Information Technology Reference
In-Depth Information
x
state-space models
.
In the following, various approaches for building stationary models of time
series are presented.
2.5 Regressive Models
Regressive models are built using
regression analysis
, which is a collection of
methods for the study of relationships between the variables and for estimation and
prediction of values of one variable using the values of other variables incorporated
in a joint time series (Drapper and Smith, 1981). For instance, to implement an
efficient predictor for a variable of interest, the measurable variables representing
the strong indicators for the same variable should first be identified.
The most popular regression models in engineering are the
x
autoregression model
(AR)
x
moving-average model
(MA)
x
ARMA model
x
ARIMA model
x
CARIMA models
.
2.5.1 Autoregression Model
Autoregression models express the current value of a time series by a finite linear
aggregate of previous values and by a shock
P
x
DD
x
2 2
...
x
DP
x
QQ
,
t
11
t
t
t
t
where D to D are the
autoregression parameters
, P is the white noise and Q is
the model order. The validity of an autoregressive model assumes that the time
series to be modeled is stationary. Also, because of some possible internal
cumulative effects, the autoregressive process will only be stable if the values of
parameters D are within a certain range.
It is common to write the autoregressive equation in terms of deviations
,
generally using the variable
Z
and its deviation
The
xx
P
ZZ
P
.
t
t
t
individual terms of the time series now become
"
, resulting in the
,
,
,
,
ZZ
Z
Z
t
t
1
t
2
t
3
autoregressive model
I
I
I
"
I
,
a
Z
Z
Z
Z
Z
t
t
1
t
2
t
3
t
p
t
1
2
3
p
where
II
are unknown parameters to be estimated from the
observation data. Introducing the autoregressive operator
2
P I
,
,
,
,...,
I V
,
1
2
3
q
a
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