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Correlation Matrices

Introduction to correlation matrices:

The operation on the correlation matrices represents the matrices which have the property of having the dimension elements in 1 or unity. Generally correlation is used in the process of the data analysis. Correlation matrix irregularity arises when a matrix that is mathematically not possible to understand in entering.  The mathematical condition is that a matrix should be positive semi-definite.  i.e All its eigenvalues should be `<= ` to zero.In this article we are going to discuss about the correlation matrices in detail for the students.

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Examples for Correlation Matrices

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Take R as an correlation matrix.

It is presented with equal number of columns and the rows.

         `R_(ij) = [[1, r_(12) ,r_(13), ..r_(1m)], [r_(21), 1 ,r_(23), ..r_(2m)], [r_(31), r_(32) ,1,..r_(3m)],[.,.,.,.],[.,.,.,.],[.,.,.,.], [r_(m1), r_(m2) ,r_(m3),..1]]`

rij = sample correlation between the ith and jth variables.

The formula is given as

Sij = `(sum_(i =1)^n (x_(ij) - bar(x_j) * (x_(ij) - bar(x_k))))/(n-1)`

`r_(ij) = (S_(ij))/((S_j) * (S_k))`

Problems for Correlation Matrices

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  • Review on finding the correlation of the given matrix the five data operators are Q, W, E, R and T.

The Q values are the 0.089, -0.02, 0.08, 3.33, 0.12, -0.02, 0.05, -0.01, 0, -0.13, 0.02

The W Values are the 0.13, 0.2, 0.05, 0.02, 0.03, -0.02, 0.25, 0.31, -0.01, 0.14, 0.07.

The E values are the 0.01, 0.01, 0, 0.07, 0.3, -0.06, 0.09, 0.03, 0.07, -0.07

The R Values are the 0.04, 0, -0.05, 0.07, 0.1, -0.08, 0.05, 0, -0.01, 0.02, 0.06.

The T values are the 0.02, 0.07, -0.1, 0.07, 0.09, -0.03, 0.05, 0.06, -0.1, 0.02, 0

Solution:

Now we finding the correlation matrix for the given data’s

Step 1:

We have to find the term `S_(ij)`  value by the formula                                

Sij = `sum_(i=1)^n (x_(ij) - bar(x_j) * (x_(ij) - bar(x_k)))/(n-1)`

`S_(ij)` = `[[0.00422,0,0,0,0],[-0.00125,0.01090,0,0,0],[0.00229,0.00034,0.00901,0,0],[0.00116,0.00064,0.00306,0.00258,0],[0.0023,0.00320,0.00329,0.00212,0.00397]]`

Step 2:

We have to find the term `R_(ij)` value using  `S_(ij)`  value by the formula                                

`S_i = [[1,0,0,0,0],[-147.36,1,0,0,0],[162.45,100.88,1,0,0],[302.41,188.28,207.222,1,0],[24.304,152.28,311.981,313.4597,1]]`

 `S_j` =` [[1,0,0,0,0],[137.16,1,0,0,0],[262.45,100.88,1,0,0],[102.41,158.28,227.222,1,0],[24.314,152.28,331.981,303.4597,1]]`

`r_(ij) = (S_(ij))/((S_i) * (S_j))`

`R_(ij) = [[1,0,0,0,0],[-0.1842,1,0,0,0],[0.3720,0.0343,1,0,0],[0.3508,0.1205,0.6341,1,0],[0.0559,0.4873,0.5498,0.6614,1]]`


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