Spectral Analysis by Stationary Time Series Modeling - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 146

computational burden. HOYW makes a good compromise between these two
aspects.

6.2.2. Estimation of ARMA parameters

The estimation of ARMA parameters [KAY 88, MAR 87] is made in two steps:
first, we estimate the AR parameters and then the MA parameters. This suboptimal
solution leads to a considerable reduction of the computational complexity with
respect to the optimal solution that would consist of estimating the AR and MA
parameters. The estimation can be efficiently performed in two steps. Starting from
equation [6.1], we show that the autocorrelation function of an ARMA process itself
follows a recursion of the AR type but starting from the lag q + 1 (see section 4.2.2,
equation [4.17]):

p

∑

() ( )

γ m =− a γ m − n for m > q

xx x xx

n = 1

The estimation of the AR parameters is done as before by taking these modified
Yule-Walker equations into account (starting from the rank q + 1).

The estimation of the MA parameters is done first by filtering the process by the
inverse AR filter, using the AR estimated parameters so that we come back to an
MA of order q, according to the principle in Figure 6.2.

Figure 6.2. Principle of the estimation of MA parameters in an ARMA

If the inverse AR filter is supposed perfectly known, we obtain (by using
equation [6.1]):

p q

() ()

∑

( )

∑

( )

yk = xk + axk n − = buk n −

n n

n = 1 n = 0

and the filtered process y ( k ) is a pure MA(q). Thus, we purely come back to a
problem of estimation of MA parameters. This model MA(q) can theoretically be
modeled by an infinite order AR by writing: