IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 2, FEBRUARY 2013 165

Decentralized Conditional Posterior Cramér–RaoLower Bound for Nonlinear Distributed Estimation

Arash Mohammadi and Amir Asif, Senior Member, IEEE

Abstract—Motivated by the decentralized adaptive resourcemanagement problems, the letter derives recursive expressionsfor online computation of the conditional decentralized posteriorCramér–Rao lower bound (PCRLB). Compared to the non-con-ditional PCRLB, the conditional PCRLB is a function of the pasthistory of observations made and, therefore, a more accurate rep-resentation of the estimator’s performance and, consequently, abetter criteria for sensor selection. Previous algorithms to computethe conditional PCRLB are limited to centralized architectures.The letter addresses this gap. Our simulations verify the optimalityof the conditional dPCRLB by comparing it with the centralizedconditional PCRLB in bearing-only tracking applications.

Index Terms—Bayesian estimation, distributed signal pro-cessing, particle filters, PCRLB, sensor resource management.

I. INTRODUCTION

S ENSOR resource management is a critical issue in de-centralized, geographically distributed networks where

the number of active sensors is constrained by power andbandwidth limitations. Being independent of the estimationmechanism, the posterior Cramér–Rao lower bound (PCRLB)[1] provides an effective criteria [2]–[4] for sensor selection.This letter derives the conditional PCRLB for sensor networksconfigured in a decentralized topology as an alternative tothe non-conditional (conventional) PCRLB. The conditionalPCRLB is an online bound and provides a more accurate rep-resentation of the systems’s performance and a better criteriafor sensor-selection. Current conditional PCRLB expressionsare limited to centralized architectures utilizing a fusion centre,which make them inappropriate for decentralized topologies.In the context of decentralized estimation, [4] has proposed

an approximate expression for computing the non-conditionaldecentralized PCRLB (dPCRLB). Our previous work [6], [7]improve on [4] by deriving the exact expressions based on theChong-Mori-Chang track-fusion theorem [5] for computingthe non-conditional dPCRLB distributively for full-order [6]and reduced-order [7] decentralized state estimation. In [8], wehave proposed a dPCRLB-based sensor selection algorithm.This letter extends our non-conditional dPCRLB framework

Manuscript received October 17, 2012; revised December 11, 2012; acceptedDecember 14, 2012. Date of publication December 19, 2012; date of current ver-sion January 09, 2013. This work was supported in part by the Natural Scienceand Engineering Research Council (NSERC) Discovery Grant 228415. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Min Dong.The authors are with the Department of Computer Science and Engineering,

York University, Toronto, ONM3J 1P3 Canada (e-mail: marash@cse.yorku.ca;asif@cse.yorku.ca).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LSP.2012.2235430

[6]–[8] to conditional dPCRLB for full-order estimation. Ex-tending the non-conditional dPCRLB to conditional dPCRLBis challenging due to the following issues: (i) The underlyingexpectations in the conditional dPCRLB are with respect to theconditional posterior, hence, the Chong-Mori-Chang theoremcan not be used directly. (ii) The recursive expressions forthe conditional Fisher information matrix (FIM), i.e., inverseof the PCRLB, utilize an auxiliary FIM corresponding to theprevious iteration (instead of its own previous value), there-fore, distributed expressions for computing the auxiliary FIMare now needed, and; (iii) In addition, recursive expressionsfor computing the predictive conditional PCRLB from theauxiliary FIM are required. The performance of the proposedconditional dPCRLB is evaluated using a distributed bearingonly tracking simulation in a connected sensor network.

II. PROBLEM FORMULATION

In this letter, we consider a full-order system given by

(1)

where is the state vector comprising of state variables.The observation model at node , for , is

(2)

The state and observation functions are nonlinearand the corresponding forcing terms can poten-tially be non-Gaussian. In the full-order estimation, each nodeestimates the entire state vector .

