Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 57

Here, x t is the observed value at the

same point of time, D is the smoothing constant value, and

is the exponentially smoothed value, ( )

x

() t

is the previous

e xt

( )

exponentially smoothed value.
The value of the smoothing constant D depends on the properties of the given
time series. Values between 0.1 and 0.3 are most commonly used because they
produce a forecast which depends on a large number of past observations. Values
close to one are rarely used because they give forecasts which depend much more
on recent observations. For instance, when smoothing constant D = 1, the forecast
is equal to the most recent observation.
The term exponential can be understood from the result of iterative calculation

of using , , , etc ., which results in

x

() t

e xt

( )

e xt

( )

e xt

( )

D xt ( ) (1 )[ ( 1) (1 ) ( 2)]

() 1 ){ ( ) 1 [ ( 2) 1 )( 3 ]}

D D xt D x t

e

D xt D D xt D D xt D x t

e

or generally

2 t

e xt

() D xt () (1 D )[ ( xt 1) (1 D ) xt ( 2) ... (1 D ) (0)] x ,

from where the exponentially decreasing value of weights is evident.
In addition, because the expression in the second term on the right-hand side of
the last equation within the bracket is equal to

e xt

( 1), it can be rewritten as

xt () D xt () (1 D ) ( xt 1)

e e

From this equation it follows that in order to estimate the smoothed value

x

( ) t of

the time series at the time point t , we need the current value x t and the estimate

()

of the smoothed value

e xt at the previous time point ( t- 1), supposing that the
value of the constant D is time invariant.
Prior to applying exponential smoothing algorithm it should be decided

( )

x how to initialize the exponential smoothing process
x how to select the value of smoothing constant D .

For simplicity, the algorithm is initialized by setting . With regard the

value of exponential smoothing constant D, it can generally be arbitrarily selected
within the interval [0, 1]. Its optimal value depends largely on the time series
pattern and on the smoothing objectives. Since the value of D determines how
strong the older observations are dampened, selection of higher D values dampens
the old values more strongly than the selection of lower D values. There is also a
direct experimental way to evaluate the optimal value of D in which the values D =
0.1, 0.2, …, 0.9 are taken and for each value the efficiency of estimation is
calculated. In this way the value of D giving the best efficiency is found.

x

(2) x (2)