Neil Gordon, John Percival and Martin Robinson. Intelligence ... Gordon,John.Percival ..... [14] W. D. Blair, G. A. Watson, T. Kirubarajan, and Y. Bar-. Shalom ...
The Kalman-Levy filter and heavy-tailed models for tracking manoeuvring targets Neil Gordon, John Percival and Martin Robinson Intelligence, Surveillance and Reconnaissance Division Defence Science and Technology Organisation PO Box 1500, Edinburgh SA 5111, Australia {Neil.Gordon,John.Percival,Martin.Robinson}@dsto.defence.gov.au Abstract – The assumption of Gaussian system and measurement noise models has been standard practice for target tracking algorithm development for many years. For problems involving manoeuvring targets or groups of interacting targets this is known to be an over-simplification and a potentially poor approximation. In this paper the use of heavy-tailed noise models coupled with simple dynamics is considered as a means of capturing the non-Gaussian behaviour of targets. A recently published approximate technique called the Kalman-Levy filter is used for track maintenance. Performance of a heavy-tailed system model implemented via the Kalman-Levy filter is compared against an IMM for two sample trajectories taken from a benchmark problem. Keywords: Tracking, filtering, estimation, non-Gaussian, heavy-tailed distribution.
1
Introduction
The models and techniques of target tracking have been almost exclusively built around the assumption of Gaussian noise processes for both system and measurement models. This is understandable from the viewpoint of analytical convenience since if Gaussian error and prior structure is assumed then a straightforward (and exact) analysis results in the well known Kalman filter relations. The expected value of the state becomes a simple linear weighted combination of measurement and predicted value and the posterior covariance of the state is independent of the observed data. Use of a non-Gaussian error model leads to analytic intractability of the full Bayesian solution. However, for many situations assuming Gaussian noise is an over-simplification and a potentially poor approximation. Concentrating first on the system model: c Commonwealth of Australia 2003.
• a military aircraft manoeuvre is more accurately described as a pilot induced shock to the speed and heading of the aircraft • systems with interacting components are known to produce heavy-tailed non-Gaussian distributions on the state evolution [1]. An example of such a system is multiple aircraft moving in formation or engaged in a dogfight. Similarly, within the measurement process, spurious outlying measurements have long been known to adversely affect the estimation procedure [2, 3, 4]. Masreliez [2] summarises the situation: “... highly non-Gaussian densities, or densities which may have Gaussian shape in the middle but possess small but potent deviations from normality in the tails, ... may seriously degrade the performance of a linear estimator”. Hence, consideration of issues of realism or robustness led to the investigation of non-normal error models. In particular, the use of heavy tailed measurement error models (such as the Student-t) to facilitate outlier accommodation has received much attention [5, 6, 7] with theoretical justification provided by Dawid [8] and O’Hagan [9] who obtain necessary conditions on the measurement model to ensure that aberrant measurements are eventually effectively ignored. If the distributions are sufficiently heavy-tailed then it may be that the second moment of the distribution does not exist and so the Central Limit Theorem (CLT) does not readily apply [10]. In [11] a CLT is developed for estimation in state space models with non-Gaussian noise distributions. Rates of convergence within the resulting CLT are developed in [12] and used to develop an algorithm for model validation. In this paper we shift the focus away from the measurement error process and consider the use of heavy-tailed non-Gaussian driving noise models for target tracking. In Section 2 power law distributions are introduced as an alternative and a recently published algorithm called the
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Kalman-Levy filter [13] is described for state estimation and extended to allow nonlinear system models. Section 3 gives details of the benchmark trajectories selected for performance comparison and a heavy-tailed system model implemented via the Kalman-Levy filter is compared against an IMM for two sample trajectories.
2
Heavy-tailed system model and filter
In this section we outline some of the properties of power-law distributions and specify the Kalman-Levy filter [13] as an approximate implementation method for recursive state estimation in state space models with powerlaw noise. The formulation is then extended via first-order Taylor series approximation to give an Extended KalmanLevy filter.
