CORRELATION OF K GROUPS OF RANDOM VARIABLES

This paper is a continuation of 4.It whs shown from various points of view that the measure defined in 4 is reasonadle.The method usad in 3 to obtain canonical variables was simplified in this paper.Several applications were also given.Let x_i be p_i×1 random vector, i=1, …, k and \Sigma=\left(\beginarraycccc\Sigma_11 & \Sigma_12 & \cdots & \Sigma_11 k \\ \Sigma_21 & \Sigma_22 & \cdots & \Sigma_2 k \\ \vdots & \vdots & & \\ \Sigma_k 11 & \Sigma_k 2 & \cdots & \Sigma_k k\endarray\right) where ∑_ij=E（x_i-Ex_i）(x_j-E_xj), i, j=1, …, k.The correlation coefficient of k groups of random variables x₁, …, x_k was defined as follows R=1-\frac|\Sigma|\prod_i=1^k\left|\Sigma_i\right| R defined above has many good properties.1.0 \leqslant R^2 \leqslant 1 2.If k=2 and p_1=p_2 then R is just the commonly defined correlation coefficient of x_1 and x_2.3.Let y_i=A_i X_i, where \left|A_i\right| \neq 0, i=1, \cdots, k.Then the correlation coefficient of y_1, \cdots, y_k is the same as that of x_1, \cdots, x_k.4.For any permutation i_1, \cdots, i_k of 1, \cdots, k the correlation ooefficient of x_h, \cdots, x_l_k is the same as that of x_1, \cdots, x_k.5.If k=2, p_1=1 and p_2 \geqslant 1, R is,just the multivariate correlation coeffioient \rho_1,2-k 6.If x_1, \cdots, x_\mathrmk are pairwise uncorrelated then R=0.The reverse is also true.Further more we proved that if R_* is any correlation coefficient defined with the following properties (i) R_*\left(x_1, x_2\right)=\rho_\text an , \boldsymbolc i s i_i (ii) R_*\left(x_1, \cdots, x_n\right)=R_*\left(x_h_1, \cdots, x_i_n\right) for any permutation \doti_1, \cdots, \dot\theta_n of 1, \cdots, k_\text ; (iii) If R_*\left(x_1, \cdots, x_j\right)=0 then R_*\left(x_1, \cdots, x_j, x_j+1\right)=\rho_\varepsilon_-1,\left(x_1, \cdots, x_j\right); (iv) If \rho_\varepsilon_j+2,\left(a, \cdots, \varepsilon_j\right)=0 then R_*\left(x_1, \cdots, x_j, x_j+1\right)=R_*\left(x_1, \cdots, x_j\right) (v) There is a function F(\cdot, \cdot) such that R_*\left(x_1, \cdots, x_k+1\right)=F\left(R_*\left(x_1, \cdots, x_k\right), \rho_x_k+1,\left(\boldsymbolc_1, \cdots, a_k\right)\right) where \rho_x_i+1,,\left(x_1, \cdots, x_i\right) is the generalized correlation coefficient of x_i+1 and ( \left.x_1^\prime, \cdots, x_i^\prime\right) 5, then R_* is equivalent to R.