Environmental Engineering Reference
In-Depth Information
be random vectors also formed by stacking.
Vector W
(
ξ
)
represents a window of a stationary white noise random sequence
{...,
w
(−
r
,
ξ
),...,
w
(
0
,
ξ
),
w
(
1
,
ξ
),...,
w
(
N
,
ξ
),
w
(
N
+
1
,
ξ
),...}
(4.126)
where E
{
w
(
k
,
ξ
)} =
0, E
{
w
(
k
,
ξ
)
w
(
j
,
ξ
)} =
0for k
=
j ,and E
{
w
(
k
,
ξ
)
w
(
k
,
ξ
)} =
σ w .
Then,
Y
(
ξ
)=
X
(
ξ
)
θ
+
W
(
ξ
)
(4.127)
is the assumed model of the plant, and
θ
Y
(
ξ
)=
X
(
ξ
)
(4.128)
is the model output. The error vector is
Y
ε
(
ξ
)=
Y
(
ξ
)−
(
ξ
).
(4.129)
Then, from (4.128) and (4.129)
θ
(
)=
(
)
+
(
).
Y
ξ
X
ξ
ε
ξ
(4.130)
ξ )
ξ )
For a given set of measurements, X
(
is a matrix of real numbers and Y
(
,
Y
ξ )
ξ )
(
(
are vectors of real numbers.
4.3.2.2 Time Averages
Let
t
n 1 +
T
1
T
1
T
+
T
A T
(
ξ
)=
z
(
i
,
ξ
)
A T
(
ξ
)=
z
(
t
,
ξ
)
dt
(4.131)
t
i
=
n 1
+
1
be the time average of a random sequence or of a random function. Since the sum
is with respect to i or the integration is with respect to t , A T
is only a function of
ξ , and therefore has the form of a random variable, which it really happens to be
[47].
It may be immediately seen that A T
(
ξ
)
(
)
{
(
,
)}
ξ
is an unbiased estimator of E
z
t
ξ
if
z
(
t
,
ξ
)
is wide sense stationary, because
t
t
+
T
+
T
1
T
1
T
E
{
A T
(
ξ
)} =
E
{
z
(
t
,
ξ
)}
dt
=
zdt
=
z
,
(4.132)
t
t
where E
z and similarly for random sequences.
This fact is illustrated in Figure 4.15, where p
{
z
(
t
,
ξ
)} =
(
A T
)
is the probability density
function of A T .
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