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.
Recall that the covariance between two random variables is defined as:
and that the variance of a random variable is defined as:
.
Further, the standard deviation is the square root of the variance: .
It follows immediately from the definitions that:
,
meaning that the variance of is the covariance of
with itself.
The correlation coefficient, , between two random variables,
and
, is defined as:
.
Obviously , due to symmetry, and from the definitions above it also follows that:
,
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, and
then the correlation coefficient matrix, denoted by
is defined as:
Notice that the diagonal elements are all 1, and that the matrix is symmetric since the correlation coefficients are symmetric.