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In-Depth Information
e
(
t
)
1
B
(
q
)
+
A
(
q
)
u
(
t
)
y(t)
FIGURE 3.2
ARX model structure.
where
q
is the delay operator and
A
(
q
) and
B
(
q
) are represented as:
()
=+
1
q
−
−
n
Aq
1
a
++
a q
n
(3.3)
s
1
s
()
=+
−
1
−
n
+
1
Bq
bbq
+…+
b
q
(3.4)
b
1
2
n
b
and
q
is known as the backward shift operator defined by
q
-1
u
(
t
) =
u
(
t
- 1). Figure 3.2
shows a graphical representation of an ARX model. In the context of this research,
u
(
t
) and
y
(
t
) may be taken as the input vertex count and output frame rate, respec-
tively. For compact notation, the following vectors are used:
T
θ=
aa bb
1
……
(3.5)
n
1
n
a
b
T
(
)
−−
(
)
…−
(
)
−−
yt yt yt n
ut nutn n
1
2
−
a
φ
()
=
(3.6)
(
)
(
)
…−−+
−
1
k
k
b
From Equations (3.5) and (3.6), Equation (3.2) can be expressed as:
()
θ
(3.7)
y
()
=
φ
T
Alternatively, we can use the following notation to highlight the dependency of
y
(
t
)
on the set of parameters in θ:
()
θ
ˆ
(
)
=
φ
T
yt
θ
|
(3.8)
We want to compute the set of parameters θ by using the least square method and
the criterion function:
N
N
1
∑
1
2
1
∑
1
2
(
)
[
]
T
2
[
()
(
)
]
=
()
φ
(
)
θ
(3.9)
N
ˆ
2
VZ
θ
,
=
yt
−
yt
|
θ
yt
−
ϕ
t
N
N
N
t
=
1
t
=
1
with the objective to get:
−
1
N
N
∑
∑
1
·
LS
(
)
min
φ φ φ
(3.10)
N
VZ
T
θ
=
arg
θ
,
=
ϕϕ
()
t
()
t
ϕ
(
t
))()
yt
N
θ
t
=
1
t
=
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