Instrumentation & Measurement Magazine 26-4 - 39

Z = QX = QAS = VS
(4)
where V is an orthogonal matrix which is easy to be estimated
since it has very few parameters. From (4), S can be determined
as follows:
SV ˆ  TT
ZV QX
relative to a cost function. The Kantorovich Distance between
the two distributions can be computed in the following manner
[9]:
(5) 

2
,
 Trb a ba
2
W A B          (11)
22
1





a
From (2) and (5), a relationship between W and V can be
expressed:
W VQ
T

(6)
The next task is to estimate the ICs, and this is possible by
utilizing a fixed-point algorithm that is based on maximizing
non-Gaussianity using negentropy approximation [9]. Once
the matrix of ICs is determined, the next task is to select the k
dominant ICs, and this is achieved using the cumulative percentage
variance (CPV) technique. The developed ICA model
is made up of systematic space that represents a model having
k dominant ICs, excluded space having m-k ignored ICs and
the residual space. The ICA-based anomaly detection strategy
consists of anomaly (abnormal event) indicators that can monitor
three parts of ICA model, and they can be represented in
the following manner:
I  TT
2
d
2
e
I  TT
XWW X
kk
XW W X
mk m k

SPEeeT
ˆ
where, e  XX
(10)
ˆ
where X Q V WX. Using the model parameters, the refer
1

k
When new data is available, these indicators are computed
and compared with the reference threshold to check for the
abnormal events. The reference threshold is computed using
kernel density estimation (KDE) technique [9]. The abnormal
event indicators provide an efficient way of detecting abnormal
situations, but they cannot determine the sensor variable
responsible for the abnormal condition. The objective of the
isolation procedure consists of identifying the root cause or
variable responsible for the abnormal condition. Here, an
isolation procedure based on contribution plots is used to determine
the variable responsible for improper sitting posture.
The residuals from the model described in (10) are plotted as a
bar graph to get the contribution plot. The relative size of the
bar graph indicates the contribution of each variable to residuals.
A high value on the bar graph indicates the variable that is
responsible for the improper sitting posture.
Kantorovich Distance: belongs to the family of optimal mass
transport theory primarily used to locate data from one distribution
to the other. For any two distributions, A and B, the
Kantorovich Distance can be defined as the mode of transferring
a mass of data from the first distribution to the second
June 2023
k
ence threshold is computed for the abnormal event detectors
using the KDE technique. Next, the abnormal sitting posture
wheel-chair data Y is obtained and normalized to have zero
mean. Using the reference ICA model parameters, the residuals
R2 are generated for the abnormal data Y as follows:
ˆ
R  YY
2
ˆ
where Y Q V WY. Next, the KD statistic is computed be
1

k
k
tween the residuals R1 and R2 using (11) and compared with
the reference threshold. Any improper sitting posture will be
highlighted whenever the value of the KD statistic exceeds the
threshold.
Results and Discussion
Data Description
This part is dedicated to evaluating the efficiency of the proposed
approach in detecting abnormal sitting poses in a
wheelchair. The experiments are accomplished through actual
data from a publicly available database provided in [7].
Two types of sensors are used to collect pressure: HC-SR04
IEEE Instrumentation & Measurement Magazine
39
(13)
(7)
(8)
(9)
ˆ
R 1  XX
(12)
and μb
correspond to the means while
2 
11 2
a
b
In this expression, μa
represent the covariance matrix of the two distributions.
The distance between distributions A and B will be small
if they are similar and large if they are dissimilar. This concept
can be taken forward to be utilized for abnormal event
detection problems that involve comparing normal data with
abnormal data. For continuous time-series data, the KD metric
can be computed using a segmentation process where data
from both distributions are stacked into different segments
followed by segment-to-segment comparison to ensure that
the minor details in both distributions are captured very efficiently.
This property has made the KD-based statistic very
effective to be applied in the domain of abnormal event detection
problems.
ICA-KD Monitoring Approach: This paper integrates the ICA
model with the KD-based detector to identify incorrect sitting
postures of wheelchair users. Initially, the normal sitting
posture wheelchair data X is normalized to zero mean, and
then a reference ICA model is developed. The developed ICA
model is used to generate the residuals R1 using the following
expression:
Σa and Σb
1
2
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