the nearest neighbor (NN) is not always the most informative measurement. Simulations corroborate our analysis. Keywords: Data association, tracking, order ...
12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009
Distributed Estimation with Data Association: Is the Nearest Neighbor The Most Informative? Paolo Braca† , Marco Guerriero∗ , Stefano Marano† , Vincenzo Matta† , and Peter Willett∗ †
∗
DIIIE, University of Salerno, Fisciano (SA), Italy ECE Department, University of Connecticut, Storrs, CT 06269, USA
combine the NN measurements from the sensors using the maximum a posteriori estimator (MAP) criterium [4]. The main result of this paper is to show that the fusion of the local NN measurements can be outperformed by fusing some optimized local order statistics. More specifically each sensor will send the “best” kth modulus order statistic (k-MOS) to the FC, where the word best is referred to the most informative measurement (maximizing the Fisher information). The remainder of the paper is organized as follows: in Section 2 we describe the model, in Section 3 we study the problem of designing the best strategy, in Section 4 we provide some simulation results, in Section 5 we discuss our work and in Section 6 we present some future directions.
Abstract – In distributed multi-sensor estimation/tracking the problem of measurement fusion arises. In large sensor networks (SN), each sensor is constrained by bandwidth to communicate only one of its observations to a Fusion Center (FC) for a global estimate. We study the problem of distributed estimation with data association, where the FC “optimally” combines the “best” measurements from the sensors, instead of suboptimally combining the local estimates. Using order statistics, we show that, surprisingly, the nearest neighbor (NN) is not always the most informative measurement. Simulations corroborate our analysis. Keywords: Data association, tracking, order statistics.
2 1
Introduction
Problem Formulation
Let us introduce some notation. Consider N sensors, each modeled as a 1-dimensional gate (see Figure 1). The observation range is (−V /2, V /2).
In this paper, we are interested in estimation fusion of distributed observations that have to be shared over a channel with bandwidth constraints. As pointed out in [1], there is recent interest in considering measurement-fusion in place of the more classical track-fusion approach for which the fusion is almost always suboptimal [2], [3]. Our prime interest is in target tracking using multiple sensors: there is a network of sensors in clutter environment, each observing a target, and each constrained by bandwidth to communicate only one of its observation to a FC, which must make a global estimate. In order to get some useful insights we do not consider dynamic targets, and we work in one dimension. The problem of measurement origin uncertainty is still preserved. In data association problems, a simple estimator for the target’s position is the NN measurement [2], that is the measurement which is closest to the predicted target’s position. A possible strategy for the FC could be to
• Each random variable (r.v.) will be denoted with upper-case letter (ex. X), the value of the r.v. will be denoted with lower-case letter (ex. x). • The uniform pdf is written as 1 x ∈ − V2 , V2 , V U(x; V ) = 0 otherwise • For the Gaussian distribution we use (x−µ)2 1 N (x; µ, σ) = √ e− 2σ2 , x ∈ R σ 2π • The true target position is indicated with θ. • The vector m = [m1 , m2 , . . . , mN ] denotes the aggregate of the number of measurements. • The measurement vector at the nth sensor is xn = [xn1 , xn2 , . . . , xnmn ], whereas the set of the measurements across all the sensors is denoted with [x1 , x2 , . . . , xN ].
∗ Marco
Guerriero and Peter Willett were supported by the Office of Naval Research under contract N00014-07-1-0429 and N00014-07-1-0055.
978-0-9824438-0-4 ©2009 ISIF
∗
780
Figure 1: Illustration of the problem. Here, three sensors send the k-th order statistics of the observations and attempt to estimate a common θ. All sensors have false alarms, and the first and the third sensors miss its detection. • The modulus ordering vector for the observations xn of the nth sensor is xord = n [xn,π(1) , xn,π(2) , . . . , xn,π(mn ) ] with |xn,π(1) − θ0 | ≤ |xn,π(2) − θ0 | · · · ≤ |xn,π(mn ) − θ0 |, where θ0 is our a priori prediction on the target position. Consequently, the k-MOS is the kth element of that vector. For example if xn = [3.1, 2.0, 0.4] and θ0 = 0 then we have xord n = [0.4, 2.0, 3.1].
standard deviation σ, which accounts for measurement error. Furthermore we will denote with Yn such a random variable. • At most one measurement per sensor corresponds to the target. • For each nth sensor, the number of false alarms (clutter) zn is Poisson distributed according to the probability mass function (pmf) µF (zn ; λ) = zn e−λV (λVzn)! , with spatial density λ. The pmf of the number of measurements mn will be obviously µM (mn ; λ) = (1 − Pd )µF (mn ; λ) + Pd µF (mn − 1; λ).
