Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 59

algorithm , which simultaneously considers the trend components by processing the
equations

Xt () D Xt () (1 D ) Xt ( 1)

e e

Xt () E [ Xt Xt () ( 1)] (1 E ) Xt ( 1)

t e e t

in which the constant Ecan (under certain boundaries) be freely selected. In this
case the resulting estimated forecasting value for Qsteps ahead is defined by

x ( t Q ) x () t Q x () t

f e t

The double exponential smoothing algorithm can also be extended to deal with the
time series containing trend and seasonal components. The extended algorithm is
the triple exponential smoothing algorithm or Holt-Winter algorithm , based on
simultaneous consideration of the following three equations:

xt () [ / xt ( )] x (1 )[ ( xt 1) xt ( 1)]

D K D

e s t e t

xt () E [ () xt xt ( 1)] (1 E ) ( xt 1)

t e e t

xt () [ / J xt () (1 J ) ( xt K )

s e s

from which the estimated value for Q steps-ahead forecast is defined as

xt ( Q ) [ xt ( ) Q xtxt ( )] ( Q K )

f e t s

Also here, the constant value J, under certain limits, can be freely selected.
For assessment of forecasting results the MAE criterion

) n MAE (1/ ne ¦

i

i 1

can be used, or alternatively the RMSE criterion

n

RMSE ¦ , e

2

i

i 1

or the MAPE criterion

e

n

MAPE (1/ n ) ¦

i

100% .

xi ()

i 1

f

For estimation of the confidence intervals for the forecast the criterion

f xt

( )