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IEEE SENSORS JOURNAL, VOL. 14, NO. 10, OCTOBER 2014 3715
Target Tracking Using Machine Learning and
Kalman Filter in Wireless Sensor Networks
Sandy Mahfouz, Farah Mourad-Chehade, Paul Honeine, Joumana Farah, and Hichem Snoussi
Abstract— This paper describes an original method for target
tracking in wireless sensor networks. The proposed method
combines machine learning with a Kalman filter to estimate
instantaneous positions of a moving target. The target’s accel-
erations, along with information from the network, are used to
obtain an accurate estimation of its position. To this end, radio-
fingerprints of received signal strength indicators (RSSIs) are first
collected over the surveillance area. The obtained database is then
used with machine learning algorithms to compute a model that
estimates the position of the target using only RSSI information.
This model leads to a first position estimate of the target under
investigation. The kernel-based ridge regression and the vector-
output regularized least squares are used in the learning process.
The Kalman filter is used afterward to combine predictions of
the target’s positions based on acceleration information with the
first estimates, leading to more accurate ones. The performance
of the method is studied for different scenarios and a thorough
comparison with well-known algorithms is also provided.
Index Terms—Radio-fingerprinting, Kalman filter, machine
learning, RSSI, target tracking, wireless sensor networks.
I. INTRODUCTION
RECENTLY, advances in radio and embedded systems
have led to the emergence of Wireless Sensor Networks
(WSNs), that have become a major research field during the
last few years. These networks are beginning to be deployed
at an accelerated pace for many applications, ranging from
home monitoring [1] to industrial monitoring [2], and covering
medical applications [3].
Target tracking [4], [5] is an interesting research and appli-
cation field in WSNs, that consists of estimating instantly the
position of a moving target. Target tracking can be viewed as
a sequential localization problem, thus requiring a real-time
location estimation algorithm. Typically, sensors broadcast
signals in the network, while targets collect these signals
for location estimation. Several types of measurements can
be considered, such as received signal strength indicators
Manuscript received January 20, 2014; accepted June 5, 2014. Date of
publication June 20, 2014; date of current version August 29, 2014. This work
was supported by the Champagne-Ardenne Region in France under Grant
WiDiD: Wireless Diffusion Detection. The associate editor coordinating the
review of this paper and approving it for publication was Dr. Anupama Kaul.
S. Mahfouz, F. Mourad-Chehade, P. Honeine, and H. Snoussi are
with the Centre National de la Recherche Scientifique, Institut Charles
Delaunay, Université de Technologie de Troyes, Troyes 10010, France
(e-mail: sandy.mahfouz@utt.fr; farah.mourad@utt.fr; paul.honeine@utt.fr;
hichem.snoussi@utt.fr).
J. Farah is with the Department of Telecommunications, Faculty of Engi-
neering, Université Saint-Esprit de Kaslik, Kaslik 05621, Lebanon (e-mail:
joumanafarah@usek.edu.lb).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSEN.2014.2332098
(RSSIs) [6], angle of arrival (AOA) [7], time difference of
arrival (TDOA) [8] and time-of-arrival (TOA) [9]. Previous
studies have shown that investigating TOA and TDOA leads
to more accurate position estimates compared to other meth-
ods [10]. However, implementing these techniques requires
high-cost hardware, making them impractical for most appli-
cations. Unlike these techniques, the RSSI-based ones achieve
acceptable performance, with no extra hardware.
Many RSSI-based tracking techniques have been proposed.
For instance, [11] proposes a target tracking technique using
a particle filter with the exact RSSI channel model. However,
such an approach is not reliable with the highly varying
RSSIs, due to the signal fading, the additive noise, etc. Also
exploring RSSIs, authors of [12] and [13] propose target
tracking methods based on connectivity measurements using
the interval analysis or the variational filter. By exploring con-
nectivity, these methods are more robust than the ones using
the exact channel model. However, the performance highly
depends on the number and the positions of the sensors in the
network. In [14], the authors propose a tracking algorithm that
works in indoor and outdoor environments. Indeed, it switches
between GPS information when the node is outdoors and an
existing Wifi-based application when the GPS signals are no
more available. This android application yields several position
estimates of the node when the Wifi is activated. Then, it
is followed by a Gaussian process regression that uses the
estimated positions, to reconstruct a smooth trajectory and
recover the missing positions. In addition to measurements,
tracking algorithms can employ a state-space model to refine
the position estimation based on its previous position. For
instance, a first-order model is used with a Kalman filter
in [15], and with a particle filter in [11], whereas [16] employs
a second-order one. However, these models are only reliable
for targets having slightly varying velocities or accelerations.
In other contexts, RSSI-based methods have been proposed
for nodes localization in WSNs. These methods aim at
location estimation by investigating observation information
without taking advantage of nodes mobility. One interesting
RSSI-based localization approach consists of radio-
fingerprinting [17], [18]. Such an approach allows taking into
consideration the stationary characteristics of the environment.
Several studies have been made for sensors localization using
RSSI-based radio-fingerprinting, such as the weighted
K-nearest neighbor (WKNN) algorithm [19]. We have
recently proposed in [20] and [21] two localization methods
using radio-fingerprinting in WSNs, by taking advantage
of kernel methods in machine learning. These methods
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3716 IEEE SENSORS JOURNAL, VOL. 14, NO. 10, OCTOBER 2014
outperform the WKNN approach. In order to perform
tracking, the authors of [22] propose to correct the WKNN
estimates using the Kalman filter with a second-order
state-space model.
In this paper, we propose a new method for target tracking
in WSNs that combines radio-fingerprinting and accelerom-
eter information. The proposed method consists of setting
reference positions along the network where RSSI measure-
ments are collected, leading to a radio-fingerprint database.