• The probability density function (pdf) of the kMOS will be denoted as fXπ(k) (x), the conditional pdf as fXπ(k) (x |θ, m ). We use the following assumptions: • A single target, whose true position is θ, is present in each sensor gate with a given probability of detection Pd , and detections are independent across sensors.
• False alarms are distributed uniformly over the gate (−V /2, V /2) and they are independent and identically distributed, both at the single-sensor level and with respect to other sensors.
• The a priori distribution of θ is Gaussian with mean θ0 and standard deviation σθ . Moreover we shall assume θ0 = 0, that is the gate is centered on the expected target position. This assumption is made only for simplicity, because the modulus ordering is symmetric respect to θ0 .
With the above assumptions, the joint pdf of the measurement set [x1 , x2 , . . . , xN ] and of the number of measurements m, for a given θ, is
• Conditioned on θ, the correspondent measurement at the nth sensor is a θ-mean Gaussian variable (independent from sensor to sensor) with
f (x1 , . . . , xN , m|θ) =
N
n=1
781
f (xn , mn |θ)
(1)
1.4
1.14
1.2
1.12
m =5 n
k=1
π(k)
Fisher Information
1.1
target position θ
0.8
fX
(x|θ,m)
1
0.6
0.4
1.06
0.2
1.02
mn=15 −4
−3
−2
−1
0
1
2
3
4
5
1 2
x
µF (mn − 1; λ)Pd mn V mn −1 +
6
8
10
12
14
Figure 3: The Fisher Information, as in Equation (6), is plotted versus the index k of the k-MOS MOS, for different values of m. Here we used V = 10.
where, for mn > 0, f (xn , mn |θ) is given by [2]: f (xn , mn |θ) =
4
kn
Figure 2: Pdf of the k-MOS for two different values of k. Here, we have used V = 10, m = 15, Pd = 0.9, σ = 0.1, θ = 1.5.
mn
k=1
xn,π(1) (the NN measurement) maximizes the association probability that the ith measurement of the vector xn is target originated [2]. In some situation (heavy clutter), transmit the first order statistic (NN measurement) is not a good strategy. It would be preferable to transmit the best k-MOS which maximizes the Fisher information.
N (xnk ; θ, σ)
µF (mn ; λ)(1 − Pd ) (2) V mn
whereas for mn = 0 we have: f (xn , mn = 0|θ) = µF (0; λ)(1 − Pd ).
3.1
(3)
Best k-MOS modulus order statistic
In distributed estimation for measurement-origin uncertain situations, a trade-off between estimation performance and communications constraints is required. The former is measured in terms of MSE, and the latter is taken into account by imposing that each nth sensor has to transmit only one measurement to the FC. Our intuition is that in heavy clutter, despite its being the most likely to be target originated, the NN might not contain a significant information about θ. Measurements from a higher MOS would show a pdf that, while more diffuse than that of the NN, would exhibit a bump at the true target location θ, see Figure 2. A good metric to measure the estimation performance at the FC is the Fisher information on the random parameter θ, I(k, m, N ) [4]:
Further we will assume that the gate probability is close to one.
