Stationary Priors for Bayesian Target Tracking

Transcription

1 Stationary Priors for Bayesian Target Tracking Brian R. La Cour Applied Research Laboratories The University of Texas at Austin Austin, Texas, U.S.A Abstract A dynamically-based birth/death process is proposed for use in Bayesian single-target tracking. The underlying dynamics are based on an integrated Ornstein-Uhlenbeck process, which is stationary in velocity but not in position. Target birth death events are defined in terms of a given spatial region of interest. Target death (or, escape) occurs when the dynamics move the target outside this region. Target birth (or, entrance) is specified statistically to achieve a desired stationary probability distribution, which then serves as a Bayesian prior. An implementation of this process for both grid particlebased distribution representations is outlined, numerical examples are provided to illustrate convergence. Keywords: Bayes procedures, stochastic processes, target detection, target tracking I. INTRODUCTION Accurate tracking relies upon modeling the underlying dynamics of moving targets ], 2]. Typically, such models take the form of stochastic differential equations, with the stochasticity representing a diffusive process noise to be added to the target s otherwise deterministic motion. The deterministic portion of the model gives rise to dispersion. Together, they allow the tracker to anticipate target motion between temporally separated measurement updates while allowing for additional flexibility in light of inevitable model mismatch. Due both to the dispersive diffusive nature of the motion, target state uncertainties tend to increase with time between measurement updates. If such updates occur frequently enough, this uncertainty remains bounded within reasonable limits. In the absence of such measurements, however, the uncertainty tends to grow without bound, the track will tend to follow rom clutter. Now, in the general Bayesian tracking problem, one is interested in estimating both the number of targets present their collective kinematic state. The formal Bayes procedure to do this consists of a series of motion measurement updates on a given initial prior distribution to obtain the current posterior distribution. The choice for the prior is often given short shrift is usually selected based merely on convenience. Justification for a particular choice of prior, when sought, is often made by appeal to information theory 3]. Hill Spall 4], for example, argue that the maximally uninformative prior for Bayesian inference over a sequency of measurements is that which maximizes the Kullback-Leibler entropy between the final posterior original prior. Of course, if one considers any other state variable in one-to-one correspondence with the kinematic variables but connected via a non-lebesgue measure preserving mapping, then an altogether different prior may be obtained. Physically, one would expect that the timeevolved posterior, in the absence of measurement updates, should revert back to the starting prior. It is perhaps more natural, then, to think of the prior distribution as instead having a dynamical origin 5]. This paper sets out to develop a target motion model which combines the continuous kinematic evolution with the discrete target number distribution. The corresponding stationary distribution, a mixture of both the discrete continuous components, is then taken to be the prior for the purposes of Bayesian inference. Specifically, consider a target tracking scenario in which, at any time, there is at most one target present in a given spatial region, R. The target s kinematic state at time t, S(t), is taken to be composed of a d- dimensional position, X(t), velocity, V (t), component; thus, S(t) X(t), V (t)]. This is similar in form to a jump Markov system for an Interacting Multiple Model (IMM) tracker 2], 6], wherein target birth death are treated as rom transitions between different target models according to preassigned probabilities. The key difference here is that the transition probabilities are derived from the target dynamics; hence, the process need not be strictly Markovian. Let p t (m) denote the probability that there are m targets present (m, ), let ρ t (s) denote the probability density that, given a target is present, S(t) s. Given an arbitrary initial distribution characterized by p (m) ρ (s), we seek a physically reasonable stochastic model for the target dynamics such that p t (m) ρ t (s) approach a unique stationary distribution as t with no measurement updates. It is this stationary distribution which we shall take to be the Bayesian prior. The approach, outline, of the paper is as follows. In Sec. II the preliminary (i.e., without birth death) target dynamics are defined. Section III extends this model to incorporate a stochastic, but dynamically induced, birth death process. A solution for the stationary distribution, given the birth statistics, is outlined in Sec. IV. Alternatively, it is shown that, for a desired stationary distribution, the required birth statistics are easily determined. A specific example is worked out wherein the stationary kinematic distribution is uniform in position Gaussian in velocity. Section V

2 concerns asymptotic convergence to the stationary distribution, with some rate estimates. Finally, Secs. VI VII consider specific implementations of the motion model the results of convergence through simulation. A summary of findings conclusions is given in Sec. VIII. II. PRELIMINARY TARGET DYNAMICS Suppose the target dynamics are given by a set of linear stochastic differential equations of the form ds(t) ΓS(t)dt + Σ dw (t), () where Γ Σ are 2d 2d matrices W (t) is a 2ddimensional Wiener (Brownian motion) process. (This is an example of a multidimensional Langevin equation.) The matrix Γ gives the basic kinematic relation, giving rise to dispersion, while Σ describes the diffusion process. The solution to this equation gives the time-evolved probability density function (PDF) ρ t (s). It can be shown that, if S() is independent of {W (t) : t }, then the solution is given by S(t) S() + t e Γ(t t ) Σ dw (t ), (2) where the distribution of S() is given by ρ (s). The target is taken to follow an Integrated Ornstein-Uhlenbeck (IOU) process 7], with Γ I γ I ] Σ ], (3) σ I where I is the d d identity matrix. Conditioned on S(), the above Itô integral is an independent Gaussian rom variable with mean A(t)S() covariance Q(t), where A(t) Q(t) are of the following form. ] I γ A(t) ( e γt ) I e γt (4) I ] q (t) I q Q(t) 2 (t) I, (5) q 2 (t) I q 22 (t) I respectively. The elements of A i (t) shall be denoted a (t), a 2 (t),..., while those of Q i (t) are given by q (t) σ2 2γ 3 2γt 4( e γt ) + e 2γt] (6a) q 2 (t) σ2 ( e γt ) 2 q2 2γ 2 (t) (6b) q 22 (t) σ2 ( e 2γt ) 2γ (6c) Being a sum of two independent rom variables, the probability density for S(t) is therefore ρ t (s) K t (s s )ρ (s ) ds, (7) where the Markov motion kernel, K t (s s ), is a Gaussian probability density with mean A(t)s covariance Q(t). 97 For large t, the velocity components of A(t) Q(t) approach σ 2 /(2γ)I, respectively. Thus, the velocity distribution of the target asymptotically approaches a stationary Gaussian distribution with zero mean a root-mean-square speed of v : σ/ 2γ. The linear diffusion term, (σ/γ) 2 t, in q (t), however, causes the position to become unbounded, with no stationary distribution possible. In the next section we will consider ways to achieve a truly stationary distribution. III. TARGET BIRTH/DEATH PROCESS Since the the motion is spatially unbounded, we need to account for the target possibly leaving the area of interest, R. To achieve stationarity, we must also suppose that a new target may enter this area. These considerations suggest that the target population should be modeled as a stochastic birth/death process, which is characterized as follows. Let Φ t ( ) denote the probability of a target remaining absent at time t, given that none was initially present. If a new target appears at time t, where none was present initially, then the probability density for state s is Φ t (s ). In a similar manner, let Φ t ( s ) be the probability that a target is not present at time t, given that it was present in state s initially. Finally, let Φ t (s s ) denote the probability density for the target to remain present be in state s at time t, given that it was present in state s initially. (Note that this is not the same as K t (s s ) above.) Considering the preliminary target dynamics of the previous section, the probability that a target in state s at time will still be in the region of interest at time t > is Φ t ( s ) K t (s s ) ds. (8) R R d Since no more than one target can be present, the population distribution at time t is p t () p () ( β t ) + p () ε t (s ) ρ (s )ds (9a) p t () p () β t + p () ε t (s )] ρ (s )ds (9b) p t (m), m > (9c) where β t : Φ t ( ) is the target birth probability ε t (s ) : Φ t ( s ) is the conditional target escape probability. Given that a target is present, the kinematic distribution is given by an IOU process, conditioned on the target being present in R at time t; thus, ρ t (s) p ()b t (s) + p () Φ t (s s )ρ (s )ds, () where b t (s) : Φ t (s ) is the kinematic PDF for a newly entered target, Φ t (s s ) K t(s s ) ε t (s ) R Rd(s) () is the conditional motion-update kernel, R R d( ) is the indicator function on the set R R d.