This database is used with machine learning algorithms to
define a kernel-based model, whose input is the RSSI vector
and whose output is the corresponding position. To esti-
mate this model, we investigate two learning algorithms: the
ridge regression (RR) and the vector-output regularized least
squares (vo-RLS). A moving target then measures its RSSIs
and instantaneous acceleration. A first position estimate is
obtained using the already-defined kernel-based model and
the measured RSSIs, then this estimate is combined with
the acceleration information, by means of a Kalman filter, to
achieve better accuracy. To this end, three different orders of
the tracking models are examined. The proposed method out-
performs existing methods, especially for hyperactive targets.
Compared to our previous works, the proposed method takes
advantage of the target’s mobility to enhance the obtained
position estimate, which is not the case in [20] and [21]. It also
proposes two kernel-based learning algorithms the RR and the
vo-RLS, compared to only the RR in [20] and [21].
The rest of the paper is organized as follows. The proposed
tracking approach is presented in Section II. Section III
describes three orders of the state-space model, while
Section IV defines the observation model and the use of
machine learning in our method. In Section V, we examine
the performance of the proposed method and compare it to
two recently derived methods. Finally, Section VI concludes
the paper.
II. TRACKING APPROACH
Consider an environment of Ddimensions, for instance
D=2 for a two-dimensional environment, and Nsstation-
ary sensors having known locations, denoted by si,i∈
{1,...,Ns}. In the following, all coordinates are D-dimension
row vectors. For the sake of clarity, and without loss of
generality, only one target with the unknown position x(k)
is considered, kbeing the current time step. Nevertheless, the
proposed method could be extended to several moving targets,
since they are tracked independently from each other, using
their accelerations and information exchanged only with the
stationary sensors.
To this end, a linear state-space model is proposed to
describe the target’s motion as follows:
x(k)=x(k−1)A+B(k)+θ(k), (1)
where x(k−1)is the target’s previous position, Ais a
D-by-Dstate transition matrix that relates the current position
of the target to its previous one, B(k)is a control-input
vector depending on the accelerations, and θ(k)is a random
vector noise whose probability distribution is assumed to be
normal, having zero mean and covariance matrix Q(k),thatis
θ(k)∼N(0,Q(k)). More details about the definitions of A,
B(k)and Q(k)are given in Section III, where different orders
of the state-space model are considered. In addition to its
accelerations, the target exchanges information in the network
with the stationary sensors at each time step. It therefore
collects a set of measurements, stored in z(k). These measure-
ments are described in detail in Section IV. Let the observation
equation be given by its linear general formulation as follows:
z(k)=x(k)H+n(k), (2)
where His the observation matrix that relates the state x(k)to
the measurement z(k)and n(k)∼N(0,R)is the observation
noise with normal distribution, zero mean and covariance
matrix R. This variable is assumed to be independent of θ(k).
The values of Hand R, as well as the choice of the
linear observation model, will also be discussed in detail in
Section IV.
Having defined both the state-space model and the observa-
tions, we now propose to solve the tracking problem by using
a Kalman filter [23], [24]. To this end, the proposed filter first
predicts the unknown position using the previous estimated
position and the state-space equation (1). Then, the predicted
position is corrected using the observation (2) in the following
step.
Now let ˆx(k−1)denote the target’s position estimated with
the Kalman filter at time step k−1. Therefore, the predicted
position can be written as:
ˆx−(k)=ˆx(k−1)A+B(k). (3)
At k=0, ˆx(0)is assumed to be known. Then, the Kalman
filter updates the D-by-Dpredicted estimation covariance as
follows:
T−(k)=AT(k−1)A+Q(k), (4)
where T(k−1)is the final covariance estimation at time step
k−1andT(0)is null since the initial state is known. Then,
the predicted quantities ˆx−(k)and T−(k)are corrected using
the observation equation (2) as follows:
ˆx(k)=ˆx−(k)+(z(k)−ˆx−(k)H)GK(k)(5)
T(k)=(ID−GK(k)H)T−(k), (6)
where IDis the D-by-Didentity matrix and GK(k)is the
optimal Kalman gain given by:
GK(k)=T−(k)H(HT
−(k)H+R)−1.(7)
In the following section, the state-space model is described in
detail by writing the model (1) in three different forms with
the corresponding covariance matrix Q(k). As for the choice
of the observation z(k)in (2), it is explained in Section IV.
III. STATE -SPAC E MODELS
This section highlights the definition of the state-space-
model of the tracking problem, where the target is assumed
to be equipped with an accelerometer, yielding at each time
step its current Daccelerations. The target is assumed to be
fixed at a known position x(0)at the beginning of the tracking.
MAHFOUZ et al.: TARGET TRACKING USING MACHINE LEARNING AND KALMAN FILTER 3717
The objective consists then of relating the current position of
the target x(k)to its previous position x(k−1), using its mea-
sured accelerations. To do this, three orders of the state-space
model are described: (i) a first-order, where, consecutively, the
velocities then the accelerations are assumed to be constant
between any two consecutive time steps; (ii) a second-order,
where the accelerations are assumed to be constant between
any two consecutive time steps, with linearly varying veloci-
ties; and finally, (iii) a third-order, where the accelerations are
assumed to vary linearly between any two consecutive time
steps. In all these cases, the target’s trajectory is described by
the equation (1). In the following, ν(k)denotes the estimated
velocity vector of the target at the time step k,γ(k)denotes
its measured acceleration vector at the time step k,andt
is the tracking period, that is the time period separating two
consecutive time steps.
A. First-Order State-Space Model
The first-order model makes two assumptions on the motion
of the target. It first assumes that the acceleration vector of the
target is constant between two consecutive time steps k−1
and k, and equal to γ(k). It thus computes the target’s velocity
vector iteratively by:
ν(k)=ν(k−1)+γ(k)t,(8)
with ν(0)null since the target is assumed to be fixed at the
beginning of the tracking. It then assumes that the velocity
between k−1andkis constant and equal to ν(k). This leads
to:
B(k)=ν(k)t,(9)
in the state-space equation (1) with Aequal to the D-by-D
identity matrix. It is obvious that this model only works for
slightly-varying-velocity targets (γ0). With more active
targets, the performance of this first-order model degrades very
fast, because of its two assumptions.
