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Statistics I Covariance and correlation coe cient
1.5 Covariance and Correlation Coe cient
- The covariance between two variables X and Y , denoted with S XY
, is a measure
of the degree of relationship (join variation) between the variables.
- A positive covariance is found when high values of one of the variables occur when
the other variable also takes high values
- A negative covariance corresponds to the case when high values of one of the
variables occur when the other variable takes low values
Statistics I Covariance and correlation coe cient
The covariance between variables X and Y can be computed as
S
XY
N
n ∑
i=
m ∑
j=
n ij
(x i
− x¯)(y j
− y¯)
and also with the formula
S
XY
N
n ∑
i=
m ∑
j=
n ij
x i
y j
− x¯y¯
One can show that the maximum and the minimum value that the covariance can
take are relate to the variances of the two variables. In this sense we have that
Statistics I Covariance and correlation coe cient
- To address this issue we use the so-called Correlation Coe cient r (or Pearson's
coe cient):
r =
S
XY
S
X
S
Y
Notice that, because of what we have found in [1], we have
− 1 ≤ r ≤ 1
A direct way to compute r is using the formulae for S XY
, S
X
and S Y
, that is,
r =
n
i=
m
j=
n ij
(x i
− x¯)(y i
− y¯)
n
i=
n i•
(x i
− x¯)
2 )(
m
j=
n
(y j
− y¯)
2 )
Statistics I Covariance and correlation coe cient
Notice:
- The sign (+ or -) of the correlation coe cient r, which is inherited from the
sign of the covariance S XY
, determines the nature (positive or negative) of the
relationship between X and Y
- The closer to 1 (or to -1) the correlation coe cient r is, the more intense the
relationship between X and Y is
- A value of r close to zero does not mean that the relationship is week (or null)
Statistics I Linear combinations of variables
- In general, we say that the variable X is a linear combination lineal of the variables
X
1
, X
2
if there exist linear coe cients a 1
, a 2
(real numbers) such that
X = a 1
X
1
X
2
- The following properties regarding the mean and variance of linear combinations
of variables are of interest:
X = a 1
X
2
X
2
S
2
X
= a
2
1
S
2
X 1
2
2
S
2
X 2
if X 1
, X
2
are independent
Statistics I Linear combinations of variables
Moreover, if
Y = b 1
Y
1
Y
2
the we have the following property for the covariance
• S
XY
= a 1
b 1
S
X 1
Y 1
b 2
S
X 1
Y 2
b 1
S
X 2
Y 1
b 2
S
X 2
Y 2
Statistics I Mean vector and Covariance matrix
Data matrix
The observations of the variables X 1
i X 2
can be jointly represented in a matrix X
with N × 2 dimensions
X =
x 11
x 21
x 12
x 22
x 1 N
x 2 N
Statistics I Mean vector and Covariance matrix
Mean vector
The mean vector
X simply consists of
X = (¯x 1
, x¯ 2
Such mean vector
X can be computed using matrix algebra using the data matrix
X. Indeed, if we denote with
1 the (row) vector with N components, all of then
equal to 1:
N
Statistics I Mean vector and Covariance matrix
Covariance matrix
Finally, the variances and covariances between X 1
and X 2
are presented in the
covariance matrix Σ
S
2
X 1
S
X 1 X 2
S
X 2
X 1
S
2
X 2
As with the mean vector, this matrix Σ can be computed using matrix algebra.
Indeed, let
T
denote the transpose of vector
1 (column vector)
T
Statistics I Mean vector and Covariance matrix
Then,
T
X =
(¯x 1
, x¯ 2
x ¯ 1
x¯ 2
x ¯ 1
x¯ 2
x ¯ 1
x¯ 2
Therefore,
X −
T
X =
x 11
x 21
x 12
x 22
x 1 N
x 2 N
x ¯ 1
x¯ 2
x ¯ 1
x¯ 2
x ¯ 1
x¯ 2
x 11 −
x¯ 1
x 21
− x¯ 2
x 12
− x¯ 1
x 22
− x¯ 2
x 1 N
− x¯ 1
x 2 N
− x¯ 2
Statistics I Mean vector and Covariance matrix
N
1
(x 1 i
− x¯ 1
2
N
i=
(x 1 i
− x¯ 1
)(x 2 i
− x¯ 2
N
i
(x 2 i
− x¯ 2
)(x 1 i
− x¯ 1
N
i=
(x 2 i
− x¯ 2
2
If we now multiply by
1
N
we nd
N
(X −
T
X)
T
(X −
T
X) =
N
1
(x 1 i
−x¯ 1
)
2
N
∑ N
i=
(x 1 i
−x¯ 1
)(x 2 i
−x¯ 2
)
N ∑ N
i
(x 2 i
−x¯ 2
)(x 1 i
−x¯ 1
)
N
∑ N
i=
(x 2 i
−x¯ 2
)
2
N
S
2
X 1
S
X 1 X 2
S
X 2
X 1
S
2
X 2
That is,
N
(X −
T
X)
T
(X −
T
¯
X)
Statistics I Mean vector and Covariance matrix
This matrix is symmetric since S X !
X 2
= S
X 2
X 1
