Environmental Engineering Reference
In-Depth Information
Moreover, considering the time series value
y
T+1
, one can update
a
o
(T)to
a
o
(T + 1) by Eq.
4.28
, and also update
ʔ
(T)to
ʔ
(T + 1) by:
Dð
T
þ
1
Þ
=
T
Dð
T
Þþ
y
T
þ
1
a
o
ð
T
Þ
j
j
ð
4
:
31
Þ
T
þ
1
Now, using Eqs.
4.28
and
4.31
in Eqs.
4.26
and
4.29
, the update forecast is
obtained.
4.14 Winter
'
s Method
Winter
is method is an exponential smoothing procedure appropriate for seasonal
data (Winters
1960
). Winter
'
'
is multiplicative model is given by:
y
t
¼
ðb
0
þ b
1
t
Þ
SN
t
þ e
t
ð
4
:
32
Þ
That, this model assumes a linear trend and a multiplicative seasonal variation.
In order to apply this method, the following calculations are required:
1. An initial estimate b
1
(0)of
ʲ
1
2. An initial estimate a
o
(0)of
ʲ
0
3. An initial estimate s
n
t (0)ofSN
t
From the historical data of the last m year, an average
y
i
is de
ned (for the i
th
year,
; ...;
m. then b
1
ð
0
Þ
=
ð
y
m
y
1
Þ=ð
m
1
Þ
s. s is the length of season),
the initial estimate for
i ¼ 1
;
2
ʲ
0
, the average level of the series at
t = 0 is given by:
y
1
2
b
1
ð
0
Þ
a
o
ð
0
Þ
=
ð
4
:
33
Þ
The initial estimate for the s seasonal factors is given by:
s
t
= y
t
=
y
i
½
ð
s
þ
1
Þ=
f
2
j
b
1
ð
0
Þ
g
ð
4
:
34
Þ
where
y
i
is the average of the observations for the year in which season t occurs (if
1
t
s, then i =1,ifs
þ
1
t
2s, then i = 2, etc.).
The letter j represents the position of season t within the year. For monthly data,
j = 1 represents January,j= 2 is February and so forth. [S
t
must be computed for
each season (month, quarter, etc.) t occurring in the year 1 through m].
Equation
4.32
yields m values for S
t
. These m distinct estimates for seasonal
factor are averaged to give
m
1
k¼0
S
t
þ
ks
1
m
sn
t
=
ð
4
:
35
Þ
t =1
;
2
; ...;
s
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