As for the model noises, the acceleration noise is assumed
to be independent with zero-mean normal distribution, having
known variances σ2
γ,d,d=1, ..., D. Their values can be esti-
mated by performing a calibration of the accelerometer before
the tracking stage. With noisy accelerations, the estimated
velocities have noise, with a zero-mean normal distribution
having the covariance matrix Qν(k)updated recursively as
follows:
Qν(k)=Qν(k−1)+t2Diag σ2
γ,(10)
where Diag σ2
γis the D-by-Ddiagonal matrix with entries
σ2
γ,d,d=1, ..., D,and Qν(0)is null. The state noise θ(k)is
then normally distributed with zero-mean and the covariance
matrix Q(k)given as follows:
Q(k)=Cov (x(k−1)+ν(k)t),
=Q(k−1)+Qν(k−1)t2,
where Q(0)is null since there is no uncertainty over the
target’s position at time step k=0. It is worth noting that the
covariance matrices of the target’s velocity and state noises
are diagonal since noises over the coordinates are assumed to
be independent.
B. Second-Order State-Space Model
The second-order model assumes that the acceleration vec-
tor is constant between two consecutive time steps k−1andk,
and equal to γ(k). The velocity vector is estimated as for
the first-order model using equation (8). However, the control
vector B(k)of the state-space equation is modified as follows:
B(k)=ν(k−1)t+γ(k)t2
2,(11)
with the transition matrix Aequal to identity.
Here, the covariance matrix Q(k)is diagonal, given by:
Q(k)=Cov x(k−1)+ν(k−1)t+γ(k)t2
2,
=Q(k−1)+Qν(k−1)t2+1
4t4Diag σ2
γ,
where Q(k)is null at time step k=0 since the target’s
position is initially known, and Qν(k)is given by (10) as
for the first-order model.
The second-order state-space model outperforms the
first-order one, since it considers less approximations and
assumptions. This model performs well with slightly varying
accelerations motions. However, it is not well-adapted to tra-
jectories with abrupt changes in accelerations, since estimates
might be significantly deviated from the exact trajectory due
to cumulative model errors over time.
C. Third-Order State-Space Model
This model considers that the target’s accelerations vary
linearly between two consecutive time steps, that is the accel-
eration vector varies from γ(k−1)at k−1toγ(k)at kwith
a slope equal to γ(k)−γ(k−1)
t. According to this assumption,
the velocity vector of the target at time step kis estimated
recursively by:
ν(k)=ν(k−1)+γ(k−1)t+γ(k)−γ(k−1)
t
t2
2,
(12)
where the target is also assumed to be fixed at the beginning
of the tracking (i.e., ν(0)=0) with null acceleration (i.e.,
γ(0)=0), and at a known position x(0). Then, the vector
B(k)in (1) is given by:
B(k)=ν(k−1)t+γ(k−1)t2
2
+γ(k)−γ(k−1)
t
t3
6,(13)
with the transition matrix Aequal to identity, as for the other
two models.
The covariance matrix Q(k)is also diagonal, given by:
Q(k)=Cov x(k−1)+ν(k−1)t
+γ(k−1)t2
2+γ(k)−γ(k−1)
t
t3
6
=Q(k−1)+Qν(k−1)t2+5
36t4Diag σ2
γ,
3718 IEEE SENSORS JOURNAL, VOL. 14, NO. 10, OCTOBER 2014
Fig. 1. Rotations in a three-dimensional environment.
where Q(0)is null since there is no uncertainty at time step
k=0, and Qν(k)is given by:
Qν(k)=Qν(k−1)+1
2t2Diag σ2
γ,
with Qν(0)also null. This model outperforms the other
models, since it brings the estimated trajectory closer to the
real one compared to the others, especially for hyperactive
targets having highly varying accelerations.
Remark 1: In the previous paragraphs, the target is
assumed to be rotationally constrained. Indeed, the accelera-
tions measured by the accelerometer in the target’s coordinate
system are used directly in the equations, as if they are
measured in the world coordinate system. However, during its
motion in real applications, the target could rotate, and the
coordinate system, where the accelerations are given, might
change. The solution to this problem is to equip each target
with a gyroscope, which yields its orientations with respect to
the world coordinate system.
Assume that the localization is performed in a three-
dimensional environment (i.e., D=3). Consider that ϑ,ϕ,
and φare the angles of the counter-clockwise rotation of the
target, given by its gyroscope at a given time step around
the third coordinate axis of the world system, the first one
and the second one respectively. The plots (a), (b) and (c)
of Fig. 1 illustrate the single rotations around the third, the
first, and the second axes respectively, 1, 2, and 3 being the
world coordinate axes, and 1’, 2’, and 3’ the target’s ones.
Let γ=(γ1γ2γ3)be the acceleration vector of the target
in the world coordinate system at the same time step and
let γ=γ
1γ
2γ
3be its measured one in its coordinate
system. Then, having the rotation angles, γis computed as
follows:
γ=γR,(14)
where the first column of the three-dimensional rotation matrix
Ris defined by:
⎛
⎝
cos ϑcos φ
−sin ϑcos φ
sin φ
⎞
⎠,(15)
its second column is defined by:
⎛
⎝
cos ϑsin ϕsin φ+sin ϑcos ϕ
−sin ϑsin ϕsin φ+cos ϑcos ϕ
−sin ϕcos φ
⎞
⎠,(16)
and its third column is defined by:
⎛
⎝
−cos ϑcos ϕsin φ+sin ϑsin ϕ
sin ϑcos ϕsin φ+cos ϑsin ϕ
cos ϕcos φ
⎞
⎠.(17)
In a two-dimensional environment (i.e., D=2), where
γ=(γ1γ2), the rotation is only possible in the plane with
the rotation angle ϑ. By setting φ=ϕ=0, one gets the
following transformation:
γ=γcos ϑsin ϑ
−sin ϑcos ϑ.(18)
During the tracking, the target measures its acceleration
vector in its coordinate system at each time step, then finds
its orientations using the gyroscope. Its accelerations in the
world coordinate system can then be computed and used in
the localization algorithm. For simplicity, we only consider
rotationally constrained targets in our paper. However, as
just explained, the computations could be easily modified to
consider the target’s rotations.
