Data-Driven Sensor Selection for Signal Estimation of Vertical Wheel Forces in Vehicles

Abstract

Sensor selection is one of the key factors that dictate the performance of estimating vertical wheel forces in vehicle durability design. To select K most relevant sensors among S candidate ones that best fit the response of one vertical wheel force, it has S!/(K!(S-K)!) possible choices to evaluate, which is not practical unless K or S is small. In order to tackle this issue, this paper proposes a data-driven method based on maximizing the marginal likelihood of the data of the vertical wheel force without knowing the dynamics of vehicle systems. Although the resulting optimization problem is a mixed-integer programming problem, it is relaxed to a convex problem with continuous variables and linear constraints. The proposed sensor selection method is flexible and easy to implement, and no additional hyper-parameters needed to be tuned using cross-validation. The feasibility and effectiveness of the proposed method are verified using experimental data in vehicle durability design. The results show that the proposed method has good performance with different data sizes and model orders, in providing sub-optimal sensor configurations for estimating vertical wheel forces in vehicles.

Full text

Xueke Zheng
UM-SJTU Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: zxk2046@sjtu.edu.cn
Ying Wang
UM-SJTU Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: dadide@sjtu.edu.cn
Le Wang
UM-SJTU Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: wangle@sjtu.edu.cn
Runze Cai
UM-SJTU Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: cairunze@sjtu.edu.cn
Mian Li
1
Professor
Fellow ASME
UM-SJTU Joint Institute,
Global Institute of Future Technology,
Department of Automation,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: mianli@sjtu.edu.cn
Yu Qiu
SAIC Motor Corporation Limited,
Shanghai 201800, China
e-mail: qiuyu_1123@163.com
Data-Driven Sensor Selection
for Signal Estimation of Vertical
Wheel Forces in Vehicles
Sensor selection is one of the key factors that dictate the performance of estimating vertical
wheel forces in vehicle durability design. To select K most relevant sensors among S can-
didate ones that best fit the response of one vertical wheel force, it has S
K

possible
choices to evaluate, which is not practical unless K or S is small. In order to tackle this
issue, this paper proposes a data-driven method based on maximizing the marginal likeli-
hood of the data of the vertical wheel force without knowing the dynamics of vehicle
systems. Although the resulting optimization problem is a mixed-integer programming
problem, it is relaxed to a convex problem with continuous variables and linear constraints.
The proposed sensor selection method is flexible and easy to implement, and the hyper-
parameters do not need to be tuned using additional validation data sets. The feasibility
and effectiveness of the proposed method are verified using numerical examples and exper-
imental data. In the results of different data sizes and model orders, the proposed method
has better fitting performance than that of the group lasso method in the sense of the 2-norm
based metric. Also, the computational time of the proposed method is much less than that of
the enumeration-based method. [DOI: 10.1115/1.4055514]
Keywords: data-driven engineering, industrial internet of things, machine learning for
engineering applications, qualification, verification and validation of computational models
1 Introduction
When designing a new vehicle, vertical wheel force (i.e., the ver-
tical force acting at a wheel center) is one of the important quantities
to assess the durability performance of prototypes. In practice, ver-
tical wheel forces are usually measured directly by the so-called
wheel force transducers (WFTs) during road load data acquisition
(RLDA) testing campaigns in which prototype vehicles are driven
on proving grounds or public roads [1,2]. However, WFTs are
expensive, intrusive, and time-consuming to install in vehicles,
especially multiple prototypes are under tested simultaneously.
Thus, it is of great interest to alternatively obtaining vertical
wheel forces without measuring them directly.
Soft sensors are combination of easy-to-measure variables in the
processing plants with models for delivering estimates of hard-to-
measure variables [3–5]. Note that the data of hard-to-measure var-
iables are usually not available in most times (e.g., the online esti-
mation process) other than a short of time (e.g., the offline training
process). The sensors for measuring easy-to-measure variables are
called easy-to-measure sensors (e.g., the accelerometers and displa-
cement sensors), and the sensors for measuring hard-to-measure
variables are called hard-to-measure sensors (e.g., vertical wheel
forces or other force sensors in vehicle durability design). Soft
sensors based on multibody simulation models have been applied
to estimate vertical wheel forces [1,6,7]. However, high-accuracy
multibody simulation models are usually difficult to obtain in
many scenarios, where data-driven soft sensors are preferred to esti-
mate vertical wheel forces [8]. Furthermore, due to good properties
such as bounded-input bounded-output (BIBO) stability, multiple-
inputs single-output (MISO) finite impulse response (FIR) models
have proven to be good candidate ones to estimate vertical wheel
forces and other hard-to-measure variables in vehicle durability
design [7,9]. Specifically, FIR models are identified in the offline
training process, and then the vertical wheel forces are estimated
using the easy-to-measure sensors and identified FIR models in
the online estimation process.
In previous work [7], a large amount of easy-to-measure sensors
are installed specifically only to estimate vertical wheel forces, and
these easy-to-measure sensors are available for severing as inputs of
FIR models. Note that some of the easy-to-measure sensors may be
irrelevant to the vertical wheel forces in identifying FIR models.
Thus, the economic cost could be further reduced if these irrelevant
1
Corresponding author.
Contributed by the Computers and Information Division of ASME for publication
in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript
received April 15, 2022; final manuscript received August 31, 2022; published
online December 9, 2022. Assoc. Editor: Bin He.
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sensors are removed, which is particularly important in RLDA
testing campaigns. Also, it has been shown by many studies that
the fitting performance of regression models can be tremendously
improved if only vital sensors are included in the development of
regression models [10,11]. However, in previous work [7], the strat-
egy of selecting relevant or vital easy-to-measure variables as inputs
of FIR models is based on physical insights or trial and error.
This work aims to effectively and efficiently select Ksensors
among Seasy-to-measure sensors that fit vertical wheel forces in
identifying MISO FIR models. The determination of Kshould be
done considering several factors, e.g., the fitting performance and
the cost of installation and maintenance. One simple method for
solving this sensor selection problem is to evaluate the performance
of all S
K

