Particle filtering with applications in networked systems: a survey

Theoretically, there are an infinite number of possible choices for the importance density \(q({x_k}|{z_{1:k}})\). In practice, some choices are superior over the others since they are closer to the optimal importance density \(p({x_k}|x_{k - 1}^i,{z_k})\) (see [17]). Here the optimality is defined in the sense that the variance of the important weights is minimized. In fact, it has been shown (see [18]) that the unconditional variance of importance weights can only increase over time, which leads to a common problem with particle filtering methods, i.e., the degeneracy phenomenon. Intuitively, we hope that all the particles are evenly weighted to guarantee the efficiency of the algorithm. The actual situation, however, is that after a few filtering iterations, only one particle will have a significant weight, while all the others play a negligible role in the representation of state posterior. Kong et al. introduced in [18] the effective sample size (ESS) as a measure of degeneracy, defined aswhere \({{\mathrm{var}} (\tilde{w}_k^i)}\) is the variance of importance weights. It is clear from (10) that a large variance of importance weights implies a small effective sample size, hence a severe degeneracy.

One way to reduce particle degeneracy is to use optimal importance density \(p({x_k}|x_{k - 1}^i,{z_k})\) so that \({{\mathrm{var}} (\tilde{w}_k^i)}\) in (10) is minimized. With very few exceptions, however, it is impossible in practice to evaluate \(p({x_k}|x_{k - 1}^i,{z_k})\) analytically. Therefore, many suboptimal schemes for importance density selection have been proposed. One idea they have in common is that current measurements should be taken into account when constructing the importance density. The proposal distribution is said to be an adapted one if the current measurements are incorporated. The bootstrap particle filter proposed by Gordon et al. [4] uses the state transition probability \({p({x_k}|{x_{k - 1}})}\) as the importance density. Even though the algorithm is simple to realize, it has ignored the current measurements, which might cause a large deviation between the predicted particles and the actual support of the posterior PDF. Compared with the true optimal importance density, the Gaussian approximation of it is much easier to evaluate. A variety of tools for the calculation of Gaussian approximation of \(p({x_k}|x_{k - 1}^i,{z_k})\) are available, including extended Kalman filter (EKF), unscented Kalman filter (UKF), etc. The procedure is rather simple: when new measurements are received, a Kalman-type propagation is performed to obtain the Gaussian approximation of \(p({x_k}|x_{k - 1}^i,{z_k})\). This approximation is then used as the importance density from which the new set of particles is drawn. The filtering method is termed as extended particle filter (EPF)/unscented particle filter (UPF) when EKF/UKF is employed to calculate the importance density function (see [19, 20], respectively).

It has been observed that the mixture structure of particle filters will cause a great increase in running time when adaptation is performed [21]. To adapt the proposal distribution without a great loss in efficiency, the auxiliary particle filter (APF) has been introduced, whose basic idea is to carry out the particle filtering algorithm in a higher dimension. The motivation is that it is wasteful to draw particles which will at last be abandoned with a large probability. In the APF algorithm, an auxiliary variable \({j^i}\), serving as the index of the particle \({\tilde{x}_{k - 1}^i}\), is weighted at the beginning of the kth iteration according to the compatibility of \({\tilde{x}_{k - 1}^i}\) given \({z_{1:k}}\). The new set of particles, \(\left\{ {\tilde{x}_k^i,i = 1,2,\ldots ,{N_s}} \right\} \), is then sampled from the modified state transition probability in which the weighted index is incorporated. It is revealed in [22] that essentially the APF method is equivalent to adding a well-designed resampling step (see details below) before each iteration of the standard SIS procedure.

Generally, the suboptimal proposal distribution should be constructed on a case by case basis. In [23], a problem-specific proposal distribution has been designed for radar tracking based on particle filter. The particle swarm optimization (PSO) has been used in [24] to optimize the proposal distribution for the simultaneous localization and mapping (SLAM) problem. In [25], the ensemble Kalman filter (EnKF) has been employed to define the proposal density of particle filter for soil moisture estimation.

Another way to reduce degeneracy is to perform resampling at each filtering iteration. Resampling is a procedure in which particles \(\left\{ {\tilde{x}_k^i,i = 1,2,\ldots ,{N_s}} \right\} \) are reselected in accordance with their weights \(\left\{ {\tilde{w}_k^i,i = 1,2,\ldots ,{N_s}} \right\} \). In this way, the particles with larger weights will have a greater number of offspring while those with negligible weights are simply discarded. The motivation is to conserve our computing resources for the particles which will play greater roles. After resampling, one gets a new set of particles which are equally weighted and distributed according to the state posterior \(p({x_k}|{z_{1:k}})\). If resampling is performed after each iteration of the SIS procedure, such algorithm is referred to as sampling importance resampling (SIR) algorithm. There are various resampling schemes, such as stratified sampling [26], residual sampling [27], systematic sampling [28], exquisite resampling [29], etc. A recent review of existing resampling algorithms could be found in [30].

One problem caused by the resampling step is the so-called sample impoverishment. It is found that when resampling is performed, all the particles will locate at the same point in the state space after a few iterations, implying that the diversity of particles is lost, which may lead to a severe deterioration in the capability of particles for representing the state posterior. Theoretically, the problem of sample impoverishment can be avoided if we are able to resample from a continuous distribution rather than a discrete one. Based on the above consideration, the regularized particle filter (RPF) has been proposed in [31] where the Kernel density is introduced to approximate the true posterior density with a continuous density function. In [32], the regularized auxiliary particle filter (RAPF) which combines the RPF and APF methods has been presented to diversify the particles.

The effect of particle impoverishment is especially significant in smoothing problems where the estimation is derived based on particle paths. In smoothing estimation, each particle denotes a complete realization of state evolution rather than the current state only, which implies that the elements of a particle, in particular those corresponding to earlier states, have to be resampled many times with the running of smoothing algorithm. As a result, most particles will share a common path, thus incapable of representing the smoothing distribution. To address this problem, we hope to obtain a different set of particles which are still distributed according to the smoothing distribution. As discussed in the previous section, this can be done by the MCMC method. A key step to implement the MCMC method is to construct a Markov chain with \(p({x_k}|{z_{1:k}})\) as its invariant distribution. In [33], a Gibbs sampler has been used to update the state of Markov chain. The Metropolis-Hasting (MH) sampler has been adopted in [34] to generate new particles. It is also shown in [34] that the support of the smoothing distribution could be improved through the MCMC procedure.