2.1
Levy or Power Law Distributions
2.2
Kalman-Levy Filter algorithm
Reference [13] proposes a method of implementing a recursive state estimation algorithm for state space models with power-law noise distributions. The method is based on the standard linear Kalman filter equations. The prediction and update equations are retained in their linear forms. At each time step, the gain in the update equation is chosen to minimise the global uncertainty in the filter error. The tail-covariance recursions are derived by writing update and prediction errors as auto-regressive processes with µ-variable inputs. The above stated properties of µvariables lead to the prediction and update relations for the tail covariances. Consider a linear discrete-time dynamic system for the state vector xk ∈ RN , xk = Ak xk−1 + ηk ,
(5)
Power-law distributions of scalar random variables are characterised by heavy tails that decay according to:
where ηk is the system noise which is assumed to follow a power-law distribution of exponent µ. Observations are assumed to be linear functions of the state vector
P (w) ' C|w|−(1+µ) ,
yk = H k x k + k
(1)
where the exponent µ controls the rate of decay of the probability and C is the scale factor controlling the global amplitude of the power law tail. For a given µ, the uncertainty of w is defined by C. For µ ≤ 2 the variance is not defined mathematically. The key properties we wish to exploit are (from [13]): • if w is a µ-variable with scale factor C, then so is aw with scale factor |a|µ C. • if w1 and w2 are two independent µ-variables with scale factors C1 and C2 then w = w1 + w2 is also a µ-variable with scale factor C = C1 + C2 . Extension to multi-variate distributions is achieved by constructed a linear sum of N independent power-law variables = Gω.
with additive noise k which is also assumed to follow a power-law distribution of exponent µ. We define the notation x ˆi|j as the estimate of the state vector at time i given measurements (y1 , . . . , yj ). Assuming an estimate is available at time step k-1: x ˆk−1|k−1 , the unbiased prediction is given by x ˆk|k−1 = Ak x ˆk−1|k−1 . (7) The error associated with this prediction can be written as an auto-regressive process x ˆk|k−1 − xk = Ak x ˆk−1|k−1 − xk−1 − ηk ; (8)
this is a linear combination of multi-variate µ-variables. Writing this in the style of (2) gives x ˆk−1|k−1 − xk−1 = Gk−1|k−1 wk−1|k−1 , ηk = Gηk wkη ,
(2)
Assuming that all of the independent random variables are power-law distributed with a common exponent µ, is characterised by this exponent and the tail covariance matrix [13] B = G[µ/2] CGT [µ/2] , (3)
[β]
(4)
Given a tail-covariance matrix B, G and C can be straightforwardly obtained via diagonalisation.
(9) (10)
with associated scale factor matrices Ck−1|k−1 and Ckη . The tail-covariance of the prediction can thus be written Bk|k−1 = (Ak Gk−1|k−1 )[µ/2] × Ck−1|k−1 (Ak Gk−1|k−1 )T [µ/2] +
where C is a diagonal matrix containing the scale factors of the N independent random variables ω and the operator {.}[β] is defined as Gij = sign(Gij )|Gij |β .
(6)
(Gηk )[µ/2] Ckη (Gηk )T [µ/2] .
(11)
The update relation for the state estimate is restricted to be of the linear Kalman update form (ie weighted combination of measurement and prediction) x ˆk|k = x ˆk|k−1 + Kk yk − Hk x ˆk|k−1 . (12)
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From this the error in the updated estimate can be written as an auto-regressive process x ˆk|k − xk = (I − Kk Hk ) x ˆk|k−1 − xk + Kk k . (13)
We can again obtain the prediction and update error recursions as auto-regressive processes x ˆk|k−1 − xk = fk (ˆ xk−1|k−1 ) − (fk (xk−1 ) + ηk ) = Fk x ˆk−1|k−1 − xk−1 − ηk , (23)
Again, this is a linear combination of µ-variables. Using x ˆk|k−1 − xk = Gk|k−1 wk|k−1 , k =
(14)
Gk wk ,
(15)
with associated scale factor matrices Ck|k−1 and Ck gives the tail-covariance of the updated estimate as Bk|k = Gk|k−1 − Kk Hk Gk|k−1
[µ/2]
Ck|k−1 Gk|k−1 − Kk Hk Gk|k−1 [µ/2] (Kk Gk )
× T [µ/2]
+
T [µ/2] Ck (Kk Gk )
.