IV. OBSERVATION MODEL
The aim of this section is to define the observation model
(i.e., z(k)in (2)), based on the information gathered by the
target from the stationary sensors in the network. The proposed
method is a radio-fingerprinting approach using the Received
Signal Strength Indicators (RSSIs) of the signals exchanged
between the target and the stationary sensors. It is worth
noting that the target is assumed to be active and cooperative
in the proposed approach, that is, it exchanges informa-
tion with its neighborhood. Based on radio-fingerprinting,
the approach needs then a configuration phase, before the
tracking. To this end, Npreference positions, denoted by p,
∈{1,...,Np}, are generated uniformly or randomly in the
studied region. All stationary sensors continuously broadcast
signals in the network at a fixed initial power, and a sensor
is placed consecutively at the reference positions to detect
the broadcasted signals and measure their RSSIs. Let ρ=
(ρs1,p... ρ
sNs,p)be the vector of RSSIs sent by all
Nssensors and received at the position p,∈{1,...,Np}.
In this way, a set of Nppairs ρ,pis obtained. This
radio-fingerprint database is considered in the estimation of
the observation model, that is z(k).
A. Definition of the Observations
While moving, the target collects the sensors signals and
measures their RSSIs. Instead of using the target’s RSSIs
as observations, the proposed approach consists of finding a
function ψ:IR Ns→ IR D, based on the radio-fingerprint data-
base, that associates to each RSSI vector ρthe corresponding
position p, with the advantage of not having to estimate the
channel model. Kernel methods in machine learning [25]–[27]
provide an elegant framework to define the function ψ(·),asit
will be shown in the following subsection. It is worth noting
that the database construction and the computation of ψ(·)
are performed only once, before the tracking phase. Once the
model is available, the target is able to perform all tracking
MAHFOUZ et al.: TARGET TRACKING USING MACHINE LEARNING AND KALMAN FILTER 3719
computations and determine its own position. Indeed, consider
the moving target collecting RSSIs in the network. At a given
time step k, it stores them into a vector ρ(k), and then uses the
defined model ψ(·)to compute its position. The first estimated
coordinates of the target at time step kare then given by
ψ(ρ(k)). This estimate is considered as an observation of the
desired value, namely
z(k)=ψ(ρ(k)). (19)
Both the observation and the state-space model (1) are then
used in the Kalman filter to compute a more accurate position
estimation, as shown in Section II. Determining the model
ψ(·)is explained in the following paragraph.
Following the definition of z(k), one can see that the
matrix Hof (2) is set to identity. As for n(k)∼N(0,R),
an approximation of the value of its covariance matrix R
is done by generating a new set of reference pairs, and by
localizing the positions according to the defined model ψ(·).
The error on the new set is computed and stored into a vector,
then the matrix Ris determined by computing the covariance
of the error vector. This matrix is considered to be constant
over time and for all targets.
B. Definition of ψ(·)Using Kernel Methods
In this paragraph, the objective is to determine the afore-
mentioned function ψ(·)that associates to each RSSI vector
ρthe corresponding position p. Determining ψ(·)requires
solving a nonlinear regression problem. We take advantage
of kernel methods [25], [26], that have been remarkably
successful for solving such problems. Let the vector-valued
function ψ(·)be decomposed into Dreal-valued functions,
namely ψ(·)=(ψ1(·) ... ψD(·)),whereψd:IR Na→
IR ,d∈{1,...,D}, estimates the d-th coordinate in p=
(p,1... p,D), for an input ρ.LetP=(p
1... p
Np).
The matrix Pis then of size Np-by-Dhaving p,dfor the
(, d)-th entry, and pfor the -th row. In the following,
we denote pby P,∗and the d-th column of Pby P∗,d.
Therefore, the vector P∗,dholds all Nppoints for a fixed
coordinate d.
Two different machine learning techniques are investigated
in the following: the ridge regression and the vector-output
regularized least squares. The kernel-based ridge regres-
sion is considered in Subsection IV-B1, where Doptimiza-
tion problems are set separately to define the Dmodels
ψ1(·),...,ψD(·). In Subsection IV-B2, we explore multi-
task learning to determine a vector-output model ψ(·)that
estimates simultaneously all Dcoordinates.
1) Ridge Regression: The kernel-based ridge regression is
considered in this paragraph to determine the Dmodels,
ψ1(·),...,ψD(·), by setting Dseparate optimization prob-
lems. Indeed, each function ψd(·)is estimated by minimizing
the mean quadratic error between the model’s outputs ψd(ρ)
and the desired outputs p,d:
min
ψd∈H
1
Np
Np
=1
(( p,d−ψd(ρ))2+ηψd2
H,(20)
where ηis a positive tunable parameter that controls the trade-
off between the fitness error and the complexity of the solution,
as measured by the norm in the Reproducing Kernel Hilbert
Space H. According to the representer theorem [26], [28], the
optimal function can be written as follows:
ψd(·)=
Np
=1
α,dκ(ρ,·), (21)
where κ:IR Ns×IR Ns→ IR is a reproducing kernel, and
α,d, ∈{1,...,Np}, are parameters to be estimated. Let α
be the Np×Dmatrix whose (, d)-th entry is α,d, and whose
d-th column is denoted by α∗,dand -th row by α,∗.
