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Causal Discovery in Symmetric Dynamic Systems with Convergent Cross Mapping

Yiting Duan, Yi Guo, Jack Yang, Ming Yin

TL;DR

This work tackles the problem that Convergent Cross Mapping (CCM) can misidentify causality in chaotic systems when the attractor exhibits symmetry, especially two-fold rotation symmetry, by producing a noninjective shadow-mapping that obscures true directional relations. It introduces Segment CCM (sCCM), which partitions the invariant shadow manifold into symmetry-adapted sub-manifolds using $k$-means clustering (with $k=2$) and then applies CCM to each pair, averaging the forecast skill to recover the correct causal structure without using information from other variables. The authors provide a theoretical account of how symmetry induces noninjective mappings and demonstrate, through extensive experiments on Lorenz63-like systems and high-dimensional Lorenz-like attractors, that sCCM corrects the failures of standard CCM and remains robust to moderate noise. While effective for two-fold symmetry, they discuss limitations for higher-fold symmetry and outline directions for refining segmentation to preserve locality and injectivity. The results have practical significance for causal inference in nonlinear, chaotic, and potentially geophysically or biologically symmetric systems where CCM is a popular tool.

Abstract

This paper systematically discusses how the inherent properties of chaotic attractors influence the results of discovering causality from time series using convergent cross mapping, particularly how convergent cross mapping misleads bidirectional causality as unidirectional when the chaotic attractor exhibits symmetry. We propose a novel method based on the k-means clustering method to address the challenges when the chaotic attractor exhibits two-fold rotation symmetry. This method is demonstrated to recover the symmetry of the latent chaotic attractor and discover the correct causality between time series without introducing information from other variables. We validate the accuracy of this method using time series derived from low-dimension and high-dimensional chaotic symmetric attractors for which convergent cross mapping may conclude erroneous results.

Causal Discovery in Symmetric Dynamic Systems with Convergent Cross Mapping

TL;DR

This work tackles the problem that Convergent Cross Mapping (CCM) can misidentify causality in chaotic systems when the attractor exhibits symmetry, especially two-fold rotation symmetry, by producing a noninjective shadow-mapping that obscures true directional relations. It introduces Segment CCM (sCCM), which partitions the invariant shadow manifold into symmetry-adapted sub-manifolds using -means clustering (with ) and then applies CCM to each pair, averaging the forecast skill to recover the correct causal structure without using information from other variables. The authors provide a theoretical account of how symmetry induces noninjective mappings and demonstrate, through extensive experiments on Lorenz63-like systems and high-dimensional Lorenz-like attractors, that sCCM corrects the failures of standard CCM and remains robust to moderate noise. While effective for two-fold symmetry, they discuss limitations for higher-fold symmetry and outline directions for refining segmentation to preserve locality and injectivity. The results have practical significance for causal inference in nonlinear, chaotic, and potentially geophysically or biologically symmetric systems where CCM is a popular tool.

Abstract

This paper systematically discusses how the inherent properties of chaotic attractors influence the results of discovering causality from time series using convergent cross mapping, particularly how convergent cross mapping misleads bidirectional causality as unidirectional when the chaotic attractor exhibits symmetry. We propose a novel method based on the k-means clustering method to address the challenges when the chaotic attractor exhibits two-fold rotation symmetry. This method is demonstrated to recover the symmetry of the latent chaotic attractor and discover the correct causality between time series without introducing information from other variables. We validate the accuracy of this method using time series derived from low-dimension and high-dimensional chaotic symmetric attractors for which convergent cross mapping may conclude erroneous results.
Paper Structure (15 sections, 3 theorems, 39 equations, 20 figures, 5 tables, 1 algorithm)

This paper contains 15 sections, 3 theorems, 39 equations, 20 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Attractor Reconstruction
  4. Convergent Cross Mapping
  5. Related Work about CCM
  6. Problem Formulation
  7. Explanation and Proposed Solution
  8. Preliminary
  9. Explanation
  10. Proposed Method
  11. Experiments
  12. Case 1: Rotation symmetric systems under order-two group $\mathbb{C}_{2}$ in $\mathbb{R}^{3}$
  13. Case 2: Rotation Symmetric Systems under order-two group $\mathbb{C}_{2}$ in High Dimension
  14. Further discussion: Rotation Symmetric Systems under order-$n$ group $\mathbb{C}_{n}$
  15. Conclusion

Key Result

Theorem 1

takens2006detecting Let $M$ be an $n$-dimensional smooth manifold. If $v$ is a vector field on $M$ with flow $\psi_{t}$ and $h$ is a measurement function on $M$, then for generic choices of $v$ and $h$, the differential mapping $F_{h,m}: M \rightarrow \mathbb{R}^{m}$ of the continuous dynamic system which is an embedding when $m = 2n + 1$, where $m$ is the embedding dimension, $\frac{d}{dt}|_{0}$

Figures (20)

  • Figure 1: Sequential figures representing the changes in reconstructed shadow manifold $\mathcal{M}_{x}$ of Lorenz63 system with increasing lag value $\tau$.
  • Figure 3: True causality versus CCM's result. a. Graphical model of causal relations within the Lorenz63 system. b. Graphical model of causal relations obtained by CCM. The Lorenz63 system is simulated using the fourth-order Runge-Kutta method, where the temporal domain is $t\in[0,100]$, the time step for discretization is $\Delta t=0.01$ which could be regarded as the sampling rate $T$ for continuous-time systems, the initial condition is $\mathbf{x}_{0} = [1,1,1]$, and $\tau = 9$, $n =3$ for CCM.
  • Figure 4: Influence of embedding parameters. Left. The influence of lag value $\tau$ when $n =3$. Right. The influence of embedding dimension $n$ when $\tau = 9$.
  • Figure 5: x-z plane projections of attractors. a. Chen $\&$ Ueta system, b. Lorenz63 system, c. Burke $\&$ Shaw system, d. three-scroll chaotic system.
  • Figure 6: A symmetric pair of attractor "kissing" of the reflection equivariant system, when a = 0.7 and two initial values $\mathbf{x_{1}} = [-2, 2, 0]$ and $\mathbf{x_{2}} = [2, 2, 0]$. Two parts of trajectories can become arbitrarily close but never cross the plane $x = 0$.
  • ...and 15 more figures

Theorems & Definitions (4)

  • Theorem 1: Takens
  • Theorem 2: Weak Whitney Embedding
  • Theorem 3: Cross
  • proof