基于动态模式分解的时间序列的因果关系分析

Causal Relationship of Time Series Based on Dynamic Mode Decomposition

摘要: 本文基于动态模式分解(DMD)分析四种气候时间序列(地表温度，海面温度，太阳黑子和碳排放)的模式之间的因果关系。通过延迟坐标的形式把每条单变量时间序列嵌入至多维，DMD捕捉了气候时间序列中不同的演化特征，对这些特征进行不同数量的截断，提取出不同快慢层面的模式。转移熵(TE)得出了模式之间的因果关系，我们观察到因果关系在各种模式间的分布。总体上看，由最大特征值提取的主导模式具有较大周期或长期趋势，它作为很强的驱动因素，与自身内部的快模式，以及其他气候对象的模式之间有着很强的关联。太阳黑子的长期趋势对海面温度的各种周期和长期趋势有很大贡献，影响程度大于地表温度。碳排放主要影响气温极慢的模式即超长期趋势，气温反过来对碳排放的影响偏弱。四个气候对象构建的因果框架显示，因果信息的传递基本平衡，海面温度比地表温度吸收了更多的信息。这些方法的联合使用可以成为从复杂系统中提取各种模式，分析内蕴性质和探索因果关系的灵活工具。

Abstract: This study explores the causal relationships between modes in four climate time series (surface temperature, sea surface temperature, sunspots and carbon emissions) using Dynamic Mode Decomposition (DMD). By embedding each univariate time series into multidimensional space using delay coordinates, DMD captures distinct evolutionary features in the climate time series. Different truncations of these features reveal modes at various timescales. Transfer Entropy (TE) is employed to quantify the causal relationships between modes, revealing the distribution of causal interactions among different modes. Overall, dominant modes extracted from the maximum eigenvalue exhibit larger cycles or long-term trends. These dominant modes serve as strong driving factors, displaying significant associations with fast internal modes and modes of other climate objects. The long-term trend in sunspots contributes significantly to various cycles and long-term trends in sea surface temperature, with a greater impact than on surface temperature. Carbon emissions predominantly influence extremely slow temperature modes, representing ultra-long-term trends, while the influence of temperature on carbon emissions is relatively weak. The causal framework constructed for the four climate objects illustrates a relatively balanced transfer of causal information, with sea surface temperature absorbing more information than surface temperature. The combined use of these methods serves as a flexible tool for extracting diverse modes from complex systems, analyzing intrinsic properties, and exploring causal relationships.