In this paper, we introduce the concept of fractional integration for spatial autoregressive models. We show that the range of the dependence can be spatially extended or diminished by introducing a further fractional integration parameter to spatial autoregressive moving average models (SARMA). This new model is called the spatial autoregressive fractionally integrated moving average model, briefly sp-ARFIMA. We show the relation to time-series ARFIMA models and also to (higher-order) spatial autoregressive models. Moreover, an estimation procedure based on the maximum-likelihood principle is introduced and analysed in a series of simulation studies. Eventually, the use of the model is illustrated by an empirical example of atmospheric fine particles, so-called aerosol optical thickness, which is important in weather, climate and environmental science.

翻译：本文提出了空间自回归模型中的分数积分概念。研究表明，通过向空间自回归移动平均模型（SARMA）引入分数积分参数，可以扩展或缩小依赖关系的空间范围。这种新模型被称为空间自回归分数整合移动平均模型，简称sp-ARFIMA。我们阐释了该模型与时间序列ARFIMA模型以及高阶空间自回归模型之间的关系。此外，基于最大似然原理的估计方法被提出，并通过一系列模拟研究进行了分析。最终，通过大气细颗粒物（即气溶胶光学厚度）的经验实例展示了该模型的应用，该参数在天气、气候及环境科学领域具有重要意义。