3). . Utilizing the identify, . valid . for all stationary and
invertible processes
Bois 1 airy s 5062)
A(z) C(z)
2 -k
with C(z) — 1 + 2 e, 7
D 1 be "A
(z) = B4 z
Here the ARMA model is described by an AR- or MA- model
of infinite order (Pukkila/ Leppálà /13/).
Taking the various considerations into account it was
decided to select and concentrate on the algorithms
published by Pukkila/Leppdld /13/ because of the high
precision of the estimated parameters, its numerical
simplicity and the short computational times.
The algorithm assumes that the infinite order models can be
approximated by models of a high order M, that the coeffi-
cients c, are estimated by other methods (i.g. Ulrych/Clay-
ton) and that the coefficients d can be determined from
the c,. Pukkila /12/ states that the c and d coefficients
j
can be expressed as functions of the a and b parameters by
eim Eps anh ED wy ly be "obse, oe
db bey app rm A 8, d, t &,
e, 71:i du 4n ji-1, M
The parameters a and b are estimated by minimizing the sum
of the squared model errors e and § with respect to the
parameters
M
S(a,b) - 2 ( ex? + 6,7 y
The precision of the estimated parameters is dependent on
the location of the zeros of the polynomials A(z) and B(z).
The parameters are indeterminable if the locations are
equally distributed. The precision is not very sensitive to
the order M. An order of M - 10 - 20 is enough if p+q < 5.
5. Process Identification
Process identification covers, in addition to parameter
estimation, the determination of the orders p,d,q of an
ARIMA process. Since, in most cases, the process order is
not known a priori, it is usual practice to postulate
several model orders and to analyse them. The order d of
the derivatives can be found out by counting the zeros of
A(z) located outside the unit circle. The determination of
the orders p and q is more difficult. One possibility is to
reverse the linear filter generating the ARMA process. Thus
the innovations e(t) - now called prediction errors - are
computed from the time series x(t). The prediction errors
of the optimum order may be represented by a white noise
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