3 IV. STATIONARY DISTRIBUTION An initial distribution p p ρ ρ will be stationary if only if the following relations are satisfied. p() p() β t + p() ε t (s )]ρ(s )ds (2) ρ(s) p() b t (s) + p() Φ t (s s )ρ(s )ds. (3) Given β t b t, one may, in theory, solve for p ρ. Equation (3) is recognized as a Fredholm integral equation of the second kind 8]. By the Fredholm Alternative, a unique solution exists for any b t, provided p(). This solution, in turn, may be approximated via a Neumann series which, to first order in p(), is ρ(s) p() b t (s) + p() Φ t (s s )b t (s )ds ]. (4) Substituting this result into Eqn. (2) retaining terms that are first order in p() gives ] p() p() β t + p() ε t (s )]b t (s )ds. (5) The resulting quadratic equation is readily solved, giving p. If, instead, a desired p ρ are given, we may readily solve for β t b t (s) to obtain β t p() p() b t (s) p() ε t (s )ρ(s )ds, (6) ] ρ(s) p() Φ t (s s )ρ(s )ds. (7) Thus, a desired stationary distribution, characterized by a given p ρ, may be achieved by selecting the characteristics of newly entering targets, described by β t b t (s), in the manner prescribed by Eqns. (6) (7). The choice of p ρ is not entirely arbitrary, however. The value of β t, for example, is guaranteed to fall within the unit interval only for p() /2. Similarly, a large value for p() could result in b t (s) taking on negative values. On the other h, for p() /2 we may approximate b t (s) by ρ(s). More directly, taking b t ρ β t as per Eqn. (6) implies a new stationary distribution, characterized by p ρ given implicitly by p() p() p() p() ρ(s) ρ(s) p() p() ε t (s )]ρ(s ) ρ(s )]ds (8) Φ t (s s ) ρ(s )ds ]. (9) For small p(), the true stationary distribution, given by p ρ, will be close to the nominal stationary distribution, given by p ρ. In practice, then, it is not necessary to determine p ρ explicitly. 97 A. Uniform/Gaussian Case Let p() /2 be given consider the following nominal stationary density. ρ(s) c R (x) g(v, v 2 I), (2) where c is a normalization constant g(, v 2 I) is a Gaussian PDF with mean covariance v 2 I. The value for β t will be given by β t p() p() ( α t), (2) where the survival probability, α t, is determined by α t : ε t (s )]ρ(s )ds K t (s s ) ds ρ(s ) ds R R d g(s As, Q) dv ρ(s ) dv dx dx. R The integral over v reduces as follows. g(s As,Q) dv ρ(s ) dv g(x x + a 2 v, q I) g(v, v 2 I) dv g (v x x, q ) a 2 a 2 I g(v, v 2 I) dv ( ( 2 x x g, v 2 + q ) ) I where a 2 g ( x x, r 2 I ), r(t) a 2 2 (22) (23) q (t) + v 2 a 2 2 (t). (24) Now suppose R R R d, where R i ξ i R, ξ i + R] R > for each i,..., d. Then α t i erf 2R ξi +R ξi +R ξ i R ξ i R g(x i x i, r(t) 2 ) dx i dx i ( 2R/r(t) ) exp ( 2R 2 /r(t) 2) 2πR/r(t) ] d. (25) A plot of α t versus t is shown in Fig. for typical parameter values appropriate for submarine tracking. Since r(t) increases with t, for small t we may approximate α t using the asymptotic expansion of the error function 9]. Thus, for t min{/γ, R/ v}, α t r(t) 2πR ] d vt 2πR ] d. (26) This shows that the survival probability is primarily a function of the ratio of the mean drift distance the region size, as given by vt R, respectively. The target birth PDF, b t (s), can be determined in a similar manner, using Eqn. (7), but the integral therein does not lend to an analytic solution. Nevertheless, one can show that

4 972.8 Exact Approximate.8.8 R 25 km α t R km α τ.4 R 5 km R km t (min) p().2 Figure. Plot of α t versus t for d 2, γ 2 6 /s, v 5 m/s, several values of R. The solid curves are the exact values, while the dashed curves are the approximation from Eqn. (26) Marginal Birth PDF (/km) t min t 3 min t 6 min 5 5 x (km) Figure 2. Plot of Monte Carlo estimates for b t(x, v) dv versus x for d, p().5, R km, γ 2 6 /s, v 5 m/s, several values of t. The black line is the nominal stationary PDF, which is uniform. the PDF separates into the product of a gaussian PDF in velocity, identical to that of Eqn. (2), a marginal PDF in position; thus, the position velocity of the birth particle are statistically independent. Monte Carlo estimates of this marginal for several values of t are shown in Fig. 2. As p(), the marginal PDF tends to a uniform distribution, corresponding to the nominal prior. For p() > t small, the PDF tends to weight regions near the boundary more heavily, corresponding intuitively to a target entering the region. V. CONVERGENCE The previous section showed that p ρ form a stationary distribution for the choices of β t b t given in Eqns. (6) (7). It remains, then, to be shown whether, for an arbitrary Figure 3. Plot of λ τ versus p() α τ. The black curve indicates the locus of points for which λ τ, where