By injecting (21) in (20), we get a dual optimization
problem in terms of α∗,d, whose solution is given by taking
its derivative with respect to α∗,dand setting it to zero. One
can easily find the following form of the solution:
α∗,d=(K+ηNpINp)−1P∗,d,(22)
where INpis the Np-by-Npidentity matrix, and Kis the
Np×Npmatrix whose (i,j)-th entry is κ(ρi,ρj),fori,j∈
{1, ..., Np}. For an appropriate value of the regularization
parameter η, the matrix between parenthesis is always non-
singular.
One can see that the same matrix (K+ηNpINp)needs to
be inverted in order to estimate each coordinate. To reduce the
computational complexity, all Destimations are collected in
a single matrix inversion problem, as follows:
α=(K+ηNpINp)−1P.(23)
We then define a model that allows us to estimate all D
coordinates at once, using equation (21) and the definition
of the vector of functions ψ(·), as follows:
ψ(·)=
Np
=1
α,∗κ(ρ,·). (24)
2) Vector-Output Regularized Least Squares: In this para-
graph, we take advantage of multi-task learning by using
the vector-output regularized least squares (vo-RLS) algorithm
[29] to estimate all Dcoordinates at once. Instead of estimat-
ing the set of functions ψd(·), we now determine a 1-by-D
vector-output function ψ(·).
In multi-task learning, ψ(·)takes the form:
ψ(·)=
Np
=1
βP,∗κ(ρ,·), (25)
where β,∈{1,...,Np}, are parameters to be defined.
As for the optimization problem, the objective stays the same.
Indeed, the function ψ(·)is determined by minimizing the
mean quadratic error between the model’s outputs ψ(ρ)and
the desired outputs P,∗, namely
min
ψ
1
Np
Np
=1
P,∗−ψ(ρ)2+ηβ2,(26)
3720 IEEE SENSORS JOURNAL, VOL. 14, NO. 10, OCTOBER 2014
where β=β1... β
Np. Substituting the expression of
ψ(·)from (25) in the optimization problem (26), we get,
in matrix form, the following problem formulation:
min
βtr(PP)−2ξβ+βGβ+ηNpββ,(27)
where tr(·)is the matrix trace operator, Gis the Np-by- Np
matrix whose (j,k)-th entry is
Pj,∗P
k,∗
Np
i=1
κ(ρj,ρi)κ(ρk,ρi),
and ξis the Np-by-1 vector whose j-th entry is
Np
k=1
Pj,∗P
k,∗κ(ρj,ρk).
By taking the gradient of the objective function in (27) with
respect to β, namely −ξ+Gβ+ηNpβ, and setting it to zero,
we obtain the final solution:
β=(G+ηNpINp)−1ξ.
V. PRACTICAL SIMULATIONS AND RESULTS
In this section, we evaluate the performance of our method
on simulated data. In the first paragraph, several trajectories
with different orders for the state-space model are examined.
In the second paragraph, we study the impact of the noises
standard deviations σγand σρon the estimation error. In the
third paragraph, we study the impact of the number of sta-
tionary sensors and the number of reference positions on the
estimation error. Finally, results are compared to ones obtained
with the WKNN algorithm combined with a Kalman filter [30]
and tracking using particle filtering [31].
The same practical setup is considered for the two following
paragraphs, given as follows. We consider a 100m×100m
area, and generate 16 stationary sensors and 100 reference
positions uniformly distributed over the area. The RSSI values
are obtained using the well-known Okumura-Hata model [32]
given by:
ρsi,p=ρ0−10 nPlog10 si−p+εi,,(28)
where ρsi,p(in dBm) is the power received from the sensor at
position siby the node at position p, that is the i-th entry of
the vector ρ,ρ0is the initial power (in dBm) set to 100, nPis
the path-loss exponent set to 4 as often given in the literature,
si−pis the Euclidian distance between the position pof
the considered node and the position siof a stationary sensor,
and εi, is the noise affecting the RSSI measures with σρits
standard deviation. We also generate a trajectory and calculate
the RSSI values collected by the moving target using (28). For
the definition of ψ(·)using kernel methods, we consider the
Gaussian kernel given by:
κ(ρu,ρu)=exp −ρu−ρu2
2σ2,
where σis its bandwidth that controls, together with the
regularization parameter η, the degree of smoothness, noise
tolerance, and generalization of the solution. The choice of
Fig. 2. Estimation of the first trajectory.
Fig. 3. Estimation of the second trajectory.
the values for ηand σis done using a grid search on
ηNp=2rwith r∈{−20,−19,··· ,−1}and σ=2r
with r∈{1,2,··· ,10}, where the corresponding error is
estimated using the 10-fold cross-validation scheme. This
scheme consists of dividing the data into 10 folds: 9 for
training the model and the remaining one for validating it [33].
A. Evaluation of Our Method on Three Trajectories
We consider three different trajectories of 100 points with
t=1s. For the trajectory illustrated in Fig. 2, the accelera-
tions are assumed equal to zero, leading to constant velocities.
As for the second and the third trajectories of Fig. 3 and
Fig. 4 respectively, their respective accelerations are given in
the top plots and in the bottom plots of Fig. 5, γ1and γ2being
the first and the second acceleration coordinates respectively.
One can see that the accelerations of the third trajectory have
MAHFOUZ et al.: TARGET TRACKING USING MACHINE LEARNING AND KALMAN FILTER 3721
Fig. 4. Estimation of the third trajectory.
Fig. 5. Acceleration signals for the second trajectory in the top plots and
for the third trajectory in the bottom plots.
more variations than the accelerations of the second trajectory.
The coordinates expressions are obtained by taking twice
the primitive integral of the accelerations. By taking these
three trajectories, the performance of the proposed method is
evaluated for different types of scenarios, considering first a
monotonously moving target, then more hyperactive ones.
