Message-Efficient Location Prediction for Mobile
Objects in Wireless Sensor Networks Using a
Maximum Likelihood Technique
Bing-Hong Liu, Min-Lun Chen, and Ming-Jer Tsai
Abstract—In the tracking system, a better prediction model can significantly reduce power consumption in a wireless sensor network because fewer redundant sensors will be activated to keep monitoring the object. The Gauss-Markov mobility model is one of the best mobility models to describe object trajectory because it can capture the correlation of object velocity in time. Traditionally, the Gauss-Markov parameters are estimated using an autocorrelation technique or a recursive least square estimation technique; either of these techniques, however, requires a large amount of historical movement information of the mobile object, which is not suitable for tracking objects in a wireless sensor network because they demand a considerable amount of message communication overhead between wireless sensors which are usually battery-powered. In this paper, we develop a Gauss-Markov parameter estimator for wireless sensor networks (GMPE MLH) using a maximum likelihood technique. The GMPE MLH model estimates the Gauss-Markov parameters with few requirements in terms of message communication overhead. Simulations demonstrate that the GMPE MLH model generates negligible differences between the actual and estimated values of the Gauss-Markov parameters and provides comparable prediction of the mobile object’s location to the Gauss-Markov parameter estimators using an autocorrelation technique or a recursive least square estimation.
Index Terms—Wireless sensor network, Gauss-Markov mobility model, Gauss-Markov parameter estimation, object tracking, message-efficient location prediction.
F
1
I
NTRODUCTIONA
WIRELESSsensor network is composed of multiple wireless sensors. Each sensor can collect, process, and store environmental information as well as com-municate with others via inter-sensor communication. The rapid development of wireless communications and embedded micro-sensing technologies has facilitated the use of wireless sensor networks in our daily lives; the study of wireless sensor networks has become one of the most important areas of research [1], [2], [10], [16], [24], [28], [29], [34], [36], [42]. A wide range of applications exist for wireless sensor networks, including environ-mental monitoring, battlefield surveillance, health care, nuclear, biological, and chemical (NBC) attack detection, intruder detection, and so on. Another application–and one of the most important areas of research–is object tracking, in which sensors monitor and report the loca-tions of mobile objects [4], [7], [20], [25], [33], [38], [45].In a wireless sensor network, sensors are usually in the sleep state to save energy to prolong the network life. The tracking system in a wireless sensor network usually includes three components: 1) a monitoring mechanism, 2) a prediction model, and 3) a recovery mechanism [14], [35], [39]. A monitoring mechanism activates selected sensors to monitor and collect the location information • The authors are with the Department of Computer Science, National Tsing
Hua University, Hsinchu, Taiwan 30013, ROC.
• E-mail: [email protected], [email protected], [email protected].
of the mobile object using acoustic signal [7], [8], [27] or images of objects [11], [15], [37]. Once the object moves away from the activated sensors, the primary sensor among the activated sensors uses a prediction model to predict the next location of the object and activates the appropriate sensors to continue monitoring the object. One of the activated sensors receives knowledge of being assigned to be the next primary sensor. If the prediction fails to track the object, the recovery mechanism activates additional sensors in order to re-capture the lost object. Therefore, a better prediction model can significantly reduce power consumption because fewer redundant sensors will be activated.
Many methods for predicting object trajectory have been proposed. The methods in [22], [23] predict object trajectory using Kalman filters. In [19], [44], extended Kalman filters are proposed because Kalman filters pro-cess non-linear variations in non-trivial systems with difficulty. In [12], [41], sequential Monte Carlo filters are adopted because the use of extended Kalman filters may lead to divergence due to the non-linear nature of the system. All of these filters, however, require storage of many parameters, which are updated by the measured location, velocity, and acceleration of the object, in order to predict the next location of the object. Therefore, these filters are not suitable for tracking objects in a wireless sensor network because multiple parameters (messages) must be transmitted between the primary sensors, placing a heavy power consumption burden on wireless sensors, which are usually battery-powered.
TABLE 1: Summary of Notations
µ A Gauss-Markov parameter used to denote the mean velocity as t→ ∞.
σ A Gauss-Markov parameter used to denote the stan-dard deviation of velocity as t→ ∞.
α A Gauss-Markov parameter used to vary the random-ness of the Gauss-Markov equation.
