Environmental Engineering Reference
In-Depth Information
⎡
⎤
⎡
⎤
ϕ
1
,
ϕ
1
···
ϕ
m
,
ϕ
1
y
,
ϕ
1
⎣
.
.
⎦
⎣
.
⎦
P
xx
=
ϕ
1
,
ϕ
k
···
ϕ
m
,
ϕ
k
P
yx
=
y
,
ϕ
k
.
(4.106)
.
.
.
ϕ
1
,
ϕ
m
···
ϕ
m
,
ϕ
m
y
,
ϕ
m
If
P
−
1
xx
exists, then
θ
P
−
1
xx
P
yx
=
.
(4.107)
For example, let the ϕ
k
and
y
be random variables, in a random variable inner
product space ℵ, where the inner product is
x
j
,
ϕ
k
=
E
{
x
j
ϕ
k
}
.Let
T
ϕ
=[
ϕ
1
···
ϕ
m
]
(4.108)
be a random vector whose elements are the random variables ϕ
i
.Then
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
ϕ
1
} ···
ϕ
1
,
ϕ
1
···
ϕ
m
,
ϕ
1
E
{
E
{
ϕ
m
ϕ
1
}
.
.
.
.
P
xx
=
ϕ
1
,
ϕ
k
···
ϕ
m
,
ϕ
k
=
E
{
ϕ
1
ϕ
k
} ···
E
{
ϕ
m
ϕ
k
}
(4.109)
.
.
.
.
ϕ
m
ϕ
1
,
ϕ
m
···
ϕ
m
,
ϕ
m
{
} ···
{
}
E
ϕ
1
ϕ
m
E
ϕϕ
T
and
P
xx
=
E
{
} =
R
ϕϕ
is the correlation matrix of random vector ϕ, while
P
xy
=
E
is the correlation vector between the random variable
y
and the
random vector ϕ. The optimal parameter vector then is
{
y
ϕ
} =
R
y
ϕ
}
{
θ
ϕϕ
T
}
−
1
E
=
E
{
{
ϕ
y
}.
(4.110)
This is the same result (4.58) obtained in Section 4.2.2 for the optimum parameter
vector θ in the case of LIP models when the necessary condition for optimality was
used in the case of random variables. However, here the optimal θ has been found
by particularizing the general result (4.107) to the case of random variables.
In most practical cases, the probability functions needed to compute expected
values are not known, so parameter vector given by (4.110) cannot be found. How-
ever, under certain assumptions (
i.e.
, wide sense stationarity, ergodicity [47]) ex-
pected values may be estimated using time averages. In such cases the estimation θ
of θ becomes a random vector having an expected value and a covariance matrix,
as shown in Section 4.3.2.
Let it be assumed that the LIP plant model is
ϕ
T
θ
y
=
+
w
,
(4.111)
where
w
is orthogonal to all elements of ϕ. Then, from (4.99),
ϕ
T
θ
ε
=
(
θ
−
)+
w
.
(4.112)
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