In this letter, we study the confounder detection problem in the linear model, where the target variable Y is predicted using its n potential causes X n = ( x 1 , … , x n ) T . Based on an assumption of a rotation-invariant generating process of the model, recent study shows that the spectral measure induced by the regression coefficient vector with respect to the covariance matrix of X n is close to a uniform measure in purely causal cases, but it differs from a uniform measure characteristically in the presence of a scalar confounder. Analyzing spectral measure patterns could help to detect confounding. In this letter, we propose to use the first moment of the spectral measure for confounder detection. We calculate the first moment of the regression vector–induced spectral measure and compare it with the first moment of a uniform spectral measure, both defined with respect to the covariance matrix of X n . The two moments coincide in nonconfounding cases and differ from each other in the presence of confounding. This statistical causal-confounding asymmetry can be used for confounder detection. Without the need to analyze the spectral measure pattern, our method avoids the difficulty of metric choice and multiple parameter optimization. Experiments on synthetic and real data show the performance of this method.

In this article, we deal with the problem of inferring causal directions when the data are on discrete domain. By considering the distribution of the cause and the conditional distribution mapping cause to effect as independent random variables, we propose to infer the causal direction by comparing the distance correlation between and with the distance correlation between and . We infer that X causes Y if the dependence coefficient between and is smaller. Experiments are performed to show the performance of the proposed method.