Since a noiseless setup is not realistic in a practical envi-
ronment, we consider that noises are present in all scenarios.
Here, we take both components of σγequal to 0.01m/s2,and
σρequal to 1dB. Let the estimation error be evaluated by the
root mean squared distance between the exact positions and the
estimated ones. Fig. 2, Fig. 3, and Fig. 4 show the estimated
trajectories when using the proposed method with the ridge
regression (RR) for the third-orderstate-space model described
in Section III. Table I shows the average over 50 simulations
of the estimation errors for the three trajectories and the three
different state-space models, using the RR and the vo-RLS in
TAB L E I
ESTIMATION ERRORS (IN METERS)FOR DIFFERENT ORDERS OF THE
STATE -SPAC E MODELS AND THE THREE TRAJECTORIES
the learning process. The three models yield almost the same
results for the first two trajectories. However, for the third
trajectory, the smallest estimation error is obtained when using
the third-order state-space model. This result is expected since
the accelerations in this trajectory have high variations, and as
explained in Section III, the third-order state-space model is
well suited for such cases.
B. Impact of σγand σρ
In this section, we will test our method using the trajectory
of Fig. 4, where the general case of a hyperactive target is con-
sidered. The third-order state-model from Section III-C is used
since it yields the best results as shown in the previous section.
Indeed, even though the first-order model and the second-order
model yield good results for the trajectories of Fig. 2 and 3,
the estimation error increases significantly compared to the
third-order model when the target is hyperactive (Fig. 4) as
showninTableI.
Let us now study the impact of the noises standard devia-
tions σγand σρon the estimation error. We first take different
percentages of the standard deviation of the acceleration, going
from 1% to 10%, along with a fixed σρequal to 5% of standard
deviation of the RSSI measures. The estimation errors are
averaged over 50 Monte-Carlo simulations. It is worth noting
that the standard deviation of the RSSI is equal to 10.79dBm;
therefore, σρis equal to 0.54dBm. The top plot of Fig. 6
shows the impact of the variation of σγon the estimation
error. One can see that the results obtained in this figure
with the ridge regression and the vo-RLS are independent
from the acceleration noise, whereas estimations using only
accelerometer information are highly affected by the variations
of σγ. The RR combined with the Kalman filter yields the best
results. In fact, the filter corrects the results, and the error is
always smaller than the error in the case of the ridge regression
alone, and around σγequal to 7% of the standard deviation
of the acceleration, the error becomes constant.
We then take several percentages of the standard deviation
of the RSSI measures, going from 0% to 50%, with σγfixed
to 1% of the standard deviation of the acceleration. σρis then
varying from 0dBm to 5.40dBm. The estimation errors are
also averaged over 50 Monte-Carlo simulations. The bottom
plot of Fig. 6 shows the impact of the variation of σρon
the estimation error. One can see that localization using only
3722 IEEE SENSORS JOURNAL, VOL. 14, NO. 10, OCTOBER 2014
Fig. 6. Estimation error as a function of the noise on the accelerations in
the top plot and as a function of the noise on the RSSI in the bottom plot.
accelerometer information is independent from the noise on
the RSSIs, which is expected. The RR and the vo-RLS are
highly affected by the noise variations, since they use these
RSSI measurements for the estimation. As for the method
combining the RR with the Kalman filter, it outperforms the
method using only accelerations. It is interesting here to see
the effectiveness of the Kalman filter. Indeed, one can see in
Fig. 6 that the RR used alone yields better results than the vo-
RLS also used alone; however, after adding the Kalman filter,
the results of the two techniques become very similar and
the error becomes almost constant for both methods when σρ
exceeds 30% of the standard deviation of the RSSI measures.
C. Impact of the Stationary Sensors and Reference Positions
In this section, we consider the trajectory of Fig. 4, with
both components of σγequal to 0.01m/s2and σρequal to
1dB. We first study the impact of the distribution of the 16
stationary sensors and the 100 reference positions on the per-
formance of the tracking method. In the previous paragraphs,
we considered a uniform distribution of the stationary sensors
and the reference positions. We now consider a random distri-
bution, instead of the uniform distribution, to see the impact
of such a choice on our method. We repeat the experiment
50 times for the ridge regression and the vo-RLS, using the
third-order state-space model. The mean estimation error is
shown in Table II, where σMSE is the standard deviation of the
TAB L E I I
ESTIMATION ERRORS IN THE CASE OF RANDOM DISTRIBUTIONS OF
STATIONARY SENSORS AND REFERENCE POSITIONS
mean estimation error. Compared to the results obtained in the
case of a uniform distributions (Table I), one can see that the
estimation error increases with the use of random distributions.
Indeed, a uniform distribution allows a better coverage of
the surveillance area, while a random distribution does not
always guarantee a good coverage of the area. Nevertheless,
the results are still satisfactory, and random distributions can
still be used for accurate tracking when uniform distributions
are not applicable.
We now study the impact of the number of stationary
sensors Nsand the number of reference positions Npon
the performance of the tracking method. We choose to use
the ridge regression in the following instead of the vo-RLS
for two reasons. First, the RR yields better results than the
vo-RLS in terms of accuracy, as one can easily see from
the previous paragraphs. Second, we would like to point out
that by combining the Dseparate optimization problems as
shown in section IV-B1, and by using equation (24), the ridge
regression’s computational complexity was reduced, and thus
it outperforms the vector-output regularized least squares in
terms of time complexity. In fact, the elapsed time for the
training phase is around 8 milliseconds for the RR and around
25 milliseconds for the vo-RLS, for simulations run on version
7.10.0.499 of Matlab on a Dell laptop with Windows 7 and
Intel Core i7 CPU. Nevertheless, it is worth noting that varying
Nsand Nphas the same impact on the tracking method if we
use the vo-RLS in the learning process.