candidate solutions. Similar works have been con-
ducted in the cases of small Kand S[12,13]. However, obviously
it is impractical in the scenarios when the value of S
K

becomes large. For example, in the estimation of one vertical
wheel force (not to mention considering other hard-to-measure var-
iables in the vehicle durability design), for K=10 and S=34, to
find the globally optimally best sensor selection by direct enumer-
ation would require evaluating the performance approximately
34
10

times, which is on the order of 10
9
times and clearly very
difficult, if not practically impossible. Also, some global optimiza-
tion techniques, such as branch and bound [14,15], can and often
run for very long time [16]. Thus, an effective and efficient
sensor selection strategy is in high demand for estimating vertical
wheel forces when the value of S
K

is large.
Prior and related work: Many researches on sensor selection or
optimal sensor placement aim at developing effective and efficient
methods to deal with specific applications, such as system monitor-
ing [17], manufacturing process control [18], health management
allocation [19], and structural design [20]. Many works focus on
sensor selection in the scenarios where the system dynamics is
known in advance. For example, optimal sensor selection strategies
are proposed in terms of system observability in the Kalman-
filtering framework [1,6]. Some heuristic methods such as generic
algorithm or particle swarm optimization technique are applied to
sensor selection field by defining objective functions based on spe-
cific properties of finite element models [21,22]. Sensor selection
methods based on the conditioning analysis of the Markov param-
eters matrix or Bayesian formulation to select groups of features are
used in the identification of the input force [23,24]. The authors in
Refs. [16,25,26] propose sensor selection strategies based on max-
imizing the norm or determinant of Fisher information matrices for
modal analysis of large-scale structures. Sensor selection based on
information entropy that minimizes the uncertainty in the model
parameters of structures is also discussed in Refs. [27,28]. These
sensor selection methods rely on (accurate) first-principle models
and have proven to be very effective and efficient. However, as
mentioned previously, the first-principle models are not often
achievable in vehicle durability design, and hence these methods
may not be directly applicable to sensor selection in fitting vertical
wheel forces.
On the other hand, some studies focus on data-driven methods for
sensor selection. Different from the model-based sensor selection
methods, most data-driven methods do not rely on first-principle
models. For example, typical machine learning methods combining
with domain knowledge are applied to automatically select an
optimal set of real and virtual sensors to improve the system
energy performance through fault detection and system health mon-
itoring [29]. Data-driven methods based on clustering algorithms,
information loss approach, and Pareto principle are used to derive
the optimal sensor placement strategies for intelligent building
energy management [30]. Due to the simple implementation
using modeling packages such as CVX [31], the group lasso
method has been proven as a popular, effective, and efficient
sensor selection strategy in many signal processing applications
[24,32–34]. In these applications, multiple coefficients of each
feature are grouped together and share similar structures, and the
group lasso method provides both inter-group and intra-group spar-
sity to remove irrelevant coefficients. In this work, the group lasso
method could be easily adopted in identifying FIR models. It is
worthwhile to note that the hyper-parameters in the group lasso
method need to be carefully tuned to balance the regularization
terms and the original cost function by using an additional round
of cross-validation [34]. Also, as mentioned in Refs. [17,35], the
results of the group lasso may not return the smallest possible set
of groups that are sufficient to obtain an accurate data-driven model.
In this work, a simple and effective sensor selection method is
developed and presented by maximizing the marginal likelihood
of the data of the output in FIR models. First, we innovatively for-
mulate the problem into a standard mixed-integer programming
problem by maximizing the marginal likelihood of the output
data. Then, we approximately relax the mixed-integer programming
problem into a convex optimization problem with continuous vari-
ables and linear constraints, which can be solved efficiently. To the
authors’best knowledge, this work is the first one in data-driven
sensor selection for estimating hard-to-measure variables in
vehicle durability design. For example, in Refs. [1,6], the authors
discuss the optimal sensor selection strategies relying on multibody
simulation models in estimating wheel center loads in vehicles. It is
believed that using the proposed method with the real test data will
facilitate the durability design of new vehicles.
Compared to the group lasso method and other data-driven
methods, the proposed method brings the benefits of
(i) small possible subsets of sensors while retaining the same
model accuracy;
(ii) no additional data sets are needed to tune hyper-parameters.
Test results of sensor selection in vehicle durability design power-
fully prove that the proposed method provides small possible
subsets of sensors to realize the balance of the fitting performance
and deployment cost of easy-to-measure sensors. Without loss of
generality, we focus on the sensor selection of the vertical wheel
force on the front left of the vehicle hereafter, the sensor selection
of the other hard-to-measure variables in vehicles can be performed
in a similar manner.
The rest content of this paper is organized as follows: In Sec. 2,
we formally define the sensor selection problem in the context of
parameter estimation. In Sec. 3, we present the proposed method
based on convex optimization that can efficiently solve this
problem. We illustrate the proposed sensor selection method with
numerical examples and experimental data in Secs. 4and 5, respec-
tively. Some concluding remarks are presented in Sec. 6.
2 Sensor Selection
In this section, preliminaries on identifying MISO FIR models
are given in Sec. 2.1. Then, the sensor selection problem is pre-
sented in Sec. 2.2.
Fig. 1 The transmissibility Tfrom the easy-to-measure vari-
ables y
I
to the hard-to-measure variable y
O
. The input uis
unknown.
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2.1 Parameter Estimation. Consider an unknown linear time
invariant (LTI) system as shown in Fig. 1, where udenotes the
unknown excitation or input to the system, and y
I
as well as y
O
are the outputs of the system. The relationship from y
I
to y
O
can
be described as a time-domain transmissibility [13,36,37],
denoted by T. Here y
I
refers to easy-to-measure variables (e.g.,
the accelerations and displacements), and y
O
refers to the
hard-to-measure variable (e.g., the vertical wheel force).
According to Refs. [7,13,37], the time-domain transmissibility T
can be estimated by a noncausal MISO FIR model
yO(k)=
S
s=1
ϕs(k)Tθs+e(k),k=d,...,N+d−1(1)
where ϕ
s
(k)=[y
I,s
(k−d)···y
I,s
(k+r)]
T
is the sth regressor associ-
ated with sth easy-to-measure variable denoted by yI,s(k)∈R,r
and dare model orders, e(k) is the residual modeled by white
noise with mean zero and variance σ
2
, and yO(k)∈Ris the
hard-to-measure variable. Let θs∈Rndenote the model parameters
associated with y
I,s
, where n=r+d+1. In the offline training
process, Eq. (1) can be extended to
Y=
S
s=1
Φsθs+E(2)
=Φϑ+E(3)
where
Y=yO(d)···yO(N+d−1)

