Abstract
This paper presents our winning entry for the EVA 2019 data competition, the aim of which is to predict Red Sea surface temperature extremes over space and time. To achieve this, we used a stochastic partial differential equation (Poisson equation) based method, improved through a regularization to penalize large magnitudes of solutions. This approach is shown to be successful according to the competition’s evaluation criterion, i.e. a threshold-weighted continuous ranked probability score. Our stochastic Poisson equation and its boundary conditions resolve the data’s non-stationarity naturally and effectively. Meanwhile, our numerical method is computationally efficient at dealing with the data’s high dimensionality, without any parameter estimation. It demonstrates the usefulness of stochastic differential equations on spatio-temporal predictions, including the extremes of the process.
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Bacro, J.-N., Gaetan, C., Opitz, T., Toulemonde, G.: Hierarchical space-time modeling of asymptotically independent exceedances with an application to precipitation data. J. Am. Stat. Assoc. 115(530), 555–569 (2020). https://doi.org/10.1080/01621459.2019.1617152
Chavez-Demoulin, V., Davison, A.C.: Generalized additive modelling of sample extremes. Journal of the Royal Statistical Society: Series C (Applied Statistics) 54(1), 207–222 (2005). https://doi.org/10.1111/j.1467-9876.2005.00479.x
Davis, R.A., Klüppelberg, C., Steinkohl, C.: Statistical inference for max-stable processes in space and time. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 75(5), 791–819 (2013). https://doi.org/10.1111/rssb.12012
Davison, A.C., Padoan, S.A., Ribatet, M.: Statistical modeling of spatial extremes. Stat. Sci. 27(2), 161–186 (2012). https://doi.org/10.1214/11-STS376
Godsil, C., Royle, G.F.: Algebraic Graph Theory. Springer-Verlag, New York (2001). https://doi.org/10.1007/978-1-4613-0163-9
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer-Verlag, New York (2009). https://doi.org/10.1007/978-0-387-84858-7
Huser, R., Davison, A.C.: Space–time modelling of extreme events. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76(2), 439–461 (2014). https://doi.org/10.1111/rssb.12035
Huser, R.: Editorial: EVA 2019 data competition on spatio-temporal prediction of Red Sea surface temperature extremes. Extremes. https://doi.org/10.1007/s10687-019-00369-9 (2020)
Knabner, P., Angerman, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer-Verlag, New York (2003). https://doi.org/10.1007/b97419
Krainski, E., Gómez-Rubio, V., Bakka, H., Lenzi, A., Castro-Camilo, D., Simpson, D., Lindgren, F., Rue, H.: Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA. Chapman and Hall/CRC, Boca Raton (2018). https://doi.org/10.1201/9780429031892
Lindgren, F., Rue, H., Lindström, J.: An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73(4), 423–498 (2011). https://doi.org/10.1111/j.1467-9868.2011.00777.x
Pérez, P, Gangnet, M., Blake, A.: Poisson image editing. ACM Trans. Graph. 22(3), 313–318 (2003). https://doi.org/10.1145/882262.882269
Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131 (1975). https://doi.org/10.1214/aos/1176343003
Reich, B.J., Shaby, B.A.: A hierarchical max-stable spatial model for extreme precipitation. Ann. Appl. Stat. 6(4), 1430–1451 (2012). https://doi.org/10.1214/12-AOAS591
Acknowledgements
We thank the organizing committee of the 11th international conference on Extreme Value Analysis, and Dr. Raphaël Huser for organizing the data challenge. Also, we thank the editors and anonymous reviewers whose suggestions helped improve and clarify this manuscript. Both authors are financially supported by China Scholarship Council for their PhD studies at TU Delft. The authors also acknowledge the support of their supervisors – they are D. Cheng’s supervisors Dr. Pasquale Cirillo, and Prof. dr. Frank Redig, and Z. Liu’s supervisors Dr. Eugeni L. Doubrovski, Prof. dr. Jo M.P. Geraedts, and Prof. Charlie C.L. Wang.
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This work was first presented on July 5th, 2019, at EVA 2019.
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Cheng, D., Liu, Z. Spatio-temporal prediction of missing temperature with stochastic Poisson equations. Extremes 24, 163–175 (2021). https://doi.org/10.1007/s10687-020-00397-w
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DOI: https://doi.org/10.1007/s10687-020-00397-w