Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 51

x ˆ [ AA Ay

T

]

1 T

,

follows.
The generalization of the least-squares estimator consists of minimization of
the linear form

ˆ ˆ

) T ( yAxByAx ) (

with the diagonal positive definite matrix

c , 2 c ,...,

c

B = diag [ ]

1

playing the role of a weighting matrix. Using the above calculations, the
corresponding generalized least-squares formulation

x ˆ [ ACA

T

]

1

ACy

T

,

is achieved.
The mathematical model on which the nonlinear regression relies has the
general form

yf (,) x DH ,

i i

where y i is the i th observation of dependent variable y , x i is the i th observation of x ,
and f is a selected nonlinear function. In practice, the polynomial

n

i

y ¦ x DH

i

i 0

is frequently selected as the nonlinear function.

2.9.4 Forecasting Using the Box-Jenkins Method

Box and Jenkins have developed a general forecasting methodology for time series
generated by a stationary autoregressive moving-average process. In the following,
the methodology is explained on regressive models described in Section 2.4.

2.9.4.1 Forecasting Using an Autoregressive Model AR(p)
The autoregressive model

p

x a X P

¦

t i t i

i 1

can be used to estimate the forecasts for any number of steps ahead. For example,