ˆ
µt The value of µ estimated at time slot t.
ˆ
σt The value of σ estimated at time slot t.
ˆ
αt The value of α estimated at time slot t.
vt The velocity of a mobile object at time slot t.
Vt The random variable of vt.
Vt−1 The random variable of vt−1. A The random variable of α.
N (µ, σ2) A normal distribution having a mean equal to µ and
a standard deviation equal to σ. ˜
αt The most likely value of α at time slot t.
¯ ˜
αt The mean of ˜α1, ˜α2, . . ., and ˜αt.
˜
αt(v, x) The evaluated value of ˜αt, given that Vt−1= vand Xt−1= x.
¯ ˜
α The mean of ˜αt(v, x)for all possible values of v and
x.
fV(v) The probability density function of random variable
V.
LVt(α) The likelihood function of α for sample vt.
Some methods for predicting object trajectory produce little message communication overhead in a wireless sensor network. The instant prediction model [21], [39], [40] and the average prediction model [14], [26], [39] predict the subsequent velocity of the object using the current velocity and the mean of previous velocity mea-surements, respectively. The exponential average predic-tion model [39] predicts the subsequent velocity of the object using the current velocity and the last estimated velocity. Although the instant, average, and exponen-tial average prediction models have little or no need for message transmission in a wireless sensor network, they do not well predict object trajectory in a more complicated mobility model, such as the Gauss-Markov mobility model [17].
The Gauss-Markov mobility model is one of the best mobility models to describe object trajectory because it can capture the correlation of object velocity in time. Ad-ditionally, the Gauss-Markov mobility model, using dif-ferent Gauss-Markov parameters, can duplicate the ob-ject mobility pattern generated by other popular mobility models, such as the random walk, the fluid flow, and the random waypoint mobility models [13], [17], [18], [43]. Estimation of the Gauss-Markov parameters is critical to correctly predict object trajectory. The Gauss-Markov parameters were estimated via an autocorrelation tech-nique in [17], [18], [43], and were estimated via a recur-sive least square estimation in [13]. Since these Gauss-Markov parameter estimators require a large amount of historical movement information of mobile objects, they are not suitable for the estimation of the Gauss-Markov parameters of mobile objects in wireless sensor networks because they demand a considerable amount of message
communication overhead, and however, wireless sensor networks are power-sensitive.
To date, to the best of our knowledge, no existing methods can accurately estimate the Gauss-Markov pa-rameters with few requirements in terms of message communication overhead in a wireless sensor network, thereby providing the motivation of this paper. The remainder of this paper is organized as follows. Related works are introduced in Section 2. In Section 3, the GMPE MLH model is proposed. In Section 4, theoretical analysis for the GMPE MLH model is provided. Section 5 gives numerical results. Finally, we conclude this paper with the discussion of future research in Section 6.
2
R
ELATEDW
ORKSWe demonstrate two Gauss-Markov parameter esti-mators: GMPE ACR [17], [18] and GMPE RLSE [13]. GMPE ACR and GMPE RLSE use the autocorrelation and recursive least square estimation techniques to esti-mate the Gauss-Markov parameters, respectively. TABLE 1 summaries the notations used in this paper.
2.1 The GMPE ACR Model
In the GMPE ACR model, n Gauss-Markov equations are used to describe the movement of an object in n-dimensional space. In each dimension, the velocity of a mobile object at time slot t, vt, is modeled by the
following Gauss-Markov equation:
vt= αvt−1+ (1− α)µ +
√
1− α2X
t−1, (1)
where Xt−1 is the (t− 1)-th random variable chosen
from a normal distribution having a mean equal to zero and a standard deviation equal to σ, α (0 ≤ α ≤ 1) denotes the parameter used to vary the randomness of Eq. 1, µ denotes the parameter used to represent the mean velocity as t → ∞, and σ denotes the parameter used to represent the standard deviation of velocity as
t → ∞. Additionally, µ, σ, and α are called
Gauss-Markov parameters. A normal distribution having a mean equal to µ and a standard deviation equal to σ is denoted by N (µ, σ2).
