The discovery of causal relationships from the observational data is an important task. To identify the unique causal structure belonging to a Markov equivalence class, a number of algorithms, such as the linear non-Gaussian acyclic model (LiNGAM), have been proposed. However, two challenges remain to be met: 1) these algorithms fail to work on the data which follow linear structural equation model with Gaussian noise and 2) they misjudge the causal direction when the data contain additional measurement errors. In this paper, we propose an entropy-based two-phase iterative algorithm for arbitrary distribution data with additional measurement errors under some mild assumptions. In the first phase of the algorithm, based on the property that entropy can measure the amount of information behind the data with arbitrary distribution, we design a general approach for the identification of exogenous variable on both Gaussian and non-Gaussian data, and we give the corresponding theoretical derivation. In the second phase, to eliminate the effects of measurement errors, we revise the value of the exogenous variable by removing its measurement error and further use the revised value to remove its effect on the remaining variables. Experimental results on real-world causal structures are presented to demonstrate the effectiveness and stability of our method. We also apply the proposed algorithm on the mobile-base-station data with measurement errors, and the results further prove the effectiveness of our algorithm.

Nonlinear causal discovery with high-dimensional data where each variable is multidimensional plays a significant role in many scientific disciplines, such as social network analysis. Previous work majorly focuses on exploiting asymmetry in the causal and anticausal directions between two high-dimensional variables (a cause-effect pair). Although there exist some works that concentrate on the causal order identification between multiple variables, i.e., more than two high-dimensional variables, they do not validate the consistency of methods through theoretical analysis on multiple-variable data. In particular, based on the asymmetry for the cause-effect pair, if model assumptions for any pair of the data are violated, the asymmetry condition will not hold, resulting in the deduction of incorrect order identification. Thus, in this article, we propose a causal functional model, namely high-dimensional deterministic model (HDDM), to identify the causal orderings among multiple high-dimensional variables. We derive two candidates' selection rules to alleviate the inconvenient effects resulted from the violated-assumption pairs. The corresponding theoretical justification is provided as well. With these theoretical results, we develop a method to infer causal orderings for nonlinear multiple-variable data. Simulations on synthetic data and real-world data are conducted to verify the efficacy of our proposed method. Since we focus on deterministic relations in our method, we also verify the robustness of the noises in simulations.

This paper investigates the problem of selecting instrumental variables relative to a target causal influence $X\to Y$ from observational data generated by linear non-Gaussian acyclic causal models in the presence of unmeasured confounders. We propose a necessary condition for detecting variables that cannot serve as instrumental variables. Unlike many existing conditions for continuous variables, i.e., that at least two or more valid instrumental variables are present in the system, our condition is designed with a single instrumental variable. We then characterize the graphical implications of our condition in linear non-Gaussian acyclic causal models. Given that the existing graphical criteria for the instrument validity are not directly testable given observational data, we further show whether and how such graphical criteria can be checked by exploiting our condition. Finally, we develop a method to select the set of candidate instrumental variables given observational data. Experimental results on both synthetic and real-world data show the effectiveness of the proposed method.