We first vary the number of stationary sensors (Ns=
12,...,152), while keeping a fixed number for the reference
positions (Np=100). The top plot of Fig. 7 shows the
evolution of the estimation error in terms of the number of
stationary sensors. We then take a fixed number of stationary
sensors equal to 16, and we vary the number of reference
positions, Np=52,...,252. The bottom plot of Fig. 7
shows the evolution of the estimation error in terms of the
number of reference positions. By comparing the obtained
results, one can notice that both the increase in the number
of stationary sensors and in the number of reference positions
yield a better estimation of the target’s positions. Indeed, the
top plot of Fig. 7 shows that when using 16 stationary sensors,
the average over 50 simulations of the estimation error is
0.90mcompared to an error of 0.77mwhen using 62=36
or 122=144 stationary sensors. The bottom plot of Fig. 7
shows that for Np=100, the average over 50 simulations of
the estimation error is 0.92mcompared to an error of 0.62m
when increasing Npto 242=576. In fact, with a higher
number of stationary sensors and reference positions, we get
better coverage and knowledge of the environment, which
MAHFOUZ et al.: TARGET TRACKING USING MACHINE LEARNING AND KALMAN FILTER 3723
Fig. 7. Estimation error as a function of the number of stationary sensors
in the top plot and as a function of the number of reference positions in the
bottom plot.
explains the improvement in the results. However, increasing
the number of stationary sensors increases the total cost in
material, while increasing the number of reference positions
induces a significant increase in the algorithm’s complexity.
Therefore, depending on the practical system constraints, a
tradeoff should be found between the algorithm’s accuracy
and the computational load.
D. Comparison to Other Tracking Techniques
The objective now is to compare the proposed method to
two recently proposed tracking methods. For the first compar-
ison, we use the method proposed in [22], that also makes
use of the Kalman filter to correct the trajectory estimated
by radio-fingerprints. We then compare our method to the
centralized version of the method described in [31], which
involves the use of a particle filter and RSSI measurements.
We consider the three trajectories described in Section V-A.
In order to have a fair comparison of our technique towards
these two methods, we consider a setup that is the closest
possible to the one the authors use in their papers. For this
purpose, we take the number of stationary sensors Ns=4,
even though taking Ns=16 gives better results in the case
of our method as one can see from the top plot of Fig. 7.
We take different values of σρand both components of σγ
equal to 0.01m/s2.
TABLE III
ESTIMATION ERROR (IN METERS)FOR DIFFERENT TRAJECTORIES
AND DIFFERENT VALUES OF σρ(INdB)
We proceed by briefly describing the method in [22]. It con-
sists of estimating the position using the weighted K-nearest
neighbor (WKNN) algorithm, then applying the Kalman filter
to enhance the estimation. A target’s first position estimate
using WKNN is given by weighted combinations of the K
nearest neighboring positions from the training database, with
the nearness indicator being based on the Euclidean distance
between RSSIs. The weight used for the WKNN algorithm in
[22] is given by:
wn=1/δn
z∈I1/δz,
where δnis the Euclidean distance between the RSSI vector
ρ(k)of the target at time step kand ρn,n∈I,andI
is the set of indices of ρof the database yielding the K
smallest distances (i.e., Knearest neighbors) δat time step k.
The estimated target’s position is then given by: n∈Iwnpn.
The number of neighbors Kistakenequalto8asinthe
simulations of [22]. As for the correction using the Kalman
filter, the authors use a second-order state-space model similar
to the one in Section III-B. The estimation errors (in meters)
obtained when using this algorithm for the three trajectories of
Section V-A and for σρ=1dB are computed 50 times for each
case, and their averages are shown in Table III. The estimation
errors using our method with the second-order and the third-
order state-space models are also stored in Table III. Table III
shows as well the mean estimation error obtained when using
the third trajectory, for different values of σρand for both
components of σγtaken equal to 0.01m/s2. Our method
clearly outperforms the one in [22]. Indeed, the estimation
error obtained with the proposed method, using a second-order
or a third-order state-space model, is significantly smaller than
the one obtained with the WKNN algorithm followed by the
Kalman filter, for all three types of trajectories.
As for the second method used for our comparison, it
employs a particle filter (PF) along with RSSI measurements
and a first-order state-space model [31]. The particle fil-
ter approximates the minimum mean-square error (MMSE)
estimate of the emitter state given all present and past
observations, i.e. RSSI measurements. It seeks to represent
the posterior distribution of the hidden states by a properly
weighted set of time-varying random samples such that, as the
number of samples go to infinity, the weighted averageof those
samples converges at each time step, in some statistical sense,
to the true global MMSE estimate of the current unknown
3724 IEEE SENSORS JOURNAL, VOL. 14, NO. 10, OCTOBER 2014
states given all present and past network measurements [31].
We used at first the first-order state-space model as described
in the authors work. However, this model did not work well for
the second and third trajectories, due to the abrupt variations
in the target’s motion. Therefore, we used the second-order
state-space model for all three trajectories. Table III shows
the mean estimation errors obtained with the method in [31],
when using different trajectories and different values of σρ.
Both components of σγare taken equal to 0.01m/s2. One can
see that our tracking method also outperforms the well-known
tracking technique based on particle filtering.
VI. CONCLUSION
In this paper, we proposed a new method for target tracking
in wireless sensor networks by combining machine learning
and Kalman filtering. For the learning process, we investigated
the use of two kernel-based machine learning algorithms:
the ridge regression and the vector-output regularized least
squares. We also described three different orders for the state-
space models to be used in the Kalman filtering, and high-
lighted the difference between them and how they can affect
the performance of the tracking procedure. Simulation results
showed that the proposed method outperforms two recently
developed approaches. The method allows accurate tracking,
and is proved to be robust in the case of noisy data, whether
the noise affects the acceleration information or the RSSI
measures. Future works will handle further improvements
of this method, such as introducing a model that estimates
distances between sensors instead of positions. Solutions to
cases where zones of the surveillance area are not covered by
all stationary sensors could also be provided.