In each dimension, given the previous westsamples of
the device velocity v1′, v2′, . . ., vw′est, the estimated values
of µ, σ, and α at time slot t, denoted by ˆµt, ˆσt, and ˆαt,
respectively, are calculated by the following equations: ˆ µt= 1 west west ∑ i=1 v′i, (2) ˆ σt2= 1 west− 1 west ∑ i=1 (v′i− ˆµt)2, (3) and ˆ αt= 1, if ˆσt≈ 0, max { 0,σˆt′2 ˆ σ2 t } , otherwise, (4) where ˆσ′2t =west1−1 ∑west−1 i=1 (vi′− ˆµt)(v′i+1− ˆµt).
t v 1 ˆt P 1 t v 1 t D 1 ˆt V 1 t Dˆt ˆt P t v t Dˆt V t
Fig. 1: The GMPE MLH model.
In each dimension, the predicted location of the mobile device at time slot t + n, ˆxt+n, is calculated by the
following equation: ˆ xt+n= xt+ 1− ˆαn t 1− ˆαt vt+ ( 1−1− ˆα n t 1− ˆαt ) ˆ µt, (5)
where xtdenotes the actual location of the mobile device
at time slot t.
2.2 The GMPE RLSE Model
In the GMPE RLSE model, the velocity and the moving direction with respect to the positive x-axis are used to describe the movement of an object in 2-dimensional space. The velocity and the moving direction of a mobile object at time slot t, denoted by vt and θt, respectively,
are modeled by the Gauss-Markov equation shown in Eq. 1.
Given the previous westsamples of the device velocity v′1, v2′, . . ., vw′est, the estimated value of µ at time slot t,
ˆ
µt, is calculated by Eq. 2. The estimated value of α at
time slot t, ˆαt, is calculated by the following recursive
least square estimation: ˆ
γ
west = ˆγwest−1− Kwest(Hwestˆγwest−1− ywest), (6)
where ˆγ west = [ ˆ αt 1− ˆαt ] , Hk = [ v′k−1 µˆt ] , y k = [ vk′ ], Kk = PkHTk, Pk = λ1 [ Pk−1 − Pk−1HTkHkPk−1 λ+HkPk−1HT k ] ,
P0 is an identity matrix, and λ is a tunable parameter.
The estimated velocity of the mobile object at time slot
t + 1, ˆvt+1, is calculated by the following equation:
ˆ
vt+1= ˆαtvt+ (1− ˆαt)ˆµt. (7)
The estimated moving direction of the mobile object at time slot t + 1, ˆθt+1, can be calculated in the manner
analogous to that of ˆvt+1. The predicted location of the
mobile object at time slot t + 1, (ˆxt+1, ˆyt+1), is calculated
by the following equations: ˆ
xt+1= xt+ ˆvt+1cos ˆθt+1 (8)
and
ˆ
yt+1= yt+ ˆvt+1sin ˆθt+1, (9)
where (xt, yt) denotes the actual location of the mobile
object at time slot t.
3
T
HEGMPE MLH M
ODELOur system model and assumptions are first demon-strated. Then, we describe how the Gauss-Markov pa-rameters µ, σ, and α are estimated in the GMPE MLH model. Finally, the prediction location of a mobile object using the estimated values of µ, σ, and α is given.
3.1 System Model and Assumptions
It is assumed that each sensor has a region of detection and is capable of measuring the velocity vectors of the target. In our system model, once an object moves into the sensing range of the wireless sensor network, a monitoring mechanism activates sensors to monitor and collect the location information of the object and selects one of the activated sensors to be the primary sensor. Once the object moves away from the activated sensors, the primary sensor uses the GMPE MLH model to predict the next location of the object, activates the ap-propriate sensors to continue monitoring the object, and designates the next primary sensor among the activated sensors.
In the Gauss-Markov mobility model, when an object moves, the future location is expected to be accurately predicted by the estimation of its Gauss-Markov parame-ters, µ, σ, and α, in n dimensions. Here, we only consider the estimation of µ, σ, and α in one of the n dimensions due to the similarities of the calculations. Fig. 1 illustrates the GMPE MLH model in one dimension. The primary sensor uses the parameter estimator to evaluate ˆµt, ˆσt, ˆαt,
and ¯˜αtafter it measures vtand receives ˆµt−1, ˆσt−1, ¯˜αt−1, vt−1, and t−1 from the previous sensor, where ˆµt, ˆσt, and
ˆ
αtdenote the estimated values of µ, σ, and α at time slot t, respectively, and ¯˜αt denotes the mean of ˜α1, ˜α2, . . .,
˜
αt, where ˜αt denotes the most likely value of α at time
slot t, as discussed later. For each dimension, 4 messages of ˆµt, ˆσt, ¯˜αt, and vt must be transmitted between the
primary sensors. Therefore, a total of 9 messages are required to be transmitted between the primary sensors in 2-dimensional space.
