Time Series Modeling - Water Quality Management: Assessment and Interpretation - page 55

4.13.1 Simple Exponential Smoothing

For a stationary series, suppose:

y t = b 0 þ e t ð 4 : 25 Þ

In general, if a o ðÞ is the estimate in time period T for the average level b 0 of
Eq. 4.25 then, a point forecast ˄ steps ahead made in time period T for y T þs , is given

by:

y T þs ð T Þ = a o ð T Þ ð 4 : 26 Þ

To start the forecasting process, a o (0) is needed which is usually the average of

the first one third or half of historical data, depending on the length of the series.

That is:

a o ð 0 Þ = X ½n = k

t¼1 y t = ½n = k

ð 4 : 27 Þ

where k = 2or3and [n/k] is the integral part of n/k. The portion used to estimate
a o (0) should not be too long or too short.
The updating equation for a o (T) is given by:

a o ð T þ 1 Þ = c y T þ 1 þð 1 cÞ a o ð T Þ ð 4 : 28 Þ

where is the smoothing constant, often taking a value between 0.01 and 0.30 in

most applications. Its best value can be obtained by trial and error via minimizing
the sum of squared forecast errors. However, the guiding principle is that the
smaller value of

ʳ

ʳ indicates that the average level of the time series does not change

much over time.
Equations 4.26 and 4.28 allow the computation the forecasting of the values for
the desired time points.
The expression for constructing interval forecasts, of giving con

dence level, for

y t þs in time period T, is given by:

½y t þs ð T Þ B ½100 ð 1 x Þ ð T Þ; y T þs ð T Þ B ½100 ð 1 x Þ ð T Þ ð 4 : 29 Þ

t þs T þs

where, B ½100 ð 1 x Þ
t þs

ð T Þ = Z a= 2 1 : 25 Dð T Þ and ʔ (T) is the average absolute forecast error

for the time period T, i.e.

Dð T Þ = X T
t¼1

j y t a o ð t 1 Þ j= T ð 4 : 30 Þ

The value of Z a= 2 is the 100(1

− ʱ

/2) the centile of standard normal distribution.