A marginalised particle filter with variational inference for non-linear state-space models with Gaussian mixture noise

This work proposes a marginalised particle filter with variational inference for non ‐ linear state ‐ space models (SSMs) with Gaussian mixture noise. A latent variable indicating the component of the Gaussian mixture considered at each time instant is introduced to specify the measurement mode of the SSM. The resulting joint posterior distribution of the state vector, the mode variable and the parameters of the Gaussian mixture noise is marginalised with respect to the noise variables. The marginalised posterior distribution of the state and mode is then approximated by using an appropriate marginalised particle filter. The noise parameters conditionally on each particle system of the state and mode variable are finally updated by using variational Bayesian inference. A simulation study is conducted to compare the proposed method with state ‐ of ‐ the ‐ art approaches in the context of positioning in urban canyons using global navigation satellite systems.


| INTRODUCTION
Non-linear state-space models (SSMs), composed of a nonlinear system and measurement equations, are applied to a wide variety of practical applications including global navigation satellite system (GNSS) positioning [1], radar target tracking [2] and communication systems [3].A central problem when using these models is recursively inferring the state vector based on a sequence of measurements.In many practical applications, the measurement noise has non-Gaussian statistics resulting from measurement outliers, for instance when the GNSS receiver is affected by multipath (MP) signals in urban canyons [4] or when there are irregular electromagnetic wave reflections from the target surface in radar target tracking [5].
The probability density function (pdf) of a non-Gaussian noise can be well-approximated by using a finite sum of Gaussian pdfs according to the density approximation theorem [6].As a consequence, different kinds of filters with noises based on Gaussian mixture models (GMMs) have been investigated in the literature for solving the state estimation problem in the presence of non-Gaussian measurement noise.One attempt to solve this problem is based on the multiple model approach, such as the Gaussian sum filter [7,8] where each component of the GMM corresponds to a possible noise distribution and the posterior estimates of the state vector can be obtained by using a bank of Kalman filters.Mixture reduction strategies such as the interactive multiple models (IMM) [9][10][11] have also been investigated to try to prevent the number of components of the joint posterior from growing exponentially over time.When the parameters of the GMM are unknown a priori, the expectation-maximisation (EM) algorithm can be embedded into the IMM for simultaneously estimating the unknown state and noise parameters [12].In order to adaptively determine the number of GMM components, variational Bayesian (VB) EM approaches have been studied in Ref [13].These approaches estimate the state vector in the VB-expectation step and then update the posterior distributions of the GMM in the VB-maximisation step.However, to our knowledge, they have only been considered in a batch manner.A flexible Bayesian non-parametric model embedded into the Rao-Blackwellized particle filter has also been considered in Ref [14], modelling the measurement noise as an infinite mixture of Gaussian vectors.However, the price to pay with this kind of approach is its high-computational complexity, which may limit its use in some practical applications.
This work studies a new marginalised particle filter (MPF) for non-linear SSMs with observation noise distributed according to a mixture of Gaussian vectors.The MPF studied in Ref [15] for mixed linear/non-linear systems is introduced for state and measurement equations depending linearly on a subset of the state vector and non-linearly on the other state variables.The idea of this technique is to marginalise the distribution of interest with respect to the state variables appearing linearly in the state and measurement equations, allowing the linear components to be processed using analytical methods (such as the Kalman filter) and to handle the non-linear components by sequential Monte Carlo (SMC) techniques [16].The MPF has been used successfully in many applications including navigation using an inertial navigation system and other aided positioning and multiple target tracking [17,18].For instance, an MPF has been proposed in Ref [19,20] to estimate jointly the number of targets and their states using a sequential algorithm.An overview of the sequential Monte Carlo methods for tracking different objects was investigated in Ref [21].In addition, the MPF has been applied to track-before-detect for a target tracking problem in Ref [22,23].The MPF was also combined with a variational Bayes approximation in Ref [24][25][26] to estimate the state and unknown measurement noise parameters in non-linear SSMs.For example, the means and covariances of the measurement noise were assumed to be slowly time-varying in Ref [24].This assumption allowed the derivation of a marginalised adaptive particle filter to infer the unknown noise parameters jointly with the state vector by using maximum entropy with a Kullback-Leibler (KL) divergence constraint.Finally, it is interesting to mention the robust particle filter proposed in Ref [25,26] to address cases where the additive measurement noise has a Student-t distribution.
This work introduces a new strategy for state estimation in non-linear SSMs inspired by the MPF.The main innovations with respect to the existing approaches are summarised below: � We introduce a latent variable indicating the component of the GMM used for the measurement noise at each time instant and specify the measurement mode of the system.� The joint posterior distribution of the state vector, the latent variable and the GMM parameters is derived and its parameters are estimated using an MPF with variational inference.
More precisely, the proposed approach is decomposed into two steps: (a) The joint posterior distribution of the state vector, the latent variable and the GMM parameters is marginalised with respect to the noise variables.The resulting marginalised posterior distribution is approximated by using an appropriate MPF; (b) the noise parameters conditionally on each particle generated in (a) are updated by using VB inference.
The work is organised as follows: The problem of state estimation for a non-linear SSM with a GMM measurement noise is presented in Section 2. Section 3 studies a new MPF with variational inference for jointly estimating the state vector, the mode variable and the GMM parameters of this kind of non-linear SSM.The performance of the proposed approach is evaluated in Section 4 and compared with the state-of-the-art approaches in the context of positioning in urban canyons using GNSS.Conclusions are finally drawn in Section 5.