T∈RN(4)
Φs=ϕs(d)···ϕs(N+d−1)

T∈RN×n(5)
E=e(d)···e(N+d−1)

T∈RN(6)
Φ=Φ1···ΦS

∈RN×Sn (7)
and ϑ=[θT
1··· θT
S]T∈RSn, by stacking all historic data together.
It is a well known fact [16] that the least squares estimator of ϑis
ˆ
ϑLS =(ΦTΦ)−1ΦTY(8)
When Φ
T
Φis close to being singular, which indicates that the
easy-to-measure variables are not persistently exciting [37,38],
the least squares estimation may cause large variances [39–41]. In
these cases, the ridge regression strikes a good balance between
the biases and variances of the estimates [40,41]. The solution of
ridge regression can be obtained from the Bayesian point of view
[41]. Under Bayesian assumption that ϑis Gaussian with zero
mean and covariance matrix λI
Sn
, i.e., ϑ∼N(0,λISn), the posterior
probability density function (pdf) is given by Bayes rule
p(ϑ∣Y)=p(Y∣ϑ)p(ϑ)
p(Y)(9)
where p(Y∣ϑ) is the likelihood function corresponding to
Y∼N(Φϑ,σ2IN), and p(ϑ) is the prior pdf corresponding to
ϑ∼N(0,λISn), and the pdf p(Y)isaϑ-independent normalization
which will be clear shortly. Apart from p(Y) and other
ϑ-independent terms, twice the negative logarithm of (9)
−2lnp(ϑ∣Y)=−2( ln p(Y∣ϑ)+ln p(ϑ)−ln p(Y))
=Y−Φϑ2
2/σ2+N(ln(2π)+ln (σ2)) (10)
+ϑ2
2/λ+Sn(ln(σ2)+ln (2π)) +2lnp(Y)(11)
=Y−Φϑ2
2/σ2+ϑ2
2/λ+constant (12)
Let
L(ϑ):= Y−Φϑ2
2/σ2+ϑ2
2/λ(13)
Minimizing Eq. (13) leads to ridge regression estimator of ϑ
ˆ
ϑR=ΦTΦ+σ2
λISn−1
ΦTY(14)
where λand σ
2
are hyper-parameters and can be estimated by
maximizing the marginal likelihood p(Y|λ,σ
2
), i.e., the probability
of Yconditioned on λand σ
2
, using empirical Bayes method
[40–42]. Assume ϑ∼N(0,λISn). According to Eq. (3),
Φϑ∼N(0,λS
s=1Ψs), where Ψs=ΦsΦT
s, is a positive semi-
definite matrix, and s=1, …S. Thus, Ywill be a Gaussian
random vector with mean zero and covariance matrix
P(λ,σ2)=λ
S
s=1
Ψs+σ2IN(15)
Thus, the marginal likelihood of Ycorresponds to
Y∼N(0,λS
s=1Ψs+σ2IN). As similar to Eq. (13), it can be ver-
ified that twice the negative logarithm of p(Y|λ,σ
2
) leads to (omit the
terms independent of λor σ
2
)
W(λ,σ2):= YTP(λ,σ2)Y+log det P(λ,σ2)(16)
and maximizing the marginal likelihood function p(Y|λ,σ
2
) w.r.t. λ
and σ
2
leads to
ˆ
λ,ˆσ2=argmin
λ>0,σ2>0
W(λ,σ2)
=argmin
λ>0,σ2>0
YTλ
S
s=1
Ψs+σ2IN−1
Y(17)
+log det λ
S
s=1
Ψs+σ2IN(18)
Problem (18) is difference of convex programming (DCP) problem
and some nonconvex optimization methods can be used to obtain
high-accuracy locally optimal solutions [40,43].
2.2 Sensor Selection Problem. One may use many
easy-to-measure sensors to estimate the hard-to-measure variable
y
O
[1,6,7]. For the sake of simplicity, one sensor corresponds to
one easy-to-measure variable. Then there may have a scenario
where some sensors are not related to the hard-to-measure variable
y
O
, that is, do not contribute to fitting the hard-to-measure variable
y
O
, and hence these irrelevant sensors are aimed to be removed from
Seasy-to-measure sensors. Whereas we do not know which
easy-to-measure sensors are irrelevant ones, since the system
dynamics is unknown.
In addition to estimating the hyper-parameters λand σ
2
, we can
further consider the situation that a subset of only Ksensors out of
Seasy-to-measure sensors (K<S) is chosen in the context of max-
imizing the marginal likelihood of Y. Following Eqs. (15)–(18), this
can be expressed by the optimization problem
minimize
c1,...,cS,λ,σ2YTλ
S
s=1
csΨs+σ2IN−1
Y
+log detλ
S
s=1
csΨs+σ2IN
subject to cs∈{0,1},s=1,...,S