3.2 Estimation of µ and σ
In the GMPE MLH model, the following recurrence exists for ˆµt: ˆ µ1= v1; ˆ µt= t− 1 t µˆt−1+ 1 tvt, if t≥ 2; (10) and, the following recurrence exists for ˆσt:
ˆ σ21= 0; ˆ σ22= (v1− ˆµ2) 2+ (v 2− ˆµ2)2 2 ; ˆ σ2t = t− 1 t ˆσ 2 t−1+ 1 t− 1(vt− ˆµt) 2, if t≥ 3. (11)
In the GMPE MLH model, once the t-th primary sensor measures vtand receives vt−1, ˆµt−1and ˆσt−1from
the previous sensor, ˆµtis first evaluated according to Eq.
D D ˃ˁˆˈˆˆˋ ˃ˁˇˇˉ˃˅ ˃ ˃ˁ˄ ˃ˁ˅ ˃ˁˆ ˃ˁˇ ˃ˁˈ ˃ˁˉ ˃ˁˊ ˃ˁˋ ˃ˁˌ ˄ ˃ ˃ˁ˄ ˃ˁ˅ ˃ˁˆ ˃ˁˇ ˃ˁˈ ˃ˁˉ ˃ˁˊ ˃ˁˋ ˃ˁˌ ˄ ˥˸˴˿ʳ˖̈̅̉˸ ˥˸˺̅˸̆̆˼̂́ʳ˖̈̅̉˸
Fig. 2: A plot of ¯˜αversus α.
3.3 Estimation of α
We evaluate ¯˜αt using a maximum likelihood technique.
Subsequently, we calculate ˆαt by showing the
relation-ship between ¯˜αand α.
B.1 Evaluation of ¯˜αt
The following recurrence exists for ¯˜αt:
¯ ˜ αt= 1 tα˜t+ t− 1 t α¯˜t−1. (12)
This leads us to calculate ˜αt, the α value having the
maximum probability that the velocity of the mobile object is changed from vt−1to vtat time slot from t−1 to t, given vt−1, vt, µ, and σ. Our idea is to evaluate ˜αtusing
a maximum likelihood estimation for α. Let A denote the random variable of α. Assume vtis a random sample of a
random variable Vt. Let LVt(α)be the likelihood function of α for sample vt. It follows that ˜αt, the maximum
likelihood estimation for α, is the α (0 ≤ α ≤ 1) value which maximizes LVt(α).
Given vt−1, vt, µ, and σ, LVt(α)is calculated as follows: Substitution into Eq. 1 yields Vt = b + aXt−1, where a = √1− α2 and b = αv
t−1 + (1 − α)µ. Because Xt−1 ∼ N(0, σ2), we have Vt ∼ N(b, a2σ2). Therefore,
the conditional probability density function of Vt given A = α, fVt|A(v|α) = 1 aσ√2πe −(v−b)2 2a2 σ2. Thus, LV t(α) is established in Eq. 13. LVt(α) = fA|Vt(α|vt) = 1 aσ√2πe −(vt−b)2 2a2 σ2 . (13)
Here, ˜αt is calculated by a simple method as follows:
The interval [0, 1] is split into a finite number of subin-tervals [α0, α1], [α1, α2], . . ., [αm−1, αm] with α0 = 0 < α1< . . . < αm= 1. Subsequently, ˜αt is set to αi+α2i+1 if
∫αi+1
αi LVt(α)dα≥ ∫αj+1
αj LVt(α)dαfor 0≤ j < m.
In the GMPE MLH model, once the t-th primary sensor has vt−1, vt, ˆµt, ˆσt, and ¯˜αt−1, the sensor first
evaluates ˜αtas the α (0≤ α ≤ 1) value which maximizes
1
√
1−α2σˆt√2πe
−(vt−αvt−1−(1−α)ˆ2(1−α2)ˆσ2 µt)2
t , and subsequently, eval-uates ¯˜αt according to Eq. 12.
