How to Calculate Correlation Matrix from Covariance Matrix

Definition:

Correlation matrix is a type of matrix, which provides the correlation between whole pairs of data sets in a matrix.

Example :

Find out the correlation matrix from the given 3 X 3 matrix?

Matrix,

1	4	7
6	5	4
9	5	1

Given:

n = 3 x̄ = (1 + 4 + 7) / 3 = 4 ȳ = (6 + 5 + 4) / 3 = 5 z̄ = (9 + 5 + 1) / 3 = 5

Solution :

Step : 1

First, let us calculate the matrix value for Sum of Squared Matrix.

Sum of Squared Matrix

= 1/ (n-1)	SSxx	SSxy	SSxz
	SSyx	SSyy	SSyz
	SSzx	SSzy	SSzz

= 1/ (3-1)	(1-4)2 + (4-4)2 + (7-4)2	(1-4)x(6-5) + (4-4)x(5-5) + (7-4)x(4-5)	(1-4)x(9-5) + (4-4)x(5-5) + (7-4)x(1-5)
	(6-5)x(1-4) + (5-5)x(4-4) + (4-5)x(7-4)	(6-5)2 + (5-5)2 + (4-5)2	(6-5)x(9-5) + (5-5)x(5-5) + (4-5)x(1-5)
	(9-5)x(1-4) + (5-5)x(4-4) + (1-5)x(7-4)	(9-5)x(6-5) + (5-5)x(5-5) + (1-5)x(4-5)	(9-5)2 + (5-5)2 + (1-5)2

= 1/2	9 + 0 + 9	-3 + 0 + -3	-12 + 0 + -12
	-3 + 0 + -3	1 + 0 + 1	4 + 0 + 4
	-12 + 0 + -12	4 + 0 + 4	16 + 0 + 16

= 1/2	18	-6	-24
	-6	2	8
	-24	8	32

=	9	-3	-12
	-3	1	4
	-12	4	16

Step : 2

Calculate the matrix value of Correlation Matrix.

=	1	Pxy	Pxz
	Pyx	1	Pyz
	Pzx	Pzy	1

=	1	-3 /√(9x1)	-12 /√(9x16)
	-3 /√(1x9)	1	4 /√(1x16)
	-12 /√(16x9)	4 /√(16x1)	1

=	1	-3/3	-12/12
	-3/3	1	4/4
	-12/12	4/4	1

=	1	-1	-1
	-1	1	1
	-1	1	1