Civil Engineering Reference
In-Depth Information
Nj
−
1
(
)
( )
(
)
∑
Cov
j
t
E x t
⎡
x t
⎤
x
x
τ
=⋅Δ =
⋅
+
τ
=
⋅
(2.20)
⎣
⎦
x
kj
k
+
Nj
−
k
1
=
from which it is seen that
j
must be considerably smaller than
N
for a meaningful
outcome of the auto covariance estimate. The same is true for the auto correlation
function in Eq. 2.14.
Example 2.2
Given a variable:
()
(
)
x t
a
sin
t
=
2
T
TnT
=
(where
n
=⋅
ω
,
ωπ
. Using the substitutions
1
1
1
1
1
ˆ
(
)
t
2
T
t
is an integer) and
=
π
, then the auto covariance of
x
is given by
1
T
T
⎡
⎤
1
1
1
()
() (
)
(
)
(
)
Cov
lim
∫
x t
x t
dt
lim
⎢
n
∫
a
sin
t
a
sin
t
dt
⎥
τ
=
⋅
+
τ
=
⋅
ω
⋅
ω
+
ω τ
x
1
1
1
1
1
T
nT
⎢
⎥
T
→∞
n
→∞
1
⎣
⎦
0
0
2
π
2
π
2
2
a
a
(
)
ˆ
ˆ
ˆ
2
ˆ
(
)
ˆ
ˆ
(
)
ˆ
1
∫
1
∫
⎡
⎤
=
sin
t
⋅
sin
t
+
ωτ
dt
=
sin
t
⋅
cos
ωτ
+
sin
t
⋅
cos
t
⋅
sin
ωτ
dt
1
⎣
1
1
⎦
2
2
π
π
0
0
2
(
)
2
2
⎡
π
π
⎤
⎥
sin
ωτ
a
(
)
2
ˆˆ
1
ˆˆ
1
cos
∫
sin
tdt
∫
sin 2
tdt
=
⎢
⎣
ωτ
+
1
2
2
π
⎢
⎥
⎦
0
0
The first of these integrals is equal to π
, while the second is zero, and thus:
2
1
a
()
(
)
Cov
cos
τ
=
ω τ
x
1
2
()
2
2
1
2
Since the variance of
x t
is
σ
=
a
(see example 2.1), then the auto covariance coefficient
x
is given by:
()
Cov
τ
()
x
(
)
cos
ρ
τ
=
=
ω τ
x
1
2
σ
x
(
)
01
As can be seen:
x
ρτ==.
Similar to the definitions above, cross correlation and cross covariance functions may be
defined between observations that have been obtained from two short term realisations
()
()
()
()
X
t
=+
xxt
and
X
t
=+
xxt
of the same process or alternatively from
1
1
1
2
2
2
realisations of two different processes:
T
1
()
()
(
)
()
(
)
∫
R
τ
=
EX t X t
⎡
⋅
+
τ
⎤
=
lim
X t X t
⋅
+
τ
dt
(2.21)
⎣
⎦
XX
1
2
1
2
12
T
T
→∞
0