| PROBLEM FORMULATION
In this work, we consider the following non-linear discretetime SSM related to a hidden state vector x t ∈ R n x and the measurement vector ð Þ is a known linear or non-linear transition function and ω t is the process noise distributed according to a zero-mean Gaussian distribution of covariance matrix Q.The non-Gaussian measurement noise is approximated using a finite GMM with density where K denotes the number of Gaussian components in the mixture, π k is the mixing coefficient for the kth component with denotes a Gaussian distribution with mean vector μ k and precision (inverse covariance) matrix Λ k for the kth component, Accordingly, the state and measurement equation in ( 1) and ( 2) can be defined by the following conditional pdfs: Since each component of y t can be considered as a measurement mode with a specified distribution, the measurement switches among different modes corresponding to the different GMM components.When working with mixture distributions, it is common to introduce a latent variable r k indicating the mode of y t at the tth time instant with a given probability u k;t defined as [27] where k ¼ 1; …; K, P K k¼1 u k;t ¼ 1 and u k;t > 0. On using these notations, the vector u t ¼ u 1;t ; …; u K;t À � T contains the probabilities that a measurement vector at the tth time instant belongs to the different GMM components, which are timevarying quantities.Moreover, the vector of mixing coefficients π ¼ π 1 ; …; π K ð Þ T contains the probabilities that the measurements in y 1:t ¼ y 1 ; …; y t f g are associated with the different GMM components, which are assumed to be constant quantities in this work.
When considering Bayesian estimators, assigning conjugate priors to π, μ and Λ simplifies the analysis.In this work, we have assigned a conjugate Dirichlet distribution Dir ⋅ ð Þ to π and a normal Wishart distribution N W ⋅ ð Þ to fμ; Λg [28] leading to where n y is the dimension of the measurement vector).Accordingly, the joint prior distribution for the parameters of the noise GMM is defined as follows: When the measurement vector y t switches between different modes, the latent variable r t is time-varying.In addition, the parameters of the GMM used for the measurement noise are generally difficult to specify a priori.Thus, the aim of this work is to infer the states x t jointly with the unknown measurement mode variables and the GMM noise parameters, leading to the joint posterior distribution of all unknown variables given the measurement sequence y 1:t , denoted as p x t ; r t ; θjy 1:t ð Þ with θ ¼ π; μ; Λ f g.This posterior distribution is studied in the next section.

| A MARGINALISED PARTICLE FILTER FOR NON-LINEAR SSMs WITH A GMM FOR THE MEASUREMENT NOISE
According to the MPF concept [15], the joint posterior distribution of the state and unknown parameters p x t ; r t ; θjy 1:t ð Þ can be factorised according to the following chain rule of probability: where the three pdfs on the right-hand side of Equation ( 10) can be calculated recursively.More precisely, we propose to approximate the posterior pdf p x t jy 1:t ð Þ by using an empirical density following the principle of particle filters.The pdf of the mode variables p r t jx t ; y 1:t ð Þ can then be computed conditionally on the state x t .Finally, the pdf of the noise parameters is shown to be a Gaussian mixture p μ; ð Λjx t ; r t ; y 1:t Þ conditionally on the state x t and the mode variable r t , which can be obtained using VB inference.The different steps required to compute these quantities are described in the next sections.