S
s=1
cs=K
λ>0,σ2>0
(19)
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which is a mixed-integer programming problem. The first term in
the objective function (19) is related to the C-optimum design
[44], i.e., minimizing the inverse of the covariance matrix
λS
s=1csΨs+σ2INin the direction of vector Y. Here c
s
encodes
whether the sth sensor is used to fit the data of the hard-to-measure
sensor. Note that the data of sth sensor (i.e., Ψ
s
) can be easily
removed from the data of Seasy-to-measure sensors by simply
setting c
s
to be zero.
3 Approximate Relaxation of Sensor Selection
In this section, we will describe the method solving the sensor
selection problem in Sec. 2approximately but efficiently, while
retaining a good accuracy.
3.1 Approximate Relaxed Optimization Problems. By
replacing the nonconvex constraints c
s
∈{0, 1} with convex con-
straints c
s
∈[0, 1] and minimizing an upper bound of the objective
function of (19), we obtain the approximate relaxation of the sensor
selection problem (19) as
minimize
c1,...,cS,λ,σ2YTλ
S
s=1
csΨs+σ2IN−1
Y
+λ
S
s=1
cstr(Ψs)+Nσ2−N
subject to 
S
s=1
cs=K
0≤cs≤1,s=1,...,S
λ>0,σ2>0
(20)
where the upper bound of the objective function of (19) is given
using the inequality log det(X)≤tr(X−IN) for any positive definite
matrix X∈RN×N. This inequality is often used to approximate the
log-determinant term, e.g., in Refs. [45,46]. The problem (20) is not
equivalent to the original sensor selection problem (19), and a sub-
optimal subset selection can be obtained in indices associated with
the Klargest values of the solution c
1
,…,c
S
to the problem (20).
Also, note that the objective function of the problem (20) is noncon-
vex w.r.t. c
1
,…,c
s
,λ,σ
2
, and thus it is difficult to obtain a global
optimum. Nevertheless, this difficulty can be eliminated by defining
new variable h
s
=λc
s
minimize
h1,...,hS,λ,σ2YT
S
s=1
hsΨs+σ2IN−1
Y
+
S
s=1
hstr(Ψs)+Nσ2−N
subject to 
S
s=1
hs=Kλ
0≤hs≤λ,s=1,...,S
λ>0,σ2>0
(21)
which comes down to a convex problem. Note that the calculation
of inverting the matrix S
s=1hsΨs+σ2INis computationally inten-
sive for large data size Nsince the size of this matrix depends on the
data size N. Also, it would be problematic to solve problem (21)
when the matrix S
s=1hsΨs+σ2INis close to being singular.
These two issues will be tackled in Sec. 3.2.
For problems with the large data size N,itisusefultoefficiently
supply the gradient and Hessian of the objective function in problem
(21). This avoids expensive computation of their finite-difference
counterparts. Fortunately, the analytical expressions of the gradient
and Hessian are tractable and are given in details in the Appendix.
In addition, we can impose the selection constraints of the sensors
as linear equalities or inequalities on the variable h=[h
1
,…,h
S
]
T
in
the approximate relaxed problem (21). For example, the following
two types of constraints are common and useful in practice.
Logical constraints: For example, the constraint that sensor ican
be chosen only when sensor jis also chosen is expressed as h
i
≤h
j
;
Budget constraints: A maximum allowed cost for the selection,
as z
T
h≤λB, where z=[z
1
···z
S
]
T
,z
i
is the cost, say, in
dollars, power, or weight associated with choosing sensor
i, and Bis the budget.
More details on these additional constraints can be found in
Ref. [16]. It is worthwhile to note that it may not be easy for
other sensor selection methods (e.g., the group lasso [34]) to take
these constraints into account.
Let h⋆=[h⋆
1···h⋆
S]Tdenote a solution to the problem (21). The
greater of the value h⋆
s, the higher likelihood that the sth sensor is
chosen. A simplest possible method to generate a sub-optimal
subset selection denoted by ˆ
Kis as follows [16]. Let h⋆
i1,...,h⋆
iS
denote the elements of h⋆sorted in descending order. The subset
selection is then
ˆ
K={i1,...,iK}(22)
where the indices are the Klargest elements of h⋆.
According to Eq. (18) and ˆ
K, the hyper-parameters λand σ
2
can
further be optimized as follows:
ˆ
λ,ˆσ2=argmin
λ>0,σ2>0
YTλ
K
s=1
Ψis+σ2IN−1
Y
+log detλ
K
s=1
Ψis+σ2IN(23)
=argmin
λ>0,σ2>0
YTλ
K
s=1
Ψis+σ2IN−1
Y
+Nlog σ2+log detλ
σ2
K
s=1
Ψis+IN(24)
=argmin
λ>0,σ2>0
YTλ
K
s=1
Ψis+σ2IN−1
Y
+Nlog σ2+log detλ
σ2ΦT
ˆ
KΦˆ
K+IKn(25)
=argmin
λ>0,σ2>0
YTλ
K
s=1
Ψis+σ2IN−1
Y
+(N−Kn) log σ2+log det(λΦT
ˆ
KΦˆ
K+σ2IKn)(26)
where Φˆ
K=[Φi1···ΦiK], and the third equality holds using Sylve-
ster’s determinant theorem [40], and the initial guess of λand σ
2
,
denoted by ˆ
λ0and ˆσ2
0respectively, is provided from the solution
to the convex problem (21). The algorithm of identifying FIR
models is as follows.
Algorithm 1 (Identification of FIR models.) The algorithm
consists of following steps.
(1) Solve the convex problem (21) to obtain sensor selection ˆ
Kas well as
initial guess ˆ
λ0and ˆσ2
0;
(2) Solve the nonconvex problem (26) according to ˆ
K,ˆ
λ0and ˆσ2
0, to obtain
high-accuracy local optimum ˆ
λand ˆσ2;
(3) Estimate ϑusing ridge regression to obtain ˆ
ϑR=ΦT
ˆ
KΦˆ
K+ˆσ2
ˆ
λIKn

