arXiv:1307.3283v1 [stat.CO] 11 Jul 2013 . un,B,Gpln,RB,Fre,JF April filte vol Systems, particle press. Electronic In and “A of Aerospace J.F. case on The Transactions Forbes, IEEE bound: R.B., lower Cram´er-Rao posterior Gopaluni, approximate B., Huang, A., oe on ntescn-re ro ME bandwith obtained (MSE) error theoretic second-order a the provides on (FIM) bound matrix lower information Fisher in the problems of complex most the of [9]. one theory remains estimation non- still the evaluating it in filters, interest approximati practical statistical great limite and the often Despite numerical is various UKF performance by and their affected EKF tracking, as in such efficient filters, are in Although unscented tracking [8]. allow and SSMs to developed non-linear (EKF) Bayesia been have filter the (UKF)) filter Kalman of Kalman extended approximation (e.g., statistical filtering and analytical [7]. density on filter posterior a constructing filter a at [5] framework, aims detection Bayesian problem fault the Within [4], nav [6]. tracking diagnosis in: fault [3], applications guidance key [2], several gation with methods, inferencing ehd,tre tracking target methods, re-entry includin at examples, tracking develope target the simulation ballistic of two of efficacy problem on be The practical illustrated noise. can is non sensor with and method systems and non-linear general state in to PCRLB Gaussian is the methods approximate method to SMC available developed used the uses of The sequence approach a particle measurements. using The states or hidden proposed. the (SMC) new estimate is a Monte-Carlo unmeasured, (PF) sequential or filters hidden on are or based states the simulations approach when in PCRLB except the approx Fur- t recursive of or available, allow the To be solution. numerical experiments. to designed not access carefully analytical using may priori it easy which a require states, approximate an approaches to based to simulation attempt themselves any lend thermore, not expectation PCR multi-dimensional do the (MSE) Computing complex, error filter. square solving state mean involves non-linear the any on with bound a obtained provides [1] in derived swt h rcs oeigadCnrlLb eateto C [email protected]). of (e-mail: V Department Canada, Columbia, Lab, British BC, Control of and University Engineering, Modeling Biological Process the with is niern,Uiest fAbra dotnTG26 Alb and T6G-2G6, Chemical Edmonton Alberta, of (e-mail: of Department University Group, Engineering, Control Process Computer h rme-a oe on CL)dfie sa inverse an as defined (CRLB) bound Cram´er-Rao lower The based algorithms tractable several decades, few last the In Bayesian important most the of one is filtering Non-linear ne Terms Index Abstract uhr’adess .Tlyn .HagadJF obsare Forbes J.F. and Huang B. Tulsyan, A. addresses: Authors’ ⋆ atcefitrapoc oapoiaeposterior approximate to approach filter particle A odne eso fti ril a enpbihdi:Tu in: published been has article this of version condensed A { usa;ba.un;fraser.forbes biao.huang; tulsyan; TepseirCram posterior —The PRB o-iersses idnsae,SMC states, hidden systems, non-linear —PCRLB, .I I. dtaTlyn ioHag .BuhnGpln n .Frase J. and Gopaluni Bhushan R. Huang, Biao Tulsyan, Aditya NTRODUCTION Cram rRolwrbud(PCRLB) bound lower er-Rao ´ } ulet.a;adRB Gopaluni R.B. and @ualberta.ca); rRolwrbound lower er-Rao ´ novrV6T-1Z3, ancouver 9 o ,2013, 4, no. 49, idnstates”. hidden prahto approach r ra Canada, erta, eia and hemical ,which s, Materials imation ihthe with phase. sensor linear lsyan, or d ons. and a g rue ing LB in al i- d n - rcigi iutos uha:maueetoii uncer- origin measurement as: allow such which ( appeared, tainty situations, also have in PCRLB tracking the of versions fied h oe on o rcigi eea o-ierSSMs, non-linear detection general of in probability tracking the with for operating bound lower be the can discussions critical other [24]. with in development along found historical PCRLB, the the of overview of applicable An is noise. it sensor st as non-Gaussian and with general SSMs th more non-linear multi-dimensional is [23], to for [1] [22], in PCRLB to formulation the Compared PCRLB compute SSMs. recursively with non-linear propos system to discrete-time, authors linear approach the suitable [1], elegant paper a an seminal of of the In matrix that noise. information with Gaussian the SSM compared non-linear authors a the [22 in where later non- much [23], d developed scalar to was for extension case Its multi-dimensional [21] noise. with by Gaussian additive proposed for with was SSMs impractical linear PCRLB the computation of recu its version a Alternatively, SSMs. renders non-linear often multi-dimensional which data, [20]. obser [19], optimal tracking and bearings-only [18]; for trajectory submarin positioning in sensor deployment [17]); sonobuoy tracking [16], other (e.g., several allocation deployment in resource resource used multi-sensor performance widely to: also pre-specified related is areas with PCRLB The systems [15]. of bounds design n terrain and non- [13]; tracking [14]; several target of ballistic comparison for performance filters include: linear PCRLB practical filter key the the the of of Some plications whether not. or practical determining are requirements (iii) opt an and of that filter; against filters non-linear (ii) of filters; performances non-linear ing different of quality the assessing [12]. o characteristics density noise prior system dynamics; and system import on: states; the an depends the characterize is only it fully it as not Nonetheless, tool, filters. does non-linear PCRLB of the accuracy result, mome a higher-order an As all non- of [11]. requires any characterization density with statistical the posterior full obtained as Gaussian A MSE [1]. defined the filter on is a linear bound (PFIM) PCRLB lower matrix a The information provides Fisher PCRLB. posterior the the of commonly inverse as is which to [10], by referred o derived class the was to estimators CRLB Bayesian of extension analogous param- An or estimator. state eter unbiased based (ML) maximum-likelihood any rbblt fflealarm false of probability h CL n[]poie eusv rcdr ocompute to procedure recursive a provides [1] in PCRLB The batch on based is [10] in formulation PCRLB original The (i) for: benchmark a as used widely been has PCRLB The Pr d 1 = and Pr f ≥ Pr 0 ⋆ 2] isddtcin( detection missed [25]; ) f Forbes r 0 = ic hn eea modi- several then, Since . Pr d 1 = avigation Pr compar- n the and non- y d radar rsive imal ≤ ap- ver ant ate eal nts nd ed ], 1 e e s 1 f f 2
and Prf =0) [26]; and cluttered environments (Prd ≤ 1 and PCRLB approximation; (ii) the method is applicable only for Prf ≥ 0) [27]. However, unlike the bound formulation given non-linear SSMs with additive Gaussian state and sensor noise; in [1], the modified versions of the lower bound are mostly (iii) convergence of the numerical solution to the theoretical for a special class of non-linear SSMs with additive Gaussian lower bound is not guaranteed; (iv) provides limited control state and sensor noise. for improving the quality of the resulting numerical solution; Notwithstanding a recursive procedure to compute the and (v) it involves long and tedious calculations of the first PCRLB in [1], obtaining a closed form solution to it is non- two moments of the assumed Gaussian densities. trivial. This is due to the involved complex, multi-dimensional Recently, [35] derived a conditional lower bound for general expectations with respect to the states and measurements, non-linear SSMs, and used an SMC based method to approxi- which do not lend themselves to an easy analytical solution, mate it in absence of the true states. Unlike the unconditional except in linear systems [12], where the Kalman filter (KF) PCRLB in [1], the conditional PCRLB can be computed provides an exact solution to the PCRLB. in real-time; however, as shown in [35], the bound in less Several attempts have been made in the past to address optimistic (or higher) compared to the unconditional PCRLB. the aforementioned issues. First, several authors considered This limits its use to applications, where real-time bound approximating the PCRLB for systems with: (i) linear state computation is far more important than obtaining a tighter dynamics with additive Gaussian noise and non-linear mea- limit on the tracking performance. However, in applications, surement model [12], [28]; (ii) linear and non-linear SSMs such as filter design and selection, where the primary focus with additive Gaussian state and sensor noise [9], [29]; and (iii) is on devising an efficient filtering strategy, the PCRLB in [1] linear SSMs with unknown measurement uncertainty [30]. The provides an optimistic measure of the filter performance. special sub-class of non-linear SSMs with additive Gaussian To the authors’ best knowledge, there are no known numer- noise allows reduction of the complex, multi-dimensional ical method to approximate the unconditional PCRLB in [1], expectations to a lower dimension, which are relatively easier when the true states are unavailable. to approximate. The following are the main contributions in this paper: (i) an SMC based method is developed to numerically approximate the unconditional PCRLB in [1], for a general stochastic II. MOTIVATION AND CONTRIBUTIONS non-linear SSMs operating with Prd =1 and Prf =0. The To obtain a reasonable approximation to the PCRLB for expectations defined originally with respect to the true states general non-linear SSMs, several authors have considered and measurements are reformulated to accommodate use of using simulation based techniques, such as the Monte Carlo the available sensor readings. This is done by first condi- (MC) method. Although a MC method makes the lower bound tioning the distribution of the true states over the sensor computations off-line, nevertheless, it is a popular approach, readings, and then using an SMC method to approximate it. since for many real-time applications in tracking and navi- (ii) Based on the above developments, a numerical method to gation, the design, selection and performance evaluation of compute the lower bound for a class of discrete-time, non- different filtering algorithms are mostly done a priori or off- linear SSMs with additive Gaussian state and sensor noise line. Furthermore, availability of huge amount of historical is derived. This is required, since several practical problems, test-data, makes MC method a viable option. An MC based especially in tracking, navigation and sensor management, are bound approximation have appeared for several systems with: often modelled as non-linear SSMs, with additive Gaussian target generated measurements [13], [28]; measurement ori- noise. (iii) Convergence results for the SMC based PCRLB gin uncertainty [25]; cluttered environments [27], [31]; and approximation is also provided. (iii) The quality of the SMC Markovian models [32], [33]. Although MC methods can be based PCRLB approximation is illustrated on two examples, effectively used to approximate the involved expectations, with which include a uni-variate, non-stationary growth model and respect to the states and measurements, it requires an ensemble a practical problem of ballistic target tracking at re-entry of the true states and measurements. While the sensor readings phase. may be available from the historical test-data, the true states The proposed simulation based method is an off-line method, may not be available, except in simulations or in carefully which can be used to deliver an efficient numerical approx- designed experiments [34]. imation to the lower bound in [1], based on the sensor To avoid having to use the true states, [34] proposed an EKF readings alone. Compared to the EKF and UKF based PCRLB and UKF based method to compute the PCRLB formulation in approximation method derived in [34], the proposed SMC [1]. To approximate the bound, [34] first assumes the densities based method: (i) is far more general as it can approximate associated with the expectations to be Gaussian, and then uses the PCRLB for a larger class of discrete-time, non-linear an EKF and UKF to approximate the Gaussian densities using SSMs with possibly non-Gaussian state and sensor noise; (ii) an estimate of the mean and covariance. Even though the avoids numerical errors arising due to the use of dynamics method proposed in [34] is fast, since it only works with the linearisation methods; and (iii) provides a far greater control first two statistical moments, there are several performance and over the quality of the resulting approximation. Moreover, applicability related issues with this numerical approach, such several theoretical results exist for the SMC methods, which as: (i) relies on the linearisation of the underlying non-linear can be used to suggest convergence of the SMC based PCRLB dynamics around the state estimates, which not only results approximation to the actual lower bound. All these features in additional numerical errors, but also introduces bias in the of the proposed method are either validated theoretically or 3 illustrated on simulation examples. below by the following matrix inequality
T −1 P , E [(Xt − X )(Xt − X ) ] < J , (2) III. PROBLEM FORMULATION t|t p(X0:t,Y1:t) t|t t|t t b b In this paper, we consider a model for a class of general where: Pt|t is a n × n matrix of MSE; tm n stochastic non-linear systems. Xt|t , Xt(Y1:t) := R → R is a point estimate of Model 3.1: Consider the following discrete-time, stochastic Xt ∈ X at time t ∈ N, given the measurement sequence b b −1 non-linear SSM {Y1:t = y1:t} , {y1,...,yt}; Jt is a n×n PFIM matrix; Jt is a n × n PCRLB matrix; p(x0:t,y1:t) is a joint probability Xt+1 =ft(Xt,ut,θ,Vt), (1a) density of the states and measurements up until time t ∈ N; T Yt =gt(Xt,ut,θ,Wt), (1b) the superscript (·) is the transpose operation; and Ep(·)[·] is the expectation operator with respect to the pdf p(·). where: X ∈X ⊆ Rn and Y ∈Y⊆ Rm are the state vari- t t Proof: See [10] for a detailed proof. ables and sensor measurements, respectively; u ∈U⊆ Rp is t −1 < Rr Inequality (2) implies that Pt|t − Jt 0 is a positive semi- input variables and θ ∈ Θ ⊆ are the model parameters. n n definite matrix for all Xt|t ∈ R and t ∈ N. (2) can also be Also: the state and sensor noise are represented as Vt ∈ R Rm written in terms of a scalar MSE (SMSE) as and Wt ∈ , respectively. ft(·) is an n-dimensional state b mapping function and gt(·) is a m-dimensional measurement S 2 −1 P , E [kXt − X k ] ≥ Tr[J ], (3) mapping function, where each being possibly non-linear in its t|t p(X0:t ,Y1:t) t|t t arguments. where Tr[·] is the trace operator, andb k·k is a 2-norm. Model 3.1 represents one of the most general classes of Lemma 3.7: For a system represented by Model 3.1 and discrete-time, stochastic non-linear SSMs. For notational sim- operating under Assumptions 3.2 through 3.5, the PFIM in plicity, explicit dependence on ut ∈U and θ ∈ Θ are not Lemma 3.6 can be recursively computed as [1], [9] shown in the rest of this article; however, all the derivations 22 12 T 11 −1 12 that appear in this paper hold with ut and θ included. As- Jt+1 = Dt − [Dt ] (Jt + Dt ) Dt , (4) sumptions on Model 3.1 are discussed next. where: Assumption 3.2: The state and sensor dynamics are defined n n m m as ft := X × R → R and gt := X × R → R , respec- 11 E Xt Dt = p(X0:t+1,Y1:t+1)[−∆X log p(Xt+1|Xt)]; (5a) tively, are at least twice differentiable with respect to X ∈ X . t t 12 Xt+1 D =E [−∆ log p(Xt+1|Xt)]; (5b) Also, the parameters θ ∈ Θ and inputs ut ∈U are assumed to t p(X0:t+1,Y1:t+1) Xt be known a priori. D22 =E [−∆Xt+1 log p(X |X ) t p(X0:t+1,Y1:t+1) Xt+1 t+1 t Assumption 3.3: Sensor measurements are target-originated, Xt+1 − ∆ log p(Yt+1|Xt+1)]; (5c) operating