Another problem brought up by the resampling step is the increased computational complexity. This is due to the fact that resampling is the only step in the particle filtering algorithm that hinders a parallel implementation. Based on this observation, it is suggested in [35] that the resampling step should be abandoned when its disadvantages outweigh advantages. In [35], the Gaussian particle filtering (GPF) method has been proposed where the state posterior is approximated by a Gaussian distribution whose mean and covariance are propagated using sequential importance sampling. Since an average is calculated in each iteration over the entire set of particles, we do not need to worry about the problem of particle degeneracy any more. Hence there is no need to resample the particles, i.e., a fully parallel implementation becomes possible.

For a given estimation \({I_{k|k}}\left( {g({x_k})} \right) \) of \({g({x_k})}\) based on \({z_{1:k}}\), we hope its variance is as small as possible. Numerous results have been reported on variance reduction for particle filtering (see [33] and references therein). In [36], the methods of SIS, SIR and APF are compared from the perspective of variance reduction. It is discovered that the resampling step in the SIR procedure has led to an increase in variance from two aspects: First, the fact that resampling is performed on a discrete distribution has introduced dependence among samples, which further leads to a larger variance compared with the fully adapted APF algorithm; Second, the randomness of the resampling step itself has produced an extra variance term. To avoid the extra variance caused by resampling while coping with particle degeneracy, a hybrid algorithm is proposed which automatically switches between SIS and APF according to whether a serious decrease in the effective sample size is detected.

For state estimation of the jump Markov linear systems (JMLS), it is suggested that the salient model structure should be made good use of [33]. It is shown that given the current state of Markov chain, the state estimation problem for JMLS reduces to Kalman filtering, which allows for a closed-form solution. Taking advantage of this property, one can draw samples from a lower dimensional distribution, where the continuous states are marginalized out. A lower dimensional distribution means a reduced number of particles and thus a lower computational complexity. Further, it is revealed that sampling from a lower dimensional distribution will result in a smaller variance. This approach has borrowed the idea of Rao-Blackwellization [37] which is especially useful for nonlinear model with a linear substructure. The well-known variance decomposition formulaforms the theoretical basis of the Rao-Blackwellization method. From (11), it is not difficult to see that \({E\{\tau (U,V)|V\}}\), in which the random variable V is integrated out, has the same mean with \({\tau (U,V)}\), yet a reduced variance. It is therefore advantageous to integrate out any redundant random variable present in the estimation. Specifically, when state can be decomposed as \(x = {[x_1^T,x_2^T]^T}\), where \({{x_1}}\) is conditionally linear given \({{x_2}}\) and measurements z, i.e., \(p({x_1}|{x_2},z)\) can be calculated analytically by using the standard Kalman filter, we can decompose the posterior density in the following wayNext we only need to approximate \(p({x_2}|z)\) with a set of particles \(\{ \tilde{x}_2^i,i = 1,2,\ldots ,{N_s}\}\). For each particle \(\tilde{x}_2^i\), a Kalman filter is designed to calculate \(p({x_1}|\tilde{x}_2^i,z)\). Particle filters applying Rao-Blackwellization method are also referred to as marginalized particle filter in some literatures [38‐40].

Up to now, we have assumed, in our particle filter design, that the system dynamics and noise statistics are precisely known to us, which might not hold in practical applications. Various methods have been put forward to robustify the particle filtering algorithm for systems with unknown statistics. The box particle filtering, which combines sequential Monte Carlo method with interval analysis, has been introduced in [41‐43]. Unlike the standard particle filtering method where particles are points in the state space and likelihood functions are defined by a statistical model, the box particle filter uses multidimensional intervals in the state space as particles and a bounded error model to evaluate the likelihood functions. The key advantage of box particle filter is the reduced number of particles required for a specified accuracy in the presence of model uncertainty.

In [44], a cost-reference particle filtering (CRPF) method has been proposed, which can be seen as a generalization of the standard particle filter. The CRPF method takes advantage of a basic fact that the methodology of particle representation and propagation can be applied to any function of the state as long as it admits a recursive decomposition. In CRPF, a user-defined cost function, instead of the state posterior as is the case with standard particle filter, is defined and minimized following a procedure similar to that of standard particle filter. CRPF has been combined with APF in [45] for target tracking in binary sensor networks without probabilistic assumptions on the model. It is shown that the CRPF and APF have a similar form when the cost functions in CRPF are considered as the generalized weights. In [46], state estimation method which combines the CRPF with \({H_\infty }\) method has been proposed for conditionally linear systems with unknown noise statistics.

When only hard bounds of the noises are available for filter design, set-membership theory is a powerful tool for guaranteed estimation, i.e., to find the smallest region in the state space that is guaranteed to enclose the possible states. The set-membership theory and particle filtering method have been blended together in [47] where the significance of each particle is evaluated according to the feasible set given by set-membership theory. For a class of noises with unknown time-varying parameters, the marginalized adaptive particle filtering approach has been studied in [48]. The predictive density of the noise parameters is approximated under the principle of maximum entropy so that the uncertainty is not underestimated. Since the conditional density of noise parameters admits an analytical expression given current states, the marginalization technique is employed in the joint estimation of state and noise parameters.

At the end of this section, we would like to discuss briefly on the implementation issues of particle filtering. It is expected that the execution time of PFs could be minimized by exploiting the parallel structure inherent in the algorithms and allocating the computational tasks of central unit (CU) to some processing elements (PEs) which run in parallel. As mentioned previously, resampling is the main obstacle to the distributed implementation of PF algorithms since all the particles have to be involved in the resampling step, i.e., it bears no natural concurrency among iterations. Two algorithms for distributing the resampling procedure, namely resampling with proportional allocation (RPA) and resampling with nonproportional allocation (RNA), have been proposed in [49] where the sample space is divided into several groups and each PE is in charge of processing one such group. Since the numbers of particles are distributed unevenly among the PEs, a particle routing scheme is required to define the architecture for exchanging particles among PEs. It is a main focus of [49] to offer a particle routing scheme in which inter-PE communication is deterministic and independent of the CU. Another scheme for distributed implementation of PF algorithms has been proposed in [50], which is based on decomposition of the state space rather than the sample space. In the proposed approach, the original state space is decomposed into two mutually orthogonal subspaces. At each filtering period, samples are drawn sequentially from the two subspaces. Through state decomposition, the original filtering problem is transformed into two nested subproblems each of which corresponds to one of the derived subspaces. The main advantage of such decomposition lies in that part of the resampling procedure can be implemented in parallel, which facilitates more efficient calculation. Note that even though the method in [50] resembles that of marginalized particle filtering, it is applicable to any system with no requirement for a tractable linear substructure.