A. Centralized Non-Conditional PCRLB

The mean square error (MSE) in the estimate of thestate vector is lower bounded [1] by the FIM

(3)where notation denotes the secondorder partial derivative with the first order partial derivative

. The FIMassociated with the estimate is obtained from the inverse ofthe right-lower square block of the inverse of .An alternative is to perform the expectation in (3) with respect tothe posterior distribution leadingto the auxiliary FIM

(4)

Reference [1] derives recursively without manipulatinglarge matrices, while [3] has derived similar recursive expres-sions for computing (the inverse of right-lower square block of the inverse of ). Similar

1070-9908/$31.00 © 2012 IEEE

166 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 2, FEBRUARY 2013

to (and ) the predictive auxiliary FIMis defined as

(5)

where predictive distribution .Our scheme extends [3] to decentralized topologies.

B. Centralized Conditional PCRLB

The conditional PCRLB provides a bound on the perfor-mance of estimating given that the past observations

are known [3]. The conditionalMSE in the estimateof the state vector is lower bounded by

(6)

where . The condi-tional FIM is defined as the the inverse of theright-lower block of . A centralized recursive ex-pression for updating is derived in [3]. For deriving theconditional dPCRLB, we need recursive expressions for com-puting the predictive conditional PCRLB defined as

(7)

where . Termis defined as the inverse of the right-lower block of

. Using the factorization

we derive the following recursive expression for computing(proof is not included here to save on space)

(8)

(9)

(10)

(11)

Next, we compute the conditional dPCRLB distributively.

III. DECENTRALIZED CONDITIONAL DPCRLB

Definition 1: The local conditional FIM corre-sponding to the local estimate , for ,is defined as follows

(12)where .The local bound on , is given by the

inverse of the right-lower block of .

Definition 2: The local predictive conditional FIMis defined as follows

(13)where . The localbound on is given by the inverse

of the right-lower block of .The centralized bound [3] can be used to compute both

and with relevant local distributionsreplacing the global ones. The local auxiliary FIMsand are derived from and

, with definitions similar to (4) and (5)except that the local distributions are used. Another format forthe local FIMs is

(14)

(15)

where the expectations are with respect to. It can be shown that in Gaussian linear systems,

and (similarly and) are equivalent.Theorem 1 provides the optimal recursive formula for com-

puting the overall conditional FIM as a function of local FIMs.Theorem 1: The sequence of the global informa-

tion sub-matrices follows the recursion

(16)where

(17)

(18)

(19)

In order to compute the conditional dPCRLB, term

is replaced with and similarly isreplaced with . Later, we derive distributedrecursive expression for . First, we extend theChong–Mori–Chang track-fusion theorem [5] to conditionalposterior.Lemma 1: Assuming that the observations conditioned on

the state variables are independent, the global posterior for a-sensor network is factorized as shown in (20) at the bottom

of the next page.Proof of Lemma 1: Using the Markovian property

(21)

MOHAMMADI AND ASIF: DECENTRALIZED CONDITIONAL POSTERIOR CRLB FOR NONLINEAR DISTRIBUTED ESTIMATION 167

Considering independent observations given the state variables,the first term on the right hand side (RHS) of (21) is

(22)

Using the Chong-Mori-Chang track-fusion theorem [5], thethird term on the RHS of (21) is factorized as follows

(23)

Finally, substituting (22) and (23) in (21), we get (20).Proof of Theorem 1: Decomposing

, (12) reduces to

(24)

Block stands for a block of all zeros. Terms ,and are defined as in (17) and (18). Terms ,

, , and are derived as follows

(25)where . Term is theinverse of the right-lower block of (25), i.e.,

(26)

Term is simplified as

(27)

Finally, using (27) and definitions (14), (15), term re-duces to (19). The information sub-matrix can be cal-culated as the inverse of the right lower sub-matrixof and (26) as follows

(28)

In general, there is no recursivemethod to calculate .An approximated centralized recursive expression is proposedin [3]. Next we derive a decentralized recursive expression forcomputing using the approximation stated in [3].Proposition 1: The global sequence of informa-

tion sub-matrices follows the approximated recursion, i.e.,

(29)

(30)

(31)

(32)

The proof of Proposition 1 is similar to that for the non-con-ditional dPCRLB as explained in [6].

