Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 57

which is the average seasonal index for each different season. Finally, the initial
estimate sn t (0)ofSN t is given by:

2
4

3
5

S

sn t ð 0 Þ ¼ sn t

S
t¼1 sn t

ð 4 : 36 Þ

t ¼ 1 ; 2 ; ...; s

The updating equations are de ned as:

y T
sn T ð T s Þ þð 1 cÞ ½a o ð T 1 Þþð T 1 Þ

a o ð T Þ ¼ c ð 4 : 37 Þ

where ʳ is a smoothing constant, 0 < ʳ <1.

b 1 ð T Þ = h ½a 0 ð T Þ a 0 ð T 1 Þ þ ð 1 hÞ b 1 ð T 1 Þ ð 4 : 38 Þ

where ʸ is a smoothing constant, 0 < ʸ <1.

y T
a 0 ð T Þ þð 1 xÞ sn T ð T S Þ

sn t ð T Þ = x ð 4 : 39 Þ

where <1.

Having the updated values for the components of Eq. ( 4.32 ), which are given in
Eqs. 4.37

ˉ is a smoothing constant, 0 < ˉ

4.39 , a point forecast made at time T for yT+τ T+ ˄ is obtained by:

-

y T þs ð T Þ = ½a 0 ð T Þþ b 1 ð T Þs sn T þs ð T þ s s Þ ð 4 : 40 Þ

Interval forecasts based on complicated formulas can be constructed which can
be found in Bowerman and O ' connell ( 1987 ).

4.15 One and Two-Parameter Double Exponential
Smoothing

The two parameter exponential smoothing is a special case of Winter s method

where SN t equated to unity for all values of t. That is, y t = b 0 þ b 1 t þ et. Thus, the
previous results in the Winter ' s method also apply in this case.
In one-parameter of exponential smoothing, the smoothing constants are related
to each other, hence one parameter suf

'

2 and 2w

1 þ w

ces. Thus let c =1 w h = where

w =1 d, and

ʴ

is a smoothing constant which is chosen to lie between the values
of 0 and 1. Therefore;

y T þs ð T Þ = a 0 ð T Þþ b 1 ð T Þs ð 4 : 41 Þ