3
m =10 n
1.04
k=3
0 −5
1.08
System Design
In a bandwidth unconstrained scenario, the FC would be demanded to estimate the random parameter θ (target position) based on the measurement set X from all the N sensors. Among different possible estimation strategies, we adopt the MAP criterium. The FC computes the MAP estimates θˆmap (x1 , . . . , xN , m). The MAP equation is the following: θˆmap (x1 , . . . , xN , m) = arg max f (θ|x1 , . . . , xN , m) θ
(4) Discussions on the properties of the MAP estimator are given in detail in [4]. A downside of this approach is the high bandwidth requirements of sending all of the measurements X to the FC. In [6] the authors have proposed some approaches for quantization where each nth sensor was constrained to quantize its own measurement vector xn . In this work we constrain each sensor to send only one measurement, possibly the most informative. It can be easily shown that the
I(k, m, N ) =
N
n=1
I(kn , mn ) +
1 σθ2
where I(kn , mn ) is given by: 2
∂ I(kn , mn ) = E log fXπ(kn ) (x|θ, mn ) ∂θ
782
(5)
(6)
1
1
10
10
0
0
10
MSE
MSE
10
−1
−1
10
10
NN − simulation Best k−MOS − simulation NN − CRLB Best k−MOS − CRLB
−2
NN − simulation Best k−MOS − simulation NN − CRLB Best k−MOS − CRLB
−2
10
1
2
10
3
10
10
10
4
2
10
4
6
8
k
16
18
20
• σθ = 1.5 (a-priori standard deviation of the r.v. θ). • σ = 1.5 (standard deviation of the r.v. Yn ). The first comparison is given in Figure 4 as function of the number of sensors N . The Cramer-Rao Lower Bound (CRLB) is defined as follows:
(7)
The FC computes the MAP estimate for the NN and for the k-MOS as follows: θˆN N
=
arg max f (θ|x1,π(1) , . . . , xN,π(1) , m)
θˆk−MOS
=
∗ ) , m) arg max f (θ|x1,π(k1∗ ) , . . . , xN,π(kN
CRLB N N = E [I(1, m, N )] CRLB
θ
best k-MOS
∗
−1
(8)
−1
(9)
= E [I(k , m, N )]
where the averaging is with respect to the pmf of the number of measurements. The evaluation of the CRLB as above is carried out by numerical integration. For the best k-MOS strategy, a sensible gain in terms of MSE is shown, especially for large N . The second comparison, provided in Figure 5, with N = 750, shows the better performance of the best k-MOS strategy when the clutter density λ is large, as discussed before.
θ
For small values of mn (which reflect a low clutter scenario), the best k-MOS coincides with the NN strategy (see Figure 3), whereas for larger values of mn (high clutter), the Fisher information attains its maximum for kn > 1. Surprisingly, we found that the NN is not the most informative measurement in terms of Fisher information. In the high clutter scenarios, infact, sending the measurement which is closest to the expected target’s position θ0 , is not a good strategy.
4
14
• λ = 15 (density of clutter).
where the averaging is with respect to the joint pdf fθ,Xπ(kn ) (θ, x|mn ) = fXπ(kn ) (x|θ, mn )N (θ; θ0 , σθ ). The expression for fXπ(kn ) (x|θ, mn ) is derived in the appendix. For the NN strategy we have kn = 1 for all the n. The best k-MOS is defined as: def
12
Figure 5: MSE of the θ estimates, and the CRLB for the best k-MOS strategy and the NN strategy versus the clutter density λ. Here, we have used 104 Monte Carlo runs.
Figure 4: MSE of the θ estimates, and the CRLB for the best k-MOS strategy and the NN strategy versus the number of sensor N . Here, we have used 104 Monte Carlo runs.
kn∗ = arg max I(k, mn ).
10
λ
N
5
Discussion
In this paper, the problem of estimation fusion of distributed observations under measurement origin uncertainty paradigm has been explored. In a distributed sensor network for target tracking applications, each sensor is constrained by bandwidth to communicate only one of its observations to a FC, which must make a global estimate. We worked in one dimension and we considered a static problem. Our results indicate that for light clutter the NN scheme is best: transmit the measurement closest to where the target is expected to be. However, as clutter becomes
Performance Analysis
In this section we present numerical results. The comparison is made in terms of MSE among the best kMOS strategy and the NN strategy. For the Figures 45 we refer to the following: • V = 10 (gate amplitude). • Pd = 0.9 (detection probability).
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heavier it appears to be of significant benefit to transmit the second (or even the third or the fourth) nearest. Our intuition is that in heavy clutter, despite its being the most likely to be target originated, the NN is nonetheless almost certainly the wrong (non-target) one. Measurements from a higher MOS would show a histogram that, while more diffuse than that of the NN, would exhibit a “bump” at the true target location θ.
