Geology Reference
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4.1.1 Autoregressive Model
An AR model expresses a time series as a linear function of its past values. The
order of the AR model indicates how many lagged past values are included. AR (1)
means the order of the model is one.
The equation for a
first-order AR (1) model is
Y t ¼ u 1 Y t 1 þ
u t
ð 4 : 1 Þ
where autoregressive constant
1 and u t is a Gaussian (white noise) error term.
In simpler terms, it can be said that AR (1) model has the form of a regression
model in which Y is regressed on its previous value. This is why the model name
autoregressive refers to the regression on self (auto).
The second-order moving average model (MA), MA (2), has one more lagged term:
jj \
Y t ¼ u 1 Y t 1 þ u 2 Y t 2 þ
u t
ð 4 : 2 Þ
4.1.2 Moving Average Model
The MR model of q order is denoted as MA (q). The MA model is a form of
ARMA model in which the time series is regarded as a MR (unevenly weighted) of
a random error series. In the case of moving average of time series processes, errors
are the average of this period
'
s random error and the last period
'
s random error. The
first-order moving average, or MA (1), model is given by
Y t ¼ c 1 e t 1 þ e t
ð 4 : 3 Þ
The second-order MR model is given by
Y t ¼
c 2 e t 2 þ
c 1 e t 1 þ
e t
ð 4 : 4 Þ
where c 2 and c 1 are moving average coef
cients and e t 2 , e t 1 , and e t are residuals
at time t
2, t
1, and t.
4.1.3 Autoregressive Moving Average Model: ARMA (P, Q)
The ARMA (p, q) process corresponds to a mixture of AR (p) and MA (q) by
concatenating both modeling systems. The model can be expressed as
Y t ¼ u 1 Y t 1 þ u 2 Y t 2 þ ...þ u p Y t p þ
e t
ð 4 : 5 Þ
þ
c 1 e t 1 þ
c 2 e t 2 þ ...þ
c q e t q
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