Environmental Engineering Reference
In-Depth Information
a multivariate normal vector (X 1 , X 2 , …, X d ) and its mean vector follows the χ-square dis-
tribution with m DOF. The Mahalanobis distance is defined as
T
1
X
X
µ
µ
σδσσ
2
δδσσ
X
X
µ
µ
1
1
1
12
12
1
d
1
d
1
1
σ
2
δ σσ
(
)
T
(
) =
2
2
2
22
d
d
2
2
XCX
1
(1.64)
Q
=−
µ
µ
d
X
µ
σ
2
X
µ
d
d
d
d
d
where μ i and σ i are the mean and standard deviation of Xi, i , and the matrix to be inverted is
the covariance matrix. In reality, the mean μ i is replaced by sample mean m i , and the covari-
ance matrix is replaced by the sample covariance matrix:
T
()
k
()
k
X
m
X
m
1
1
1
1
n
1
ˆ
C =
×
(1.65)
n
1
k
=
1
()
k
()
k
X
m
X
m
d
d
d
d
For the dataset in Table 1.9 , ( m 1 , m 2 , m 3 ) = (0.092, 0.29, 0.072), and the sample covari-
ance matrix is
142088 099
108047
102
.
.
.
ˆ
C =
.
.
(1.66)
sym
.
.
Therefore, the Mahalanobis distance for the k th sample in Table 1.9 can be determined as
T
1
()
k
()
k
X
X
X
0 092
029
0 072
.
142088 0099
108047
102
.
.
.
X
X
X
0 092
029
.
1
1
()
k
()
k
()
k
(1.67)
Q
=
.
.
.
.
2
2
d
()
k
(
k )
.
sym
.
.
0 072
.
3
3
Table 1.11 shows the calculated Mahalanobis distances for the dataset in Table 1.9.
In contrast to Equation 1.46 , the CDF for Q d does not have an elegant analytical form.
However, the following equation still holds:
()
FF
χ
q
=
q
(1.68)
2
d
d
2
χ
d
d
where F χ 2 is the χ-square CDF with d DOFs. If we replace the F χ 2 inside the square bracket
by the ECDF, we get
()
FF
χ d
q
q
(1.69)
nd
d
2
() can be
evaluated using MATLAB function chi2inv( p , d ). If Q d is indeed d -DOF χ-square, it is clear
The ECDF can be estimated using Equations 1.11 or 1.12 , whereas the F χ d
p
2
 
 
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