Abstract We study a model where one target variable Y$Y$ is correlated with a vector X:=(X1,…,Xd)$\textbf{X}:=(X_1,\dots,X_d)$ of predictor variables being potential causes of Y$Y$. We describe a method that infers to what extent the statistical dependences between X$\textbf{X}$ and Y$Y$ are due to the influence of X$\textbf{X}$ on Y$Y$ and to what extent due to a hidden common cause (confounder) of X$\textbf{X}$ and Y$Y$. The method relies on concentration of measure results for large dimensions d$d$ and an independence assumption stating that, in the absence of confounding, the vector of regression coefficients describing the influence of each X$\textbf{X}$ on Y$Y$ typically has ‘generic orientation’ relative to the eigenspaces of the covariance matrix of X$\textbf{X}$. For the special case of a scalar confounder we show that confounding typically spoils this generic orientation in a characteristic way that can be used to quantitatively estimate the amount of confounding (subject to our idealized model assumptions).

中文翻译：

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通过谱分析检测多元线性模型中的混杂

摘要 我们研究了一个模型，其中一个目标变量 Y$Y$ 与向量 X:=(X1,…,Xd)$\textbf{X}:=(X_1,\dots,X_d)$ 的预测变量是潜在的Y$Y$ 的原因。我们描述了一种方法来推断 X$\textbf{X}$ 和 Y$Y$ 之间的统计依赖在多大程度上是由于 X$\textbf{X}$ 对 Y$Y$ 的影响以及在多大程度上是由于X$\textbf{X}$ 和 Y$Y$ 隐藏的共同原因（混杂因素）。该方法依赖于大维度 d$d$ 的测量结果的集中和独立假设，该假设说明在没有混杂的情况下，描述每个 X$\textbf{X}$ 对 Y$Y 的影响的回归系数向量$ 通常具有相对于 X$\textbf{X}$ 的协方差矩阵的特征空间的“通用方向”。