(16)
The gain of the filter Kk is obtained by minimising the average uncertainty of x ˆk|k (i.e. trace(Bk|k )). This means solving N sets of non-linear simultaneous equations ![µ−1] N L X X f G G f Gip − Kim Hmp Hjp Cp 0=µ − p=1
+
L X q=1
L X
m=1
m=1
Kim Gmq
![µ−1]
Gjq Cq
for each i, j where Gf = Gk|k−1 , C f = Ck|k−1 , and H G = Hk Gk|k−1 . Here N and L are the dimensions of the state vector and measurement vector respectively and the subscripts refer to matrix elements.
2.3
x ˆk|k − xk = x ˆk|k−1 + Kk yk − hk (ˆ xk|k−1 ) − xk = (I − Kk Hk ) x ˆk|k−1 − xk + Kk k . (24)
The update and prediction process for the Extended Kalman-Levy filter is thus just the standard Kalman-Levy filter relations (11) and (16) with the usual Jacobian substitution.
3
Benchmark tracking problem
The benchmark problem [14, 15] is an easily available standard framework for comparing the performance of manoeuvring target tracking algorithms. This section outlines the benchmark problem, the IMM algorithm of [16] which we use for our baseline performance comparison and the implementation of the KL filter.
3.1
Outline of benchmark target trajectories
The benchmark problem [15] contains a range of highly manoeuvring and evasive target trajectories designed to cover a wide range of target types and manoeuvre capabilities. Here we consider two of these trajectories. The first represents a fighter/attack aircraft undergoing a series of 6 or 7-g turns. The trajectory is shown in Figure 1. The second trajectory represents an anti-ship missile and is shown in Figure 2. Here the missile undergoes a succession of 7-g weaves in an attempt to deceive shipboard defences.
Extended Kalman-Levy filter 4
Now consider the situation where the system and measurement models are given by
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T=0s
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(17)
2
yk = hk (xk ) + k .
(18)
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xk = fk (xk−1 ) + ηk ,
An Extended Kalman-Levy filter can be derived by using first-order Taylor series approximations
1
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fk (xk−1 ) ≈ fk (ˆ xk−1|k−1 ) + Fk (xk−1 − x ˆk−1|k−1 ), (19) hk (xk ) ≈ hk (ˆ xk|k−1 ) + Hk (xk − x ˆk|k−1 ),
x 10
0
(20)
−0.5 T=188s
where
and
∂fk (xk−1 ) Fk = ∂xk−1
−1 3.5
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5 5.5 X Position (m)
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7 4
x 10
(21) xk−1 =ˆ xk−1|k−1
∂hk (xk ) Hk = ∂xk
. xk =ˆ xk|k−1
Figure 1: Target trajectory of fighter (22)
These trajectories exhibit the type of features we consider important for the system model to capture. Relatively long
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4
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The rest of the transition matrix is defined as follows:
x 10
T=0s 2
Y Position (m)
1.5
1
P12 = 0.1(1 − P11 ), P21 = 0.1(1 − P22 ),
P13 = 0.9(1 − P11 ), P23 = 0.9(1 − P22 ),
P31 = 0.3(1 − P33 ),
P32 = 0.7(1 − P33 ).
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3.3
The Kalman-Levy filter
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Figure 2: Target trajectory of missile
periods of quiescent motion interspersed with high acceleration turns. This should be well matched to the heavy-tailed system noise models. Throughout the simulations the time step between measurements is fixed at 2 seconds. This is in line with the average update time selected by the tracking filter in [16]. Note that we have not considered the altitude component of the target trajectory since this is usually estimated with a decoupled filter.
