Instrumentation & Measurement Magazine 26-4 - 34

educationI&M continued
in
Fig. 10. ARIMA(3,1,2) model. (a) Pole (x)-zero (o) map; (b) Power spectrum.
hence uncorrelated as a function of time. Based on the estimated
innovations, we can resample the sequence. Consider
u
a resampled innovation sequence if its elements () are
un
uniformly drawn (with replacement) from the estimated innovations
ˆˆ ˆ
u(0), (1), , ( 1) .
u uN
To simulate the future as well as the past, we consider a
resampling of length N + Nh implying Nh samples in the future.
The resampled sequence u is then ARIMA-filtered by the
identified model to obtain the simulated time series y whose
elements are given by:
 ˆˆ ˆp (
yn 12a yn a yn
( )       
( 1)
  
ˆˆ ˆ
( )
bun bun bun q
 
[ 1]
mm m
q
 1
[ 1]
( 1)   )
[ 1]
(
Note that simulating an entire record of length N + Nh
plies a smooth transition from past to future.
imBackcasting
the Past
Simulating the time series provides many time series which
do not match the observations measured. If a correct projection
of the future must be made, we consider these
simulation which sufficiently match the historical time series.
Only these matches will give a reasonable continuation
into the future. The backcast error of simulation k is the root
mean squared error of the simulated time series on the historical
record 0 ≤ n ≤ N - 1:
1
v 
N
kk
n
N 0
 ( ) ()
 
yn y n
These backcast errors can be sorted where the user
can choose a threshold where a certain fraction of the best
simulations are used. One can use the best 1% simulations corresponding
well to the measured observations. The average of
34
2
Pre ()diction
( 2)
a yn p)
the retained 1% best simulations as well as the variance provides
the mean prediction into the future together with its
uncertainty interval.
Consider that the index set I consists of the retained simulations.
As a result, the mean prediction into the future together
with the prediction uncertainty can be estimated by:
1
ˆ
y
Prediction() 
n yn N n N N
I
||  () for
j
jI

ˆ
2

Prediction n) 
( ˆ
|| 1   ()yn yprediction n N n N Nh
() for
1
I
j
jI
(13)
where |I| denotes the size of the set I. The larger set I becomes
the smoother the estimate ˆ
yn becomes. This procedure
is illustrated in Fig. 11. where the time horizon is the entire year
2021 where the set I consists of 100 possible paths which align
to the past observations. Although the validation observations
in Fig. 11 fall within the prediction uncertainty interval, the observations
in April '21 follow a more extreme path running up
to the upper limit of the 95% prediction interval. The situation
of April '21 corresponds to one of the final Covid waves of the
Alpha variant striking Belgium. This mid-term wave is predicted
by the model but in general far less extreme than what
was observed. The upward trend during the fall period is also
corresponding to a Covid wave which is accordingly to the
model persistently present in November 2021 implying a significant
increase.
Conclusion
In this paper an introduction is given to ARIMA modeling.
This included some descriptive tools to explore data as well as
analytical methods to identify such ARIMA models. Finally,
IEEE Instrumentation & Measurement Magazine
June 2023
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Instrumentation & Measurement Magazine 26-4

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