Civil Engineering Reference
In-Depth Information
T
1
()
()
()
()
Cov
Extxt
⎡
⎤
lim
∫
xtxtdt
=
⋅
=
⋅
(2.11)
⎣
⎦
xx
1
2
1
2
12
T
T
→∞
0
Similarly, given two data sets of
N
individual and equally probable realisations that have
been extracted from two random variables,
X
and
X
, then the ensemble correlation
2
and covariance are defined by:
N
1
[
]
∑
R
EX X
lim
X
X
=
⋅
=
⋅
(2.12)
xx
12
1
2
12
N
k
k
N
→∞
k
1
=
(
) (
)
Cov
E
X
x
X
x
=
⎡
−
⋅
−
⎤
⎣
⎦
xx
11
2 2
12
N
1
(
) (
)
∑
(2.13)
lim
X
xX x
=
−
⋅
−
1
1
2
2
N
k
k
N
→∞
k
=
1
However, correlation and covariance estimates may also be taken on the process variable
itself. Thus, defining an arbitrary time lag
τ
, the time domain auto correlation and auto
covariance functions are defined by
T
1
()
()
(
)
()
(
)
R
EXtXt
lim
∫
XtXt
dt
τ
=
⎡
⋅
+
τ
⎤
=
⋅
+
τ
(2.14)
⎣
⎦
x
T
T
→∞
0
T
1
()
() (
)
() (
)
Cov
E xt xt
lim
∫
xt xt
dt
τ
=
⎡
⋅
+
τ
⎤
=
⋅
+
τ
(2.15)
⎣
⎦
x
T
T
→∞
0
These are defined as functions because
τ
is perceived as a continuous variable. As long
as
τ
is considerably smaller than
T
()
(
)
E
Xt
EXt
x
(2.16)
⎡ ⎤ ⎡
=
+
τ
⎤
=
⎣ ⎦ ⎣
⎦
R
and
Cov
is the following
and thus, the relationship between
x
{
}
{
}
()
()
(
)
()
2
Cov
E
⎡
X t
x
X t
x
⎤
R
x
τ
=
−
⋅
+
τ
−
=
τ
−
(2.17)
⎣
⎦
x
x