3.2
The choice of system model for the KL filter is very important. Our rationale is to develop a model where a manoeuvre can be considered as independent heavy-tailed shocks to one or more of the state variables. For example, in an aircraft the transition from straight motion into a turn could be seen as a shock to the heading. Similarly linear accelerations could be seen as shocks to the speed. Following [17], which compares models for tracking turning aircraft, we take our state vector to be Cartesian position, speed, heading and rate of change of heading: xk = (x, y, v, θ, φ)k . The differential equation for the state evolution is then x˙ 0 v cos θ y˙ v sin θ 0 v˙ = 0 + η v , θ˙ φ 0 0 ηφ φ˙
Specification of the IMM
In [16] an IMM algorithm is specified for the benchmark tracking problem. The filter has been well tuned to the application so should be considered as a near optimal tracking solution. The IMM used the following three models: Benign Motion Model: A constant velocity model with a low process noise level (σ1 = 3m/s1.5 ).
aclevel
Reference [18] derives a first-order approximation to the constant turn rate model: xk = f (xk−1 ) + ηk ,
x + v∆tk (cos θ − φ∆tk sin(θ)/2) y + v∆tk (sin θ + φ∆tk cos(θ)/2) , v f (x) = θ + φ∆tk φ
(25)
where ui = 0.9 and li = 0.1 are the upper and lower limits for the ith diagonal element and τi is the expected sojourn time for each of the modes (τ1 = 15s, τ2 = 4s and τ3 = 2s).
(28)
with
The Markov chain transition matrix P is principally defined by the three probabilities on the diagonal (Pii , i = 1, 2, 3), given by Pii = min ui , max li , 1 − ∆tτi−1 ,
(27)
where η v and η φ are taken to be independent power-law random µ-variables.
Maneuver Model: A constant velocity model with a high process noise level (σ2 = 35m/s1.5 ). Maneuver Detection Model: A constant celeration model with a high noise (σ3 = min(25∆t, 70)m/s2.5 ).
(26)
(29)
where ∆tk = tk − tk−1 is the time between measurements and ηk is the driving noise. This noise is power law distributed with exponent µ and tail-covariance matrix Bkη . The tail-covariance matrix is obtained from Q by diagonal-
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isation. The non-zero and unique elements of Q are Q11 = σv2 cos2 (θ)∆t3k /3 + σφ2 v 2 sin2 (θ)∆t5k /20 Q22 = Q33 = Q44 = Q55 = Q12 = Q13 = Q14 = Q15 = Q23 = Q24 = Q25 = Q45 =
σv2 sin2 (θ)∆t3k /3 σv2 ∆tk σφ2 ∆t3k /3 σφ2 ∆tk
+
σφ2 v 2
cos
2
(θ)∆t5k /20
(30) (31) (32) (33) (34)
σv2 ∆t3k /3 − σφ2 v 2 ∆t5k /20 σv2 ∆t2k cos(θ)/2 −σφ2 v sin(θ)∆t4k /8 −σφ2 v sin(θ)∆t3k /6 σv2 sin(θ)∆t2k /2 σφ2 v cos(θ)∆t4k /8 σφ2 v cos(θ)∆t3k /6 σφ2 ∆t2k /2
cos(θ) sin(θ)
(35) (36) (37) (38)
coordinates. Thus B(1, 1) B(1, 2) C= , B(2, 1) B(2, 2) B0 (i, j)∆t−1 B0 (i, j) , B(i, j) = B0 (i, j)∆t−1 2B0 (i, j)∆t−2 1 0 0 0 0 0 1 0 , J = 0 x/s ˙ 0 y/s ˙ 0 −y/s ˙ 2 0 x/s ˙ 2 p s = x˙ 2 + y˙ 2 , JCJ T 0 B0|0 = 2 . 0 ( (0.1)∗π 180 )
(40)
55
1 0
0 1
0 , 0
x0 , y0 ,
q
45
x˙ 20 + y˙ 02 , tan−1 (y˙ 0 /x˙ 0 ), 0 ,
40
(44)
(45)
with σr2 = 1000m2 . The filter is initialised using two-point differencing: x ˆ0|0 =
(51)
50
(43)
where k is the measurement noise vector. This noise is power law distributed with exponent µ and tail-covariance matrix Bk which is obtained from the covariance matrix of the measurement noise
(50)
Whole Traj Manuevering non−Manuevering
The relationship between covariance and tail covariance comes directly from the definition of the tail covariance matrix in (3). Following [19] the values σv = 2.5m/s1.5 and σφ = 0.03rad/s1.5 are assumed. The measurements are taken to be Cartesian, so
R = σr2
(49)
65
(41) 60
0 0
(48)
(39)
(42)
zk = Hxk + k , 1 0 0 0 H= 0 1 0 0
(47)
(46)
where x ˆk|k is the track estimate at time step k given measurements up to and including time step tk , xk and yk are the x and y measurements respectively at time tk and x˙ 0 = (x0 − x−1 )∆t−1 0 with y˙ 0 similar. The tail-covariance matrix B0 of the measurement noise is used to calculate the initial tail-covariance matrix. This is done by taking the initial covariance matrix in rectangular coordinates and using a Jacobian to change this to polar