Time delay is quite common in networked systems where there is limited access to the transmission medium. Time delay in networked systems may result from the following factors:There are various delay models, including constant delay, random delay which is independent of previous transmission, and random delay with the probability distribution governed by a Markov chain. It is widely understood that time delay can cause performance degradation or even instability to the system. Existing research results have addressed the time delay in networked systems from two aspects: one is to determine an admissible upper bound of the time delay for which a prescribed performance can be guaranteed; the other is to design filtering algorithm for a given time delay to meet certain performance requirements.

As another network-induced phenomenon, packet dropout occurs frequently in inter-node communication. Typically the network has its own mechanism to keep the packet dropout at a low level so that the system performance will not be affected. Once the loss rate approach five per cent or higher (the threshold depends on the applications, for real-time ones, it should be lower), the user will begin to notice the presence of communication problems. There are various reasons for packet dropout, including:Typically, a Bernoulli distribution is used to describe the randomness of packet loss, where the probability of loss is assumed to be fixed. This assumption can effectively simplify the filter design, and the results obtained have been extended to the cases where packet loss follows a more general distribution. Similarly to time delay, packet dropout can also have adverse effects on state estimation, the severity of which depend on the loss ratio. Existing studies on packet dropout in networked systems have been focused on the determination of maximum admissible packet loss rate and filter design in the presence of packet dropout. Note that with the maximum admissible loss rate, we can discard some redundant data artificially to save the network bandwidth without violating the performance requirement.

Analog signals have infinitely variable amplitude and therefore have to be quantized before they are transmitted through the network. Quantization is involved in almost all digital signal processing. Examples of quantization processes include rounding and truncation. As a many-to-one mapping, quantization is inherently a lossy process. Considerable research efforts have been devoted to the selection of information that can be discarded without significant loss in performance. The module that realizes the quantization procedure is called quantizer. Existing types of quantizer include logarithmic quantizer and uniform quantizer.

Another common phenomenon in network communication is that the strength of received signals may vary over time, an effect also referred to as signal fading. It is closely related to multipath, a propagation phenomenon that results in signals reaching the receiving antenna by two or more paths. Multipath propagation can cause fading and phase shifting of the received signals. Movements of transmitter or receiver may also give rise to signal fading. Fading can occur in many forms. When all the frequency components transmitted are attenuated to the same degree, this type of fading is called flat fading. Otherwise we say there is a frequency-selective fading. Existing mathematical descriptions for the channel fading phenomena include analog erasure channel model, Rice fading channel model and Rayleigh channel model [54].

Time delay Early results on filter design for time delay systems can be found in [55‐57]. In [55], the standard Kalman–Bucy filter has been extended to include delayed measurements. The ordinary differential equations in Kalman filtering are replaced by partial differential equations together with boundary conditions which may not have an explicit solution. Note that this result is more closely related to the smoothing problem where the past state is estimated using current measurements. In [56], orthogonal projection has been employed to derive the optimal filter for discrete systems with multiple time delays. In [57], the same result as that in [56] has been obtained via maximum likelihood estimation for an augmented state. Despite the straightforwardness, this method is valid only when the random processes considered follow Gaussian distribution.

The \({H_\infty }\) filter for linear continuous systems with delayed measurements has been provided in [58]. Like \({H_\infty }\) filter design without delay, the filtering problem is transformed to seeking a bounded solution of a Riccati differential equation. In [59], a less conservative bounded real lemma (BRL) has been employed in the filter design for systems with known state delay to achieve a smaller overdesign. A robust \({H_\infty }\) filtering approach has been proposed in [60] for a type of uncertain discrete time-delay systems whose parameter matrices are assumed to belong to a convex bounded polytope. Similarly, the filtering method proposed in [61] also addresses parameter uncertainty and time delay, but in this case continuous-time systems are considered and the delay is assumed to be unknown (only the upper bound is available). A similar result has been obtained in [62] where an exponentially stable filter is designed for time-delay systems with norm-bounded parameter uncertainties. Quadratic matrix inequalities are adopted in the analysis and design of the filter. Note that robust filtering approaches in [60‐62] have a common presumption that the original system is stable, which may to some extent limit the application scope of the results obtained. As an extension to [59], a robust \({H_\infty }\) filter has been proposed in [63] to deal with time-varying delay and polytopic type uncertainties using a more efficient BRL. The fact that the BRL is applied to the resulting error system can remove the requirement on a stable system matrix.

Robust filtering for nonlinear time-delay systems have been investigated in [64‐66]. In [64], the nonlinearities are assumed to satisfy global Lipschitz conditions. A delay-dependent robust \({L_2}/{L_\infty }\) filter has been designed based on linear matrix inequalities (LMIs). In [65], full-order filter has been derived for a general class of nonlinear time-delay systems with guaranteed mean square boundedness of the error dynamics. A general class of nonlinear systems with randomly varying sensor delays have been considered in [66] where conditions for guaranteed \({H_\infty }\) performance are provided in terms of Hamilton–Jacobi–Isaacs (HJI) inequality.

Recently, extensive studies have been devoted to address the conservatism of \({H_\infty }\) filtering design due mainly to inequality scaling in the derivation. The conservatism of certain filtering method can be measured by the maximum admissible delay or the \({H_\infty }\) performance. In [67], the free-weighing matrix method has been adopted in the \({H_\infty }\) filter design for systems with time-varying interval delays to reduce conservatism of the existing results. In [68], a novel integral inequality has been used to establish the LMI conditions for the existence of \({H_\infty }\) filter without resorting to model transformation or bounding technique for cross terms both of which are sources of conservatism.

Recent results on Kalman filtering for time-delay systems can be found in [69‐72]. The novel idea in [69] and [70] is to reorganize the measurements from different channels as a delay free system. This reorganized innovation is combined with the orthogonal projection formula to derive the optimal filter. It is shown that for systems with m delays, the obtained solution consists of m standard Kalman filters with the same dimension as the original systems. To reduce the computational complexity of reorganization innovation, the equivalent delay free system has been obtained in [71] by directly solving stochastic equations. The optimal filter and error propagation formula are then derived through Itô differentials of the state expectation conditioned on observation processes. In [72], a new sub-optimal filter has been proposed in the minimum variance framework. In this method, only instantaneous terms are used, thereby avoiding the computation of distributed terms. Also, the filter derived in [72] can be applied to any bounded delay function including non-continuous delays.