−1
ˆ
KTY.
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The estimate of the model parameter ϑcan also be obtained by least
squares method once the sensor selection ˆ
Kis chosen. The results of
the comparative study between the least squares method and the
ridge regression will be shown in Sec. 5.
3.2 Computation of σ2IN+S
s=1hsΨs−1.In this subsec-
tion, we consider how to deal with the computational issue of cal-
culating σ2IN+S
s=1hsΨs−1in the problem (21), when the data
size Nis very large or the matrix σ2IN+S
s=1hsΨs, denoted by
X
S
, is close to being singular.
The computational burden of the convex problem (21) increases
vastly as the data size Ngoes to infinity. The computational com-
plexity of calculating σ2IN+S
s=1hsΨs−1in the problem (21)
is O(N
3
+SN
2
), which could be problematic when Nis large,
e.g., Nis several thousand or even larger.
We attempt to reduce the computational complexity of calculat-
ing X
S
, such that solving the problem (21) could be possibly effi-
cient when Nis large. Here we calculate the X
S
in a recursive
way. Denote X
0
=σ
2
I
N
, and X
s
=h
s
Ψ
s
+X
s−1
, where s=1, …,S.
Then, we can compute X−1
sfrom X−1
s−1using matrix inversion lemma
X−1
s=X−1
s−1−hsX−1
s−1ΦsIn+hsΦT
sX−1
s−1Φs−1ΦT
sX−1
s−1(27)
where s=1, …,Sand Ψs=ΦsΦT
s. The computational complexity
of computing X−1
sfrom X−1
s−1is O(N
2
n+Nn
2
+n
3
). Thus, the com-
putational complexity of recursively computing X−1
Sfrom X−1
0is
O(SN
2
n+SN n
2
+Sn
3
), which could be far more efficient than
directly computing X−1
Sif N≫Sn.
Now we tackle the issue when X
S
is close to being singular. Note
that the magnitude of σ
2
I
N
could be small compared to that of
S
s=1hsΨs. In this case, XS=σ2IN+S
s=1hsΨscan be ill-
conditioned if S
s=1hsΨsis very ill-conditioned, which can cause
numerical issues, for example, the failure or inaccuracy of the cal-
culation of X−1
S.
According to matrix inversion lemma, we have
X−1
S=σ2IN+
S
s=1
hsΨs−1
=σ2IN+ΦCΦT−1
=σ−2IN−σ−2ΦC1/2(σ2ISn +C1/2ΦTΦC1/2)−1C1/2ΦT
(28)
where C1/2=diag( h
√1,...,
h
√S)⊗In, and ⊗is the Kronecker
product. Then, we resort to the thin QR factorization [40,47] to cal-
culate (σ
2
I
Sn
+C
1/2
Φ
T
ΦC
1/2
)
−1
more accurately, since it also could
be close to being singular. More specifically, consider the thin QR
factorization of
ΦC1/2
σISn

(N+Sn)×Sn
=Q1
Q2

R(29)
where Q1∈RN×Sn,Q2∈RSn×Sn , and R∈RSn×Sn is an upper trian-
gular matrix and is nonsingular. Note that
QT
1Q1+QT
2Q2=ISn (30)
ΦC1/2=Q1R(31)
σISn =Q2R(32)
σ2ISn +C1/2ΦTΦC1/2=RTR(33)
Substituting Eqs. (31) and (33) into Eq. (28) yields
X−1
S=σ−2IN−σ−2Q1R(RTR)−1RTQT
1
=σ−2(IN−Q1QT
1)(34)
such that X−1
Scan be accurately obtained without explicitly invert-
ing R.
The major computation complexity from Eq. (34) relies on the QR
factorization (29), which depends on the data size N. Note that Φis
fixed when solving the numerical optimization problem (21).Thus,
we can make use of the observation to compute QR factorization
(29) inamoreefficient way (to make the computational complexity
independent of N). Now consider the thin QR factorization of
Φ=QdRd(35)
where Qd∈RN×Sn whose columns are orthogonal unit vectors such
that
QT
dQd=ISn (36)
and Rd∈RSn×Sn is an upper triangular matrix. Now consider further
another thin QR factorization of
RdC1/2
σISn

2Sn×Sn
=Qc1
Qc2

Rc(37)
where Qc1,Qc2∈RSn×Sn ,andRc∈RSn×Sn is an upper triangular
matrix. Note that the matrix in factorization (37) is 2Sn-by-Sn dimen-
sional, which is independent of data size N. Also, we have
QT
c1Qc1+QT
c2Qc2=ISn (38)
RdC1/2=Qc1Rc(39)
σISn =Qc2Rc(40)
AccordingtoEqs.(35),(39),and(40),wehave
ΦC1/2
σISn

