g13aff
g13aff
© Numerical Algorithms Group, 2002.
Purpose
G13AFF Univariate time series, estimation, seasonal ARIMA model
(easy-to-use)
Synopsis
[par,s,ndf,sd,cm,st,nst,itc,isf,res,nres,c,ifail] = g13aff(mr,par,x<,c,kfc,...
kpiv,nit,ifail>)
Description
The time series x ,x ,...,x supplied to the routine is assumed
1 2 n
to follow a seasonal autoregressive integrated moving average
(ARIMA) model defined as follows:
d D
(nabla) (nabla) x -c=w
s t t
d D
where (nabla) (nabla) x is the result of applying non-seasonal
s t
differencing of order d and seasonal differencing of seasonality
s and order D to the series x , as outlined in the description of
t
G13AAF. The differenced series is then of length N=n-d', where
d'=d+(D*s) is the generalized order of differencing. The scalar c
is the expected value of the differenced series, and the series
w ,w ,...,w follows a zero-mean stationary autoregressive moving
1 2 N
average (ARMA) model defined by a pair of recurrence equations.
These express w in terms of an uncorrelated series a , via an
t t
intermediate series e . The first equation describes the seasonal
t
structure:
w = (Phi) w +(Phi) w +...+(Phi) w +e -(Theta) e
t 1 t-s 2 t-2*s P t-P*s t 1 t-s
-(Theta) e -...-(Theta) e .
2 t-2*s Q t-Q*s
The second equation describes the non-seasonal structure. If the
model is purely non-seasonal the first equation is redundant and
e above is equated with w :
t t
e =(phi) e +(phi) e +...+(phi) e +a -(theta) a
t 1 t-1 2 t-2 p t-p t 1 t-1
-(theta) a -...-(theta) a .
2 t-2 q t-q
Estimates of the model parameters defined by
(phi) ,(phi) ,...,(phi) ,(theta) ,(theta) ,...,(theta) ,
1 2 p 1 2 q
(Phi) ,(Phi) ,...,(Phi) ,(Theta) ,(Theta) ,...,(Theta)
1 2 P 1 2 Q
and (optionally) c are obtained by minimizing a quadratic form in
T
the vector w=(w ,w ,...,w ) .
1 2 N
The final values of the residual sum of squares and the parameter
estimates are used to obtain asymptotic approximations to the
standard deviations of the parameters, and the correlation matrix
for the parameters. The 'state set' array of information required
by forecasting is also returned.
Note: if the maximum number of iterations are performed without
convergence, these quantities may not be reliable. In this case,
the sequence of iterates should be checked, using the optional
monitoring routine, to verify that convergence is adequate for
practical purposes.
Parameters
g13aff
Required Input Arguments:
mr (7) integer
par (:) real
x (:) real
Optional Input Arguments: <Default>
c real 0
kfc integer 1
kpiv integer 0
nit integer 50
ifail integer -1
Output Arguments:
par (:) real
s real
ndf integer
sd (:) real
cm (:,:) real
st (:) real
nst integer
itc integer
isf (4) integer
res (:) real
nres integer
c real
ifail integer