with probability of false alarm Prf =0 and proba- Xt+1 bility of detection Pr =1. The target states X ∈ X are hid- d t and: ∆ is a Laplacian operator such that ∆Y , ∇ ∇T with den Markov process, observed only through the measurement X X Y , ∂ ∇X ∂X being a gradient operator, evaluated at the true process Yt ∈ Y. X0 states. Also, J0 = Ep(X )[−∆ log p(X0)]. Assumption 3.4: V , W and X are mutually independent 0 X0 t t 0 Proof: See [1] for a complete proof. sequences of independent random variables described by the For Model 3.1, obtaining a closed-form solution to the PFIM probability density functions (pdfs) p(vt), p(wt) and p(x0), respectively. These pdfs are known in their classes (e.g., or PCRLB is non-trivial. This is due to the complex integrals Gaussian; uniform) and are parametrized by a known and finite involved in (5), which do not lend themselves to an easy number of moments (e.g., mean; variance). analytical solution. The main problem addressed in this paper is discussed next. Assumption 3.5: For a random realization (xt+1, xt, vt) ∈ n m Problem 3.8: Compute a numerical solution to the PCRLB X×X× R and (yt, xt, wt) ∈ Y×X× R satis- T T given in Lemma 3.6 for systems represented by Model 3.1 and fying Model 3.1, ∇vt ft (xt, vt) and ∇wt gt (xt, wt) have rank n and m, such that using implicit function theorem, operating under Assumptions 3.2 through 3.5. Use of simulation based methods in addressing Problem 3.8 p(xt+1|xt)= p(Vt = f˜t(xt, xt+1)) and p(yt|xt) = p(Wt = is discussed next. g˜t(xt,yt)) do not involve Dirac delta functions.
A. Posterior Cramer-Rao´ lower bound IV. APPROXIMATING PCRLB The conventional CRLB provides a lower bound on the MSE MC method is a popular approach, which can be used to of any ML based estimator. An analogous extension of the approximate the PCRLB; however, as discussed in Section II, CRLB to the class of Bayesian estimators was derived by [10], MC method requires an ensemble of true states and sensor and is referred to as the PCRLB inequality. Extension of the measurements. While sensor readings may be available from PCRLB to non-linear tracking was provided by [1], and is the historical test-data, the true states may not be available in given next. practice. To allow the use of sensor readings in approximating Lemma 3.6: Let {Y1:t}t∈N be a sequence from Model 3.1, the PCRLB, this paper reformulates the integrals in (5) as then MSE of any tracking filter at t ∈ N is bounded from given below. 4
Proposition 4.1: The complex, multi-dimensional expecta- Theorem 4.3: For any bounded test function t+1 tions in (5), with respect to the density p(x0:t+1,y1:t+1) can φt : X → R, there exists Ct,p < ∞, such that for be reformulated, and written as follows: any p> 0, N ≥ 1 and t ≥ 1, the following inequality holds
1 11 Xt p E (6a) p ¯ It = p(X0:t+1|Y1:t+1)[−∆Xt log p(Xt+1|Xt)]; Ct,pφt E φt(x0:t)ǫt(dx0:t|y1:t) ≤ , (9) 12 Xt+1 1/2 I =E [−∆ log p(X |X )]; (6b) ZX t+1 N t p(X0:t+1|Y1:t+1) Xt t+1 t 22,a Xt+1 I =E [−∆ log p(X |X )]; (6c) where ǫt(dx 0:t|y1:t)=˜p(dx0:t|y1:t) − p(dx0:t|y1:t) is the N- t p(X0:t+1|Y1:t+1) Xt+1 t+1 t particle approximation error, ¯ t+1 , and 22,b Xt+1 φt = supx0:t∈X |φt(x0:t)| I =E [−∆ log p(Yt+1|Xt+1)], (6d) t p(X0:t+1|Y1:t+1) Xt+1 the expectation is with respect to the particle realizations. where: Proof: See Theorem 2 in [38] for a detailed proof. Remark 4.4: The result in Theorem 4.3 is weak, since 11 E 11 (6e) Dt = p(Y1:t+1)[It ]; Ct,p ∈ R being a function of t ∈ N, grows exponen- 12 E 12 tially/polynomially with time [39]. To guarantee a fixed pre- Dt = p(Y1:t+1)[It ]; (6f) 22,a 22,b cision of the approximation in (8), has to increase with D22 = E [I + I ]. (6g) N t p(Y1:t+1) t t t. The result in Theorem 4.3 is not surprising, since (7) Proof: The proof is based on decomposition of the requires sampling from the pdf p(x0:t|y1:t), whose dimension pdf p(x0:t+1,y1:t+1) in (5), using the probability condition increases as n(t + 1). In literature Theorem 4.3 is referred to p(x0:t+1,y1:t+1)= p(y1:t+1)p(x0:t+1|y1:t+1). as the sample path degeneracy problem. This is a fundamental Remark 4.2: In Proposition 4.1 the integrals are with re- limitation of SMC methods; wherein, for N ∈ N, the quality spect to p(y1:t+1) and p(x0:t+1|y1:t+1). The advantage of of the approximation of p(dx0:t|y1:t) deteriorates with time. representing (5) as (6) is evident: using historical test-data, The motivation to use SMC methods to approximate the expectations with respect to p(y1:t+1) can be approximated complex, multi-dimensional integrals in Proposition 4.1 is using MC, while that defined with respect to p(x0:t+1|y1:t+1) based on the fact that encouraging results can be obtained can be approximated using an SMC method. under the exponential forgetting assumption on Model 3.1. Since θ ∈ Θ is assumed to be known (see Assumption 3.2), A. SMC based PCRLB approximation the forgetting property in Model 3.1 holds. With the forgetting property, it is possible to establish results of the form given It is not our aim here to review SMC methods in details, in the next theorem. but to simply highlight their role in approximating the multi- Theorem 4.5: For an integer L > 0, and any bounded test dimensional integrals in Proposition 4.1. For a detailed exposi- function φ : X L → R, there exists D < ∞, such that for tion on SMC methods, see [7], [11]. The essential idea behind L L,p any p> 0, N ≥ 1 and t ≥ 1, the following inequality holds SMC methods is to generate a large set of random particles p 1 (samples) from the target pdf, with respect to which the p ¯ E DL,pφL integrals are defined. The target pdf of interest in Proposition φL(xt−L+1:t)ǫL(dxt−L+1:t|y1:t) ≤ 1/2 , Z L N X 4.1 is p(x0:t|y1:t). Using SMC methods, the target distribution, (10) defined as p(dx0:t+1|y1:t+1) , p(x0:t+1|y1:t+1)dx0:t+1 can be approximated as given below. where ǫL(dxt−L+1:t|y1:t)= X t−L+1 ǫt(dx0:t|y1:t). Proof: See Theorem 2 inR [38] for a detailed proof. N R N i Remark 4.6: Since DL,p ∈ is independent of t ∈ , The- p˜(dx0:t+1|y1:t+1)= W0:t+1|t+1δXi (dx0:t+1), (7) 0:t+1|t+1 orem 4.5 suggests that an SMC based approximation of the Xi=1 most recent marginal posterior pdf p(xt−L+1:t|y1:t), over a where: p˜(dx0:t+1|y1:t+1) is an N-particle SMC approx- fixed horizon L> 0 does not result in the error accumulation. imation of the target distribution p(dx0:t+1|y1:t+1) and For our purposes, to make the SMC based PCRLB approxi- i i N {X0:t+1|t+1; W0:t+1|t+1}i=1 are the N pairs of particle mation effective, the dimension of the integrals in Proposition realizations and their associated weights distributed according 4.1 needs to be reduced. An SMC based approximation of the N i to p(x0:t+1|y1:t+1), such that i=1 W0:t|t =1. Using (7), an PCRLB over a reduced dimensional state-space is discussed SMC approximation of (6a), forP example, can be computed as next. N Lemma 4.7: For a system represented by Model 3.1, using ˜11 i Xt i i (8) the Markov property of the target states in Assumptions 3.3, It = W0:t+1|t+1[−∆Xt log p(Xt+1|t+1|Xt|t+1)]. Xi=1 Proposition 4.1 can be written as follows: where I˜11 is an SMC estimate of I11 and the Laplacian is 11 E Xt t t It = p(Xt:t+1 |Y1:t+1)[−∆X log p(Xt+1|Xt)]; (11a) i N t evaluated at {Xt:t+1|t+1}i=1. 12 Xt+1 I =E [−∆ log p(Xt+1|Xt)]; (11b) The convergence of (8) to (6a) depends on (7). Many sharp t p(Xt:t+1 |Y1:t+1) Xt I22,a =E [−∆Xt+1 log p(X |X )]; (11c) results on convergence of SMC methods are available (see t p(Xt:t+1 |Y1:t+1) Xt+1 t+1 t [36] for a survey paper and [37] for a book length review). 22,b Xt+1 I =Ep(X |Y )[−∆ log p(Yt+1|Xt+1)]. (11d) A selection of these results highlighting the difficulties in ap- t t+1 1:t+1 Xt+1 proximating p(dx0:t|y1:t) with an SMC method are presented Proof: The proof is based on a straightforward use of the below. definition of expectation and Markov property of Model 3.1. 5