=QdRdC1/2
σISn

=QdQc1Rc
Qc2Rc

=QdQc1
Qc2

Rc(41)
AccordingtoEqs.(36) and (38),wehave
QT
c1QT
dQT
c2

QdQc1
Qc2

=QT
c1QT
dQdQc1+QT
c2Qc2=ISn (42)
which indicates that R=R
c
,and
Q1
Q2

=QdQc1
Qc2
 (43)
AccordingtoEq.(43),wehave
Q1=QdQc1(44)
Substituting Eq. (44) into Eq. (34) yields
X−1
s=σ−2(IN−QdQc1QT
c1QT
d)(45)
Therefore, we obtain an efficient and accurate way to calculate X−1
S=
(σ2IN+S
s=1hsΨs)−1using Eqs. (35),(37),and(45). The algorithm
of calculating (σ2IN+S
s=1hsΨs)−1is as follows.
Algorithm 2 Assume that the thin QR factorization (35) has
been computed, and Qd,Rdis available. Then the algorithm consists
of following steps to compute (σ2IN+S
s=1hsΨs)−1.
(Computation of (σ2IN+S
s=1hsΨs)−1).
(1) Compute the diagonal matrix C1/2;
(2) Compute Qc1according to QR factorization (37);
(3) Compute X−1
S=(σ2IN+S
s=1hsΨs)−1according to (45).
In MATLAB, the command qr(·,0) can be used to compute the thin QR
factorization (37), which costs O(S3n3)without involving the data size
N[48]. The total computational cost of calculating X−1
Sis
O(S3n3+NS2n2+N2Sn). Note that the computation of (σ2IN+
S
s=1hsΨs)−1above is performed when N≥Sn, the computation can be
performed in a similar way with minor revision when N<Sn.
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4 Numerical Validation
In this section, we demonstrate the effectiveness of the proposed
sensor selection method through numerical examples.
All systems are multiple-inputs multiple-outputs (MIMO)
systems with n
u
=4 inputs and n
o
=9 outputs. Note that the
inputs are generated using the command idinput which produces
random Gaussian signals. The first eight outputs serve as relevant
easy-to-measure variables, and the last output serves as the
hard-to-measure variable aimed to be estimated. In addition, 32
random Gaussian signals are generated using Matlab command
randn, to serve as irrelevant easy-to-measure variables. Thus,
there are 40 signals available where eight ones are relevant
easy-to-measure variables and the remaining 32 ones are irrelevant
easy-to-measure variables.
All n
o
=9 outputs are added by zero-mean white Gaussian noise
using the command awgn in MATLAB, and the signal-to-noise ratio
(SNR) is 40 dB. All data samples are normalized using the
command normalize, the training data size is N=1000, the
test data size for model validation is 500, and the FIR model
orders are set as r=d=30. The performance of the proposed
sensor selection method is compared with that of the group lasso
method [17,24,33,34]. The group lasso could be adopted in the
identification of FIR models by simply grouping the model param-
eters of each sensor together
minimize
θ1,...θS
Y−
S
s=1
Φsθs









2
2
+κ
S
s=1θs2(46)
The group lasso (46) is a convex problem and can be efficiently
solved. Here θ
s
,Φ
1
,…,Φ
S
, and Yare shown in Eq. (2). The
group lasso requires the tuning of a tradeoff hyper-parameter
denoted by κbetween a penalty term of the form S
s=1θs2and
the sum of squares of the training error. This hyper-parameter κis
usually tuned in the experiments by using an additional round of
cross-validation. In this work, the tradeoff hyper-parameter is
selected from the candidate set {1 × 10
−5
,5×10
−5
,…,1,…,5×
10
4
,1×10
5
}. Once the hyper-parameter and the model parameters
are determined, the rank order of sensors in the group lasso is deter-
mined using the norm of the model parameters for each
easy-to-measure variable, i.e., by sorting the easy-to-measure vari-
ables according to θs2, where s=1, …,S. After that, the associ-
ated model parameters of selected easy-to-measure variables are
re-calculated by least square methods.
The trained models are cross-validated using test (unseen) data
sets. To quantitatively evaluate the performance of the methods in
fitting the hard-to-measure variable, we define the measure of FIT
(i.e., 2-norm based metric) as [49]
FIT =100 ·1−Y−Φˆ
K
ˆ
ϑ


