Instrumentation & Measurement Magazine 26-4 - 32

educationI&M continued
in
average part, it only serves as an initial estimate. This approach
gives access to the model residuals:
u [0] n yn a yn a yn( 2)  p
[0]
ˆ ˆˆ ˆ
() () ( 1)

 1

 2
[0]
ˆ
a yn p)
[0]
( 
These estimates serve as the first estimates denoted by
which gives estimates of the parameters [0]
[1,0,0, ,0]
e  
1
as well as the innovation [0]
uˆ
 ae [ ˆ ,] with
. In this approach
[0]
1
one initializes the parameters. Now the estimates can be iteratively
refined by solving the following least squares problem:
yn 12a yn a yn pq( 2)
 
( )         0
( 1)
ˆ
leading to [] [
ˆ ˆˆ ˆ
( )
u[] n 
m mm m
p
1
ˆ m
b
[] 12

yn a y n a yn
( ) 
b1
m u n  bq
( 1)   )
[] ( 2)
ˆˆ )qˆˆ( 1)
[] [ 1]m
[] []
m u nm ( 
for iteration step m. The disadvantage of this procedure is
that it heavily relies on the initialization by the autoregressive
process. This procedure fails when the data generating
system exhibits zeros close to the unit circle. In order to improve
this approach Durbin's method improves the approach
considerably.
Autoregressive Moving-average Model
Identification: Durbin's method
Durbin's method tries to improve the initialization procedure.
Assuming an autoregression as a starting procedure may imply
convergence to a wrong local solution. As a result, an
improved prior estimation of the unknown innovation process
will avoid getting trapped in a wrong solution.
The idea of Durbin implies to estimate the innovations in a
" non-parametric " way. Non-parametric models imply models
a yn p
[ ]
(
ˆ
 ab[,ˆ
m mm
] []
] and hence:

(
)
a yn p b u n b u nmm m
ˆ
[1]
( )  1
ˆ
[1]
( 1)  
b u n q)
ˆ
[1]
(
which are infinite dimensional since the number of parameters
increases with its record length. As a result, Durbin's method
assumes the following initialization to model yby an AR( n )
model. Note that the order n used is the preferred choice of
the authors while any other order AR(p) can be used as well, as
long as the dimensionality p is sufficiently high. One can argue
that this is only valid if such a time series ycan be represented
by an infinite autoregression. Indeed, a few additional technical
assumption are required since this assumption does not
hold for any weakly stationary process.
Theorem: A weakly stationary processes y(n) such that the impulse
response h(n) satisfies

m

n 0 n hn| ( )| then y(n) can be
modeled by an infinite autoregression:
1
()  ( 
m
y n a y n m un


) ( )
As a result, Durbin's method proposes to initialize the iterative
least squares by:
ˆ ˆˆ ˆ
(
u n y n 12a yn a yn
[0]
)
( 1)
( 2)
) (      
n
a y(n n)
Note that these only initialize the innovations and not the
parameters as the AR-part is overparametrized. These estimates
can then be further refined through the same procedure
described in the previous section.
Example: Consider the following ARMA(4,2) system
where there are two complex conjugate pole pairs

zi zi 
[1]
p  0.9exp 9 / 40


zz  i 
and 
[2]
p  0.6exp 24 / 40 a sharper
and weaker resonance, respectively. In between both resonances
we consider a transmission zero pair (i.e., a zero on
the unit circle) exp 15 / 40
. In Fig. 6 a simulated time
series sampled at 10 sps, its innovation and the power spectrum
are shown. One can see in the right plot that the power
Fig. 6. ARMA(4,2) simulation example. (a) Time series and innovation; (b) Power spectrum.
32
IEEE Instrumentation & Measurement Magazine
June 2023
Instrumentation & Measurement Magazine 26-4

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