相关矩阵半正定性推出相关系数范围

PSD Constraint for Correlation

专题: Algorithmic Programming / 算法编程
难度: L4
来源: QuantQuestion

题目详情

随机变量 $x,y,z$ 满足 Corr $(x,y)=0.8$ ，Corr $(x,z)=0.8$ 。问 Corr $(y,z)$ 的取值范围。

A real symmetric matrix $A$ is PSD if $x^T A x\ge0$ for all $x$ , or equivalently all eigenvalues nonnegative. If strictly positive, it is PD.

Question: Three random variables $x,y,z$ , with Corr( $x,y$ )=0.8, Corr( $x,z$ )=0.8. The correlation between $y$ and $z$ is ?

解析

构造相关矩阵

P=\begin{bmatrix} 1&0.8&0.8\\ 0.8&1&\rho\\ 0.8&\rho&1 \end{bmatrix}.

必要条件为 $P\succeq 0$ ，等价于 $\det(P)\ge 0$ （并结合对角为 1 的约束），解得

0.28\le \rho\le 1.

Original Explanation

Build the correlation matrix $P = \begin{bmatrix} 1 & 0.8 & 0.8\\ 0.8 & 1 & \rho\\ 0.8 & \rho & 1 \end{bmatrix},$ then $\det(P)\ge0.$ Expanding leads to $0.28\le\rho\le1.$