时空平稳性的概念与STARMA模型

时空相关性

时空相关性反映了序列对时空的依赖,一般会讨论到时空自相关性和时空偏相关性两个概念。

时空平稳性

时空平稳性(Space-time Stationarity)的概念,是说如果是平稳的,那意味着在时空范畴内不存在任何趋势或者模式。在统计上,表现为均值、方差、协方差等不随时间和空间的变化而变化。在实际应用中,很少能够满足统计上绝对的不变值,因此只要求近似为常数即可认为时空序列平稳。

因此,对于时空平稳的时空变量来说,

时空非平稳性,可能是大尺度的趋势,也可能是局部效应。

  1. 大尺度的非平稳:描述均值的总体变化。通常称为“一阶”效应。
  2. 局部的非平稳:是对时时空相关局部特征的描述。通常称为“二阶”效应。对于时间序列分析,一般用自相关移动平均描述因时间依赖造成的局部效应;对于空间数据分析,一般用空间自相关描述其局部效应。

STARMA模型

严格说来,STARMA模型是由时间序列模型模型ARMA(自回归-移动平均)向空间层面扩展而来。该模型在时间序列的基础上,增加考虑了空间相邻位置的影响,因此从时空结合角度讲,更为准确。模型有两个基本假设:

  1. 序列是平稳的
  2. 随机误差符合正态分布

因此,随之而来的处理方式中,首先要解决序列的平稳化问题,过程中需要通过观察空间自相关系数来发现非平稳的存在,并可能会用到各种手段来实现平稳化,例如进行周期性差分处理。

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