Sun Boxin, Deng Jinxian, Scheel Norman, Zhu David C, Ren Jian, Zhang Rong, Li Tongtong
Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA.
Department of Radiology, Michigan State University, East Lansing, MI 48824, USA.
Entropy (Basel). 2024 Jun 24;26(7):539. doi: 10.3390/e26070539.
Rooted in dynamic systems theory, convergent cross mapping (CCM) has attracted increased attention recently due to its capability in detecting linear and nonlinear causal coupling in both random and deterministic settings. One limitation with CCM is that it uses both past and future values to predict the current value, which is inconsistent with the widely accepted definition of causality, where it is assumed that the future values of one process cannot influence the past of another. To overcome this obstacle, in our previous research, we introduced the concept of causalized convergent cross mapping (cCCM), where future values are no longer used to predict the current value. In this paper, we focus on the implementation of cCCM in causality analysis. More specifically, we demonstrate the effectiveness of cCCM in identifying both linear and nonlinear causal coupling in various settings through a large number of examples, including Gaussian random variables with additive noise, sinusoidal waveforms, autoregressive models, stochastic processes with a dominant spectral component embedded in noise, deterministic chaotic maps, and systems with memory, as well as experimental fMRI data. In particular, we analyze the impact of shadow manifold construction on the performance of cCCM and provide detailed guidelines on how to configure the key parameters of cCCM in different applications. Overall, our analysis indicates that cCCM is a promising and easy-to-implement tool for causality analysis in a wide spectrum of applications.
收敛交叉映射（CCM）植根于动态系统理论，由于其能够在随机和确定性环境中检测线性和非线性因果耦合，近年来受到了越来越多的关注。CCM的一个局限性在于它使用过去值和未来值来预测当前值，这与广泛接受的因果关系定义不一致，在因果关系定义中，假设一个过程的未来值不会影响另一个过程的过去。为了克服这一障碍，在我们之前的研究中，我们引入了因果化收敛交叉映射（cCCM）的概念，其中不再使用未来值来预测当前值。在本文中，我们专注于cCCM在因果分析中的实现。更具体地说，我们通过大量示例证明了cCCM在识别各种环境中的线性和非线性因果耦合方面的有效性，这些示例包括具有加性噪声的高斯随机变量、正弦波形、自回归模型、嵌入噪声中的具有主导频谱成分的随机过程、确定性混沌映射以及具有记忆的系统，还有实验性功能磁共振成像数据。特别是，我们分析了影子流形构建对cCCM性能的影响，并提供了在不同应用中如何配置cCCM关键参数的详细指南。总体而言，我们的分析表明，cCCM是一种在广泛应用中进行因果分析的有前景且易于实现的工具。