35 1.1
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Figure 3: RMS position error vs µ for fighter trajectory The final parameter to be set is the tail index of the system and measurement noise distributions µ. The best value is chosen via a simulation exercise. Figure 3 shows RMS position errors (500 Monte Carlo runs) against µ for the fighter trajectory. The three lines correspond to the manoeuvring segments of the trajectory, the non-manoeuvring segments of the trajectory and the whole trajectory. When the target is not manoeuvring a Gaussian noise model is more appropriate and so we would expect the RMS position error to tend to a minimum at µ = 2. During manoeuvres there should be benefit from the heavier tailed models – note that large reduction as µ decreases from 2. From the overall trajectory curve µ = 1.45 is chosen. All results presented for the two target trajectories have been obtained by averaging 500 Monte Carlo runs.
3.4
Performance comparison : Fighter trajectory
In Figure 4 RMS position error is shown. The IMM errors are in agreement with those presented in [16]. The Kalman-Levy filter errors are comparable but noticeably
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larger. This is not unexpected during periods of nomanoeuvre but the large peaks at on-set of manoeuvre are disappointing. Figure 5 shows the corresponding RMS speed errors. The Kalman-Levy filter errors are marginally lower in non-manoeuvring segments and comparable during manoeuvres. The Kalman-Levy results show a significant dip in the middle of manoeuvre, presumably because the turn model used is well matched at this point. The RMS errors are noticeably larger during the period 130150 seconds. During this time period the target undergoes a constant acceleration speed change. The Kalman-Levy system model only allows for random walk behaviour in speed so is not well matched. The inclusion of an extra derivative of speed might improve performance. Focusing on tail behaviour rather than RMS summary statistics, Figure 6 shows the empirical cumulative distribution function (cdf) for the estimation error. Also shown is the cdf of position measurement error to confirm that the filters are improving on raw measurement error. The IMM is seen to be marginally better than the Kalman-Levy filter.
3.5
Performance comparison : Missile trajectory
The results for the weaving missile trajectory follow the same broad pattern as the manoeuvring fighter aircraft. Figure 7 shows RMS position errors, and again the KalmanLevy filter gives higher error than the IMM during the nonmanoeuvre segment but is comparable during the weave section. Figure 8 shows RMS speed errors and the two filters give equal performance in the non-manoeuvre segment. The Kalman-Levy has large peak errors during the weave section. Figure 9 shows the empirical cdf of estimation error. This confirms that both filters improve upon raw measurement error with the IMM being marginally better.
4
Conclusions
In this paper we have considered the use of heavy tailed distributions within the system model as a means of improving upon the standard Gaussian assumption. These models are particularly appropriate for situations involving manoeuvring and/or interacting targets. The KalmanLevy filter of [13] has been used for state estimation with the heavy-tailed model and performance has been compared with the IMM for standard benchmark trajectories. It was demonstrated that the simple heavy-tailed model could achieve comparable performance to a well designed three model IMM filter. Of course, for real systems, data association is of key importance and it remains to be seen how the Kalman-Levy filter will perform in this context. It is suggested that the highest density gate approach of [20] should be investigated. It is also suggested that better motion models could be developed to benefit from the heavy-tailed assumptions.
The model chosen should be validated via the approach of [12].