| Updating x t ; r t ð Þ samples based on the marginalised particle filter
As explained before, the joint posterior pdf of ðx t ; r t Þ can be factorised as follows: where the mode variable r t has been marginalised out in the second term of the right hand side.This work proposes to approximate p x t jy 1:t ð Þ by using a particle filter, that is, where N is the number of particles, δ ⋅ ð Þ is the Dirac delta function, x i t is the ith particle and ω i t is the corresponding weight at the tth time instant.According to Equation ( 6), p r t jx t ; y 1:t ð Þ can be obtained by computing the mode probabilities conditionally on the state samples where k ¼ 1; …; K and i ¼ 1; …; N. As a consequence, the joint pdf p x t ; r t jy 1:t ð Þ is approximated by using a set of weighted particles, leading to ω i t ; , whose update is described in the rest of this section.
According to the theory of particle filters [29], the weight ω i t can be updated as follows: is an importance distribution, which is chosen as the bootstrap proposal [16] in this work, that is, CHENG ET AL. -3 Accordingly, Equation ( 14) can be rewritten as follows: where p y t jx i t ; y 1:t−1 À � can be computed after the following marginalizations: where b u i k;tjt−1 denotes the predicted probability of the kth measurement mode for the ith particle at the tth time instant.It can be assumed that the mode sequence has the Markov property.Thus, the joint pdf of the mode sequence can be defined as follows [12,30]: where Π lk is the transition probability from the lth to the kth measurement mode.By marginalising the above expression over r t−1 , b u i k;tjt−1 can be expressed as follows: t−1 denotes the estimated probability of the lth measurement mode for the ith particle at the ðt − 1Þth time instant.After replacing Equation (18) into Equation (17), the following result is obtained: The conditional pdf of the measurement associated with the kth GMM component can be determined from Equation (5), leading to Since the prior distribution for μ k ; Λ k f g follows a normal Wishart distribution as defined in Equation ( 8), the predictive distribution p μ i k ; Λ i k jy 1:t−1 À � can be defined as follows: Since integration over the product of a normal Wishart prior and a Gaussian distribution leads to a Student-t distribution St ⋅ ð Þ [31], inserting Equations ( 23) and ( 24) into Equation ( 22) leads to � is a Student-t distribution defined as follows: 4 - with Þ, the probability of the kth measurement mode for the ith particle at the tth time instant can be obtained as follows: As a consequence, the maximum a posteriori (MAP) estimator of the indicator r t for the ith particle at the tth time is 3.2 | Calculating p θjx i t ; r i t ; y 1:t

À � using VB inference
According to the mean-field theory in VB inference [32], the joint pdf of the noise parameters q θ ð Þ can be factorised into single-variable factors, that is, According to the factorised approximation, the logarithm of the marginal likelihood ln p y t jx i t ; r i t ; θ; y 1:t−1 À � can be expressed as follows [33]: where L is a variational objective function used in VB inference, KL qkp ð Þ is the Kullback-Leibler (KL) divergence between the true posterior and its approximation.Accordingly, the joint pdf p y t ; θjx i t ; r i t ; y 1:t−1 À � can be expressed, according to Equations ( 5), ( 7) and ( 8), as follows: Considering that the KL divergence is non-negative, minimisation the KL divergence can be achieved by maximising the lower bound L, which results in the computation of expectations with respect to where E θ t nX denotes the expectation with respect to the variational distributions of all variables in θ, except those contained in X .Finally, the variational distributions can be approximated as follows: where C 1 is a constant summarising the terms independent of π and C 2 contains the terms independent of μ and Λ.
The hyperparameters α i k , m i k , β i k , W i k , ν i k for the variational distributions of the GMM noise parameters conditionally on the ith weighted particle ω i t ; x i t ; u i t � � can be finally updated at the tth time instant as follows: where k ¼ 1; …; K and i ¼ 1; …; N. According to the estimate of the indicating variable b r i t in Equation ( 31), the indicator appearing in the above equations is defined as follows: According to Equations (40)(41)(42)(43)(44), the estimation accuracy for the hyperparameters associated with the GMM noise parameters depends on the estimated indicator r t (i.e. the identification accuracy of the measurement mode), while an accurate estimation for the noise parameters facilitates the identification of the measurement mode according to Equations ( 30) and (31).
Remark 1: As for particle filters, resampling is conducted to rejuvenate the particles and reduce the effects of degeneracy.
Resampling does not have to be performed at every run of the proposed approach.In this work, the resampling procedure is implemented only when the estimated effective sample size b is below the user-defined threshold N thres (for more details, see Ref [16,34,44]).

