Environmental Engineering Reference
In-Depth Information
u
(
t
−
1
)
u
2
T
u
ϕ
(
t
,
μ
)=[
y
(
t
−
1
)
y
(
t
−
2
)
y
(
t
−
3
)
u
(
t
−
1
)
(
t
−
1
)
l
(
t
)]
b
5
−
(
t
−
1
)
(4.49)
so ϕ
depends on parameter
b
5
. Then this is a NLIP (nonlinear in the parame-
ters) model and, in general, a NARX model [1].
(
t
,
μ
)
(
,
)
Example 4.
In the case of the single layer perceptron ϕ
k
t
μ
k
contains parameters
(weights) in vector β
k
, and the bias γ
k
:
d
T
β
k
ϕ
k
=
f
(
+
γ
k
),
(4.50)
where
f
is,
e.g.
, a sigmoid function (Figure 4.4). This model is NLIP.
4.2.2.1 Optimal Parameter Vector
The optimal parameter vector is obtained through minimization of functional (4.2)
using the necessary condition (4.7)
(
|
−
,
ˆ
κ
)
∂
y
t
t
1
E
{[
y
(
t
)−
y
(
t
|
t
−
1
,
ˆκ
)]
} =
0
.
(4.51)
∂κ
From (4.7)
m
k
=
1
∂θ
k
ϕ
k
ˆ
κ
∂
y
(
t
|
t
−
1
,
)
(
ξ
(
t
),
ˆμ
k
)
=
.
(4.52)
∂κ
∂
ˆ
κ
For LIP models (
e.g.
, ARX, NARX),
m
k
=
1
θ
θ
k
ϕ
k
θ
ϕ
T
ˆ
κ
y
(
t
|
t
−
1
,
)=
y
(
t
|
t
−
1
,
)=
(
t
)=
(
t
)
(4.53)
and
ˆ
κ
ˆ
κ
∂
y
(
t
|
t
−
1
,
)
∂
y
(
t
|
t
−
1
,
)
ϕ
T
=
=
(
t
),
(4.54)
∂θ
∂κ
since no parameters appear in ϕ
(
t
)
.
From (4.7), (4.53) and (4.54)
ϕ
T
θ
ϕ
T
ϕ
T
ϕ
T
ˆ
θϕ
T
E
{[
y
(
t
)−
(
t
)
]
(
t
)} =
0
,
E
{
(
t
)
y
(
t
)} =
E
{
(
t
)
(
t
)},
(4.55)
θ
θ
ϕ
T
ϕ
T
)}
−
1
E
E
{
ϕ
(
t
)
y
(
t
)} =
E
{
ϕ
(
t
)
(
t
)}
,
=
E
{
ϕ
(
t
)
(
t
{
ϕ
(
t
)
y
(
t
)}.
(4.56)
Assuming ϕ
is a wide sense stationary stochastic vector,
i.e.
, its elements are
stochastic processes (random sequences) [47], the correlation matrices in (4.56) are
independent of time
t
and may be written
(
t
)
ϕϕ
T
R
ϕϕ
=
E
{
},
R
ϕ
y
=
E
{
ϕ
y
}.
(4.57)
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