A multivariate additive noise model for complete causal discovery

Explaining causal reasoning in the form of directed acyclic graphs (DAGs) yields nodal structures with multivariate relationships. In real-world phenomena, these effects can be seen as multiple feature dependency with unmeasured external influences or noises. The bivariate models for causal discovery simply miss to find the multiple feature dependency criteria in the causal models. Here, we propose a multivariate additive noise model (MANM) to solve these issues while analyzing and presenting a multi-nodal causal structure. We introduce new criteria of causal independence for qualitative analysis of causal models and causal influence factor (CIF) for the successful discovery of causal directions in the multivariate system. The scores of CIF provide the information for the goodness of casual inference. The identifiability of the proposed model to discover linear, non-linear causal relations is verified in simulated, real-world datasets and the ability to construct the complete causal model. In comparison test, MANM has out performed Independent Component Analysis based Linear Non-Gaussian Acyclic Model (ICA-LiNGAM), Greedy DAG Search (GDS) and Regression with Sub-sequent Independent Test (RESIT), and performed better for Gaussian and non-Gaussian mixture models with both correlated and uncorrelated feature relations. In performance test, different model fitting errors which occur during causal model construction are discussed and the performance of MANM in comparison to ICA-LiNGAM, GDS and RESIT is provided. Results show that MANM has better causal model construction ability, producing few extra sets of direction with no missing or wrong directions and can estimate every possible causal direction in complex feature sets.