Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 53

or in final form as

f

ˆ (

yt k ¦ ) WZ . (2.3)

j t j

j 0

The next objective is to estimate the mean squared forecast error (MSE) from the
difference

ˆ

2

MSE H {[ yt k ( ) yt k ( )] }

f f

2

MSE H {[ ¦ T Zt k i ( ) ¦ WZt ( j ] }

i j

i 0 j 0

or,

k 1 f

MSE H {[ T Zt k i ( ) ( T W kZt ) ( j i )] }.

2

¦ ¦

i i i

i 0 i k

Assuming that the

Z are mutually independent with a mean of zero and

2 V the last equation of mean square error becomes

variance

k 1 f

2 2 2

.

MSE VT T [ ¦¦ ( W ) ]

i i i k

i 0 i k

From this it follows that the mean square error is minimized by taking

( T W ) 0,

i i k

wherefrom it follows that

W T .

i i k

This, when introduced into the k- step forecast (or k -step prediction) equation (2.3),
gives

f

yt k ˆ( ¦ ) T Z

ik ti

i 1

which can also be expressed as

f

ˆ (

yt k ¦ ) T . (2.4)

i t ki

ik