极端热点推断的贝叶斯时空间隙填充：对红海表面温度的应用
Bayesian space-time gap filling for inference on extreme hot-spots: an application to Red Sea surface temperatures
摘要：
我们开发了一种时空框架中极值热点的概率预测方法，该方法适用于包含重要缺口的大数据集。在这种情况下，直接从数据中计算汇总（例如空间-时间域上的最小值）是不可能的。为了获得这样的聚类摘要的预测分布，我们提出了一个两步的方法。我们首先对边缘分布进行建模，重点是对右尾进行准确建模，然后将数据转换为标准高斯尺度后，我们估计在时域中本地定义的高斯时空依赖模型，用于我们想要预测的时空子区域。在第一步中，我们对数据的平均值和标准差进行去趋势化，并拟合空间分辨的广义帕累托分布，以应用上尾校正。为了确保估计趋势的空间平滑性，我们要么使用最近邻技术汇集数据，要么应用广义加性回归模型。为了处理数据的高时空分辨率，局部高斯模型使用基于随机偏微分方程（SPDE）方法的Matérn相关函数的马尔可夫表示。在第二步中，它们通过在R-INLA中实现的集成嵌套拉普拉斯近似在贝叶斯框架中拟合。最后，通过Monte-Carlo估计生成后验样本以提供统计推断。受2019年极值分析数据挑战的启发，我们使用具有人工生成间隙的网格化数据集（11315天，16703像素）来说明我们预测红海表面温度异常中局部时空最小值分布的方法。特别是，我们展示了我们的两步方法在没有尾部变换的纯高斯模型上的改进性能。
英文摘要：
We develop a method for probabilistic prediction of extreme value hot-spots in a spatio-temporal framework, tailored to big datasets containing important gaps. In this setting, direct calculation of summaries from data, such as the minimum over a space-time domain, is not possible. To obtain predictive distributions for such cluster summaries, we propose a two-step approach. We first model marginal distributions with a focus on accurate modeling of the right tail and then, after transforming the data to a standard Gaussian scale, we estimate a Gaussian space-time dependence model defined locally in the time domain for the space-time subregions where we want to predict. In the first step, we detrend the mean and standard deviation of the data and fit a spatially resolved generalized Pareto distribution to apply a correction of the upper tail. To ensure spatial smoothness of the estimated trends, we either pool data using nearest-neighbor techniques, or apply generalized additive regression modeling. To cope with high space-time resolution of data, the local Gaussian models use a Markov representation of the Matérn correlation function based on the stochastic partial differential equations (SPDE) approach. In the second step, they are fitted in a Bayesian framework through the integrated nested Laplace approximation implemented in R-INLA. Finally, posterior samples are generated to provide statistical inferences through Monte-Carlo estimation. Motivated by the 2019 Extreme Value Analysis data challenge, we illustrate our approach to predict the distribution of local space-time minima in anomalies of Red Sea surface temperatures, using a gridded dataset (11315 days, 16703 pixels) with artificially generated gaps. In particular, we show the improved performance of our two-step approach over a purely Gaussian model without tail transformations.
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