Calculating Covariance and Properties of Covariance
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This is a 4x3 matrix, 4 rows, and 3 columns, and we want to calculate the covariance.
Step-1: Firstly, we have to calculate the mean of each column, i.e, X1, X2, and X3.
The formula for calculating mean:
Now we have to calculate the deviate scores. This means we have to subtract the respective column mean from each element of the column.
Now take the transpose of the deviate scores. Taking transpose means turning the columns of a matrix into rows.
where D' is called D prime. This inverted comma on D ' is called prime.
We have to premultiply the matrix D by D', that is by its transpose. How to multiply matrices see here.
Note that D'D≠DD' because matrix multiplication is not commutative or interchangeable.
Now divide D'D by n-1 (degrees of freedom). since n=4 because we have 4 rows, so, n-1= 4-1=3
( we divide by n-1 only when know that we have a sample, not the population. In case of population, we divide by N only.
Properties of a Covariance Matrix:
Covariance matrix is symmetric. This means the lower left half is the mirror image of the upper right half.
On the diagonals of the covariance matrix, we have variances of the variables and; the off-diagonal elements show how the two variable covary with each other i.e., average deviation cross-product sum.
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