2
Y−
Y2%(47)
where Yis the data of the test data set, 
Yis the mean of all entries of
Y,Φˆ
Kis the regressor matrix generated by the selection of
easy-to-measure variables in test data. Also, ˆ
ϑis the estimate
from the least squares method or the ridge regression after the
subset selection ˆ
Kis determined. The performance gets better as
the value of FIT gets larger.
Figure 2shows the fitting performance versus the number of
sensors selected. All results are averaged over 100 independent
runs of the experiment. Note that the “proposed method with least
squares”refers to the fitting performance obtained from least
squares method and the proposed sensor selection method, and
the “proposed method with ridge regression”refers to the algorithm
1 shown in Sec. 3.1. The “ideal FIT with least squares”refers to the
fitting performance obtained using eight relevant easy-to-measure
variables and least squares method (as a benchmark). It is clear that
(i) The overall results of the proposed methods are better than
those from the group lasso method in the sense of the fitting
performance.
(ii) The results based on ridge regression is slightly better than
those based on least squares method, which indicates the
small advantage of the ridge regression over ordinary
linear regression in the numerical examples.
(iii) When K=8, the fitting performance of the proposed method
with least squares is very close to the ideal FIT with least
squares, which indicates that the proposed algorithm
enables to correctly choose the relevant easy-to-measure
variables.
Thus, the proposed methods show the applicability of sensor selec-
tion and have the stronger capability of choosing the most relevant
easy-to-measure variables to fit the hard-to-measure variable in
comparison with the group lasso method. This conclusion can
also be verified in more details through the real-world application
in the next section.
5 Experimental Validation
In this section, we demonstrate the applicability and effectiveness
of the proposed sensor selection method through the real data in
fitting the vertical wheel force.
Figure 3shows six easy-to-measure accelerometers installed in
the vehicle body. In Fig. 4, an accelerometer is installed close to
one of four wheel centers, and a displacement sensor is also
installed to measure the deflection of the shock absorber. Note
that all these accelerometers measure the accelerations in X,Y,Z
directions. Eighteen accelerations measured by six accelerometers
in the vehicle body are denoted by BX
i,BY
i,BZ
i, where i=1, …,6.
Four deflections measured by displacement sensors are denoted
by D
1
,…,D
4
. Twelve accelerations measured by four accelerome-
ters near the wheel centers are denoted by WX
i,WY
i,WZ
i, where
i=1,...,4. In this test the vehicle is running at the speed of
30 kmh
−1
straightly on a Belgium block road, and over 1 × 10
4
data samples are collected. The sampling rate is 512 Hz.
5.1 The Comparative Study Between the Proposed Method
and the Group Lasso Method. Figures 5and 6show the fitting
performance as the number of sensors selected is varied within
K=1, 2, …, 10 in different scenarios. In Fig. 5, we investigate
the influence of the training data size (N=1000, 1300, 1600,
2000) on the fitting performance given the model orders r=0,
d=50 (in Ref. [7], it is reported that it is sufficient to use causal
FIR models to fit the vertical wheel force). In Fig. 6, we investigate
the influence of the model order (r=0, d=30, 40, 50, 60) on the
fitting performance given the training data size N=1000. It is
Fig. 2 The fitting performance (FIT) versus the number of
sensors selected
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clear to see that the overall results of the proposed methods are
better than those from the group lasso in different training data
sizes and the model orders. Furthermore, it can be seen that the pro-
posed methods work robustly when the data size and the model
order are varied, whereas the performance of the group lasso is sen-
sitive to the variation of the data size and the mode order. The
results indicate that the proposed methods can return smaller possi-
ble subsets of sensors while retaining the same model accuracy,
compared to the group lasso method. Therefore, it clearly validates
the effectiveness and applicability of the proposed methods when
the training data size and the model order are varied.
From Figs. 5and 6, we also notice that the fitting performance is
not improved significantly when K≥4. The possible reason is that
some related easy-to-measure sensors are redundant when K≥4
(the corresponding easy-to-measure variables can be estimated by
others), and these additional sensors involved do not provide
more information for fitting the vertical wheel force. Also, the
overall performance of the ridge regression is marginally better
than that of least squares method, which indicates that it may not
be necessary using ridge regression after sensor selection.
Particularly, we list the selected sensors based on the results of
the proposed method in the scenario when N=1000 and d=50
(which corresponds to Fig. 5(a)), to see whether the results of
sensor selection coincide with the physical insights. Table 1
shows the selected sensors when K=3, 6:
(i) When K=3, the proposed method selects the displacement
sensors D
1
,D
2
and the acceleration WZ
1, on the front of the
vehicle. Selecting the acceleration WZ
1coincides with the
physical insights since the vertical wheel force and the accel-
eration WZ
1both are measured in the Zdirection, and the
locations of associated sensors are very close to each other
(both are on the front left of the vehicle). Selecting D
2
also
improves the result, which indicates that the two suspensions
on the front may be coupled with each other, i.e., the vibra-
tion of the suspension on the front right may also influence
the vibration of the suspension on the front left, and vice
versa (this was verified when we fit the vertical wheel
force on the front right of the vehicle).
(ii) When K=6, the deflections and the accelerations in Xand Z
directions on the front of the vehicle are involved in improv-
ing the fitting performance of the vertical wheel force on the
front left of the vehicle. Note that the accelerations in the Y
direction is not selected, which coincides with the physical
insight of the vehicle dynamics because the accelerations
in the Ydirection may not contribute to fitting the vertical
wheel force when the vehicle is running straightly. Also,
the sensors on the rear right are not selected. That is
because the locations of these sensors on the rear right are
far from the wheel on the front left, and hence are less
likely to be relevant to the vertical wheel force on the front
left.
From the results of Table 1, we see that the results of the proposed
method coincide with the physical insights and hence are
explainable.
5.2 The Comparative Study Between the Proposed Method
and Enumeration-Based Method. Furthermore, the proposed
method is compared with the enumeration-based method for
small K, i.e., K=1, …, 6. Here the enumeration-based method
refers to the method based on enumerating all S
K

candidate
choices: Once a candidate choice according enumeration is given,
Fig. 4 The locations of the wheel force transducer and the
accelerometers: (a) the wheel force transducer measuring
WCLs is installed on the wheels and (b) the accelerometer is
installed close to the wheel center
Fig. 3 The locations of the accelerometers and the wheel force
transducer (a) the locations of accelerometers installed in the
vehicle body (3D demonstration) and (b) the locations of acceler-
ometers installed in the vehicle body (top view)
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Fig. 5 The fitting performance (FIT) versus the number of sensors selected when r=50: (a) data size N=1000,
(b) data size N=1300, (c) data size N=1600, and (d) data size N=2000.
Fig. 6 The fitting performance (FIT) versus the number of sensors selected when the training data size N=1000: (a)
model order r=0, d=30, (b) model order r=0, d=40, (c) model order r=0, d=50, and (d) model order r=0, d=60.
031010-8 / Vol. 23, JUNE 2023 Transactions of the ASME
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the fitting performance corresponding to the candidate choice is
cross-validated using least squares in the additional validation
data set. Then, the optimal sensor selection can be obtained after
going through all candidate sensor selection choices. Although
the enumeration-based method is time-consuming when the value
of S
K

