Nested Kriging predictions for datasets with a large number of observations
- 586 Downloads
This work falls within the context of predicting the value of a real function at some input locations given a limited number of observations of this function. The Kriging interpolation technique (or Gaussian process regression) is often considered to tackle such a problem, but the method suffers from its computational burden when the number of observation points is large. We introduce in this article nested Kriging predictors which are constructed by aggregating sub-models based on subsets of observation points. This approach is proven to have better theoretical properties than other aggregation methods that can be found in the literature. Contrarily to some other methods it can be shown that the proposed aggregation method is consistent. Finally, the practical interest of the proposed method is illustrated on simulated datasets and on an industrial test case with \(10^4\) observations in a 6-dimensional space.
KeywordsGaussian process regression Big data Aggregation methods Best linear unbiased predictor Spatial processes
Part of this research was conducted within the frame of the Chair in Applied Mathematics OQUAIDO, gathering partners in technological research (BRGM, CEA, IFPEN, IRSN, Safran, Storengy) and academia (Ecole Centrale de Lyon, Mines Saint-Etienne, University of Grenoble, University of Nice, University of Toulouse and CNRS) around advanced methods for Computer Experiments. The authors would like to warmly thank Dr. Géraud Blatman and EDF R&D for providing us the industrial test case. They also thank both editor and reviewers for very precise and constructive comments on this paper. This paper has been finished during a stay of D. Rullière at Vietnam Institute for Advanced Study in Mathematics, the latter author thanks the VIASM institute and DAMI research chair (Data Analytics & Models for Insurance) for their support.
- Bachoc, F., Durrande, N., Rullière, D., Chevalier, C.: Some properties of nested Kriging predictors. Technical report hal-01561747 (2017)Google Scholar
- Cao, Y., Fleet, D.J.: Generalized product of experts for automatic and principled fusion of Gaussian process predictions. arXiv preprint arXiv:1410.7827v2, CoRR, abs/1410.7827:1–5. Modern Nonparametrics 3: Automating the Learning Pipeline workshop at NIPS, Montreal (2014)
- Deisenroth, M.P., Ng, J.W.: Distributed Gaussian processes. In: Proceedings of the 32nd International Conference on Machine Learning, Lille, France. JMLR: W & CP vol. 37 (2015)Google Scholar
- Hensman, J., Fusi, N., Lawrence, N.D.: Gaussian Processes for Big Data. Uncertainty in Artificial Intelligence Conference. Paper Id 244 (2013)Google Scholar
- Nickson, T., Gunter, T., Lloyd, C., Osborne, M.A., Roberts, S.: Blitzkriging: Kronecker-structured stochastic Gaussian processes. arXiv preprint arXiv:1510.07965v2, pp 1–13 (2015)
- Roustant, O., Ginsbourger, D., Deville, Y.: DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Softw. 51(1), 1–55 (2012)Google Scholar
- Scott, S.L., Blocker, A.W., Bonassi, F.V., Chipman, H.A., George, E.I., McCulloch, R.E.: Bayes and big data: the consensus monte carlo algorithm. Int. J. Manag. Sci. Eng. Manag. 11(2), 78–88 (2016)Google Scholar
- Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer, Berlin (2012)Google Scholar
- van Stein, B., Wang, H., Kowalczyk, W., Bäck, T., Emmerich, M.: Optimally weighted cluster Kriging for big data regression. In: International Symposium on Intelligent Data Analysis, pp. 310–321. Springer (2015)Google Scholar
- Zhang, B., Sang, H., Huang, J.Z.: Full-scale approximations of spatio-temporal covariance models for large datasets. Stat. Sinica 25(1), 99–114 (2015)Google Scholar