Information Technology Reference
In-Depth Information
I
B
B
2
3
"
p
1
II I
I
BB
B
1
2
3
p
the autoregressive model can be written in the compact form
I
B
.
a
Z
t
t
The model contains (
p+
2) unknown parameters,
i.e
.
p
internal parameters and two
additional parameters: the variance V
a
2
and the white noise
a
t
.
A crucial problem in modelling of autoregressive time series is the selection of
the order of the model to be built. A useful approach in this case is the analysis of
the related
partial autocorrelation function
and the
inverse autocorrelation
function
, because using the autocorrelation function itself is computationally
complicated in the case of building of higher order models. Alternatively, fitting
the time series shape by models of progressively higher order can be used, along
with the analysis of the
residual sum of squares
for each order.
2.5.2 Moving-average Model
Another approach frequently used in modelling of univariate time series is based
on the
moving-average model
!
Z
aa
T
T
a
T
a
T
a
t
11
t
2 2
t
3 3
t
q
t
q
t
which expresses
Z
in terms of an infinite weighted linear sum of
t
Introducing the moving-average operator of order
q
a
,
a
,
a
, ...,
a
.
t
t
1
t
2
t
q
T
B
B
2
3
!
q
1
TT T
BB
T
B
1
2
3
q
the moving-average model can be written in the compact form as
z
T
()
Ba
t
t
The model contains (
q+
2) unknown parameters
2
to be
PT
TT TV
,
1
,
,
,
!
,
,
23
q
a
estimated from the observation data.
2.5.3 ARMA Model
The combination of the AR and MA models makes up the ARMA model
ZZ Z
I
I
...
I
Za
T
a
T
a
...
T
a
.
t
1
t
1
2
t
2
p
t
p
t
1
t
1
2
t
2
q
t
q
Rewriting the model as
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