For example, the integrals in (6a) can be written as Algorithm 1 SMC based posterior density approximation Input: Given Model 3.1, satisfying Assumptions 3.2 I11 = [−∆xt log p(x |x )]p(dx |y ), (12a) t xt t+1 t 0:t+1 1:t+1 through 3.5, assume a prior pdf on X0, such that ZX t+2 X0 ∼ p(x0). Also, select algorithm parameter N. xt = [−∆xt log p(xt+1|xt)]p(dxt:t+1|y1:t+1), (12b) Output: Recursive SMC approximation of the posterior ZX 2 p(dxt|y1:t) for all t ∈ N. =E [−∆Xt log p(X |X )], (12c) p(Xt:t+1|Y1:t+1) Xt t+1 t 1: Generate N independent and identically distributed parti- i N cles {X } ∼ p(x0) and set the associated weights where p(dx0:t+1|y1:t+1) , p(x0:t+1|y1:t+1)dx0:t+1, and in 0|−1 i=1 t i −1 N (12c), since the integrand is independent of x0:t−1 ∈ X , it is to {W0|−1 = N }i=1. Set t ← 1. i N marginalized out of the integral. Equations (11b) through (11d) 2: Sample {Xt|t−1}i=1 ∼ p(xt|y1:t−1). Set i −1 N can be derived based on similar arguments, which completes {Wt|t−1 = N }i=1. the proof. 3: while t ∈ N do Remark 4.8: The dimension of the expectations in (6a) 4: Use {Yt = yt} and compute the importance weights i N through (6c) reduces from n(t + 2) to 2n; whereas, in (6d), {Wt|t}i=1 using it reduces from n(t + 2) to n for all t ∈ N. Moreover, since i i W p(yt|X ) expectations in Lemma 4.7 are with respect to p(xt:t+1|y1:t+1) i t|t−1 t|t−1 Wt|t = N j . (18) and p(xt+1|y1:t+1), an SMC method can be effectively used i j=1 Wt|t−1p(yt|Xt|t−1) with a finite number of particles (see Theorem 4.5). P j 5: N Resample the particle set {Xt|t}j=1 with replacement B. General non-linear SSMs i N from {Xt|t−1}i=1, such that To approximate the multi-dimensional integrals in Lemma Pr(Xj = Xi )= W i , (19) 4.7 for Model 3.1, a set of randomly generated samples from t|t t|t−1 t|t the target distribution p(dxt:t+1|y1:t+1) is required. First note i where P r(·) is a probability measure. Set {Wt|t = that the target pdf p(xt:t+1|y1:t+1) can alternatively be written −1 N N }i=1. as given below. i N 6: Sample {X } ∼ p(xt+1|y1:t) using (57). Set Lemma 4.9: The target pdf p(x |y ), with respect t+1|t i=1 t:t+1 1:t+1 {W i = N −1}N . to which the integrals in Lemma 4.7 are defined can be t+1|t i=1 7: Set t ← t +1. decomposed, and written as 8: end while p(xt+1|xt)p(xt|y1:t)p(xt+1|y1:t+1) p(xt:t+1|y1:t+1)= . X p(xt+1|xt)p(dxt|y1:t) R (13) Remark 4.10: The procedure for generating random parti- cles from densities, such as the uniform or Gaussian, is well Proof: First note that the target pdf p(xt:t+1|y1:t+1) can be written as described in literature; however, due to the multi-variate, and non-Gaussian nature of the target pdf, generating random p(xt:t+1|y1:t+1)= p(xt|xt+1,y1:t,yt+1)p(xt+1|y1:t+1). particles from p(xt:t+1|y1:t+1) is a non-trivial problem. An (14) alternative idea is to employ an importance sampling function From the Markov property of (1), and from the Bayes’ (ISF), from which random particles are easier to generate [7]. theorem, (14) can be written as In this paper, the product of two pdfs in (13) is selected as the ISF, such that p(xt:t+1|y1:t+1) p(y |x , x ,y )p(x |x ,y )p(x |y ) q(xt:t+1|y1:t+1) , p(xt|y1:t)p(xt+1|y1:t+1), (17) = t+1 t t+1 1:t t t+1 1:t t+1 1:t+1 , p(yt+1|xt+1,y1:t) where q(x |y ) is a non-negative ISF on X 2, such that (15a) t:t+1 1:t+1 supp q(xt:t+1|y1:t+1) ⊇ supp p(xt:t+1|y1:t+1). Choice of an p(y |x ,y )p(x |x ,y )p(x |y ) = t+1 t+1 1:t t t+1 1:t t+1 1:t+1 , (15b) ISF similar to (17) was also employed in [40], [41] to develop p(yt+1|xt+1,y1:t) a particle smoothing algorithm for discrete-time, non-linear =p(xt|xt+1,y1:t)p(xt+1|y1:t+1). (15c) SSMs. Thus to be able to generate random samples from (17), samples from the two posteriors p(x |y ) and p(x |y ) Applying Bayes’ theorem again in (15c) yields t 1:t t+1 1:t+1 need to be generated first. Again, using the principles of p(xt:t+1|y1:t+1) ISF, particles from the posterior pdf can be generated using any advanced SMC methods (e.g., ASIR [42], resample-move p(xt+1|xt,y1:t)p(xt|y1:t)p(xt+1|y1:t+1) = , (16a) algorithm [43], block sampling strategy [44]) or for example, p(xt+1|y1:t) using the method in [41], [45]. The method described in p(x |x )p(x |y )p(x |y ) = t+1 t t 1:t t+1 1:t+1 , (16b) [41], [45] is outlined in Algorithm 1. It is important to p(x |x )p(dx |y ) X t+1 t t 1:t note that in importance sampling, degeneracy is a common where in (16b),R the Law of Total Probability is used, which problem; wherein, after a few time instances, the density of completes the proof. the weights in (18) become skewed. The resampling step in 6
N N j i (18) is crucial in limiting the effects of degeneracy. Finally j=1 i=1 p(Xt+1|t+1|Xt|t)δXi ,Xj (dxt:t+1) = t|t t+1|t+1 , using Algorithm 1., the particle representation of p(dxt|y1:t) P P N j m N m=1 p(Xt+1|t+1|Xt|t) and p(dxt+1|y1:t+1) are given by P (24c) N 1 N N p˜(dxt|y1:t)= δ i (dxt), (20a) i,j Xt|t = W δ i j (dxt:t+1), (24d) N t|t,t+1|t+1 Xt|t,Xt+1|t+1 Xi=1 Xj=1 Xi=1 N 1 where p˜(dxt+1|y1:t+1)= δXj (dxt+1). (20b) N t+1|t+1 j i Xj=1 p(X |Xt|t) W i,j , t+1|t+1 , (25) i N t|t,t+1|t+1 N N p(Xj |Xm ) Here {Xt|t}i=1 ∼ p˜(xt|y1:t) and m=1 t+1|t+1 t|t i N {X } ∼ p˜(xt+1|y1:t+1) are the N pairs of resampled P t+1|t+1 i=1 Equation (24d) is an SMC approximation of p(dxt:t+1|y1:t+1). i.i.d. samples from p˜(xt|y1:t) and p˜(xt+1|y1:t+1), respectively. The computational complexity of the weights in (25) is of the order O(N 2). As suggested in [45], without significant Remark 4.11: Uniform convergence in time of (20) has been loss in the quality of the approximation, the complexity can established by [37], [46]. Although these results rely on strong be reduced to the order O(N) by replacing (24d) with (22), mixing assumptions of Model 3.1, uniform convergence has which completes the proof. been observed in numerical studies for a wide class of non- The distribution of weights in (22) becomes skewed after linear time-series models, where the mixing assumptions are a few time instances. To avoid this, the particles in (22) are not satisfied. resampled using systematic resampling, such that Substituting (20) into (17), yields an SMC approximation of j i i i the ISF, i.e., Pr(Xt:t+1|t+1 = {Xt|t; Xt+1|t+1})= Wt|t,t+1|t+1, (26) N N where {Xi }N ∼ p˜(x |y ) are resampled i.i.d. 1 t:t+1|t+1 i=1 t:t+1 1:t+1 i j (21) q˜(dxt:t+1|y1:t+1)= 2 δX ,X (dxt:t+1), particles. With resampling, the SMC approximation of the N t|t t+1|t+1 Xj=1 Xi=1 target distribution in (22) can be represented as
2 N where q˜(dxt:t+1|y1:t+1) is an N -particle SMC approx- 1 p˜(dxt:t+1|y1:t+1)= δXi (dxt:t+1). (27) imation of the ISF distribution q(dxt:t+1|y1:t+1) and N t:t+1|t+1 i j N,N Xi=1 {Xt|t; Xt+1|t+1}i=1,j=1 ∼ q˜(xt:t+1|y1:t+1) are particles from the ISF. Expectation in Lemma 4.7, with respect to the marginalized Lemma 4.12: An SMC approximation of the target distri- pdf p(xt|y1:t+1) (see (11d)) can also be approximated using SMC methods as given in the next lemma. bution p(dxt:t+1|y1:t+1) can be computed using the SMC Lemma 4.13: Let i N in (27) be i.i.d. resam- approximation of q(dxt:t+1|y1:t+1) given in (21), such that {Xt:t+1|t+1}i=1 pled particles distributed according to p˜(xt:t+1|y1:t+1) then N i an SMC approximation of p(dxt|y1:t+1) is given by p˜(dxt:t+1|y1:t+1)= Wt|t,t+1|t+1δXi ,Xi (dxt:t+1), t|t t+1|t+1 N Xi=1 1 (22) p˜(dxt|y1:t+1)= δXi (dxt), (28) N t|t+1 Xi=1 where: where p˜(dxt|y1:t+1) is an SMC approximation of i ζ p(dxt|y1:t+1) and δ i (·) is a marginalized Dirac i t|t,t+1|t+1 Xt|t+1 Wt|t,t+1|t+1 , ; (23a) N ζj delta function in dxt, centred around the random particle j=1 t|t,t+1|t+1 Xi . P i i t|t+1 p(Xt+1|t+1|Xt|t) Proof: See [47] for the proof. ζi , ; (23b) t|t,t+1|t+1 N i m Lemma 4.13 gives a procedure for computing an SMC N m=1 p(Xt+1|t+1|Xt|t) P approximation of p(dxt|y1:t+1), using the particles from the and p˜(dxt:t+1|y1:t+1) is an SMC approximation of the target SMC approximation of p(dxt:t+1|y1:t+1). Expectation with distribution p(dxt:t+1|y1:t+1). respect to p(y1:t+1) in Proposition 4.1 can be approximated Proof: Substituting (21) into (13) followed by several using MC method, such that algebraic manipulations yields an SMC approximation of 1 M p(dxt:t+1|y1:t+1), denoted by p˜(dxt:t+1|y1:t+1), such that p˜(dy1:t+1)= δY j (dy1:t+1), (29) M 1:t+1 Xj=1 p˜(dxt:t+1|y1:t+1) where p˜(dy1:t+1) is an MC approximation of p(dy1:t+1), p(xt+1|xt)˜q(dxt:t+1|y1:t+1) = , (24a) and M is the total number of i.i.d. measurement sequences p(x |x )˜p(dx |y ) X t+1 t t 1:t obtained from the historical test-data. Note that the approxima- R N N Np(xt+1|xt) j=1 i=1 δXi ,Xj (dxt:t+1) tion in (29) is possible only under Assumption 3.2; however, = t|t t+1|t+1 , (24b) 2 P P N in general, estimating the marginalized likelihood function N X p(xt+1|xt) m=1 δXi (dxt) t|t p(y1:t+1) is non-trivial [39]. R P 7
