Geoscience Reference
In-Depth Information
Table 5.1.
Guidelines for identifying model parameters based on the
characteristics of ACF and PACF
Sl. No.
Model parameter
Characteristics of
Characteristics of
ACF
PACF
1
One autoregressive (
p
)
Exponential decay
Spike at lag 1, no correlations
for other lags
2
Two autoregressive (
p
)
A sine-wave shape
Spikes at lags 1 and 2, no
pattern or a set of
correlation for other lags
exponential decays
3
One moving average (
q
)
Spike at lag 1, no
correlation for other
Damps out exponentially
lags
4
Two moving average (
q
)
Spikes at lags 1 and 2,
A sine-wave shape pattern or
no correlation for other
a set of exponential decays
lags
5
One autoregressive (
p
)
Exponential decay
Exponential decay starting at
and one moving
starting at lag 1
lag 1
average (
q
)
One of the common functions is Akaike Information Criterion (AIC) (Brockwell
and Davis, 1991), which is expressed as:
pq
n
1
p,q
ˆ
AIC (
p,q
) =
log (
V
)
2
(6)
1
One minimization function is Bayesian Information Criterion (Brockwell
and Davis, 1991) expressed as:
(
pq
) log(
n
1)
ˆ
2
p,q
log (
V
)
BIC (
p,q
) =
(7)
n
1
Another criterion is proposed by Hannan and Quinn (1979), which is
given as:
^
`
2 (
pqc
)
log
log(
n
1)
ˆ
p,q
log (
V
)
HQ (
p,q
) =
with
c
>1
(8)
n
1
p,q
V
) becomes small as
(
p
+
q
) increases. Hence, the additive terms in the above criteria serve as penalties
for large values of
p
and
q
, and help to prevent over-fitting of the data by
selecting
p
and
q
too large. There is no specific reason to use a certain criterion
for a specific condition. However, it is to be noted that AIC has the tendency
not to underestimate the model order and the BIC is generally to be preferred
for larger samples (Schlittgen and Streitberg, 2001).
It is worth mentioning that the variance estimate (
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