is large, it can be regarded as a benchmark when Kand
Sare small.
From the results in Sec. 5.1, it can be seen that the
easy-to-measure sensors on the front of vehicles may be related to
the vertical wheel force on the front left. Thus, 14 easy-to-measure
sensors, i.e., BX
i,BY
i,BZ
i,WX
i,WY
i,WZ
i,D
i
, where i=1, 2, are
selected as candidate ones. Figure 7shows the fitting performance
of the proposed method using least squares and the enumeration-
based method in the case of K=1, …,6,S=14, N=2000, and
d=50. It can be seen that the fitting performance of the proposed
method gets closer to that of the enumeration-based method as K
gets larger, especially when K≥3. This indicates it is possible to
obtain the optimal sensor selection using the proposed method.
Note that the performance of the proposed method is poor when
K=1, 2. This may indicate the limitation of the proposed method
on small K, which will be further addressed in the future work.
5.3 The Computational Time. Now we compare the compu-
tational efficiencies of the proposed method, the group lasso, and
the enumeration-based method when K=1, …6 and S=14. The
experiments are conducted with a 2.50 GHz Intel Core i5-7200U
processor, and all implementations are done in Matlab 2021b.
The group lasso is solved using the MATLAB package CVX
[31,50], and the convex problem (21) is solved using Matlab
command fmincon with default setting. In Fig 8, we report the
run-time of the proposed method and the enumeration-based
method when the training data size N=2000 and the model order
r=0, d=50. Also, the run-time of the group lasso is 36.5 s. We
can see that the group lasso is faster than the proposed method
for going through all results when K=1, 2, …, 6. The disadvantage
of the proposed method is that it needs to solve the optimization
problem for each K, which indicates that the overall run-time gets
larger as Kgets larger. On the contrary, the run-time of the group
lasso is irrelevant to the value of K. Nevertheless, given the value
of K, the convex optimization (which takes a few seconds) is
much faster than the group lasso method, since the hyper-
parameters of the group lasso method need to be tuned using addi-
tional round of cross-validation and is time-consuming. Further-
more, it can be seen that the run-time of the enumeration-based
method gets larger vastly as Kgets larger, since the it needs to enu-
merate all possible candidate solutions. That is very time-
consuming when the value of S
K

is large or there are many
hard-to-measure variables in the vehicle durability design.
6 Conclusion
The problem of choosing Ksensors, among a large number of
easy-to-measure sensors, to identify the best FIR model is a very
important and difficult combinatorial problem. This work presents
a novel data-driven method to remove the irrelevant sensors to
the vertical wheel force in vehicle durability design. The problem
is formulated into a mixed-integer programming problem, which
is relaxed into a convex optimization problem with continuous
design variables and linear constraints, such that the sensor selec-
tion problem can be solved effectively and efficiently.
The numerical examples and real data in the vehicle durability
design are used to verify the applicability and effectiveness of the
proposed method. It has been shown that the proposed method is
capable of solving sensor selection problems, and working robustly
with different training data sizes and model orders. Compared to the
results from the group lasso or other data-driven methods, the pro-
posed method has two advantages:
(i) the results of the proposed method return smaller possible
subsets of sensors while retaining the same model accuracy,
which brings benefits to the applications where the cost of
sensors is one of the key issues;
(ii) no additional data sets are needed to tune hyper-parameters,
which brings benefits to the scenarios where data are limited
to achieve.
The proposed method presented is simple to implement and may
be applicable in various other signal processing applications where
multiple coefficients of each feature are grouped together and share
the similar structures [24,34].
Acknowledgment
The work presented here is supported in part by VMAP program
of VMware. Such support does not constitute an endorsement by
the funding agency of the opinions expressed in the article.
Table 1 Sensor selection results of the proposed in fitting the
vertical wheel force on the front left shown in Fig. 4(a)
K=3K=6
D
1
D
2
WZ
1D
1
D
2
WZ
1WX
1BZ
1BZ
2
Fig. 7 The fitting performance of the proposed method and the
enumeration-based method
Fig. 8 The run-time of the proposed method and the
enumeration-based method.
Journal of Computing and Information Science in Engineering JUNE 2023, Vol. 23 / 031010-9
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Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article
are obtainable from the corresponding author upon reasonable
request.
Appendix: Computation of the Gradient and Hessian
in the Problem (21)
We give the analytic expression of the gradient and the Hessian
of the objective function in the problem (21) w.r.t. h
1
,…,h
S
,σ
2
for
numerical optimization. Let f:RS+1→R
f(h1,...,hS,σ2)=YTX−1
SY+
S
s=1
hstr(Ψs)+Nσ2−N(A1)
where Y∈RN, and
XS=σ2IN+
S
s=1
hsΨs,hs≥0(A2)
is a positive definite matrix. Then, we have (to simplify the notation,
we omit the subscript Sin X
S
)
∂X−1
∂hs
=−X−1∂X
∂hs
X−1=−X−1ΨsX−1,s=1,...,S(A3)
∂X−1
∂σ2=−X−1∂X
∂σ2X−1=−X−2(A4)
The gradient of fw.r.t. h
1
,…,h
S
,σ
2
is
∇f=−
YTX−1Ψ1X−1Y−tr(Ψ1)
.
.
.
YTX−1ΨSX−1Y−tr(ΨS)
YTX−2Y−N
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
(A5)
=−IS+1⊗(YTX−1)˜
ΨX−1Y+η(A6)
where ˜
Ψ=[Ψ1··· ΨSIN]T,η=[tr(Ψ1)···tr(ΨS)N]T, and ⊗
denotes the Kronecker product. Furthermore, we have
∂2YTX−1Y
∂hs∂hℓ
=−YT∂X−1
∂hℓ
ΨsX−1+X−1Ψs
∂X−1
∂hℓY
=2YTX−1ΨsX−1ΨsX−1Y(A7)
and
∂2YTX−1Y
∂hs∂σ2=−YT∂X−1
∂σ2ΨsX−1+X−1Ψs
∂X−1
∂hℓY
=2YTX−1ΨsX−2Y
(A8)
According to Eqs. (A7) and (A8), the Hessian matrix is
H=2IS+1⊗(YTX−1)˜
ΨX−1˜
ΨT(IS+1⊗(X−1Y)) (A9)
which is the positive semi-definite.
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