Abstract
In causal discovery, non-Gaussianity has been used to characterize the complete configuration of a linear non-Gaussian acyclic model (LiNGAM), encompassing both the causal ordering of variables and their respective connection strengths. However, LiNGAM can only deal with the finite-dimensional case. To expand this concept, we extend the notion of variables to encompass vectors and even functions, leading to the functional linear non-Gaussian acyclic model (Func-LiNGAM). Our motivation stems from the desire to identify causal relationships in brain-effective connectivity tasks involving, for example, fMRI and EEG datasets. We demonstrate why the original LiNGAM fails to handle these inherently infinite-dimensional datasets and explain the availability of functional data analysis from both empirical and theoretical perspectives. We establish theoretical guarantees of the identifiability of the causal relationship among non-Gaussian random vectors and even random functions in infinite-dimensional Hilbert spaces. To address the issue of sparsity in discrete time points within intrinsic infinite-dimensional functional data, we propose optimizing the coordinates of the vectors using functional principal component analysis. Experimental results on synthetic data verify the ability of the proposed framework to identify causal relationships among multivariate functions using the observed samples. For real data, we focus on analyzing the brain connectivity patterns derived from fMRI data.
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Notes
\(X_1\perp \!\!\!\perp X_2\) denotes the independence of X and Y.
Suppose Z and W be binary taking \(\pm 1\) equiprobably and zero-mean Gaussian. Then, ZW and Z are not jointly Gaussian. Even though \({{\mathbb {E}}}[ZW\cdot Z]={{\mathbb {E}}}[W]\cdot {{\mathbb {E}}}[Z^2]=0\) but they are not independent.
See the definition of the Bochner integral in Hsing and Eubank (2015).
We define a bounded linear operator \(T: {\mathscr {H}}_1\rightarrow {\mathscr {H}}_2\) to be compact if, for any bounded infinite sequence \(\{f_n\}\) in \({\mathscr {H}}_1\), the sequence \(\{Tf_n\}\) has a convergent subsequence in \({\mathscr {H}}_2\).
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Funding
TLY was supported by JST, the establishment of university fellowships toward the creation of science technology innovation, under Grant No. JPMJFS2125. KYL was partially supported by the NSF under Grant No. CIF-2102243. JS was supported by the Grants-in-Aid for Scientific Research (C) under Grant No. 22K11931.
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TLY: methodology, theory, coding, and writing; KYL: theory validation; KZ: formal analysis and supervision; JS: methodology, theory validation, and supervision. All authors discussed the theoretical and experimental results and contributed to the final manuscript.
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Communicated by SHIMIZU Shohei.
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Yang, TL., Lee, KY., Zhang, K. et al. Functional linear non-Gaussian acyclic model for causal discovery. Behaviormetrika (2024). https://doi.org/10.1007/s41237-024-00226-5
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DOI: https://doi.org/10.1007/s41237-024-00226-5