Packet dropout Filtering problems for networked systems with packet dropout have also received considerable research attention. Many elegant results have been obtained regarding this direction. Due to space limitation, we will only introduce a small portion of them. A more comprehensive review can be found in the excellent survey paper [73]. Intuitively, the missing probability will not affect the boundedness of the error covariance until it reaches a certain critical value. This value is identified in [74] based on novel matrix decomposition techniques. The optimal \({H_2}\) filtering for networked system with multiple packet dropouts has been considered in [75] where stochastic packed loss is assumed to follow a Bernoulli distribution. The stochastic \({H_2}\) norm is defined, which generalizes the norm of systems with both deterministic and stochastic inputs. The filter which minimizes such a norm is derived based on the solution of a set of LMIs. The phenomenon of multiple packet dropouts with a more general form of missing probability has been treated in [76] where random measurement loss is allowed to follow any discrete distribution taking values over the interval [0,1] with known occurrence probability. The extended Kalman filter is derived through minimizing the upper bound for the filtering error covariance. For multi-rate sensor fusion with missing measurements, an unknown input observer has been proposed in [99] to minimize the mean square error.

It is known that packet loss can cause instability to the filtering system. Stability analysis of Kalman filter for networked systems with random packet losses has been provided in, to name just a few, [77‐81]. It is shown in [77] that there exists a critical value of observation arrival probability under which the expected error covariance is likely to grow unbounded. In [78], necessary and sufficient conditions have been obtained for stability analysis of networked systems with random packet losses characterized by a binary Markov chain. The Markovian packet loss has also been studied in [79] where the notion of stability in stopping times is introduced and its equivalence with the stability in sampling times is established, which simplifies the subsequent stability analysis. The necessary and sufficient conditions for mean square stability have been derived, respectively, for second order systems with different structures and high order systems with a certain structure. In [80], convergence of the error covariance has been analyzed for a rather general class of packet dropping models and alternative performance measures are introduced when the expectation of error covariance cannot be well defined. The asymptotic behavior of the random Riccati equations (RRE) which describe the evolution of estimation error covariance in Kalman filter has been studied in [81] where sufficient conditions for the existence and uniqueness of the invariant distribution for the RRE are derived.

Quantization State estimation for systems with quantized measurements has been studied in [82] where the quantizer and the estimator are designed jointly. A logarithmic quantizer is employed whose resulting quantization error can be regarded as a multiplicative noise. Choices of quantization density or number of quantization levels are discussed. Furthermore, a dynamic scaling parameter for the quantizer is introduced to ensure the convergence of estimation error in unstable systems. In [83], a quantized filtering scheme using decentralized Kalman filter has been proposed for linear discrete systems with multiple sensors. The innovation process of each local sensor, instead of local estimation, is quantized to avoid saturation of the quantizer. It is proved that stability of the filter can be achieved under sufficiently high bit rate even for unstable systems. The trade-off between quantization rate and state estimation error is analyzed. The problem of rate allocation among different sensors is also considered to enhance the asymptotic behavior of estimation error. The quantized gossip-based interactive Kalman filtering approach has been studied in [84] where it is proved that the error covariance sequence at a randomly selected sensor can converge weakly to a unique invariant measure even with information loss caused by quantization. In [85], a recursive filter is designed for power systems with quantized nonlinear measurements by minimizing the upper bound of the error covariance.

Signal fading Kalman filter for networked systems where local measurements are sent to the fusion center via fading wireless channels has been investigated in [86]. The expected error covariance of Kalman filter is proved to be bounded and converge to a steady state value. Exact recursive formulas are provided to calculate the upper bounds of error covariance which may serve as an alternative index to be optimized when the expression of error covariance is unavailable. Kalman filter with measurements transmitted over fading channels has been considered in [87] where it is assumed that the transmission power can be adjusted to alleviate the effects of fading channels. Sufficient conditions are obtained to ensure the boundedness of error covariance. These conditions are then used for power allocation to minimize the total power consumed by the network. In [88], the envelop-constrained \({H_\infty }\) filter has been presented for a class of time-varying discrete systems. The finite horizon case is considered and a novel envelope-constrained performance criterion is proposed to define transition performance of error dynamics. Borrowing idea from the set-membership filtering method, an ellipsoid description of estimation error has been utilized in [88] to transform the envelop constraints into a set of matrix inequalities solvable using standard software package.

Apparently, the network-induced phenomena mentioned above will coexist in a system. A large number of papers have been published on filter design for systems with multiple network-induced phenomena. For example, the simultaneous presence of time delay and packet loss has been addressed in [89‐91]; both quantization and packet dropout have been taken into consideration in [92]; a filtering scheme has been recently proposed in [93] which is robust against both channel fadings and gain variations. For more related results, the readers are referred to [94‐101].

Another research direction of state estimation theory for networked systems is distributed estimation methods which aim to maintain an accurate estimation of certain network states at each sensor node using measurements from all sensor nodes in the network. Besides estimation accuracy, communication overhead and computational complexity also pose constraints on the filtering scheme to be designed due to the limited bandwidth and power supply of the network. Early treatments to the distributed estimation problem can be found in [102‐104]. In [102], both measurements and local estimates are shared among neighboring nodes. It is shown that asymptotic agreement on the estimates can be achieved through infinitely frequent data exchange among sensors which form a communication ring. In [103], sufficient statistics are extracted from local measurements and transmitted to a fusion center where the centralized conditional distribution is exactly reconstructed. Sufficient conditions have been presented in [104] under which global sufficient statistics can be expressed as a function of the local ones.

For distributed estimation, consensus is an important concept and a lot of research efforts have been devoted to it. Generally, consensus refers to agreement among all the members of a group. In the specific case of state estimation for networked systems, we say the network reaches consensus if all the nodes hold an identical estimate of a certain quantity of interest. As a fully distributed framework, consensus based distributed estimation allows for cooperation over the network without the participation of a fusion center, thereby avoiding over reliance on a certain node. Various forms of consensus algorithms have been developed to establish the rule following which inter-node communications are implemented to reach an agreement among nodes. Average consensus and gossip consensus are the two major consensus strategies that have been investigated extensively in recent years. Readers are referred to [105] to obtain a full understanding of consensus algorithm for networked systems.

A fully distributed Kalman filter has been proposed in [106] for sparsely connected, large-scale systems. The global dynamic model is decomposed into low-dimensional subsystems for which local filters are designed. These subsystems overlap and the common states shared by several nodes are estimated using fusion algorithm. The centralized error covariance is derived through a distributed computation algorithm for matrix inversion, called distributed iterate-collapse inversion algorithm, which assimilates local error covariances with computation complexity independent of the system dimension. In the proposed method, each sensor node only needs to deal with a portion of the system state, which has significantly reduced the communication and computation requirements.

