29. ICA LingD β€” Cyclic LiNGAM (Lacerda et al.)

Type: Non-Gaussian / Moment-based / ICA-based
Output: Directed cyclic graph (DCG)
Assumptions: Linear SEM with independent, non-Gaussian noise; cycles allowed

IcaLingD is the cyclic extension of the LiNGAM framework proposed by Lacerda et al. It generalizes ICA-LiNGAM so that feedback cycles are permitted. Instead of assuming a strictly acyclic causal ordering, IcaLingD recovers an almost-directed structure that can include strongly connected components (SCCs) corresponding to causal feedback loops.

It is one of the earliest practical methods capable of learning linear, non-Gaussian feedback systems.

29.1. Key Idea

Classic LiNGAM uses ICA to estimate a mixing matrix for an acyclic model.
In the cyclic case, the mixing matrix does not factor cleanly into a triangular form.

IcaLingD proceeds by:

  1. Running ICA to obtain an estimated mixing matrix.

  2. Analyzing the absolute values of this matrix to identify directed influence strengths.

  3. Constructing a weighted directed graph from these influences.

  4. Identifying cycles through strongly connected components (SCC decomposition).

  5. Orienting edges wherever the ICA information yields a clear sign of influence.

  6. Representing SCCs as feedback components rather than trying to force a DAG orientation.

The result is a directed cyclic graph:

  • Outside cycles, edges reflect directed causal influence.

  • Inside cycles, the algorithm outputs SCCs indicating feedback relationships.

29.2. When to Use

Use IcaLingD when:

  • The underlying system may involve feedback loops.

  • The structural equations are linear.

  • The noise terms are independent and non-Gaussian.

  • You want to detect cyclic components, not just a DAG.

  • DirectLiNGAM or ICA-LiNGAM fails due to the presence of cycles.

Typical domains:

  • Econometrics (supply-and-demand feedback)

  • Neuroscience or physiology (mutually regulating components)

  • Engineering control systems

  • Any domain where reciprocal causation is plausible

29.3. Prior Knowledge Support

The Tetrad implementation respects:

  • Required edges

  • Forbidden edges

  • Tier constraints (although tiers interact with cycles in a limited way)

Knowledge constraints help restrict the search space for the ICA permutation stage and for the cycle decomposition.

29.4. Strengths

  • Handles cycles, which LiNGAM and DirectLiNGAM cannot.

  • Produces explicit feedback components via SCC detection.

  • Non-Gaussianity makes the system identifiable in many cases where Gaussian SEMs fail.

  • A natural extension of ICA-LiNGAM.

29.5. Limitations

  • Less statistically stable than DirectLiNGAM and ICA-LiNGAM.

  • Interpretation can be harder when cycles are large.

  • Assumes linearity; does not handle nonlinear feedback.

  • Still sensitive to ICA numerical issues.

29.6. Key Parameters in Tetrad

Parameter (camelCase)

Description

numRestarts

ICA restarts to stabilize the decomposition.

maxIterations

Maximum ICA iterations.

icaAlgorithm

Choice of ICA backend.

threshold

Cutoff for treating ICA coefficients as edges.

detectCycles

Whether to run SCC decomposition explicitly.

verbose

Show details of ICA and SCC steps.

29.7. Reference

Lacerda, G., Spirtes, P., Ramsey, J., & Hoyer, P. (2008).
Discovering cyclic causal models by independent component analysis.
Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence (UAI).

29.8. Summary

IcaLingD generalizes LiNGAM to feedback systems.
It applies ICA to linear, non-Gaussian data, identifies directed influences and strongly connected components, and outputs a cyclic causal graph that can reveal feedback loops.