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
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.
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
p
2
Search WWH ::
Custom Search