In [107], the authors consider a two-stage distributed Kalman filter which consists of a measurement update step and a fusion step using consensus algorithm. The interaction between the filter gain, the consensus matrix and the number of communications is analyzed in depth. It is proved that the common practice of minimizing the spectral radius of consensus matrix for fastest convergence is not necessarily the optimal strategy when only a small number of communications are available between two consecutive samples. It is also shown that the joint optimization of filter gain and consensus matrix is non convex and can be analytically characterized only in some special cases.

A similar two-stage distributed filter has been investigated in [108]. Sufficient conditions have been obtained to judge the distributed detectability of the networked system, i.e., the existence of filter gains which ensure an asymptotically stable error dynamic given a specific choice of consensus weights. A sub-optimal filtering scheme is then developed through minimizing an upper bound on a quadratic cost, and convergence analysis has been carried out for the time-invariant case.

One significant feature of the two-stage filtering schemes discussed above is that the consensus communication occurs at a much shorter time scale than the operation of local filters, i.e., it is assumed that there is sufficient time for the network to achieve consensus through intensive inter-node communications before the arrival of the next observation. This assumption, however, does not apply to the cases where there is a fast target dynamic and/or a high sampling rate. Besides, the high rate of consensus communication has blurred the line between distributed estimation and centralized estimation, as argued in [110]. To address this problem, Kar et al. proposes the gossip interactive Kalman filtering (GIKF) where consensus communication and observations take place at the same time scale. As a communication protocol inspired by social network phenomena, the gossip protocol has found broad applications, especially in networks with large scale or inconvenient structures. Readers can refer to [109] for a detailed overview of recent studies on gossip algorithms. In [110], the convergence of the GIKF scheme has been analyzed and the error covariance is shown to evolve according to a switched system of random Riccati operators where the switching is governed by a Markov chain determined by the network topology. Stochastic boundedness of estimation error and weak consensus of the error covariance have been established under weak assumptions about detectability and connectivity of the networked system. The method in [110] requires transmission of the error covariance, which may be burdensome for high-dimensional systems. An alternative approach has been given in [111] based on dynamic consensus on pseudo-observations (DCPO). The network tracking capacity (NTC) is used to characterize the influence of network topology and observation models on the stabilizability of DCPO error process. An explicit expression of the NTC is derived and asymptotic stability of DCPO error dynamics is established. The averaged pseudo-observations obtained are then used to construct local filters whose gains are designed to minimize the mean square error (MSE). It is shown that the method in [111] can achieve lower MSE while maintaining the major advantage of GIKF, i.e., the inter-node communication occurs no more frequently than sensor sampling.

Consensus-based distributed least square (LS) estimation problems have been studied in [112, 113]. The authors of [112] consider the total least square (TLS) estimation for overdetermined systems where both the input data matrix and the data vector are assumed to be noisy. A semi-definite relaxation technique is used to transform the nonconvex TLS problem into an equivalent convex semidefinite program (SDP). At this point, the dual based subgradient algorithm (DBSA) can be used to solve the distributed TLS problem without reliance on the computationally expensive SDP optimization procedure. In [113], underdetermined least square estimation problem has been considered. The requirement for consensus is expressed as constraints where an auxiliary variable is introduced to facilitate parallel processing, and the resulting constrained optimization problem is solved under the Augmented Lagrangian framework.

Distributed estimation schemes based on consensus algorithms aim to reach an agreement among all the nodes in the network. However, the ultimate purpose of state estimation problems is to achieve at each node an estimate that minimizes a predefined cost function, which does not necessarily require that all nodes provide the same result. Moreover, it is shown in [115] that the consensus network can become unstable even if all the local filters are stable, i.e., cooperation by means of consensus algorithms may lead to disastrous consequences. Motivated by such observations, the estimation schemes based on diffusion strategies have been proposed. As another class of fully distributed estimation methods, diffusion algorithms make several key improvements upon the consensus ones, among which the fundamental one is that agreement is no longer the goal. In the diffusion Kalman filtering proposed in [114], each node first adapt its local estimate using measurements from neighboring sensors, obtaining an intermediate estimate, which is refer to the incremental update step. Then a diffusion update step is performed by combining (through calculating a weighted average) the intermediate estimates received from neighboring nodes. Diffusion filter belongs to the single time-scale estimation scheme, i.e., the communication requirement of distributed filter is comparable to that of the gossip filter, but diffusion networks can achieve faster convergence rate and lower MSE than consensus networks. In addition, it is proved in [115] that the stability of local filters is sufficient to guarantee global stability of network under the diffusion framework, regardless of the choice of combination weights.

Generally speaking, there are two categories of methods for missing data problems: guaranteed cost ones and data imputation based ones. The methods introduced in Sect. 2 obviously belong to the former category. Next we will introduce some data imputation based approaches which are more commonly used in Monte Carlo methods. The basic idea of data imputation is to generate some data which obey the same distribution as the missing ones. These artificially generated data will then be regarded as real measurements in the subsequent processing. It is typically difficult to calculate the distribution of missing data conditioned on the observed ones. This being the case, one has to resort to sampling-based approaches such as the MCMC method introduced in Sect. 2 of this paper.

As a special form of the MCMC approach, Gibbs Sampler has attracted extensive attention among researchers since its introduction in [116]. Essentially, Gibbs sampler is an iterative algorithm to draw samples from an unknown distribution without direct calculation of the density. The pseudo code of Gibbs Sampler is given in Algorithm 3 where the case of three random variables, x, y and z, is considered and the marginal distribution p(x) is of interest to us. The readers can also refer to [117] for an explanation of the underlying principle of Gibbs sampler.

Kong et al. extended the Gibbs sampler to sequential imputation which does not require iterations, thus reducing the computational burden [18]. The sequential imputation uses samples and associated weights to approximate the unknown distribution in the presence of missing data, and thus can be seen as a combination of Gibbs sampler and sequential importance sampling. As new data arrive, a new sample is drawn and the augmented data set is updated to include this sample. The corresponding weight is then determined according to the quality of the augmented data set. Some related problems, such as effective sample size, the order of imputation, the behavior of weights, and sensitivity to the choice of prior distribution, are also analyzed in detail. An interesting finding, which is also illustrated by simulation results, is that the order of imputation can have a huge impact on the approximation accuracy. To be brief, a “good” data set cannot play its due role if it is processed after the “bad” ones because the trail distribution has already been corrupted by early imputations.