B.2 Evaluation of ˆαt
Let Vt−1 denote the random variable of vt−1, fVt−1,Xt−1(v, x) denote the probability density function of Vt−1 and Xt−1, and ˜αt(v, x) denote the α (0 ≤ α ≤ 1) value which maximizes LVt(α) given that
Vt−1 = v and Xt−1 = x. Then, ¯˜α, the mean of
˜
αt(v, x) for all possible values of v and x, is equal
to ∫−∞∞ ∫−∞∞ fVt−1,Xt−1(v, x) ˜αt(v, x)dxdv. Theorem 1, in
Section 4.1, shows the following equation exists for ¯˜α
of a mobile object having the Gauss-Markov parameter
α = α1: ¯ ˜ α = ∫ ∞ −∞ ∫ ∞ −∞ 1 2πe −v2 +x2 2 α˜t(v, x)dxdv, (14)
where ˜αt(v, x)is equal to the α (0≤ α ≤ 1) value which
maximizes √ 1 2π(1−α2)e − ( α1v+√1−α21x−αv)2 2(1−α2) .
We evaluate the relationship between ¯˜α and α using a regression method. First, ¯˜α is evaluated as α1 (= α)
varies from 0 to 1, increased by 0.0001, as illustrated in Fig. 2. Let the regression model function be
G(x) = 1 θ1 ln(0.2 θ2x − 1 θ2 ),
where θ1 and θ2 are model parameters. The model
function with θ1 = 53.151 and θ2 = −0.43403 best
fits the curve plotted by the samples with α1 =
0, 0.0001, 0.0002,· · · , 0.005, and the model function with
θ1= 0.37492and θ2=−0.55069 best fits the curve
plot-ted by the samples with α1= 0.005, 0.0051, 0.0052,· · · , 1.
A function of the regression curve which best fits the plot in Fig. 2 is derived, as shown in the following equation:
α = α1= 0, if ¯˜α≤ 0.35338; 1 53.151ln ( 0.2 −0.43403¯˜α− 1 −0.43403), if 0.35338 < ¯˜α≤ 0.44602; 1 0.37492ln ( 0.2 −0.55069¯˜α− 1 −0.55069), if 0.44602 < ¯˜α≤ 1; 1, if ¯˜α > 1. (15)
Therefore, in the GMPE MLH model, once the t-th pri-mary sensor has ¯˜αt, the sensor evaluates ˆαt by Eq. 16.
ˆ αt= 0, if ¯˜αt≤ 0.35338; 1 53.151ln ( 0.2 −0.43403¯˜αt − 1 −0.43403), if 0.35338 < ¯˜αt≤ 0.44602; 1 0.37492ln ( 0.2 −0.55069¯˜αt − 1 −0.55069), if 0.44602 < ¯˜αt≤ 1; 1, if ¯˜αt> 1. (16) 3.4 Location Prediction
We calculate the predicted location of the mobile object at time slot t + 1, ˆxt+1, as xt+ E[ˆvt+1], where E[ˆvt+1]
denotes the expected value of ˆvt+1. Because ˆvt+1= ˆαtvt+
(1− ˆαt)ˆµt+
√ 1− ˆα2
tXtby Eq. 1 and E[Xt] = 0(because Xt∼ N(0, σ2)), ˆxt+1can be calculated by the following
equation: ˆ
t ˆt P P ˃ ˄ ˅ ˆ ˇ ˈ ˉ ˊ ˋ ˌ ˄˃ ˄˃˃ ˅˃˃ ˆ˃˃ ˇ˃˃ ˈ˃˃ ˉ˃˃ ˊ˃˃ ˋ˃˃ ˌ˃˃ ˄˃˃˃ (a) t ˆt V V ˃ ˄ ˅ ˆ ˇ ˈ ˉ ˄˃˃ ˅˃˃ ˆ˃˃ ˇ˃˃ ˈ˃˃ ˉ˃˃ ˊ˃˃ ˋ˃˃ ˌ˃˃ ˄˃˃˃ (b) t ˆt D D ˃ ˃ˁ˃ˆ ˃ˁ˃ˉ ˃ˁ˃ˌ ˃ˁ˄˅ ˃ˁ˄ˈ ˄˃˃ ˅˃˃ ˆ˃˃ ˇ˃˃ ˈ˃˃ ˉ˃˃ ˊ˃˃ ˋ˃˃ ˌ˃˃ ˄˃˃˃ (c) ˚̅̂̈̃ʳ˔ ˚̅̂̈̃ʳ˕ ˚̅̂̈̃ʳ˖ ˚̅̂̈̃ʳ˗ ˚̅̂̈̃ʳ˘ ˚̅̂̈̃ʳ˙ ˚̅̂̈̃ʳ˚ ˚̅̂̈̃ʳ˛
Fig. 3: The differences between the actual and estimated values of parameters (a) µ, (b) σ, and (c) α for the GMPE MLH model.