References [1] V. M. Eguiluz and M. G. Zimmermann, “Transmission of information and herd behaviour: an application to financial markets,” Physical review letters, vol. 85, no. 26, pp. 5659–5662, December 2000. [2] C. J. Masreliez, “Approximate non-Gaussian filtering with linear state and observation relations,” IEEE Trans. on Automatic Control, vol. AC-20, pp. 107– 110, 1975. [3] R. J. Meinhold and N. D. Singpurwalla, “Robustification of Kalman filter models,” Journal of the American Statistical Association, vol. 84, 1989. [4] P. J. Huber, Robust statistical procedures. 1972.
SIAM,
[5] L. Simar, “Protecting against gross errors: the aid of Bayesian methods,” in Specifying statistical models, J. P. F. et al., Ed. New York: Springer-Verlag, 1983. [6] A. F. M. Smith, “Bayesian approaches to outliers and robustness,” in Specifying statistical models, J. P. F. et al., Ed. New York: Springer-Verlag, 1983. [7] A. O’Hagan, “Modelling with heavy tails,” in Bayesian Statistics 3, J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, Eds. Oxford University Press, 1988, pp. 345–359. [8] A. P. Dawid, “Posterior expectations for large observations,” Biometrika, vol. 60, pp. 664–667, 1979. [9] A. O’Hagan, “On outlier rejection phenomena in Bayes inference,” Journal of the Royal Statistical Society, Series B, vol. 41, pp. 358–367, 1979. [10] R. L. Fante, “Central limit theorem: use with caution,” IEEE Trans. on Aerospace and Electronic Systems, vol. 37, no. 2, pp. 739–740, April 2001. [11] J. C. Spall and K. D. Wall, “Asymptotic distribution theory for the Kalman filter state estimator,” Comunications in Statistical Theory and Methodology, vol. 13, no. 16, pp. 1981–2003, 1984. [12] F. A. Aliev and L. Ozbek, “Evaluation of convergence rate in the central limit theorem for the Kalman filter,” IEEE Trans. on Automatic Control, vol. 44, no. 10, pp. 1905–1909, October 1999. [13] D. Sornette and K. Ide, “The Kalman-Levy filter,” Physica D, vol. 151, pp. 142–174, 2001.
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[14] W. D. Blair, G. A. Watson, T. Kirubarajan, and Y. BarShalom, “Benchmark for radar allocation and tracking in ECM,” IEEE Trans. on Aerospace and Electronic Systems, vol. 34, no. 4, pp. 1097–1114, October 1998. [15] G. A. Watson and D. H. McCabe, “Benchmark problem with a multisensor framework for radar resource allocation and the tracking of highly manoeuvring targets,” NSWC, Tech. Rep. TR-99/32, 1999. [16] T. Kirubarajan, Y. Bar-Shalom, W. D. Blair, and G. A. Watson, “IMMPDAF for radar management and tracking benchmark with ECM,” IEEE Trans. on Aerospace and Electronic Systems, vol. 34, no. 4, pp. 1115–1134, October 1998. [17] F. Gustafsson and A. J. Isaksson, “Best choice of coordinate system for tracking coordinated turns,” in IEEE Conf. on Decision and Control, December 1996, pp. 3145–3150. [18] S. R. Maskell, “Using Laplace transforms to solve the differential equations for turning models,” Submitted to: IEEE Trans. on Aerospace and Electronic Systems, 2002. [19] S. Blackman and R. Popoli, Design and analysis of modern tracking systems. Artech House, 1999. [20] F. J. Breidt and A. L. Carriquiry, “Highest density gates for target tracking,” IEEE Trans. on Aerospace and Electronic Systems, vol. 36, no. 1, pp. 47–55, January 2000.
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Figure 4: Comparison of RMS position estimation errors for fighter trajectory
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Figure 7: Comparison of RMS position estimation errors for missile trajectory
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Figure 5: Comparison of RMS speed estimation errors for fighter trajectory
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Figure 8: Comparison of RMS speed estimation errors for missile trajectory
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Figure 6: CDF of position estimation errors for fighter trajectory
Figure 9: CDF of position estimation errors for missile trajectory
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