28. ICA Lingam — ICA-Based LiNGAM

Type: Non-Gaussian / Moment-based / ICA-based
Output: DAG
Assumptions: Linear SEM with independent, non-Gaussian errors

ICA-LiNGAM is the original LiNGAM algorithm. It uses Independent Component Analysis (ICA) to estimate the mixing matrix of a linear, non-Gaussian SEM. After ICA identifies independent components, the algorithm determines a causal ordering and constructs a fully directed DAG.

28.1. Key Idea

ICA-LiNGAM assumes:

  • The system is linear.

  • The error terms are independent and non-Gaussian.

  • The causal structure is acyclic.

Under these assumptions, the structural equation model
X = B X + e
can be rewritten as
X = A e,
where A is the mixing matrix.

Independent Component Analysis can estimate A up to:

  • permutation of rows, and

  • scaling of rows.

ICA-LiNGAM:

  1. Runs ICA to estimate a mixing matrix.

  2. Resolves the permutation that matches each independent component to the correct observed variable.

  3. Rescales so diagonal entries take a standard form.

  4. Converts the matrix to a causal ordering.

  5. Uses that ordering to regress each variable on its predecessors, giving the DAG.

28.2. When to Use

ICA-LiNGAM is appropriate when:

  • The system is linear.

  • The error terms are non-Gaussian.

  • You want a fully oriented DAG.

  • You are comfortable with ICA’s numerical properties.

  • You want a historically established baseline method.

It is often used for:

  • Biological and neuroimaging data

  • Econometric or social-science modeling

  • Benchmarking newer non-Gaussian methods

28.3. Prior Knowledge Support

The Tetrad implementation supports:

  • Required edges

  • Forbidden edges

  • Tier-based temporal ordering

These constraints limit which orderings ICA-LiNGAM is allowed to accept after the permutation step.

28.4. Strengths

  • Produces a fully directed DAG.

  • The classic and most historically cited version of LiNGAM.

  • Useful as a baseline for later non-Gaussian methods.

  • Works well when ICA converges cleanly.

28.5. Limitations

  • Sensitive to ICA numerical instability.

  • ICA results are only identifiable up to permutation and scaling, so extra steps are required to recover a correct ordering.

  • Assumes no latent confounders.

  • Performance degrades if noise is close to Gaussian.

  • Slower and less stable than DirectLiNGAM.

28.6. Key Parameters in Tetrad

Parameter (camelCase)

Meaning

numRestarts

ICA restarts used to stabilize the solution.

maxIterations

Maximum ICA iterations.

icaAlgorithm

Choice of ICA backend (FastICA, extended-infomax, etc.).

useBootstrap

Whether to compute bootstrap DAGs.

knowledge

Required/forbidden edges and tiers.

verbose

Detailed diagnostic output.

Parameters differ depending on which ICA backend is installed.

28.7. Reference

Shimizu, S., Hoyer, P. O., Hyvärinen, A., & Kerminen, A. (2006).
A Linear Non-Gaussian Acyclic Model for causal discovery.
Journal of Machine Learning Research, 7, 2003–2030.

This is the original LiNGAM paper and the definitive source for ICA-based causal discovery.

28.8. Summary

ICA-LiNGAM is the original non-Gaussian causal discovery method.
It uses Independent Component Analysis to recover the mixing matrix of a linear SEM, resolves permutation and scaling ambiguities, produces a causal order, and constructs a fully oriented DAG.