  In this paper, 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 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.
Then, analyzing spectral measure pattern could help to detect confounding. In
this paper, 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 non-confounding 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 of analyzing the
spectral measure pattern, our method does avoid the difficulty of metric choice
and multiple parameter optimization. Experiments on synthetic and real data
show the performance of this method.
