6.6 Correlation Coefficient Matrix

In the video we will demonstrate the use of the correlation coefficient matrix and the concept of correlation coefficient. A brief overview of the theory is given below.

Play media comment.

Recall that the covariance between two random variables is defined as: 

$Cov(X,Y) = E[(X - E(X))(Y - E(Y))]$

and that the variance of a random variable is defined as: 

$Var(X) = E[ (X - E(X))^2]$.

Further, the standard deviation is the square root of the variance: $D(X) = \sqrt{Var(X)}$. 

It follows immediately from the definitions that: 

$Var(X) = Cov(X,X)$,

meaning that the variance of $X$ is the covariance of $X$ with itself. 

The correlation coefficient, $\rho_{X,Y}$ , between two random variables, $X$ and $Y$, is defined as: 

$\rho_{X,Y} = \frac{Cov(X,Y)}{D(X)D(Y)}$.

Obviously $\rho_{X,Y} = \rho_{Y,X}$ , due to symmetry, and from the definitions above it also follows that: 

$\rho_{X,X} = \frac{Cov(X,X)}{D(X)D(X)} = \frac{Var(X)}{Var(X)} = 1$,

In other words, the correlation coefficient between a random variable with itself is 1. 

In the video we also use the correlation coefficient matrix. For example, if we have three random variables, $X, Y$and $Z$ then the correlation coefficient matrix, denoted by $Corr.matrix(X,Y,Z)$is defined as: 

$Corr.matrix(X,Y,Z) =$ $\begin{bmatrix} \rho_{X,X} & \color{blue} \rho_{X,Y} & \color{green} \rho_{X,Z}\\ \color{blue} \rho_{Y,X} & \rho_{Y,Y} & \color{red} \rho_{Y,Z} \\ \color{green} \rho_{Z,X} & \color{red} \rho_{Z,Y} & \ \rho_{Z,Z} \end{bmatrix} = \begin{bmatrix} 1 & \color{blue} \rho_{X,Y} & \color{green} \rho_{X,Z}\\ \color{blue} \rho_{Y,X} & 1& \color{red} \rho_{Y,Z} \\ \color{green} \rho_{Z,X} & \color{red} \rho_{Z,Y} & \ 1 \end{bmatrix}$

Notice that the diagonal elements are all 1, and that the matrix is symmetric since the correlation coefficients are symmetric.