How to Calculate Correlation Matrix from Covariance Matrix

How to Calculate Correlation Matrix - Definition, Formula, Example

Definition:

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

Formula:

1) Sum of Squared Matrix
1/ (n-1) SSxx SSxy SSxz
SSyx SSyy SSyz
SSzx SSzy SSzz
Where,

n = N x N Matrix Value SSxx = ∑(xi - x̄)2 SSxy = ∑(xi - x̄) X (yi - ȳ) |||ly, SSyz = ∑(yi - ȳ) X (zi - z̄)

2) Correlation Matrix
1 Pxy Pxz
Pyx 1 Pyz
Pzx Pzy 1
Where,

n = N x N Matrix Value Pxy = SSxy / √(SSxx X SSyy)

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

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