Spatio-temporal prediction of missing temperature with stochastic Poisson equations
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Spatio-temporal prediction of missing temperature with stochastic Poisson equations The LC2019 team winning entry for the EVA 2019 data competition Dan Cheng1
· Zishun Liu1
Received: 31 March 2020 / Revised: 13 October 2020 / Accepted: 20 October 2020 / © The Author(s) 2020
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. Keywords Data competition · EVA 2019 Conference · Prediction · Poisson equation · Spatio-temporal data · Temperature data AMS 2000 Subject Classifications 62P12 · 62H11 · 62M30 · 35Q62
1 Introduction The aim of EVA 2019 data competition is to predict spatio-temporal extremes of Red Sea surface temperature (Huser 2020). The dataset of the competition consists of daily sea surface temperature (SST) anomalies for the entire Red Sea from 1985 to This work was first presented on July 5th, 2019, at EVA 2019. Zishun Liu
[email protected] 1
Delft University of Technology, Delft, The Netherlands
D. Cheng, Z. Liu
2015 whilst part of the data is masked. The main goal of this data competition is to predict the distribution of X(s, t) =
min
(˜s ,t˜)∈N (s,t)
ˆ s , t˜), A(˜
(1)
ˆ ·) is the predicted SST anomafor space-time validation points (s, t), where A(·, lies. The competition evaluates the performance via a threshold-weighted continuous ranked probability score (twCRPS). The main challenge is the data’s high dimensionality (16703 locations and 11315 days) and strong non-stationarity in space and time (Section 3.3 of Huser (2020)). More details of the data competition, including the definition of the anomaly, plots of the data along with some exploratory analysis, as well as the competition’s rules and goals, are available in the Editorial (Huser 2020). To predict the extremes on the spatio-temporal domain, it is sufficient to preˆ ·) instead of dict the whole field of values on it. That is to say, we can predict A(·, ˆ directly predicting X(·, ·) in Eq. (1). Given the partial data of A(·, t) for someday t, the predicted data in the unobserved region should comply with some continuity constraints. According to Fourier’s analytical theory of heat, it is reasonable to assume a temperature field has a second order derivative, implying that the anomaly
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