Expectation Maximization (EM) algorithm is another powerful tool to address missing data problems. An excellent introduction of it can be found in [118]. The EM algorithm consists of an expectation (E) step and a maximization (M) step. These two steps are executed alternately: in the E step, expected values of the missing data are computed, which will then be used in the M step; while in the M step, the maximum likelihood estimate of unknown parameters are calculated, to be used in the next E step. It is guaranteed that the likelihood function will not decrease after every such iteration. This implies that only local maximum can be reached, which may be seen as a deficiency of the EM algorithm. The basic EM procedure for estimating unknown parameter \(\theta \) in the presence of missing data \(z_{k}\) is illustrated in Fig. 1 below.

In [119], three EM algorithms have been adopted to treat the maximum a posteriori (MAP) state estimation for JMLS. Both the Markov chain and continuous states are unknown and to be determined. The first algorithm addresses MAP estimation of the Markov chain. The unknown continuous states are regarded as missing data and estimated with a fixed-interval Kalman smoother in the E step. Assuming the continuous states to be known, the MAP estimation of Markov chain is then obtained through dynamic programming in the M step. The second algorithm aims to estimate the continuous states with the unknown Markov chain viewed as missing data. A forward and backward recursion is applied in the E step to calculate the probability of Markov chain and a Kalman smoother is used in the M step to compute MAP estimation of the continuous states. The last algorithm deals with joint estimation of Markov chain and continuous states, which is realized through an alternate execution of fixed-interval Kalman smoother and dynamic programming. To overcome local convergence of the EM method, stochastic sampling based algorithms have been proposed in [120] for state estimation of JMLS. Three data augmentation (DA) algorithms (DA, stochastic annealing DA, and Metropolis-Hasting DA) are employed and an acceptable computational cost is achieved. As a special form of MCMC method, the DA algorithm can ensure convergence to the globally optimal solution. In [121], the EM algorithm is applied to estimate missing data in the Moderate Resolution Imaging Spectroradiometer (MODIS) time series for forest growth prediction.

Multiple imputation particle filter (MIPF) has been introduced in [122] where particle approximation is utilized to perform multiple imputation. This method resembles both the Gibbs sampler and EM algorithm. First, multiple imputations are drawn from a proposal distribution where the true states are replaced with particle representation which is calculated regardless of the missing observations. Next, for each imputation, a particle filter is performed to obtain an approximation of the state posterior. We then combine the approximations derived from different particle filters to give the final particle representation of the target density. Almost sure convergence of the MIPF method is established in a later work [123].

As mentioned in Sect. 2, transmission delay occurs frequently in a networked environment, which gives rise to the so-called out-of-sequence measurements (OOSMs). A great deal of research has been done to address this phenomenon, but mostly within the framework of Kalman filtering. The particle filtering algorithm in the presence of OOSMs has been studied in [124], where the basic idea is to rerun the particle filter to incorporate the delayed measurements. A major drawback of this method is that we have to store the particles and corresponding weights at each sampling period, which poses severe challenges to the storage capability of the processor, especially when the required number of particles is large. The proposed method in [124] also suffers from the problem particle degeneracy, but this can be mitigated via an MCMC step [125]. In [126, 127], a backward information filter has been adopted to retrodict particles corresponding to the delayed measurements. These particles are then used to recalculate the current weights. The implementation of backward information filter, however, involves intensive computation, which may be formidable in some practical applications.

The above-mentioned methods suffer from either excessive memory requirement or huge computational burden. A storage efficient particle filter has been proposed in [128] where only mean and covariance of the particle set need to be stored. The memory requirement of this method is dramatically decreased compared with that of [124], but the problem of particle degeneracy remains, especially when the OOSMs contain such a great amount of information that the original support is not enough to describe the filtering distribution. A selective procedure has been proposed in [129] to distinguish the OOSMs according to their utility. At every sampling period, a threshold for measurement selection is firstly calculated through a constrained optimization problem. The measurements whose utility exceeds the threshold are identified as informative and processed in the subsequent filtering step, while those with utility below the threshold are simply discarded. To reduce computational cost associated with solving the optimization problem, Gaussian approximation and linearization are employed for a rapid prediction of the mean square error (MSE) reduction brought by each delayed measurement. In addition, another threshold test is conducted to detect particle degeneracy. Once the effective sample size is found to drop dramatically, which implies that current support can no longer give an accurate description of the filtering distribution, the OOSMs are reprocessed through another filtering procedure which allows for simultaneous adjustment of the weights and locations of particles, thus avoiding particle degeneracy.

The exact Bayesian solution to the filtering problem for OOSMs has been derived in [130]. Different from the storage efficient particle filter [128] whose performance degrades when the target states do not follow a unimodal distribution, the exact Bayesian algorithm uses all the past particles and weights to achieve optimal performance. The cost of optimality, however, is a huge computational overhead, which limits the application scope of the exact Bayesian algorithm to low-dimensional problems or offline processing. For nonlinear model containing a linear substructure, two Rao-Blackwellized particle filtering algorithms have been presented in [131] to yield efficient execution and high accuracy, respectively.

Particle filtering problem for target tracking in the presence of signal propagation delay has been considered in [132]. Due to the interaction between target dynamics and propagation delay model, neither kinematic state of the target nor the propagation delay can be determined independently, which substantially complicates the problem. To tackle this difficulty, an augmented state vector is defined which includes both the time delay and the target state with a stochastic time stamp. The key of this method is to solve the unknown delay from an implicit equation. It is shown that iterative techniques can be used to obtain an approximate solution to the implicit equation as long as a fairly weak convergence condition is satisfied. The bootstrap particle filter is employed where iterations are incorporated in the time-update step to predict the time delay of current measurements. The resultant particles have different time stamps with each other, therefore a time synchronization is performed before the final estimate is derived.

Particle filtering algorithm with one-step delayed measurements has been studied in [133]. The standard particle filtering algorithm is modified to take the probable delay into account. When the latency probability is unknown to the designer, a maximum likelihood algorithm is proposed to identify it. The result has been extended to handle multiple-step randomly delayed measurements in [134], but only the case of known latency probability is considered.

For estimation problems with quantized measurements, the Cramer-Rao lower bound (CRLB) has been derived in [39], giving an indication on the information loss caused by signal quantization. Both Kalman filtering and particle filtering algorithms are developed to handle measurement quantization, and the superiority of particle filtering over Kalman filtering is demonstrated through numerical experiments. Measurement quantization will induce big error to the filter system when the values of observed data are large. The filtering problem with innovation quantization has been studied in [135]. A counterexample is constructed to show that the Kalman filtering may perform below expectation or even diverge in the presence of quantized innovation. The particle filter, instead, seems capable of approximating the optimal filter in the same situation.