4
A
NALYSIS OF THEGMPE MLH M
ODELThe evaluation of ¯˜α described in Eq. 14 is first shown to be correct. Subsequently, the convergence rates of the estimation of µ, σ, and α are studied. Finally, the accuracy of the estimation of µ, σ, and α is provided. In the paper, all lemmas and theorems are proved in the appendix.
4.1 Correctness of Evaluation of ¯α˜
Lemma 1 demonstrates the evaluation of fVt−1(v), fXt−1(x), and ˜αt(v, x), where fVt−1(v) and fXt−1(x)
de-note the probability density functions of Vt−1 and Xt−1,
respectively. Lemma 1 is necessary for the proof of Lemma 2, in which it is shown that ¯˜αis invariant with respect to µ and σ of the mobile object. Then, with the help of Lemma 2, ¯˜α of the mobile object having the Gauss-Markov parameters µ, σ, and α = α1 can be
evaluated by setting µ = 0 and σ = 1, as concluded in Theorem 1.
Lemma 1: Let O1 be a mobile object having the
Gauss-Markov parameters µ = µ1, σ = σ1, and α = α1. For object O1, fVt−1(v) = √2πσ1 1e −(v−µ1)2 2σ21 , fXt−1(x) = √2πσ1 1e −x2 2σ21, and ˜α t(v, x) is equal to
the α (0 ≤ α ≤ 1) value which maximizes
1 √ 2π(1−α2)σ1e − ( α1v+(1−α1)µ1+√1−α21x−αv−(1−α)µ1)2 2(1−α2)σ21 .
Lemma 2: Let O1and O2be two mobile objects having
the Gauss-Markov parameters µ = µ1, σ = σ1, and α = α1 and the Gauss-Markov parameters µ = µ2, σ = σ2,
and α = α2, respectively. If α1 = α2, ¯˜α of object O1 is
equal to ¯˜αof object O2.
Theorem 1: Let O1 be a mobile object having the
Gauss-Markov parameter α = α1. Then, ¯˜αof object O1is
equal to ∫−∞∞ ∫−∞∞ 1 2πe−
v2 +x2
2 α˜t(v, x)dxdv, where ˜αt(v, x)
is equal to the α (0 ≤ α ≤ 1) value which maximizes
1 √ 2π(1−α2)e − ( α1v+√1−α21x−αv )2 2(1−α2) .
4.2 Convergence Rates of Estimation of µ, σ, and α
Theorem 2 shows that the convergence rates of ˆµt and
ˆ
σt are 1/√n, where n denotes the number of samples,
using a similar argument described in [32]. In addition, because ˆαtis evaluated by ¯˜αtby Eq. 16, we demonstrate
that the convergence rate of ¯˜αt is 1/ √
n in Theorem 3. Let Wi denote a random variable, ˜αt(Vt−1, Xt−1). To
prove Theorem 3, it is sufficient to show 1
n ∑n i=1Wi ∼ N ( ¯α, (˜ √σ n) 2) as n → ∞. Let X 1, X2,· · · , Xn denote a
sequence of specific random variables, and let Fn(x)
and Φ(x) be the cumulative distribution functions of random variable Z = Tn−nµ
σ√n and N (0, 1), respectively,
where Tn =
∑n
i=1Xi, and µ and σ2 denote the mean
and variance of X1, respectively. Lemma 3, as proved in
[32], describes that the least upper bound of the absolute difference of Fn(x) and Φ(x) for x ∈ R is bounded by O(1/√n), implying that the distribution of Tn=
∑n i=1Xi
approximates to N (nµ, (σ√n)2) on the order of 1/√n.