Finally, an SMC approximation of the PCRLB for systems C. Non-linear SSMs with additive Gaussian noise represented by Model 3.1 and operating under Assumptions Many practical applications in tracking (e.g., ballistic target 3.2 through 3.5 is summarized in the next lemma. tracking [13], bearings-only tracking [48], range-only tracking Lemma 4.14: Let a general stochastic non-linear system be [49], multi-sensor resource deployment [17] and other navi- represented by Model 3.1, such that it satisfies Assumption 3.2 gation problems [50]) can be described by non-linear SSMs j M N through 3.5. Let {Y1:t = y1:t}j=1 be M ∈ i.i.d. measure- with additive Gaussian noise. Since the class of practical ment sequences generated from Model 3.1, then the matrices problems with additive Gaussian noise is extensive, especially (5a) through (5c) in Lemma 3.7 can be recursively approxi- in tracking, navigation and sensor management, an SMC based mated as follows: numerical method for approximating the PCRLB for such class M N 1 of non-linear systems is presented. D˜ 11 = − [∆Xt log p(Xi,j |Xi,j )]; t MN Xt t+1|t+1 t|t+1 Model 4.15: Consider the class of non-linear SSMs with Xj=1 Xi=1 additive Gaussian noise (30a) M N Xt+1 =ft(Xt)+ Vt, (33a) 1 Xt+1 i,j i,j D˜ 12 = − [∆ log p(X |X )]; Y =g (X )+ W , (33b) t MN Xt t+1|t+1 t|t+1 t t t t Xj=1 Xi=1 Rn Rm (30b) where Vt ∈ and Wt ∈ are mutually indepen- M N dent sequences from the Gaussian distribution, such that 1 D˜ 22 = − [∆Xt+1 log p(Xi,j |Xi,j ) Vt ∼ N (vt|0,Qt) and Wt ∼ N (wt|0, Rt). t MN Xt+1 t+1|t+1 t|t+1 Note that Model 4.15 can also be represented as Xj=1 Xi=1 X + ∆ t+1 log p(Y j |Xi,j )]; (30c) 1 T −1 Xt+1 t+1 t+1|t+1 log[p(X |X )] = c − [X − f (X )] Q t+1 t 1 2 t+1 t t t i,j N j and {Xt:t+1|t+1}i=1 ∼ p(xt:t+1|y1:t+1) is a set of N × [Xt+1 − ft(Xt)], (34a) resampled particles from (27), distributed according to 1 T −1 j j M log[p(Yt+1|Xt+1)] = c2 − [Yt+1 − gt+1(Xt+1)] R p(xt:t+1|y1:t+1) for all {Y1:t+1 = y1:t+1}j=1. 2 t+1 j Proof: For a measurement sequence {Y1:t = y1:t}, an × [Yt+1 − gt+1(Xt+1)], (34b) SMC approximation of the target distribution in (27) can be R R written as where c1 ∈ + and c2 ∈ + are normalizing constant and R N + := [0, ∞). j 1 Result 4.16: The first and second order partial derivative of p˜(dxt:t+1|y1:t+1)= δXi,j (dxt:t+1), (31) N t:t+1|t+1 Xi=1 (34a) is given by i,j j T −1 where Xt:t+1|t+1 ∼ p(xt:t+1|y1:t+1) are resampled particles. ∇Xt log[p(Xt+1|Xt)] =[∇Xt ft (Xt)]Qt [Xt+1 − ft(Xt)], Substituting (31) into Lemma 4.7, an SMC approximation of (35a) (11a) through (11d) can be obtained as follows: Xt T −1 ∆Xt log[p(Xt+1|Xt)] = − [∇Xt ft (Xt)]Qt [∇Xt ft(Xt)] N Xt T −1 1 i,j i,j + [∆ f (Xt)]Λ ΨX , (35b) I˜11 = −∆Xt log p(X |X ); (32a) Xt t Xt t t N Xt t+1|t+1 t|t+1 Xi=1 and the first with respect to Xt+1 ∈ X and the second with N 1 respect to Xt ∈ X is given by ˜12 Xt+1 i,j i,j It = −∆ log p(X |X ); (32b) N Xt t+1|t+1 t|t+1 ∆Xt+1 log[p(X |X )] =[∇ f T (X )]Q−1, (35c) Xi=1 Xt t+1 t Xt t t t N −1 −1 1 X where: Λ = Q I 2 2 ; ΨX = [Xt+1 − ft(Xt)]I 2 ; I˜22,a = −∆ t+1 log p(Xi,j |Xi,j ); (32c) Xt t n ×n t n ×n t Xt+1 t+1|t+1 t|t+1 2 2 2 N In2×n2 , and In2×n are n × n and n × n identity matrix, Xi=1 respectively. Also: [∇ f T (X )] and [∆Xt f T (X )] are N Xt t t Xt t t 22,b 1 Xt+1 j i,j I˜ = −∆ log p(Y |X ), (32d) T (1) (n) t Xt+1 t+1 t+1|t+1 , N [∇Xt ft (Xt)] [∇Xt ft (Xt), ··· , ∇Xt ft (Xt)]n×n, Xi=1 (36a) ˜ where It is an SMC approximation of It. Substituting (32) and Xt T Xt (1) Xt (n) [∆ f (xt)] ,[∆ f (Xt), ··· , ∆ f (Xt)] 2 , (29) into (6e) through (6g) yields (30a) through (30c), which Xt t Xt t Xt t n×n (36b) completes the proof. Lemma 4.14 gives an SMC based numerical method to approx- (1) (n) T where ft(Xt) , [ft (Xt), ··· ,ft (Xt)] is a n × 1 vector imate the complex, multi-dimensional integrals in Lemma 3.7. valued function in (33a). Note that since Lemma 4.14 is valid for a general non-linear Result 4.17: The second order partial derivative of (34a) and SSMs, the derivatives of the logarithms of the pdfs in (30a) (34b) is given by through (30c) are left in its original form, but can be computed ∆Xt+1 log[p(X |X )] = −Q−1 (37a) for a given system. Xt+1 t+1 t t Building on the developments in this section, an SMC Xt+1 Xt+1 T −1 ∆ log[p(Yt+1|Xt+1)] = [∆ g (Xt+1)]Λ ΨY approximation of the PCRLB for a class of non-linear SSMs Xt+1 Xt+1 t+1 Yt+1 t+1 T −1 (37b) with additive Gaussian noise is presented next. − [∇Xt+1 gt+1(Xt+1)]Rt+1[∇Xt+1 gt+1(Xt+1)] 8
−1 −1 where: Λ = R I 2 2 ; Substituting (37b) into (45) yields Yt+1 t+1 n ×n 2 2 2 2 ΨYt+1 = [Yt+1 − gt+1(Xt+1)]In ×n; In ×n , and In ×n are 22,b E p Y E E 2 2 2 It = ( 1:t) p(Xt+1 |Y1:t) p(Yt+1|Xt+1) n × n and n × n identity matrix. Also: [∇Xt+1 gt+1(Xt+1)] p(Y1:t+1) Xt+1 and [∆ gt+1(Xt+1)] are Xt+1 T −1 Xt+1 [−∆ g (X )]Λ Ψ Xt+1 t+1 t+1 Yt+1 Yt+1 T h [∇Xt+1 gt+1(Xt+1)] = T −1 +[∇Xt+1 gt+1(Xt+1)]Rt+1[∇Xt+1 gt+1(Xt+1)] . (46) = [∇ g(1) (X ),..., ∇ g(m)(X )] ; (38a) Xt+1 t+1 t+1 Xt+1 t+1 t+1 m×m Noting the following two conditions Xt+1 T [∆X gt+1(Xt+1)] t+1 E [∇ gT (X )]R−1 [∇ g (X )] Xt+1 (1) Xt+1 (m) p(Yt+1|Xt+1) Xt+1 t+1 t+1 t+1 Xt+1 t+1 t+1 = [∆ g (X ),..., ∆ g (X )] 2 ; (38b) Xt+1 t+1 t+1 Xt+1 t+1 t+1 m×m T −1 = [∇Xt+1 gt+1(Xt+1)]Rt+1[∇Xt+1 gt+1(Xt+1)], (47) (1) (n) T Xt+1 T −1 where gt+1(Xt+1) , [gt+1(Xt+1), ··· ,gt+1(Xt+1)] is a E p(Yt+1|Xt+1)[∆X gt+1(Xt+1)]ΛY ΨYt+1 ] m × 1 vector function in (33b). t+1 t+1 = ∆Xt+1 gT (X )Λ−1 E [Ψ ]=0, (48) Lemma 4.18: For a system given by Model 4.15, under Xt+1 t+1 t+1 Yt+1 p(Yt+1|Xt+1) Yt+1 Assumptions 3.2 through 3.5 the matrices (11a) through (11d) and substituting (47) and (48) into (46) yields (39d), which in Lemma 4.7 can be written as: completes the proof. 11 E T −1 Using the results of Lemma 4.18, an SMC approximation of It = p(Xt |Y1:t+1)[∇Xt ft (Xt)]Qt [∇Xt ft(Xt)]; (39a) 12 T −1 the PCRLB for Model 4.15 can be subsequently computed, as I =E [−∇X f (Xt)]Q ; (39b) t p(Xt |Y1:t+1) t t t discussed in the next lemma. 22,a −1 It =Qt ; (39c) Lemma 4.19: Let a stochastic non-linear system with ad- 22,b T −1 E p(Y ) E ditive Gaussian state and sensor noise be represented by It = 1:t p(Xt+1|Y1:t)[∇Xt+1 gt+1(Xt+1)]Rt+1 p(Y1:t+1) Model 4.15, such that it satisfies Assumption 3.2 through 3.5. T j × [∇X g (Xt+1)]. (39d) M N t+1 t+1 Let {Y1:t = y1:t}j=1 be M ∈ i.i.d. measurement sequences Proof: (39a): Substituting (35b) into (11a) yields generated from Model 4.15, then (5a) through (5c) in Lemma 3.7 can be recursively approximated as follows: 11 E T −1 It = p(Xt:t+1|Y1:t+1)[[∇Xt ft (Xt)]Qt [∇Xt ft(Xt)] M N Xt T −1 − [∆ f (x )]Λ Ψ ], (40a) 11 1 T i,j −1 i,j Xt t t Xt Xt ˜ Dt = [∇Xt ft (Xt|t+1)]Qt [∇Xt ft(Xt|t+1)]; E E T −1 MN = p(Xt|Y1:t+1) p(Xt+1 |Xt,Y1:t+1)[[∇Xt ft (Xt)]Qt Xj=1 Xi=1 (49a) Xt T −1 (40b) × [∇Xt ft(Xt)] − [∆Xt ft (Xt)]ΛXt ΨXt ], 1 M N where (40b) is obtained by substituting the probability relation D˜ 12 = −[∇ f T (Xi,j )]Q−1; (49b) t MN Xt t t|t+1 t p(xt:t+1|y1:t+1) = p(xt+1|xt,y1:t+1)p(xt|y1:t+1) into (40a). Xj=1 Xi=1 Finally, by noting the following two conditions M N 22 −1 1 T i,j −1 T −1 ˜ E Dt =Qt + [∇Xt+1 gt+1(Xt+1|t)]Rt+1 p(Xt+1 |Xt,Y1:t+1)[∇Xt ft (Xt)]Qt [∇Xt ft(Xt)] MN T −1 Xj=1 Xi=1 = [∇Xt ft (Xt)]Qt [∇Xt ft(Xt)], (41) T i,j × [∇Xt+1 gt+1(X )]; (49c) E Xt T −1 t+1|t p(Xt+1 |Xt,Y1:t+1)[∆Xt ft (Xt)]ΛXt ΨXt i,j j Xt T −1 N E (42) and {X }i=1 ∼ p(xt|y1:t+1) and = [∆Xt ft (Xt)]ΛXt p(Xt+1|Xt,Y1:t+1)[Ψxt ]=0, t|t+1 {Xi,j }N ∼ p(x |yj ) are sets of N resampled and substituting (41) and (42) into (40b) yields (39a). t+1|t i=1 t+1 1:t particles from Lemma 4.13 and Algorithm 1, respectively, for (39b): Substituting (35c) into (11b) yields j M all {Y1:t+1 = y1:t+1}j=1. 