journal-title":"Advances in Neural Information Processing Systems"},{"volume-title":"Proceedings of the 2017 Winter Simulation Conference (WSC\u201917)","author":"Dutordoir Vincent","key":"e_1_2_1_11_1"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0952-1976(03)00043-5"},{"volume-title":"et\u00a0al","year":"2008","author":"Forrester Alexander","key":"e_1_2_1_13_1"},{"key":"e_1_2_1_16_1","article-title":"A Surrogate modeling and adaptive sampling toolbox for computer based design","author":"Gorissen Dirk","year":"2010","journal-title":"Journal of Machine Learning Research 11"},{"volume-title":"Karel Crombecq, and Tom Dhaene.","year":"2009","author":"Gorissen Dirk","key":"e_1_2_1_17_1"},{"volume-title":"IEEE Congress on Evolutionary Computation. 989--996","year":"2008","author":"Gorissen D.","key":"e_1_2_1_18_1"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1198\/016214508000000689"},{"key":"e_1_2_1_20_1","unstructured":"Markus Heinonen Henrik Mannerstr\u00f6m Juho Rousu Samuel Kaski and Harri L\u00e4hdesm\u00e4ki. 2016. Non-stationary Gaussian process regression with Hamiltonian Monte Carlo. In Artificial Intelligence and Statistics. 732--740.  Markus Heinonen Henrik Mannerstr\u00f6m Juho Rousu Samuel Kaski and Harri L\u00e4hdesm\u00e4ki. 2016. Non-stationary Gaussian process regression with Hamiltonian Monte Carlo. In Artificial Intelligence and Statistics. 732--740."},{"volume-title":"Conference on Uncertainty in Artificial Intellegence. auai.org, 282--290","author":"Hensman James","key":"e_1_2_1_21_1"},{"key":"e_1_2_1_22_1","unstructured":"James Hensman and Neil D. Lawrence. 2014. Nested Variational Compression in Deep Gaussian Processes. Retrieved 12\/03\/2016 from https:\/\/arxiv.org\/pdf\/1412.1370.pdf.  James Hensman and Neil D. Lawrence. 2014. Nested Variational Compression in Deep Gaussian Processes. Retrieved 12\/03\/2016 from https:\/\/arxiv.org\/pdf\/1412.1370.pdf."},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.2014.881749"},{"volume-title":"International Conference on Learning Representations (ICLR\u201915)","author":"Diederick","key":"e_1_2_1_24_1"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1002\/asmb.2300"},{"key":"e_1_2_1_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2007.10.013"},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1002\/mmce.10032"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1109\/TAES.2006.314576"},{"key":"e_1_2_1_29_1","first-page":"1","article-title":"GPflow: A Gaussian process library using tensorflow","volume":"18","author":"Matthews Alexander G.","year":"2017","journal-title":"Journal of Machine Learning Research"},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.1080\/01605682.2017.1409154"},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.1016\/0378-3758(94)90115-5"},{"key":"e_1_2_1_32_1","article-title":"A unifying view of sparse approximate Gaussian process regression","author":"Candela Joaquin Qui\u00f1onero","year":"2005","journal-title":"Journal of Machine Learning Research 6"},{"volume-title":"Williams","year":"2006","author":"Rasmussen Carl Edward","key":"e_1_2_1_33_1"},{"key":"e_1_2_1_34_1","first-page":"I","article-title":"Non-stationary spectral kernels","volume":"30","author":"Remes Sami","year":"2017","journal-title":"Advances in Neural Information Processing Systems"},{"key":"e_1_2_1_35_1","volume-title":"Proceedings of the 32nd International Conference on Machine Learning (Proceedings of Machine Learning Research), Francis Bach and David Blei (Eds.)","volume":"37","author":"Rezende Danilo","year":"2015"},{"key":"e_1_2_1_36_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF03184811"},{"volume-title":"Notz","year":"2013","author":"Santner Thomas J.","key":"e_1_2_1_37_1"},{"volume-title":"Workshop on AI and Statistics 09","author":"Seeger Matthias","key":"e_1_2_1_38_1"},{"volume-title":"Advances in Neural Information Processing Systems 18","author":"Snelson Edward","key":"e_1_2_1_39_1"},{"key":"e_1_2_1_40_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018628609742"},{"key":"e_1_2_1_41_1","unstructured":"Michalis K. Titsias. 2009. Variational learning of inducing variables in sparse Gaussian processes. Artificial Intelligence and Statistics 12. 567--574.  Michalis K. Titsias. 2009. Variational learning of inducing variables in sparse Gaussian processes. 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