Instrumentation & Measurement Magazine 26-4 - 31

Fig. 5. Jamar dynamometer. (a) Differenced time series; (b) Periodogram analysis.
One can immediately imagine to solve this regression problem
by the least squares approach. However, one must be
cautious as the regular assumption which is required by the
linear regression framework is violated. Indeed, the regression
framework implies the following assumption:
y N Xa I n )
 
(
 y y yn
y
,
where X∈nxp is the regression matrix and a = [a1, a2, ..., ap]t
y [ (1), (2), , ( )]t
while finally:
X y p( 1) ( 2) ( 3) ( 4)


y p 










y(0)
yy(0)


(0) 000 0
(1)

y p y p( 1) ( 2) ( 3)

( )  

y p 
y p

y p
y p

yn y n y n yn







y(1)

yn p

( 1) (2) (3) (4)  (
Dynamical measurements y [ (1), (2), , ( )]t
)
 y y yn are autocorrelated
such that the whiteness assumption of the classical
least squares is violated. Indeed, the joint distribution of y is
given by:
y N Xa  2


( ,)

Nonetheless, by virtue of the theorem of Yule-Walker, the
least squares solution remains valid for autoregression problems
[6].
Theorem: Let y [ (1), (2), , ( )]t
 y y yn
AR(p) process such that:
12
yn a yn a yn
() ( 1)
June 2023
( 2)
be a weakly stationary
      ) ()
a yn p u n
p
(
where the aforementioned limit denotes convergence almost
surely.
For the proof, we refer to [6]. As a result, the estimation
of AR-models can be immediately implemented as it boils
down to solving a system of equations within the Lapack routines
where the most common numerical scheme is using the
QR-decomposition.
Autoregressive Moving-average Model
Identification: Re-weighted Iterative Least
Squares
Consider an ARMA(p,q) model such that:
yn a yn a yn
1
( 1)
2
( )        0
(
)
( 2) pq )
a yn p b u n b1u n( 1)
( )    
b u(n q
wherefore we wish to extend the least squares approach which
suggests solving the following regression problem:
yn a yn a yn
1
( 1)
2
( )        ) 01
(
( 2) pq )
a yn p b u n b u n( 1)
( )    
b u(n q
Unfortunately this regression problem is readily solvable
as the innovation sequence u(0), u(1), ..., u(n) are unobserved.
In order to circumvent this problem, one can initialize the
identification by the Yule-Walker Least Squares. Although
this incorrectly assumes an AR-model by ignoring the moving
IEEE Instrumentation & Measurement Magazine
31
and
2
with innovations uN I(0,
2



n ) such that these are a Gaussian
white process then the following holds:
tt
  


EX X a EX y


such that the least squares estimator is asymptotically
consistent
lim( )tt
XX X y a

1
 
n
Instrumentation & Measurement Magazine 26-4

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