12 T −1 j I = E [−[∇X f (Xt)]Q ]. (43) t p(Xt:t+1|Y1:t+1) t t t Proof: For {Y1:t = y1:t}, the SMC approximation in (28) can be written as Substituting the probability relation p(xt:t+1|y1:t+1) = p(xt+1|xt,y1:t+1)p(xt|y1:t+1) into (43), followed by taking N j 1 independent terms out of the integral yields (39b). p˜(dxt|y1:t+1)= δXi,j (dxt), (50) N t|t+1 (39c): Substituting (37a) into (11c) yields (39c). Xi=1 (39d): Using Bayes’ rule, the expectation in (11d) can be i,j j where Xt|t+1 ∼ p(xt|y1:t+1). Substituting (50) into (39a) and rewritten as (39b) yields 22,b Xt+1 I = E p(X ,Y ) [−∆ log p(Y |X )]. (44) t t+1 1:t+1 Xt+1 t+1 t+1 N p(Y1:t+1) 1 I˜11 = [∇ f T (Xi,j )]Q−1[∇ f (Xi,j )], (51a) Now using the probability condition t N Xt t t|t+1 t Xt t t|t+1 Xi=1 p(xt+1,y1:t+1)= p(yt+1|xt+1)p(xt+1|y1:t)p(y1:t), the N expectation in (44) can further be decomposed and written as ˜12 1 T i,j −1 It = − [∇Xt ft (Xt|t+1)]Qt , (51b) 22,b N E p Y E E Xi=1 It = ( 1:t) p(Xt+1|Y1:t) p(Yt+1|Xt+1) p(Y1:t+1) ˜ Xt+1 where It is an SMC approximations of It. Substituting (51) × [−∆ log p(Yt+1|Xt+1)]. (45) Xt+1 and (29) into (6e) and (6f) yields (49a) and (49b), respectively. 9
22 Algorithm 2 SMC based PCRLB for Model 3.1 Computing an SMC approximation of Dt in (6g) for Model 4.15 requires a slightly different approach. Substituting (39c) Input: Given Model 3.1, satisfying Assumptions 3.2 and (39d) into (6g) yields through 3.5, assume a prior pdf on X0, such that 22 −1 X0 ∼ p(x0). Also, select algorithm parameters- T , N and E E p Y E Dt = p(Y1:t+1)[Qt + ( 1:t) p(Xt+1|Y1:t) p(Y1:t+1) M. T −1 T Output: SMC approximation of the PCRLB for Model × [∇Xt+1 gt+1(Xt+1)]Rt+1[∇Xt+1 gt+1(Xt+1)]], (52a) 3.1. 1: Generate and store M i.i.d. sequences =Q−1 + E E [∇ gT (X )]R−1 t p(Y1:t) p(Xt+1 |Y1:t) Xt+1 t+1 t+1 t+1 {Y j }M ∼ p(y ) of length T, by simulating T 1:T j=1 1:T × [∇Xt+1 gt+1(Xt+1)], (52b) Model 3.1, M times starting at M i.i.d. initial states i M −1 {X } ∼ p(x0). where Qt is independent of the measurement sequence. 0|−1 j=1 j 2: E E p Y E for j =1 to M do Also, p(Y1:t+1) ( 1:t) [·] = p(Y1:t)[·]. For {Y1:t = y1:t}, p(Y1:t+1) 3: for t =1 to T do i,j N j i,j j random samples {X } ∼ p(xt+1|y ) from Algorithm 4: N t+1|t i=1 1:t Store resampled particles {Xt|t }i=1 ∼ p(xt|y1:t) us- j 1 delivers an SMC approximation of p(dxt+1|y1:t) given as ing Algorithm 1. N 5: Store resampled particles j 1 i,j N j i,j {X }i=1 ∼ p(xt−1:t|y1:t) using Lemma p˜(dxt+1|y1:t)= δX (dxt+1) (53) t−1:t|t N t+1|t Xi=1 4.12. 6: end for where p˜(dx |yj ) is an SMC approximation of t+1 1:t 7: end for p(dx |yj ). Substituting (53) and (29) into (52b) yields t+1 1:t 8: Compute PFIM J at t =0 based on the initial target (49c), which completes the proof. 0 state pdf X ∼ p(x ). If X ∼ N (x |C , P ) then from Result 4.20: An SMC approximation of the PFIM for Model 0 0 0 0 x0 0|0 Lemma 3.7, J = P −1. 4.15 is obtained by substituting (49a) through (49c) in Lemma 0 0|0 9: for t =0 to T − 1 do 4.19 into (4) in Lemma 3.7, such that 10: Compute an SMC estimate (30a) through (30c) in ˜ ˜ 22 ˜ 12 T ˜ ˜ 11 −1 ˜ 12 Jt+1 = Dt − [Dt ] (Jt + Dt ) Dt , (54) Lemma 4.14. 11: Compute PCRLB J˜−1 by substituting (30a) through where ˜ is an SMC approximation of . Applying ma- t+1 Jt+1 Jt+1 (30c) into (55). trix inversion lemma [51] in (54) gives an SMC approximation 12: end for of the PCRLB, such that ˜−1 ˜ 22 −1 ˜ 22 −1 ˜ 12 T Jt+1 = [Dt ] − [Dt ] [Dt ] −1 ˜ 12 ˜ 22 −1 ˜ 12 T ˜ ˜ 11 ˜ 12 ˜ 22 −1 VI. CONVERGENCE × Dt [Dt ] [Dt ] − (Jt + Dt ) Dt [Dt ] , (55) h i Computing the PCRLB in Lemma 3.6 involves solving ˜−1 −1 where Jt+1 is an SMC approximation of Jt+1 in (2). the complex, multi-dimensional integrals; however, as stated earlier, for Models 3.1 and 4.15 the PCRLB cannot be solved V. FINAL ALGORITHM in closed form. Algorithms 2 and 3 gives a N particle and Algorithms 2 and 3 give the procedure for computing an M simulation based SMC approximation of the PCRLB for SMC approximation of the PCRLB for Models 3.1 and 4.15, Models 3.1 and 4.15, respectively. It is therefore natural to respectively. question the convergence properties of the proposed numerical Remark 5.1: In practice, an ensemble of M measurement method. In this regard, results such as Theorem 4.5 and j M sequences {Y1:T = y1:T }j=1 required by Algorithms 2 and 3 Remark 4.11 are important as it ensures that the proposed are obtained from historical process data; however, in simula- numerical solution does not result in accumulation of errors. tions, it can be generated by simulating Models 3.1 and 4.15, It is emphasized that although Theorem 4.5 and Remark 4.11 M times starting at i.i.d. initial states drawn from X0 ∼ p(x0). not necessarily imply convergence of the SMC based PCRLB Note that this procedure also requires simulation of the true and MSE to its theoretical values, nevertheless, it provides a states; however, true states are not used in Algorithms 2 and strong theoretical basis for the numerous approximations used 3. in Algorithms 2 and 3. For illustrative purposes, to assess the numerical reliability of From an application perspective, it is instructive to highlight Algorithms 2 and 3, a quality measure is defined as follows that the numerical quality of the SMC based PCRLB approxi- T mation in Algorithms 2 and 3 can be made accurate by simply 1 −1 ˜−1 −1 ˜−1 ΛJ = [J − J ] ◦ [J − J ], (56) increasing the number of particles (N) and the MC simulations T t t t t Xt=1 (M). The choice of N and M are user defined, which can where ΛJ is the average sum of square of errors in approximat- be selected based on the required numerical accuracy, and ing the PCRLB and ◦ is the Hadamard product. ΛJ is a n × n available computing speed. It is important to emphasize that matrix, with diagonal element ΛJ (j, j) as the average sum of due to the multiple approximations involved in deriving a square of errors accumulated in approximating the PCRLB for tractable solution, for practical purposes, with a finite N and ˜−1 state j, where 1 ≤ j ≤ n. M, the condition Pt|t − Jt < 0 is not guaranteed to hold for 10
Algorithm 3 SMC based PCRLB for Model 4.15 section, the quality of Algorithm 3 is validated on a practical Input: Given Model 4.15, satisfying Assumptions 3.2 problem of ballistic target tracking at re-entry phase. This par- through 3.5, assume a prior on X0, such that X0 ∼ p(x0). ticular problem has attracted a lot of attention from researchers Also, select algorithm parameters- T , N and M. for both theoretical and practical reasons. See [52] and the Output: SMC approximation of the PCRLB for Model references cited therein for a detailed survey on the ballistic 4.15. target tracking. 1: Generate and store M i.i.d. sequences 1) Model setup: Consider a target launched along a ballistic j M {Y1:T }j=1 ∼ p(y1:T ) of length T, by simulating flight whose kinematics are described in a 2D Cartesian coor- Model 4.15, M times starting at M i.i.d. initial dinate system. This particular description of the kinematics i M assumes that the only forces acting on the target at any states {X0|−1}j=1 ∼ p(x0). 