Geology Reference
In-Depth Information
which can be rewritten, using the summations as
X
p
i¼1
u
i
Y
t
i
þ
e
t
þ
X
q
Y
t
¼
c
j
e
t
j
ð
4
:
6
Þ
i¼1
In the ARMA (p, q) model, the condition for stationarity has to deal with the AR
(p) part of the speci
cation only. The simplest ARMA model is
rst-order auto-
regressive and
first-order moving average, and can be expressed as ARMA (1, 1).
4.1.4 Autoregressive Moving Integrated Average Model:
ARIMA (P, Q)
In some situations, in order to induce stationarity in the time series, it is necessary to
detrend the raw data through a process called differencing. If Y
t
has an ARIMA (p,
1, q) model representation, this indicates an ARMA (p, q) is presented with one
differencing. The ARIMA model can be represented as
d
Y
t
ð
u
2
L
2
u
p
L
p
c
2
L
2
c
p
L
q
1
u
1
L
Þ
¼
ð
1
c
1
L
Þ
e
t
ð
4
:
7
Þ
D
Identi
cation of the best model can be achieved through three Box
Jenkins
-
model selection stages, namely (1) identi
cation (through examining Autocorre-
lation and Partial Autocorrelation functions of selected model orders), (2) estima-
tion (different models are compared using AIC and BIC), and (3) diagnostic
checking (e.g.,
fitness of model).
4.1.5 AutoRegressive with eXogenous Input (ARX) Model
The modeling signal y
i
(t) can be written as
y
i
ð
t
Þ
¼
s
i
ð
t
Þþ
n
i
ð
t
Þ
ð
4
:
8
Þ
where s
i
(t) is the useful signal component and n
i
(t) is the noise component. When
an ARX model is used to describe interaction of signal, it can be written as
X
p
q
þ
d
1
X
y
i
ð
t
Þ
¼
a
j
y
i
ð
t
j
Þþ
b
k
u
ð
t
k
Þþ
e
i
ð
t
Þ
ð
4
:
9
Þ
j¼1
k¼o
where p and q are the orders of the autoregressive and moving average parts,
respectively, a
j
s and b
j
is are the model coef
cients, u(t) is a reference signal, e
i
(t)isa
white noise process, and d is temporal delay.
