Graphics Reference
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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