It is revealed in [136] that the state conditional on quantized observations can be decomposed into the sum of two independent random variables, one of whom follows Gaussian distribution while the other is a truncated Gaussian random vector. The authors of [136] point out that we only need to propagate the truncated Gaussian variable, rather than the sum, since the truncated Gaussian variable has a probability density whose covariance is much smaller than that of the conditional state density. Taking advantage of the Gaussian properties, the authors design a Kalman-like particle filter (KLPF) where a group of Kalman filters are processing in parallel to obtain minimum mean square estimate of the state conditioned on perfect observations. One major advantage of KLPF is that the required number of particles is dramatically reduced compared with directly using particle filter as in [135].

Theoretically, the CPF can give the optimal state estimate if we ignore the error caused by particle representation of the continuous posterior distribution. The optimality, however, comes with a high cost, both in communication burden and computation complexity. The requirement on communication could be formidable in sensor networks where each node has limited power supply and thus limited communication capability. In [138, 139], a CPF method based on state partition and parallel EKF has been proposed for target tracking using collaborative acoustic sensors. The major computation task is efficiently done in the fusion center, thus freeing sensor nodes from local data processing. To obtain a proposal distribution which is closer to the state posterior, a bank of EKFs are used to process data from all activated sensors concurrently, and a weight sum of these EKF estimations is calculated and taken as the proposal distribution. An efficient way to store particles has been introduced in [140] where a compression step is taken before storing particle states. Simulation results suggest that this scheme can significantly reduce memory requirement with minimal performance loss.

In a sensor network where the communication capability is limited, data are always quantized at each sensor node before transmission. This, plus the imperfect nature of the communication channels, should be taken into account by the fusion center to gain better performance. A channel-aware particle filtering scheme is put forward in [141] to address the quantized measurements and fading channels simultaneously. The likelihood function, in which both data quantization and channel imperfection are considered, is calculated in three different scenarios. The posterior CRLBs for the proposed method are also derived. When there is a constraint on the total number of bits that can be transmitted, bit allocation becomes necessary. In [142], a dynamic bit allocation algorithm based on approximate dynamic programming has been presented. It is shown that the proposed algorithm can save much of the computational cost while achieving comparable accuracy with other existing methods. The amount of transmitted data can be significantly reduced if sensor nodes are able to distinguish informative measurements from uninformative ones. The data censoring performed at each sensor node does exactly this. The particle filtering with censored data has been studied in [143] where it is pointed out that even though uninformative measurements are not to be transmitted to the fusion center, the fact that they are uninformative also delivers some useful information for data processing. This information is exploited in the proposed filtering method through a full likelihood function, which contributes to an enhanced performance. Strictly speaking, the method in [143] does not belong to CPF since KF update is run at each node to obtain the variance of local innovation based on which the censoring threshold is identified. However, we introduce it here because, like other CPF methods, it requires transmission of raw measurements to eliminate the dependence between local data.

In the remainder of this section, we will focus on DPF for state estimation in agent networks. Hlinka et al. presented a detailed classification of the existing DPF methods (see [144]). A fundamental distinction between different DPF methods is whether a fusion center (FC) is present. In the FC-based DPF scheme, each agent processes its own measurements with a local PF and reports the obtained posterior to the FC according to a predefined communication protocol [145, 146]. This scheme is suitable for those applications where global knowledge is required only in the FC [147]. However, it suffers from two major drawbacks: (1) the filtering performance relies highly on the FC, which implies a poor robustness against the FC faults; (2) the communication path is highly dependent on the network topology, once the topology changes, which is very common in mobile networks, the total route table has to be re-established; (3) excessive communication burden is imposed on the nodes which are closer to the FC. To reduce long-distance communication, a two-tiered network structure has been proposed in [148] where some selected nodes, referred to as cluster heads (CHs), are responsible for processing raw measurements of the nearby sensors and sending the obtained local estimate to the FC for a further fusion. In this way, only the CHs are required to be capable of directly communicating with the FC.

DPF schemes without an FC is also referred to as decentralized particle filtering. We can further classify various decentralized particle filtering methods according to whether all the agents run the particle filter simultaneously or only a portion of them are in charge of data processing. We refer to the schemes where a portion of activated agents take the charge of global estimation as leader agent (LA)-based DPF (see, for example, [149‐151]), and term those with all agents performing particle filtering algorithm consensus-based DPF (see, for example, [156, 158, 164]). In the LA-based schemes, a sequence of adjacent nodes form a LA path along which the local estimation is accumulated. Typically, this LA path is constructed dynamically and adaptively, i.e., current LA is in charge of selecting the next LA among its neighbors based on the assessment of their informativeness. The selection scheme can affect the estimation accuracy and energy usage to a large degree [152]. Compared with the LA-based schemes, consensus-based DPFs can achieve enhanced scalability and robustness against changes in network topology or node failures. The price for these advantages is a heavier demand on communication and the likely delay due to consensus iterations. In view of the fact that a detailed introduction of both LA-based and consensus-based schemes has already been provided in [144], we will, in the following, investigate from a different perspective, i.e., we will focus on how different decentralized particle filtering methods make a reasonable trade-off between multiple performance index including accuracy, communication burden and computation complexity.

Ideally, we hope the performance of decentralized particle filtering methods can reach that of the centralized ones which is theoretically optimal. Decentralized particle filtering for blind equalization has been studied in [153] where each node evaluates the likelihood function of its local observations and then broadcasts it to the entire network. The local PF performed at each node thereby has access to the global likelihood function, and is guaranteed to converge to the optimal filter asymptotically. Synchronization is required to ensure that an identical set of particles are generated at different nodes. It is also shown that the filtering performance could be enhanced via the optimal importance function, which, however, asks for an extra amount of broadcast. A modified method has been proposed in [154] to reduce inter-node communication by employing parametric approximations of the remote likelihood function. This method achieves significant communication reduction with only moderate performance degradation. The communication requirement could be further cut down through a protocol where inter-node connection for message exchange is established randomly and each node transmits its local data to only one remote sensor at each time step [155].

