Instrumentation & Measurement Magazine 26-4 - 33

Fig. 7. ARMA(4,2) model: Durbin's method (red), LS method (green) and true
system (blue).
spectrum partitioned in 8 segments reveals the zero at 15π/40
rad corresponding to the root of the moving average part.
The sharpest resonance around 9π/40 rad is visible while the
weaker resonance at 24π/40 rad is more difficult to pin-point.
Modeling the ARMA data with both algorithms gives very
similar results. In Fig. 7, the modeled power spectra by the
two methods are shown with their confidence intervals computed
over 100 simulation runs. The resonances are for both
algorithms similarly estimated while the zero is clearly different.
The zero can be best modelled by Durbin's method. Note
that this is a challenging example due to the sharp minimum
present in the true system's power spectrum. As a result, it is
important that the starting values are of the highest quality
which is best represented in Durbin's method.
ARIMA Model Prediction
Besides gaining insight in the data generating process, one applies
time series models in order to predict the future. Model
prediction boils down to simulating the future while restricting
the possibilities to what is most likely.
In this section, we use a time series consisting of the number
of intensive care beds used per day at the university hospital
of Brussels in Belgium. We model the time series as an ARIMA
process with the aim to predict its future. The observed hospital
beds is illustrated in Fig. 8. For modeling the time series
we use the data coming from the period January 1, 2020 up to
December 31, 2020, while we predict the entire year 2021. The
time series is not weakly stationary as there is a slowly varying
drift. The linear drift is sufficiently piecewise linear such that
differencing the time series turns its behavior sufficiently stationary
as illustrated in Fig. 9.
The dynamics of the time series in Fig. 9 is modeled as an
ARIMA process. The best model order by maximizing the
Akaike's Information Criterion (AIC) implies an ARIMA(3,1,2)
process. In Fig. 10, the pole-zero map is shown of the estimated
ARIMA model. The pole at location z = 1 is due to the first order
differentiation. The complex conjugate poles are given by:
June 2023
Fig. 8. Intensive care beds (university hospital Brussels): data (green), ARIMA
model (red).
z 0.2434876 0.6358455 0.6808713exp( 2 0.1917955)ii




As a result, the natural resonance exhibits a periodicity of
5.213885 samples which is consistent to the week-weekend effect.
There is also one remaining real pole identified at -0.1339892
which is not immediately easy to interpret. The zeros are given by:
z 0.2989688 0.4857888 0.5704148exp( 2 0.1621962)ii




The zero marks the start of the bandwidth of the complex
resonance at 2π × i0.1917955 radians as a local minimum in the
spectrum as shown in the right plot of Fig. 10b.
Simulating the Time Series: Resampled
Innovations
Through modeling the time series by an AR(I)MA model, we
can estimate the innovations denoted by
u ˆ . The innovations
as an input sequence are assumed to be a white process and
Fig. 9. Intensive care beds (university hospital Brussels): differenced time
series (red).
IEEE Instrumentation & Measurement Magazine
33
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