Because W1, W2,· · · , Wn are a sequence of specific
ran-dom variables described in Lemma 3 and E[W1] = ¯α˜,
∑n
i=1Wi approximates to N (n¯˜α, (σ √
n)2) on the order of 1/√n. This implies that 1
n
∑n
i=1Wi ∼ N(¯˜α, (√σn)2) as n→ ∞, and hence Theorem 3 follows.
Theorem 2: The convergence rates of ˆµt and ˆσt are
1/√n, where n denotes the number of samples.
Lemma 3: Let {Xn} be a sequence of independent
and identically-distributed (i.i.d.) random variables with
E[X1] = µ, variance of X1equal to σ2and suppose that E[|X1− µ|2+δ] = ν2+δ < ∞ for some 0 < δ ≤ 1. Also
let Tn=
∑n
i=1Xi, Fn(x) = Pr{ Tn−nµ
σ√n ≤ x}, x ∈ R. Then
there exists a constant C such that ∆n= sup
x∈R
|Fn(x)− Φ(x)| ≤ C
ν2+δn−δ/2
σ2+δ , (18)
where supx∈Rg(x) denotes the least upper bound or
supremum of g(x) for x∈ R and Φ(x) is the cumulative distribution function of N (0, 1).
Theorem 3: The convergence rate of ¯˜αtis 1/ √
n, where
İȝ c o n v _ tȝ ˃ ˄˃˃ ˅˃˃ ˆ˃˃ ˇ˃˃ ˈ˃˃ ˉ˃˃ ˊ˃˃ ˋ˃˃ ˌ˃˃ ˄˃˃˃ ˃ˁ˄ ˃ˁ˅ ˃ˁˆ ˃ˁˇ ˃ˁˈ ˃ˁˉ ˃ˁˊ ˃ˁˋ ˃ˁˌ ˄ (a) İı c o n v _ tı ˃ ˄˃˃ ˅˃˃ ˆ˃˃ ˇ˃˃ ˈ˃˃ ˉ˃˃ ˊ˃˃ ˋ˃˃ ˌ˃˃ ˄˃˃˃ ˃ˁ˄ ˃ˁ˅ ˃ˁˆ ˃ˁˇ ˃ˁˈ ˃ˁˉ ˃ˁˊ ˃ˁˋ ˃ˁˌ ˄ (b) İĮ c o n v _ tĮ ˃ ˄˃˃ ˅˃˃ ˆ˃˃ ˇ˃˃ ˈ˃˃ ˉ˃˃ ˊ˃˃ ˋ˃˃ ˌ˃˃ ˄˃˃˃ ˃ˁ˄ ˃ˁ˅ ˃ˁˆ ˃ˁˇ ˃ˁˈ ˃ˁˉ ˃ˁˊ ˃ˁˋ ˃ˁˌ ˄ (c) ˚ˠˣ˘˲˔˖˥ʳʻ˪˸̆̇ː˄˃ʼ ˚ˠˣ˘˲˔˖˥ʳʻ˪˸̆̇ː˄˃˃ʼ ˚ˠˣ˘˲˔˖˥ʳʻ˪˸̆̇ːЌʼ ˚ˠˣ˘˲ˠ˟˛ ˚ˠˣ˘˲˥˟˦˘ʳʻ˪˸̆̇ː˄˃ʼ ˚ˠˣ˘˲˥˟˦˘ʳʻ˪˸̆̇ː˄˃˃ʼ ˚ˠˣ˘˲˥˟˦˘ʳʻ˪˸̆̇ːЌʼ
Fig. 4: Convergence rates of the estimated values of (a) µ, (b) σ, and (c) α for the GMPE MLH model, the GMPE ACR model, and the GMPE RLSE model in the Gauss-Markov mobility model having parameters of Group A.
