Environmental Engineering Reference
In-Depth Information
If the SACF of the time series values Z
b
,Z
b+1
,
Zn either cuts off fairly fast or
dies down fairly quickly, then this series should be considered as stationary.
…
If the SACF of the time series dies down extremely slowly then the series should
be considered as non-stationary.
finding a stationary time series is to compute SACF of
the original series and then examine its behavior. If it is not stationary, by methods
such as differencing or taking logs, the data is transformed. The SACF for the
transformed series is computed and the new SACF is examined. The procedure is
continued until stationarity is reached.
The SPACF is another important tool for identi
Thus the procedure for
cation of time series models.
The sample partial autocorrelation at lag k is de
ned with:
k =1
)
r
kk
= r
1
!,
1
k
i
!
r
k
k
i
ð
4
:
8
Þ
k =2
;
3
; ...)
r
kk
=
j¼1
r
ð
k
1
Þ;
j
:
r
k
j
j¼1
r
ð
k
1
Þ;
j
:
r
k
j
where
r
kj
= r
ð
k
1
Þ
j
r
kk
r
ð
k
1
Þ;ð
k
j
Þ
j =1
ð
4
:
9
Þ
;
; ...; ð
k
1
Þ
2
The standard error of risk is de
ned as:
0
:
5
1
ð
n
b
þ
1
Þ
S
r
kk
=
ð
4
:
10
Þ
'
st
kk
—
And the student
statistic is given by:
t
r
kk
= t
r
kk
=
s
r
kk
ð
4
:
11
Þ
The precise interpretation of the SPACF at lag k is rather complicated. However,
this quantity can intuitively be thought of as the sample autocorrelation of time
series observations, separated by a lag of k time units, with the effects of the
intervening observations eliminated.
As with SACF, one must examine the behavior of SPACF and classify it to
provide guidelines for identi
cation of time series models: the SPACF can also
display a variety of different behaviors. First the SPACF can cut off. This can occur
when r
kk
is not statistically signi
cant beyond some number K. In general, K is at
most equal to 3. In such a case t
r
k
j j
would be small and generally less than 2 for
k < K. Second, the SPACF may die down if this function does not cut off but rather
decreases in a
. This function may exhibit exponential decay,
damped sine-wave or as a mixture of them, Bowerman and O
“
steady manner
”
'
Connell (
1987
).
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