A. Practical Application of the Conditional dPCRLB

A sensor network with two different types of nodes are con-sidered [4]: (i) Sensor nodes: with limited power used only torecord measurements, and; (ii) Processing nodes: responsiblefor sensor-selection within their neighbourhoods and for per-forming decentralized estimation. Below, Case 1 [8] is near-op-timal but requires high communication overhead. By computingthe dPCRLB within local neighbourhoods, Case 2 [4] does notrequire consensus and has a reduced overhead.Case 1: The global submatrix and local submatrices

are assumed available from iteration at node . Iter-ation ) for computing conditional dPCRLB is as follows.Step 1: Compute , , and using (17) and

(18), and terms , , and using (30) and

(20)

168 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 2, FEBRUARY 2013

Fig. 1. Case 1: Comparison of the proposed conditional dPCRLB, centralizedconditional PCRLB and an approximate conditional dPCRLB (similar to [4]).

Fig. 2. Same as Fig. 1 except for Case 2 (local fusion without consensus):PCRLB comparison between two randomly selected nodes and from Case 1.

(31). These terms are global but based on the state model andcomputed locally.Step 2: Compute the local FIMs and

and local auxiliary FIMs and asexplained in Section III.Step 3: Compute using (19). Term

is computed distributively across the networkusing consensus. Similarly, in (32) includes a summa-tion term that is also requires consensus.Step 4: Theorem 1 computes the conditional dPCRLB and

Proposition 1 computes for .Under Case 1, Step 3 involves communication overhead. If

average consensus is used to distributively compute the sum-mation terms, the communication overhead at each processingnode is of , where is number of states,the number of nodes in the neighbourhood of node , and isthe number of consensus iterations. In our case, this overhead isrestricted to the processing nodes.Case 2: fuses local conditional PCRLBs within local neigh-

bourhoods [4] for sensor-selection. Consensus is not needed thatreduces overhead. Steps 1 and 2 are the same as in Case 1.Step 3: Processing node computes

over local neighbourhoods .Step 4: Theorem 1 is used at processing node to compute the

conditional dPCRLB within local neighbourhoods . Propo-sition 1 computes within local neighbourhoods.In Case 2, the communication overhead at each processing

node is of , an improvement of a factor of overCase 1.

IV. SIMULATION

A bearing-only target tracking application [8] based on asensor network of nodes is simulated. Each node com-municates within its local neighbourhood ofunit radius. A nonlinear clockwise coordinate turn (CT) motion

model [8] is considered for the target. Node ’s observation isthe target’s bearings, i.e.,

(33)

where are the coordinates of node . Both processand observation noises are normally distributed with the obser-vation noise model assumed to be state dependent such that thebearing noise variance at node depends on the distance be-tween the observer and target. Our simulations considers Cases1 and 2 as described in Section III-A. For Case 1, Fig. 1 com-pares the proposed conditional dPCRLB (obtained from The-orem 1), the centralized conditional PCRLB (using the cen-tralized bound [3]), and the approximated conditional dPCRLBbased only on the first two terms on the RHS of (19) (similar to[4]). It is observed that the proposed conditional dPCRLB andthe centralized bound overlap across various iterations. The ap-proximated PCRLB fluctuates widely over time. Having justi-fied that the proposed dPCRLB is an accurate representation ofits centralized counterpart, Fig. 2 plots the conditional dPCRLBresults for Case 2 (local fusion with no consensus). Resultsfrom two randomly selected nodes are plotted in Fig. 2. Dueto localized fusion in Case 2, some variation in the conditionaldPCRLBs is observed at the two nodes but the proposed boundis still superior to the approximated bound as plotted in Fig. 1.

V. DISCUSSION AND CONCLUSION

The letter derives recursive expressions for computing theonline conditional dPCRLB for sensor-selection in decentral-ized estimation as an alternative to the offline non-conditionaldPCRLB. Previous conditional PCRLB algorithms are limitedto centralized architectures using a fusion centre which makesthem inappropriate for decentralized sensor management. Sincethe conditional dPCRLB is a function of the past observationsmade, it is a more reliable criteria for decentralized applications.

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