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Discovering Cyclic Causal Models by Independent Components Analysis

Gustavo Lacerda, Peter L. Spirtes, Joseph Ramsey, Patrik O. Hoyer

TL;DR

This work generalizes LiNGAM to linear non-Gaussian (LiNG) SEMs that may contain cycles, by extending ICA-based discovery into LiNG-D and outputting the distribution-entailment equivalence class of SEMs. The method leverages ICA to estimate a mixing matrix, imposes a row-permutation constraint to recover the coefficient matrix $B$ while avoiding self-loops, and searches the space of admissible models via constrained n-rooks and non-local optimization. Theoretical results characterize when LiNG-D yields a unique stable model (notably with disjoint cycles under stability) and formalize the relationships among various notions of graph equivalence under non-Gaussian errors. This approach improves over prior cyclic-discovery methods by outputting a distribution-focused equivalence class and by relaxing faithfulness, with practical implications for identifying cyclic causal structures from non-Gaussian observational data.

Abstract

We generalize Shimizu et al's (2006) ICA-based approach for discovering linear non-Gaussian acyclic (LiNGAM) Structural Equation Models (SEMs) from causally sufficient, continuous-valued observational data. By relaxing the assumption that the generating SEM's graph is acyclic, we solve the more general problem of linear non-Gaussian (LiNG) SEM discovery. LiNG discovery algorithms output the distribution equivalence class of SEMs which, in the large sample limit, represents the population distribution. We apply a LiNG discovery algorithm to simulated data. Finally, we give sufficient conditions under which only one of the SEMs in the output class is 'stable'.

Discovering Cyclic Causal Models by Independent Components Analysis

TL;DR

This work generalizes LiNGAM to linear non-Gaussian (LiNG) SEMs that may contain cycles, by extending ICA-based discovery into LiNG-D and outputting the distribution-entailment equivalence class of SEMs. The method leverages ICA to estimate a mixing matrix, imposes a row-permutation constraint to recover the coefficient matrix while avoiding self-loops, and searches the space of admissible models via constrained n-rooks and non-local optimization. Theoretical results characterize when LiNG-D yields a unique stable model (notably with disjoint cycles under stability) and formalize the relationships among various notions of graph equivalence under non-Gaussian errors. This approach improves over prior cyclic-discovery methods by outputting a distribution-focused equivalence class and by relaxing faithfulness, with practical implications for identifying cyclic causal structures from non-Gaussian observational data.

Abstract

We generalize Shimizu et al's (2006) ICA-based approach for discovering linear non-Gaussian acyclic (LiNGAM) Structural Equation Models (SEMs) from causally sufficient, continuous-valued observational data. By relaxing the assumption that the generating SEM's graph is acyclic, we solve the more general problem of linear non-Gaussian (LiNG) SEM discovery. LiNG discovery algorithms output the distribution equivalence class of SEMs which, in the large sample limit, represents the population distribution. We apply a LiNG discovery algorithm to simulated data. Finally, we give sufficient conditions under which only one of the SEMs in the output class is 'stable'.

Paper Structure

This paper contains 22 sections, 7 equations, 3 figures.

Table of Contents

  1. Linear SEMs
  2. The model, with an illustration
  3. Interpretating linear SEMs
  4. Dealing with self-loops
  5. Self-loops with coefficient 1
  6. The problem and its history
  7. The problem of DG causal discovery
  8. Richardson's Cyclic Causal Discovery (CCD) Algorithm
  9. Shimizu et al's approach for discovering LiNGAM SEMs
  10. Independent Components Analysis (ICA)
  11. The LiNGAM discovery algorithm
  12. Discovering LiNG SEMs
  13. Prune and solve Constrained n-Rooks
  14. Deciding which entries are zero
  15. Constrained n-Rooks: the problem and an algorithm
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Example 1
  • Figure 2: After removing the edges whose coefficients are statistically indistinguishable from zero: (a) the raw$W_{I C A}$ matrix output by ICA on a SEM whose graph is $\mathbf{x}_{2} \rightarrow \mathbf{x}_{1} \leftarrow \mathbf{x}_{3}$ (b) the corresponding $\widetilde{W}$ matrix, obtained by permuting the error terms in $W_{I C A}$
  • Figure 3: The output of LiNG-D: Candidate$\# 1$ and Candidate #2