Acknowledging a considerable literature on modeling daily temperature data, we propose a multi-level spatiotemporal model which introduces several innovations in order to explain the daily maximum temperature in the summer period over 60 years in a region containing Aragón, Spain. The model operates over continuous space but adopts two discrete temporal scales, year and day within year. It captures temporal dependence through autoregression on days within year and also on years. Spatial dependence is captured through spatial process modeling of intercepts, slope coefficients, variances, and autocorrelations. The model is expressed in a form which separates fixed effects from random effects and also separates space, years, and days for each type of effect. Motivated by exploratory data analysis, fixed effects to capture the influence of elevation, seasonality, and a linear trend are employed. Pure errors are introduced for years, for locations within years, and for locations at days within years. The performance of the model is checked using a leave-one-out cross-validation. Applications of the model are presented including prediction of the daily temperature series at unobserved or partially observed sites and inference to investigate climate change comparison.

Supplementary materials accompanying this paper appear online.

承认有大量关于模拟每日温度数据的文献，我们提出了一个多层次的时空模型，该模型引入了一些创新，以解释西班牙阿拉贡地区 60 多年来夏季的每日最高温度。该模型在连续空间上运行，但采用了两个离散的时间尺度，即年和年中的日。它通过对一年中的天数以及对年数的自回归来捕获时间依赖性。通过截距、斜率系数、方差和自相关的空间过程建模来捕获空间依赖性。该模型以一种将固定效应与随机效应分开的形式表示，并且还将每种效应的空间、年份和天数分开。受探索性数据分析的启发，固定效应以捕捉高程的影响，季节性和线性趋势被采用。纯误差会在几年内引入，对于几年内的位置，以及对于几年内几天的位置。使用留一法交叉验证检查模型的性能。介绍了该模型的应用，包括预测未观察到或部分观察到的地点的每日温度序列，以及研究气候变化比较的推断。

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