convergence is most rapid. choice of p ρ, it is true that p t ρ t converge to p ρ, respectively. Consider the case in which ρ ρ. Considering p t to be a column vector, we may write p t M t p, where ] βt α M t : t. (27) β t α t We immediately note that, since α t as t, lim p t() p() p () t p() lim p t() p() p () t p() (28a) (28b) Clearly, p t does not converge to p unless p p. If, however, the motion model is applied iteratively with, say, t nτ M nτ approximated by Mτ n, a different result is found. An eigenvalue decomposition of M τ yields two eigenvalues: unity, corresponding to the stationary state, λ τ : α τ β τ, (29) which, since λ τ < for α τ <, gives the rate of convergence to the stationary distribution. Specifically, the time scale for convergence will be about τ/ log λ τ. Convergence will be fastest when λ τ ; i.e., when p() α τ. Figure 3 illustrates the dependence of λ τ on the prior target probability, p(), the survival probability, α τ. Note that there is no convergence (i.e., λ τ ) when p() (+α τ )/2. Of course, the true time evolution of p t is complicated by the fact that it is coupled to that of ρ t, which varies in time even in the case that ρ ρ. Thus, λ τ should be viewed only as a rough guide to the rate of convergence, valid in the case that ρ t ρ. VI. IMPLEMENTATION For the practitioner, it is useful to detail how one might efficiently implement the above theoretical formulation. We

5 suppose two possible representations of ρ t (s). The first is a grid-based representation, wherein the PDF is approximated on a set of discrete points. The second is a particle representation, wherein the PDF is represented by a set of state values associated weights. A. Grid-based Representation In the grid-based representation, the PDF is approximated on a set {C i } of disjoint grid cells covering the spatial region of interest the velocity state space. Thus, we write ρ t (s) ρ t (j) Cj (s)/ ds, (3) j where the probability mass function (PMF), ρ t ( ), is ρ t (j) : ρ t (s)ds. (3) In terms of the initial PMF the target birth distribution, we find where ρ t (j) p () b t (j) + p () i Φ t (j i) : Φ t (j i) ρ (i), (32) bt (j) : b t (s)ds (33) Φ t (s s )ds ds/ ds. (34) C i C i Suppose the grid cells take the following form. a i, b i ) u i, v i ), (35) i where a i < b i u i < v i are the bounds in position velocity, respectively, for the given cell. For the uniform/gaussian stationary distribution defined in Sec. IV-A, the birth PMF takes the following form. bt (j) ρ(s)ds bi vi dx i g(v 2R i, v 2 )dv i i a i ( ) d 4R i u i (b i a i ) ( ) ( )] vi ui erf erf, 2 v 2 v (36) where we have assumed that b i > ξ i R/2 a i < ξ i +R/2. The motion update for targets which do not escape is given by the transition probability Φ t (j i). Although this may be computed directly via numerical integration, the discrete form of the transition probability suggests the following Monte Carlo approach. First, generate a uniformly rom set {s n} Ni n of points in C i. Next, apply the motion model to each point in this set, resulting in s n A(t)s n + Q(t) /2 z n, where z n are independent identically distributed stard Gaussian rom variables. Let N ij denote the number of 973 points in {s n } N i n contained in. Now, Ni j N ij will be the number of points that do not escape from the region of interest. Since some points may escape from the spatial region of interest, it will generally be true that Ni N i. For N i sufficiently large, the motion transition probability is Φ t (j i) N ij N i (b i a i )(v i u i )], (37) i the escape probability is ε t (s )ρ (s )ds ij B. Particle Representation N ij N i ρ (i) (38) In the particle representation, the PDF is approximated by a set of N target states, s n, associated weights, w n. Thus, ρ t (s) N w n δ(s s n ), (39) n where δ( ) is the impulse function. In what follows we shall assume that the particles have been renormalized so that the weights are, in fact, uniform (i.e., w n /N). A straightforward application of the motion model to the original particle set, {s n} N n, leads to a new set, {s n } N n. Of these, a subset {s n} N n will remain in the spatial region of interest. The escape probability is therefore ε t (s )ρ (s )ds N (4) N Removal of the escaped particles gives a sample representative of the conditional motion model; i.e., Φ t (s s )ρ (s )ds. To obtain a sample representative of ρ t (s), we must draw from this latter sample in proportion to p () augment with samples from the birth distribution, in proportion to p (). This may be achieved as follows. Let N p ()N N N N. If N N, then remove the last N N particles from {s n} N n, resulting in {s n} N n. If N < N, then draw N N particles romly from {s n} N n add these to the sample, resulting in {s n} N n {s n k } N N k. Finally, draw N samples from the birth distribution add the resulting sample, {ŝ n } N n, to the aforementioned sample. The final sample is therefore if N N, otherwise {s n } N n {s n} N n {ŝ n} N n, (4) {s n } N n {s n} N n {s n k } N N k {ŝ n } N n. (42) VII. SIMULATION EXAMPLE As an illustration of convergence to the stationary distribution, consider the following example. Let p().5 let ρ be given by Eqn. (2) with d 2, R 2 km, v 5 m/s. The initial distribution is taken to represent a postmeasurement detection, with p (). ρ as ρ but with R 2 km v.5 m/s. The time-evolved distribution, p t

6 974 Entropy Stationary Prior σ.5 m/s 3/2 σ. m/s 3/2 σ.2 m/s 3/ Time (min) Figure 4. Plot of the entropy, Hρ t ], as a function of time. The green, blue, red curves correspond to σ.5,.,.2 m/s 3/2, respectively. The black curve is the equilibrium value. Probability Target Present Stationary Prior σ.5 m/s 3/2 σ. m/s 3/2 σ.2 m/s 3/ Time (min) Figure 5. Plot of the probability that a target is present, p t (), as a function of time. The green, blue, red curves correspond to σ.5,.,.2 m/s 3/2, respectively. The black curve is the equilibrium value. ρ t, will be determined iteratively with intervals of length τ 6 sec. Several values of σ will be considered. When ρ t is in equilibrium, convergence of p t () will occur on the time scale τ/ log λ τ. For the above parameters, α τ.988, so λ τ.977. This implies a convergence time scale on the order of 3 min. Convergence is expected to occur after about three times this value, i.e., in about 9 min. To quantify the convergence of ρ t the Boltzmann-Shannon entropy is used, where Hρ t ] : ρ t (s) log ρ t (s) ds. (43) For the equilibrium case, it can be shown that Hρ] d log(2r) + d 2 log ( 2π v 2) + d 2. (44) Convergence will imply that Hρ t ] Hρ] as t. The results of a simulation using 5 particles over a fourhour period are shown in Figures 4 5. Three values of σ were used:.5,.,.2 (in units of m/s 3/2 ), which are shown in green, blue, red, respectively. Figures 4 5 show Hρ t ] p t (), respectively, versus time. From the figures we see that the convergence of p t lags that of ρ t, with convergence of the latter occurring in the first hour of elapsed time. The target probability, p t () converged about 9 min later, in rough agreement with the theoretical prediction. Increasing the value of σ, which controls the rate of diffusion, tended to increase the rate of convergence for the entropy, hence, for the target probability as well. VIII. CONCLUSION This paper introduced a single-target stochastic birth/death model based on the Integrated Ornstein-Uhlenbeck process the notion that target presence is defined in terms of a given spatial region of interest. The probability of a new target entering the region, its initial kinematic distribution, were chosen so as to obtain a specified stationary prior distribution. The resulting coupled birth/death kinematic time evolution process is, in general, not Markovian, but theoretical numerical analysis suggests that the stationary distribution, for typical model parameters, is indeed asymptotically stable. These investigations suggest that it is necessary for the kinematic portion of the distribution to converge, through the motion-induced dispersion, before the target probability converges to its equilibrium (i.e., prior) value. ACKNOWLEDGMENT This work was support by the Office of Naval Research under Contract No. N4-6-G REFERENCES ] Y. Bar-Shalom T. E. Fortmann, Tracking Data Association. San Diego: Academic Press, ] Y. Bar-Shalom X.-R. Li, Multitarget-Multisensor Tracking: Principles Techniques. Storrs, CT: YBS Publishing, ] E. T. Jaynes, Information theory statistical mechanics, Physical Review, vol. 6, no. 4, pp , ] S. D. Hill J. C. Spall, Noninformative Bayesian priors for large samples based on Shannon information theory, in Proceedings of the 26th Conference on Decision Control. Los Angeles, CA: IEEE, December 987, pp ] A. Lasota M. C. Mackey, Chaos, Fractals, Noise. New York: Springer-Verlag, ] C. Andrieu, M. Davy, A. Doucet, Efficient particle filtering for jump Markov systems: Application to time-varying autoregressions, IEEE Transactions on Signal Processing, vol. 5, no. 7, 23. 7] L. D. Stone, C. A. Barlow, T. L. Corwin, Bayesian Multiple Target Tracking. Boston: Artech House, ] D. Porter D. S. G. Stirling, Integral Equations. Cambridge: Cambridge University Press, 99. 9] M. Abramowitz I. A. Stegun, Hbook of Mathematical Functions with Formulas, Graphs, Mathematical Tables, 9th ed. New York: Dover, 964.