2: for j =1 to M do given time are the forces due to gravity and drag. All other 3: for t =1 to T do forces such as: centrifugal acceleration, Coriolis acceleration, i,j N j wind, lift force and spinning motion are assumed to have a 4: Store predicted particles {Xt|t−1}i=1 ∼ p(xt|y1:t−1) using Algorithm 1. small effect on the target trajectory. With the position and the i,j N j velocity of the target at time t ∈ N described in 2D Cartesian 5: Store resampled particles {Xt|t }i=1 ∼ p(xt|y1:t) us- ing Algorithm 1. coordinate system as (Xt, Ht) and (X˙ t, H˙ t), respectively, its 6: Store resampled particles motion in the re-entry phase can be described by the following i,j N j discrete-time non-linear SSM [13] {Xt−1|t}i=1 ∼ p(xt−1:t|y1:t) using Lemma 4.13. 7: end for 0 8: end for Xt+1 = AXt + GFt(Xt)+ G. + Vt, (57) −g 9: Compute PFIM J0 at t =0 based on the initial target , ˙ ˙ T state pdf X0 ∼ p(x0). If X0 ∼ N (x0|Cx0 , P0|0) then from where the states Xt [Xt Xt Ht Ht] . Also, the matri- −1 ces A and G are as follows Lemma 3.7, J0 = P0|0 . 10: for t =0 to T − 1 do ∆T 2 0 11: Compute an SMC estimate (49a) through (49c) in 1 ∆T 0 0 2 Lemma 4.19. 0 1 0 0 ∆T 0 ˜−1 A , , G , 2 , (58) 12: Compute PCRLB Jt+1 by substituting (49a) through 0 0 1∆T ∆T 0 (49c) into using (55). 0 0 0 1 2 0 ∆T 13: end for where ∆T is the time interval between two consecutive radar measurements. N all t ∈ . In (57) Ft(Xt) models the drag force, which acts in a The quality of the SMC based PCRLB solution is validated direction opposite to the target velocity. In terms of the states, next via simulation. Ft(Xt) can be modelled as
gρ(H ) 2 2 ˙ VII. SIMULATION EXAMPLES t ˙ ˙ Xt Ft(Xt)= − Xt + Ht , (59) 2β q H˙ t In this section, two simulation examples are presented to demonstrate the utility and performance of the proposed SMC where: g is the acceleration due to gravity; β is the ballistic based PCRLB solution. The first example is a ballistic target coefficient whose value depends on the shape, mass and the tracking problem at re-entry phase. The aim of this study is cross sectional area of the target [11]; and ρ(Ht) is the density three fold: first to demonstrate the performance and utility of the air, defined as an exponentially decaying function of Ht, of the proposed method on a practical problem; second, to such that demonstrate the quality of the bound approximation for a ρ(H )= α e(−α2Ht) (60) range of target state and sensor noise variances; and third, t 1 −3 −4 −1 to study the sensitivity of the involved SMC approximations where: α1 = 1.227 kg·m , α2 = 1.09310 × 10 m for −3 to the number of particles used. Ht < 9144m; and α1 = 1.754 kg·m , α2 = 1.4910 × −4 −1 The performance of the SMC based PCRLB solution on a 10 m for Ht ≥ 9144m. Note that the drag force, Ft(Xt) is second example involving a uni-variate, non-stationary growth the only non-linear term in the state equation. In (57) the state 4 model, which is a standard non-linear, and bimodal benchmark noise Vt ∈ R is a i.i.d. sequence of multi-variate Gaussian model is then illustrated. This example is profiled to demon- random vector represented as Vt ∼ N (vt|0,Qt), with zero strate the accuracy of the SMC based PCRLB solution for mean and covariance matrix Qt given as highly non-linear SSMs with non-Gaussian noise. ∆T 3 ∆T 2 Q = γI ⊗ Θ, Θ= 3 2 , (61) A. Example 1: Ballistic target tracking at re-entry t 2×2 ∆T 2 ∆T In Section IV-C, an SMC based method for approximating 2 the PCRLB was presented for non-linear SSMs with additive where: γ ∈ R+; I2×2 is a 2 × 2 identity matrix; and ⊗ is the Gaussian state and sensor noise (See Algorithm 3). In this Kronecker product. The intensity of the state noise, determined 11 by γ, accounts for all the forces neglected in (57), including TABLE II: Cases considered for Example 1. any deviations arising due to system-model mismatch. The Case γ σr σǫ target measurements are collected by a conventional radar 1 1.0 100m 0.017rad 2 5.0 100m 0.017rad (e.g., dish radar) assumed to be stationed at the origin. The 3 1.0 500m 0.085rad sensor readings are measured in the natural sensor coordinate 4 5.0 500m 0.085rad system, which include range (Rt) and elevation (Et) of the T target. The radar readings Yt = [Rt Et] are related to the TABLE III: Variable values used in Example 1. states Xt through a non-linear observation model given below. Process variables Symbol values state noise Vt Vt ∼N (vt|0,Qt) 2 2 sensor noise Wt Wt ∼N (wt|0, Rt) Xt + Ht noise parameters γ, σr, σe see Table II Yt = q + Wt. (62) Ht initial states X0 X0 ∼N (x0|C ,P0 0) arctan x0 | Xt 232 km 2.290 cos (1900) km/s C 2 x0 88 km In (62) Wt ∈ R is an i.i.d. sequence of multi-variate Gaussian 2.290 sin(190o) km/s random vector represented as Wt ∼ N (wt|0, Rt), with zero 1km 0 0 0 mean and non-singular covariance matrix Rt given as 1/2 0 20m/s 0 0 [P0|0] 0 0 1km 0 2 0 0 0 20m/s σr 0 Rt = 2 , (63) Number of particles N 1000 0 σe MC simulations M 200 where σr ∈ R+ and σe ∈ R+ are the standard deviation as- sociated with range and elevation measurements. In (62), it Figure 1 shows a sample trajectory of the target in the is assumed that the true target elevation angle lies between 0 X − H plane along with its velocity map as a function of and π/2 radians; otherwise, it suffices to add π radians to the time, generated using Case 1 (see Table II). arctan term in (62). 3) Results: The kinematics of the ballistic target consist Remark 7.1: To avoid use of a non-linear sensor model, of nonlinear state and sensor models with additive Gaussian some authors [13], [34] considered transforming the radar noise, for which the PCRLB can be approximated using measurements in (62) into the Cartesian coordinate system, Algorithm 3. First, the state and sensor models in (57) and wherein the sensor dynamics manifest themselves into a (62), respectively, are defined as linear model. Even though this strategy eliminates the need to handle non-linearity in sensor measurements, tracking in 0 ft(Xt)= AXt + GFt(Xt)+ G. , (64a) Cartesian coordinates couples the sensor noise across two −g coordinate systems and makes the noise non-Gaussian and X2 + H2 state dependent [53]. Since the proposed method can deal with t+1 t+1 gt+1(Xt+1)= q H . (64b) strong state and sensor non-linearities, the radar readings are arctan t+1 X monitored in natural sensor coordinates alone. t+1
2) Simulation setup: For simulation, the model parameters To compute the required gradients ∇Xt ft(Xt) and are selected as given in Table I. The aim of this study is ∇Xt+1 gt+1(Xt+1), differentiating (57) with respect to to evaluate the quality of the SMC based PCRLB solution Xt, and (62) with respect to Xt+1, yields for a range of target state and sensor noise variances. This ∇ f (X )= A + GM (X ), (65a) allows full investigation of the quality of the SMC based Xt t t t t approximation for a range of noise characteristics. The cases ∇Xt+1 gt+1(Xt+1)= Nt+1(Xt+1), (65b) considered here are given in Table II. From Assumption 3.2, where: Mt(Xt) and Nt+1(Xt+1) in (65a) and (65b), respec- β is assumed to be fixed and known a priori. tively, are 2 × 4 matrices, whose entries are:
TABLE I: Parameter values used in Example 1. Mt(Xt)[1, 1]=0, (66a) Process variables Symbol values Mt(Xt)[2, 1]=0, (66b) 2 accel. due to gravity g 9.8 m/s ˙ 2 ˙ 2 ballistic coefficient β 40000 kg.m−1 · s−2 g 2Xt + Ht Mt(Xt)[1, 2] = − ρ(Ht) , (66c) radar sampling time ∆T 2 s 2β 2 2 X˙ + H˙ total tracking time T 120 s t t state noise Vt Vt ∼N (vt|0,Qt) q sensor noise Wt Wt ∼N (wt|0, Rt) g X˙ tH˙ t noise parameters γ, σr, σe see Table II Mt(Xt)[2, 2] = − ρ(Ht) , (66d) 232 km 2β ˙ 2 ˙ 2 0 Xt + Ht ⋆ 2.290 cos (190 ) km/s initial states X0 q 88 km gα2 2 2 o Mt(Xt)[1, 3] = ρ(Ht) X˙ + H˙ X˙ t, (66e) 2.290 sin(190 ) km/s 2β q t t probability of detection Prd 1 probability of false alarm Prf 0 gα2 2 2 Mt(Xt)[2, 3] = ρ(Ht) X˙ + H˙ H˙ t, (66f) 2β q t t 12
90 2.5 80
70 2
60 (in km) (in t 50 1.5
40
30 Target position: y Target position: Target velocity (in km/sec) (in Target velocity 1
20
10 0.5
0 0 50 100 150 200 0 20 40 60 80 100 120 Target position: x (in km) Time (in sec) t Fig. 1: Sample trajectory showing position and velocity of the target at re-entry phase.