The methods mentioned in the previous paragraph rely on broadcasting and are thus suitable only for fully connected networks. Two consensus-based methods have been provided in [155] where inter-node communications are limited within adjacent nodes. Both methods involve evaluating the global likelihood function. The first one uses average consensus to calculate the global likelihood at each node and a quantization step to eliminate the discrepancy between particle sets at different nodes. To avoid performance degradation caused by quantization, the second method adopts a modified minimum consensus algorithm, which borrows the idea of flooding scheme, to obtain an ordered list of local likelihood functions shared by all the sensor nodes. The merit of this method lies in that it, unlike the first one, does not require an infinite number of consensus iterations for a guaranteed performance.

The consensus-based schemes reduce inter-node communication at the cost of heavier local computation. This is desirable in most applications since communication between nodes typically consumes more energy than local computation. Similar ideas have been discussed in [156] where a likelihood consensus (LC)-based DPF method is developed. The basic idea of LC-based DPF is that the underlying sufficient statistics, rather than raw measurements, should be exploited in the DPF to avoid redundant communication. A key step in the proposed algorithm is to obtain a parametric representation of the local likelihood functions. Specifically, local likelihood functions are approximated by the sum of a series of basis functions multiplied by respective coefficients. In this way, the useful information contained in local measurements is compressed into certain coefficients. Since the basis functions are known to all the nodes, the knowledge of corresponding coefficients is sufficient to reconstruct the likelihood functions. With this approximation, a consensus procedure is carried out to calculate the sum of the exponential term of global likelihood function which can be expressed as the product of all the local likelihood functions. The pseudo code of LC-based DPF is presented in Algorithm 4. An attractive feature of the LC-based DPF method is that its communication cost does not depend on measurement dimensions, which makes it particularly suitable in applications which involve high-dimensional measurements. Also note that synchronization is no longer a requirement in the LC-based method since the likelihood functions exchanged between nodes are in a parametric form rather than represented by discrete values.

One shortcoming of the method proposed in [156] is that current measurements have not been incorporated in the proposal density. This problem has been treated in [157] via proposal adaptation which is implemented in a distributed manner. Gaussian distribution is used to approximate both local and global posteriors, and EKF/UKF is employed to incorporate local measurements in the local posterior. Fusing local information to obtain a global proposal density now amounts to running consensus algorithms to calculate two sums, for global mean and global covariance, respectively, over all sensor nodes. The benefit gained from the adapted proposal density is twofold: on the one hand, particles are used with higher efficiency since they are located in a region of higher likelihood; on the other hand, the least square approximation in the likelihood consensus will also have an improved accuracy since the local likelihood functions are approximated in a smaller region.

Another efficient way to implement proposal adaptation has been proposed in [158] where a set-membership constrained particle filtering scheme is developed. It is argued that the importance sampling step can be implemented in a distributed fashion if particles are drawn from the prior distribution. The prior distribution, however, is a non-adapted one which has not taken current measurements into account. This implies that we have to use a large number of particles to ensure that the posterior density is well characterized by these particles because it is possible that the likelihood function is a very peaked one compared with the prior PDF. A large number of particles will in turn reduce the efficiency of DPF algorithm because the computational complexity is positively correlated with the number of particles in a consensus-based scheme. To overcome this dilemma, we seek to develop a proposal adaptation scheme with affordable distributed implementation. The set-membership based adaptation proposed in [158] does exactly this. In this method, each sensor first determines a local set which approximates the local posterior density. Local sets at different nodes are then combined using consensus algorithms to construct the global set which will serve as the global proposal density in the subsequent importance sampling. The consensus algorithms are guaranteed to converge in finite iterations, which further reduces the overall communication cost. The pseudo code of set-membership constrained DPF approach is presented in Algorithm 5.

In [159], a Gaussian mixture model (GMM) has been employed to represent the posterior PDF to circumvent the transmission of a huge number of particles. In this method, each node obtains the global statistics through an average consensus which diffuses local statistics over the network. Based on the global statistics, an EM algorithm is performed to estimate the global GMM (see also [160]). The transmission of GMM representations, however, can be inefficient for high-dimensional systems since the amount of transmitted data grows with the volume of covariance matrix which is proportional to the square of state dimension. To achieve better scalability, a Markov chain distributed particle filter (MCDPF) has been proposed in [161, 162] based on random walks on the graph. In the MCDPF method, particles traverse the network along a random path and update the associated weights according to the local measurements at each node they pass by. The communication overhead of MCDPF approach depends linearly on the state dimension, which is particularly suitable for high-dimensional systems. Note that although data exchange is limited within neighboring nodes, the MCDPF does not belong to the consensus-based approaches since no consensus iterations are required. However, convergence to the optimal filter can only be established when the number of particles and the length of the random path both goes to infinity. It is also pointed out in [162] that, for low dimensional systems, the MCDPF algorithm is inefficient and GMM representation may be a better choice. Therefore, one needs to select the most suitable scheme on a case-by-case basis.

Gaussian mixture model has also been employed in [163] to develop a soft-data-constrained DPF. In this method, the global GMM is calculated from local ones using the consensus propagation algorithm. Instead of representing the global posterior from which the new sets of particles are drawn, the global GMM is used to pose soft-data-constraints according to which the local particles are reweighed. The resultant local particles at each node represent a local posterior closer to the global one, which implies an enhanced robust against noise and failures.

Up to now, the consensus-based DPF methods mentioned have a basic requirement that consensus be achieved before the arrival of next measurement. In a network with intermittent communication connectivity, however, convergence of the consensus algorithm between every two consecutive measurements cannot be guaranteed, which may lead to severe performance degradation. A consensus/fusion based DPF method has been presented in [164] to address this problem. In this method, an extra filter, referred to as the fusion filter, is employed at each node in addition to the local particle filter to diffuse local estimate and reach consensus across the entire network. Note that the fusion filter is allowed to run at a different rate from the local one, thereby removing the requirement on convergence between successive measurements. Another function of the fusion filter is to compensate for the common information contained in different local estimates which is a general problem for DPF schemes where local estimates rather than raw measurements are diffused [165].

A constrained sufficient statistic-(CSS) based DPF method has been provided in [166]. Similarly to the LC-based approach [156], this method also seeks to fuse local sufficient statistics (LSS) to the global one. However, no approximation of the global sufficient statistics (GSS) is involved, which implies an enhanced accuracy. Communication overhead per iteration is no longer dependent on the state dimensions (as is the case with [159, 167]) or the number of particles (as is the case with [158]). It is, instead, proportional to the number of GSS parameters which is much lower compared with either scenario mentioned above. To adapt the proposed method to error prone networks with intermittent connectivity, the authors of [166] further combine the CSS based DPF with distributed unscented particle filtering (DUPF) to achieve a guaranteed performance with fewer number of iterations per consensus run.