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MAHFOUZ et al.: TARGET TRACKING USING MACHINE LEARNING AND KALMAN FILTER 3725
Sandy Mahfouz was born in Fidar, Lebanon, in
1989. She received the Diploma degree in com-
puter and communication engineering (major in
telecommunications) from the Holy Spirit Univer-
sity of Kaslik, Lebanon, in 2012. She is currently
pursuing the Ph.D. degree in systems optimization
and security at the University of Technology of
Troyes, Troyes, France. Her current research inter-
ests include wireless and mobile sensor networks,
machine learning, and signal processing.
Farah Mourad-Chehade was born in 1984.
She received the Diploma degree in electrical
engineering from the Faculty of Engineering,
Lebanese University, Tripoli, Lebanon, in 2006, and
the master’s and Ph.D. degrees in systems optimiza-
tion and security from the University of Technology
of Troyes (UTT), Troyes, France, in 2007 and 2010,
respectively. Since 2011, she has been an Associate
Professor at UTT. She has supervised two Ph.D.
theses. She serves as a Reviewer for several journals
(the IEEE TRANSACTIONS ON SIGNAL PROCES S-
ING, the IEEE TRANSACTIONS ON ROBOTICS, the IEEE TRANSACTIONS
ON VEHICULAR TECHNOLOGY, and Elsevier’s Signal Processing) and con-
ferences (EUSIPCO, WOSSPA, and ROADEF). Her research interests include
wireless and mobile sensor networks, nonlinear signal analysis, machine
learning, and biomedical applications.
Paul Honeine (M’07) was born in Beirut, Lebanon,
in 1977. He received the Dipl.Ing. degree in mechan-
ical engineering and the M.Sc. degree in industrial
control from the Faculty of Engineering, Lebanese
University, Tripoli, Lebanon, in 2002 and 2003,
respectively, and the Ph.D. degree in systems opti-
mization and security from the University of Tech-
nology of Troyes, Troyes, France, 2007, where he
was a Post-Doctoral Research Associate with the
Systems Modeling and Dependability Laboratory
from 2007 to 2008. Since 2008, he has been an
Assistant Professor with the University of Technology of Troyes. His research
interests include nonstationary signal analysis and classification, nonlinear and
statistical signal processing, sparse representations, and machine learning. His
particular interests are applications to (wireless) sensor networks, biomedical
signal processing, hyperspectral imagery, and nonlinear adaptive system
identification. He has co-authored (with C. Richard) the 2009 Best Paper
Award at the IEEE Workshop on Machine Learning for Signal Processing.
Over the past five years, he has authored more than 100 peer-reviewed
papers.
Joumana Farah received the B.E. degree in electri-
cal engineering from Lebanese University, Tripoli,
Lebanon, in 1998, and the M.E. degree in sig-
nal, image, and speech processing and the Ph.D.
degree in third generation mobile communication
systems from the Polytechnic Institute of Grenoble,
Grenoble, France, in 1999 and 2002, respectively.
Since January 2010, she holds an Accreditation
to Supervise Research (HDR) from the University
Pierre and Marie Curie (Paris VI), France. She is a
Full Professor with the Department of Telecommu-
nications Engineering, Holy Spirit University of Kaslik (USEK), Lebanon.
She has supervised a large number of master’s and Ph.D. theses. She was a
recipient of several research grants from the Lebanese National Council for
Scientific Research, the Franco-Lebanese CEDRE program, and the Scientific
Research Center of USEK. She holds four registered patents and software,
and has co-authored a research book and more than 70 papers in international
journals and conferences. She was the General Chair of the 19th International
Conference on Telecommunications (ICT 2012), and serves as a TPC Member
and a Reviewer for several journals (the IEEE JOURNAL ON SELECTED
AREAS IN COMMUNICATIONS, the IEEE COMMUNICATIONS LETTERS,
Signal Processing: Image Communication,Digital Signal Processing,Annals
of Telecommunications) and conferences (the IEEE VTC, the IEEE Globecom,
the IEEE ICECS, EUSIPCO, and ICT). Her current research interests include
channel coding techniques, multicarrier systems, cooperative and wireless
sensor networks, resource allocation techniques, and distributed and multiview
video coding.
Hichem Snoussi was born in Bizerta, Tunisia, in 1976. He received the
Diploma degree in electrical engineering from Ecole Superieure d’Electricite
(Supelec), Gif-sur-Yvette, France, in 2000, the D.E.A. degree and the Ph.D.
degree in signal processing from the University of Paris-Sud, Orsay, France,
in 2000 and 2003, respectively, and the H.D.R. degree from the University of
Technology of Compiègne, Compiègne, France, in 2009.
He was a Post-Doctoral Researcher with the Institut de Recherches en
Communications et Cybernétiques de Nantes, Nantes, France, from 2003 and
2004. He has spent short periods as a Visiting Scientist with the Brain Science
Institute, RIKEN, Saitama, Japan, and the Olin Neuropsychiatry Research
Center, Institute of Living, Hartford, CT, USA. From 2005 to 2010, he was
an Associate Professor with the University of Technology of Troyes, Troyes,
France, where he has been a Full Professor since 2010. He was in charge
of the regional research program System Security and Safety of the CPER
from 2007 to 2013 and the CapSec platform (wireless embedded sensors
for security). He is the Principal Investigator of an ANR-Blanc project (mv-
EMD), a CRCA project (new partnership and new technologies), and a GDR-
ISIS young researcher project. He is a partner of many ANR projects, GIS,
and strategic UTT programs. He was a recipient of the National Doctoral and
Research Supervising Award from 2008 to 2012, and received the Scientific
Excellence Award for the period from 2013 to 2017.