6
which can be easily computed by the pdf of uniform order statistic for the kth element [5]. Then, for m = 1, k = 1, it is not difficult to show: fXπ(1) (x |θ, m = 1 ) = Pd N (x; θ, σ) + For 1 = k < m, we have: fXπ(1) (x|θ, m) =
Future Work
|x| + θ |x| − θ Pd fUπ(1) (x, m − 1) Q + Q + σ σ m−1 |x| N (x; θ, σ) 1 − + (1 − Pd ) fUπ(1) (x, m) . V /2 (16)
A final version of this manuscript would include, in addition to the previous material, the following: • We will show plots of k ∗ (the best MOS to transmit) versus m (the number of measurements at the sensor) for various σ/V , λ and V .
Let us generalize to the other case 1 < k < m, we have:
• We will extend our results to the case that two measurements (say, the NN and the second nearest) be transmitted. This involves joint order statistic distributions, and is somewhat involved. However, preliminary results indicate (for example), that, in some situations, it would be preferable to transmit two measurements each from 50 sensors than one each from 100 sensors.
fXπ(k) (x|θ, m) = Pd fUπ(k−1) (x, m − 1)+
fUπ(k) (x, m − 1) − fUπ(k−1) (x, m − 1) |x| + θ |x| − θ × Q +Q + N (x; θ, σ) σ σ m−k k−1 |x| m−1 |x| + × 1− k−1 V /2 V /2
• We will exemplify our “intuition” as above.
A
(1 − Pd ) . (15) V
Appendix
Let D and Dc be the event of target detection and its complement, respectively. We denote Uπ(k) the modulus order statistic of the clutter measurements. For m = 1, Xπ(1) is given by: Uπ(1) if D Xπ(1) = (10) Y if Dc .
(1 − Pd ) fUπ(k) (x, m) .
(17)
Finally for k = m > 1 we have:
fXπ(m) (x|θ, m) = Pd fUπ(m−1) (x, m − 1) |x| + θ |x| − θ × 1−Q −Q + σ σ m−1 |x| + (1 − Pd ) fUπ(m) (x, m) . N (x; θ, σ) V /2
For 1 = k < m we have: Uπ(1) if |Uπ(1) | ≤ |Y| ∩ D ∪ Dc Xπ(1) = Y if |Y | ≤ |Uπ(1) | ∩ D, (11) for 1 < k < m:
(18)
Uπ(k) U = π(k−1) Y
if |Uπ(k) | ≤ |Y | ∩ D ∪ Dc Xπ(k) if |Y | ≤ |Uπ(k−1) | ∩ D References if |Uπ(k−1) | ≤ |Y | ≤ |Uπ(k) | ∩ D, [1] F. Palmieri, S. Marano, and P. Willett, “Mea(12) surement Fusion for Target Tracking Under BandFor 1 < k = m we have: width Constraint”, Proceedings of the 2001 IEEE Uπ(m−1) if |Uπ(m) | ≤ |Y | ∩ D Aerospace Conference, Big Sky MT, March 2001. Uπ(m) if Dc Xπ(m) = Y if |Uπ(m) | ≤ |Y | ∩ D. [2] Y. Bar-Shalom, and X. Li, Multitarget-Multisensor (13) Tracking: Principles and Techniques, Storrs, CT: Let us define the pdf of Uπ(k) : YBS Publishers, 1995. m−k k−1 |x| k m |x| [3] S. Blackman, and R. Popoli, Design and AnalfUπ(k) (x, m) = 1− ysis of Modern Tracking Systems, Artech House V V /2 V /2 k (14) (Boston), 1999.
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[4] H. Van Trees, Detection, Estimation and Modulation Theory, Part I, Wiley, 1968. [5] B.C. Arnold, N. Balakrishnan, H.N. Nagaraja, A First Course in Order Statistics, John Wiley & Sons, New York, 1992 . [6] S. Marano, V. Matta, and P. Willett, “Some Approaches to Quantization for Distributed Estimation With Data Association”, IEEE Trans. Signal Process., vol. 53, no. 3, pp. 885-895, March. 2005.
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