4.3 Accuracy of Estimation of µ, σ, and α
Theorem 4 shows that ˆµt = µ and ˆσt = σ as t → ∞,
and Theorem 5 demonstrates the asymptotic confidence interval for α. As n → ∞, because n1
∑n
i=1Wi ∼ N ( ¯α, (˜ √σn)2), ¯˜α
t= ¯α˜, where Wi denotes a random
vari-able, ˜αt(Vt−1, Xt−1). In addition, ˆαt= G( ¯α˜t). Therefore,
as t → ∞, the difference between α and ˆαt can be
obtained by evaluating the difference between the real curve and the regression curve in Fig. 2. Lemma 4, as proved in [31], provides a statistical analysis of the data of these two curves, and helps us to complete the proof of Theorem 5.
Theorem 4: limt→∞µˆt= µand limt→∞σˆt= σ.
Lemma 4: Assume that yi = f (xi;θ) + ϵi and ˆyi = f (xi; ˆθ) for given n samples (x1, y1), (x2, y2), · · · ,
(xn, yn), where θ = (θ1, θ2, . . . , θp)T is the true value of
the parameter vector, ϵi are independent and
identically-distributed (i.i.d.) N (0, σ2), and ˆθ = ( ˆθ
1, ˆθ2, . . . , ˆθp)T. The
approximate 95% confidence interval for y at x = x0 is
given by ˆ y0± t0.025k s √ 1 + fT 0(FTF)−1f0, (19)
where ˆy0 = f (x0; ˆθ), t0.025k = 1.645 denotes the
up-per 0.025 critical value of the t-distribution with k degrees of freedom, s = √∑ni=1(yi− ˆyi)2/k, fi =
(∂f (xi;θ) ∂θ1 , ∂f (xi;θ) ∂θ2 , . . . , ∂f (xi;θ) ∂θp )T for 0 ≤ i ≤ n, and F = (f1, f2, . . . , fn)T.
Theorem 5: The approximate 95% confidence interval
for α is within 0.00812 of ˆαtas t→ ∞.
5
P
ERFORMANCEE
VALUATIONTo evaluate the performance of the GMPE MLH model, the absolute differences between the actual and esti-mated values of µ, σ, and α with mobile objects in the Gauss-Markov mobility model were investigated. In addition, the convergence rates of the estimated values of µ, σ, and α were studied for the GMPE MLH model, the GMPE ACR model, and the GMPE RLSE model.
Moreover, the root mean square error (RMSE) [44] of the GMPE MLH model, the GMPE ACR model, and the GMPE RLSE model were measured in the Gauss-Markov mobility model. The value of RMSE is used to indicate the differences between the actual and es-timated object trajectories. We also measured RMSE of the GMPE MLH model, the GMPE ACR model, and the GMPE RLSE model in various mobility models. Nevertheless, the GMPE MLH Dynamic model, an ex-tention of the GMPE MLH model, was compared with the GMPE ACR model and the GMPE RLSE model in the dynamic mobility models in terms of RMSE. In our simulations, 1000 mobile objects were generated having parameters randomly chosen from the parameter ranges of the mobility models. The plotted results were obtained by averaging the data of 1000 mobile objects.
5.1 Accuracy of Estimated Parameters
In the simulation, 1000 objects move in the Gauss-Markov mobility model, having different values of µ, σ, and α randomly chosen from the intervals [−d, d], [0, e], and [0, f ], respectively. To study the performance of the GMPE MLH model in groups of objects with different expected mean, variance, and randomness of velocities, we simulate objects having 8 groups of parameters (d, e,
f): A) (100, 100, 1), B) (100, 100, 0.5), C) (100, 50, 1), D) (100, 50, 0.5), E) (50, 100, 1), F) (50, 100, 0.5), G) (50, 50, 1), and H) (50, 50, 0.5).
As illustrated in Fig. 3, the larger the value of t, the smaller were the average values observed for |µ − ˆµt|, |σ − ˆσt|, and |α − ˆαt|. That is, the GMPE MLH model
provides better estimation of the values of µ, σ, and α as the value of t increases. This observation is reasonable because, as the value of t increases, more information concerning object trajectory is used to estimate the values of µ, σ, and α. The value of d was noted to have a negligible effect on the average values of|µ− ˆµt|, |σ−ˆσt|,
and |α − ˆαt|. Additionally, it was observed that the
smaller the value of e, the smaller the average value of
|µ − ˆµt|. If an object has the smaller expected value of σ
(as the value of e is smaller), the expected absolute dif-ference between vtand µ is smaller, making the absolute
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