TABLE IV: Average sum of square of errors in approximating Mt(Xt)[1, 4] = Mt(Xt)[2, 2], (66g) the PCRLB for the states in Example 1, under the cases in ˙ 2 ˙ 2 g Xt +2Ht Table II. Mt(Xt)[2, 4] = − ρ(Ht) ; (66h) 2β 2 2 ˙ ˙ ΛJ values Case 1 Case 2 Case 3 Case 4 Xt + Ht −6 q ΛJ (1, 1) (×10 ) 9.30 50.7 5.87 130 −11 and: ΛJ (2, 2) (×10 ) 4.50 2.06 7.08 46.2 −5 ΛJ (3, 3) (×10 ) 3.56 23.1 2.96 100 X 13 t+1 Λ (4, 4) (×10− ) 8.63 24.8 19.6 122 Nt+1(Xt+1)[1, 1] = , (67a) J 2 2 X˙ + H˙ q t+1 t+1 Ht+1 Nt+1(Xt+1)[2, 1] = , (67b) ground. ˙ 2 ˙ 2 Xt+1 + Ht+1 Nt+1(Xt+1)[1, 2]=0, (67c) Case 2: In this case the state noise intensity is increased five Nt+1(Xt+1)[2, 2]=0, (67d) fold and the sensor noise is kept at a small value (see Table Ht+1 II). Notwithstanding the increased noise variance, the PCRLB Nt+1(Xt+1)[1, 3] = , (67e) approximation is almost exact at all tracking time instants. The ˙ 2 ˙ 2 Xt+1 + Ht+1 results for Case 2 are shown in Figure 2(b). Table IV compares q Xt+1 the ΛJ values for Case 2 computed using (56). Based on Table Nt+1(Xt+1)[2, 3] = , (67f) ˙ 2 ˙ 2 IV, the results from Cases 1 and 2 closely compare in terms Xt+1 + Ht+1 of the order of the Λ values. To allow further comparison N (X )[1, 4]=0, (67g) J t+1 t+1 with Case 1, the square root of the approximate PCRLBs for Nt+1(Xt+1)[2, 4]=0. (67h) Cases 1 and 2 are compared in Figure 2(e). In terms of the To evaluate the numerical quality of Algorithm 1, we compare magnitude, the PCRLB for Case 2 is higher than that for Case the SMC based PCRLB solution against the theoretical values. 1, suggesting tracking difficulties with larger noise intensity. The theoretical bound is computed using an ensemble of the Case 3: Again for Case 3, performance similar to Figure true state trajectories, simulated using (57) (see [11], [13] 2(a) is obtained as given in Figure 2(c). The same is evident for further details). Here we compare the square root of from Table IV, where the average sum of square of error in −1 approximating the PCRLB for Cases 1 and 3 are of the same the diagonal elements of the theoretical PCRLB matrix Jt ˜−1 order. and its approximation Jt for all t ∈ [0,T ]. The results are summarized next for the cases given in Table II. For Case 4: Results for Case 4 is given in Figure 2(d). Higher fair comparison of all the cases, the parameters required by values of the PCRLB for Case 4 in Figure 2(e) reaffirms Algorithm 3 are specified as given in Table III. the estimation issues associated with larger noise variances. Case 1: Figure 2(a) compares the square root of the SMC Similar conclusions can be drawn based on Table IV, where based approximate bound against the theoretical PCRLB. the ΛJ values for Case 4 are the highest compared to the Clearly, the approximate bound for both the position and previous cases. Nevertheless, the errors are bounded and velocity of the target in both X and H coordinates accurately within a few orders of the ΛJ values reported for Case 1. follows the theoretical bound at all tracking time instants. All the above case studies suggest that the proposed ap- Note that the high values of the PCRLB in Figure2(a) proach is accurate in approximating the theoretical PCRLB highlights tracking difficulties as the target approaches the under large state and sensor noise variances. 13
1000 25 1000 30
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400 10 400 15 for position forx m) (in position for position x (in m) for position Square root of Square PCRLB root of Square PCRLB Square root of PCRLB Square 200 root of PCRLB Square 5 200 10 for velocity in x direction (in m/s) (in x for in direction velocity for velocity in x direction (in (in m/s) x for in direction velocity 0 0 0 5 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (in sec) Time (in sec) Time (in sec) Time (in sec)
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1200 20 20 1000 1000 800 15 15 600 10 500 400 for position h (in m) for position for position h (in m) for position 10 Square root of Square PCRLB root of Square PCRLB Square root Square of PCRLB root Square of PCRLB 5 200 for vel oci ty in h direct ion (in m/s) (in ion h direct in for vel ty oci for vel oci ty in h direct ion (in m/s) (in ion h direct in for vel ty oci 0 0 0 5 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (in sec) Time (in sec) Time (in sec) Time (in sec) (a) Square root of the theoretical (solid line with marker) and approximate(b) Square root of the theoretical (solid line with marker) and approximate PCRLB (solid line) for all the target states under Case 1 PCRLB (solid line) for all the target states under Case 2
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400 10 400 20 15 for position x (in m) for position x (in m) for position
Square root of PCRLB Square 200 root of PCRLB Square 5 root of PCRLB Square 200 root of PCRLB Square 10 for velocity in x direction (in m/s) in x for velocity direction (in m/s) in x for velocity direction 0 0 0 5 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (in sec) Time (in sec) Time (in sec) Time (in sec)
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for position h (in h m) (in for position h m) (in for position 400 10 Square root of PCRLB Square root of PCRLB Square 5 root of PCRLB Square root of PCRLB Square 200 for vel oci ty in h direct ion (in m/s) (in ion h direct in for vel ty oci m/s) (in ion h direct in for vel ty oci 0 0 0 5 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (in sec) Time (in sec) Time (in sec) Time (in sec) (c) Square root of the theoretical (solid line with marker) and approximate(d) Square root of the theoretical (solid line with marker) and approximate PCRLB (solid line) for all the target states under Case 3 PCRLB (solid line) for all the target states under Case 4
1000 40 600 35 800 580 36 30 560 600 25 34 540 20 520 32 400 15 500 for position x (in m) 10 for position x (in m) Square root of PCRLB Square root of PCRLB Square root of PCRLB 200 Square root of PCRLB 30 5 480 for velocity in x direction (in m/s) for velocity in x direction (in m/s) 0 0 460 28 0 20 40 60 80 100 120 0 20 40 60 80 100 120 82 84 86 88 90 92 94 96 98 82 84 86 88 90 92 94 96 98 Time (in sec) Time (in sec) Time (in sec) Time (in sec)
1600 25 850 19
1400 800 18 20 1200 750 17 1000 15 700 16 800 650 15 600 10 Case 1 PCRLB 600 14 for position h (in m) 400 Case 2 for position h (in m) N=10 Square root of PCRLB Square root of PCRLB 5 Square root of PCRLB Square root of PCRLB Case 3 N=100 200 550 13 for velocity in h direction (in m/s) for velocity in h direction (in m/s) Case 4 N=500 0 0 500 12 0 20 40 60 80 100 120 0 20 40 60 80 100 120 85 86 87 88 89 90 90 92 94 96 98 100 Time (in sec) Time (in sec) Time (in sec) Time (in sec) (e) Square root of the approximate PCRLBs for the target states under the(f) Square root of the theoretical and approximate PCRLBs for different cases listed in Table II. values of N in Example 1, Case 4. Note that all the sub-figures have been appropriately scaled up allow clear illustration of the effect of N on the quality of approximation.
Fig. 2: Results for Simulation Example 1. 14
0.01 0.01 Remark 7.2: Note that in [34], a similar ballistic target PCRLB 0.009 Approximation 0.009 tracking problem at re-entry phase was considered to illustrate 0.008 0.008 the use of an EKF and UKF based method in approximating 0.007 0.007 the theoretical PCRLB. Unlike the non-linear sensor model 0.006 0.006 considered here (see (62)), [34] used the change of coordinates 0.005 0.005 PCRLB PCRLB method to obtain a linear sensor model representation. It is 0.004 0.004 important to highlight that even with a linear sensor model, 0.003 0.003 the EKF and UKF based method yields a biased estimate of 0.002 0.002 the PCRLB for the target states (see Figures 4 through 7 in 0.001 0.001 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 [34]). Whereas, under a more challenging situation, as one Time (in sec) Time (in sec) considered here, the SMC based method yields an unbiased estimate of the PCRLB (see Figures 2(a) through 2(d), and Fig. 3: Comparing the approximate PCRLB against the the- Table IV). This highlights the advantages of the SMC based oretical PCRLB in Example 2 under Gaussian (left) and method (both in terms of the accuracy and applicability) over Rayleigh (right) sensor noise distributions. the EKF and UKF based PCRLB in presence of strong system or sensor non-linearities. Next we study the sensitivity of the involved SMC approx- i.i.d. sequence following a Gaussian distribution, such that R imations to the number of particles used. In Figure 2(f), ap- Wt ∼ N (wt|0, Rt), while for Case 2, Wt ∈ is again proximate PCRLB bounds are compared against the theoretical an i.i.d sequence, but follows a Rayleigh distribution, such that . For both the cases, the sensor noise PCRLB for different values of N. The results are obtained by Wt ∼ R(wt|Rt) variance −3 is considered. Here Case varying N in Algorithm 1. From Figure 2(f), it is clear that by Rt =1 × 10 ∀t ∈ [1,T ] 2 represents a much more challenging situation, where esti- simply increasing N, which is a tuning parameter in Algorithm 1, the quality of the SMC approximations can be significantly mation is considered under a non-Gaussian sensor noise. For improved. For all the simulation cases, the number of Monte fair comparison, M = 200 and N = 100 are selected. 3) Results: Case 1: Comparison of the approximate and Carlo simulations was selected as M = 200 (see Table III). Computation of a single Monte Carlo simulation took 0.69 the theoretical PCRLB for the Gaussian sensor noise case seconds on a 3.33 GHz Intel Core i5 processor running on is given in Figure 3. The results suggest that for the cho- Windows 7. Note that the reported absolute execution time is sen N, the approximate PCRLB almost exactly follows the solely for instructive purposes and is not intended to reflect on theoretical PCRLB at all filtering time instants. The same is reflected in the error value computed using (56), which is the true computational complexity of the proposed algorithm. −9 Collectively, from Figures 2(a) through 2(f), it is evident ΛJ =4.19 × 10 . that the SMC based method is accurate in approximating the Case 2: Figure 3 compares the approximate PCRLB solution theoretical PCRLB for a range of target state and sensor noise against the theoretical PCRLB for the Rayleigh sensor noise variances. case. Although the approximation almost exactly follows the theoretical solution, compared to Case 1, the approximation is relatively coarser at certain time instants. This highlights B. Example 2: A non-linear and non-Gaussian system the issues associated with estimation under non-Gaussian The aim of this study is to demonstrate the effectiveness of noise with limited N. Finally, the ΛJ value for Case 2 is −8 the proposed SMC based method in approximating the PCRLB 4.62 × 10 , which is within an order of the value reported in presence of a non-Gaussian noise. for Case 1. 1) Model setup: A more challenging situation is considered The simulation study clearly illustrates the efficacy of the in this section that involves the following discrete-time, uni- proposed method in approximating the PCRLB for non-linear variate non-stationary growth model SSMs with non-Gaussian noise. X 25X t t VIII. DISCUSSIONS Xt+1 = + 2 + 8cos(1.2t)+ Vt, (68a) 2 1+ Xt 2 The simulation results in Section VII demonstrate the utility Xt Yt = + Wt, (68b) and performance of the SMC based PCRLB approximation 20 method developed in this paper. It is important to highlight where Vt ∈ R is an i.i.d. sequence following a Gaussian that despite of the many convergence results discussed in distribution, such that Vt ∼ N (vt|0,Qt). The noise variance is Section VI, the choice of an SMC method plays a crucial −3 defined as Qt =5 × 10 ∀t ∈ [1,T ], where T is 30 seconds. role in determining the quality of the PCRLB approximation. Also, the initial state is modelled as X0 ∼ N (x0|0, 0.01). This Here, the use of a sequential-importance-resampling (SIR) example has been profiled due to it being acknowledged as a filter of [41], [45] is motivated by the fact that it is relatively benchmark problem in non-linear state estimation in several less sensitive to large state noise and is computationally less previous studies [7], [17]. expensive. 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