Instrumentation & Measurement Magazine 26-4 - 30

educationI&M continued
in
time domain as in the frequency domain by its periodogram.
Fig. 5 shows the differenced Jamar dynamometer time series in
the left plot and the periodograms per segment in the right plot.
One concludes that the stationarity condition is sufficiently applicable
for the time series x(n). Note that a first order difference
is used in the current example since the observed trend in Fig. 4
was linearly decreasing. Consider a quadratic trend which will
require a second order difference operator instead. As a result,
an ARIMA(p,d,q) is a way to model non-stationary time series
which can be transformed into a stationary time series by applying
a d-th order difference operation. Nonetheless, the usual
interpretation for the remaining parameters is kept.
ARIMA Model: Identification Routines
Fig. 3. Periodogram consisting of 1 reference segment (solid red line) and
its 95%-uncertainty bounds (red shaded area) as well as the other segments
(colored dots).
strength decays over time due to muscle acidity the time series
reveals a linear trend violating (6). The periodogram analysis
shows a vertical shift in the right plot of Fig. 4b due to the decrease
in power per segment. This decrease is significant in
terms of the periodogram uncertainty associated to the first
segment. If a violation is detected, one typically modifies the
model structure to cope with non-stationary behavior. Indeed,
one applies an Autoregressive Integrated Moving Average
time series model. This model extends (5) to the following
model structure:
 ( )       



x n a 1x n a x n( 2)
( 1)
2 pq yn 12a yn a yn
( ) ( ) ( 1)
x n yn yn
a x(n p b0un b1un
 
)
( )    
( 1)
(
b un q)
(11)
In order to assess the suitability of this ARIMA(p,1,q)
model, where the parameter 1 implies that a first order difference
is used, one can visualize the time series x(n) both in the
() ( 1)
and Model Configuration
Different routines are available for the parameter identification
of ARIMA models where most methods are either
belonging to the least squares framework or to the likelihood
framework. For a full overview we refer to [11]. In this tutorial
we restrict the presented methodology to the least squares
framework. Considering the easier case of an AR-model the
identification boils down to the classical least squares whereas
ARMA as well as ARIMA models requires an iterative reweighted
least squares [12].
Autoregressive Model Identification: YuleWalker's
Least Squares
Consider an autoregressive model AR(p) of the form:
      ) ()
( 2)
a yn p u n
p
(
This equation can be easily written in a regression framework
as follows:
yn 12a yn a yn
( 1)
( )        ) ( )
( 2)
a yn p u n
p
(
Fig. 4. Jamar dynamometer. (a) Time series; (b) Periodogram analysis.
30
IEEE Instrumentation & Measurement Magazine
June 2023
Instrumentation & Measurement Magazine 26-4

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