Statistical Correlation
المؤلف:
المرجع الالكتروني للمعلوماتيه
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الجزء والصفحة:
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30-3-2021
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Statistical Correlation
For two random variates 
and 
, the correlation is defined bY
 |
(1)
|
where 
denotes standard deviation and 
is the covariance of these two variables. For the general case of variables 
and 
, where 
, 2, ..., 
,
 |
(2)
|
where 
are elements of the covariance matrix. In general, a correlation gives the strength of the relationship between variables. For 
,
 |
(3)
|
The variance of any quantity is always nonnegative by definition, so
 |
(4)
|
From a property of variances, the sum can be expanded
 |
(5)
|
 |
(6)
|
 |
(7)
|
Therefore,
 |
(8)
|
Similarly,
 |
(9)
|
 |
(10)
|
 |
(11)
|
 |
(12)
|
Therefore,
 |
(13)
|
so 
.
For a linear combination of two variables,
Examine the cases where 
,
 |
(18)
|
 |
(19)
|
The variance will be zero if 
, which requires that the argument of the variance is a constant. Therefore, 
, so 
. If 
, 
is either perfectly correlated (
) or perfectly anticorrelated (
) with 
.
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