Algorithm 1 Proposed marginalised particle filter for non-linear SSMs with Gaussian mixture noise
Inputs: Outputs: 1: for i ¼ 1; …; N do 2: Generate x i t and compute b u i tjt−1 according to Equations ( 15) and ( 21), respectively.

3: Compute
according to Equation ( 25).4: Compute the weight ω i t for the ith particle x i t according to Equations ( 16) and ( 27).
Finally, the GMM noise parameters at the tth time instant can be obtained as follows: where k ¼ 1; …; K.The proposed MPF for non-linear SSMs with a GMM measurement noise is summarised in Algorithm 1.

| EXPERIMENTS
This section validates the proposed MPF for non-linear SSMs with a GMM measurement noise in the context of GNSS-based positioning in urban canyons.The pseudorange (PR) measurement noise in urban canyons is considered as a non-Gaussian stochastic process due to the fact that the tracking loop within the GNSS receiver is impacted by interferences, for example, due to multipath (MP) signals that severely impair the GNSS-based positioning accuracy [36].One important remaining challenge for the application of GNSS in urban environments is reduction of the impact of MP signals on positioning methods.Generally, reflected MP signals affecting the received direct GNSS signal can be divided into two kinds of signals: (a) MP interferences defined as the sum of the direct signal and the delayed reflections handled by the GNSS receiver; (b) non-line-of-sight (NLOS) signals resulting from a unique reflected signal received and tracked by the GNSS receiver [37].In addition, an important property of MP signals is that they not only depend on the relative position of the receiver (which is generally moving) and GNSS satellites but also on the environment where the receiver is located, especially in urban canyons [38].Accordingly, these two reception situations can occur both separately or jointly inside the tracking loop of the GNSS receiver, leading to a PR measurement model that switches between different modes [39].

| Simulation scenario
Since the GNSS positioning solution depends on the dynamic level of the vehicle, a second-order model (i.e. a constant velocity model) is used to describe the dynamics of the vehicle in the earth-centred earth-fixed (ECEF) frame.Moreover, the GNSS receiver clock offset and its drift are taken into account.Therefore, the state vector considered in this simulation is defined as follows [40]:  where t ¼ 1; …; ∞ is the tth sampling time instant, p t ¼ ðx t ; y t ; z t Þ T and v t ¼ ðv x;t ; v y;t ; v z;t Þ T are the vehicle position and velocity in the ECEF frame (Cartesian coordinates), and b t and d t are the GNSS receiver clock offset and drift.The velocity can be reasonably modelled as a random walk, for example, x ̈¼ e x , where e x is a zero-mean Gaussian noise of variance σ 2 a .For short-term applications in which the periodical clock resets of the GNSS receiver are not taken into account, the GNSS receiver clock offset b t and its drift d t can also be modelled as random walks, that is, b ̇t ¼ d t þ e b and d ̇t ¼ e d , where e b and e d are zero-mean Gaussian white noises of variances σ 2 b and σ 2 d .Based on the above assumptions, the discrete-time state model that describes the propagation of the vehicle state x t can be formulated as follows: where e t ¼ e x ; e y ; e z ; e b ; e d À � T is the zero-mean Gaussian white noise vector of covariance matrix Q, that is, Expressions of the matrices F tjt−1 and Q can be found in Ref [41].
Considering that the atmospheric propagation errors can be compensated within the GNSS receivers, the PR measurement model of the sth in-view satellite at the tth time is defined as (referred to as a nominal value).For different reception situations, in the presence of MP signals, two possible models are considered according to the availability of the direct signal, that is, variance jumps affecting the zero-mean Gaussian white noise in the context of MP interferences and mean value jumps for NLOS signals [42,43].Thus, the measurement noise in the presence of MP signals has a Gaussian distribution with nonzero μ s;2 and variance σ 2 s;2 .Accordingly, the PR measurement noise v s;t can be modelled as a GMM with two components, that is, The state space model defined in Equation (50) has been simulated with the following parameters: process noise standard deviation σ a ¼ 0:1 m=s 2 , clock offset deviation σ b ¼ 3c � 10 −10 m and drift standard deviation σ d ¼ 2πc � 10 −10 m=s, where c ¼ 3 � 10 8 m=s denotes the velocity of light.The simulation is conducted using N s ¼ 4 in-view satellites.The noise-free GNSS PR measurements have been computed based on an almanac file including all useful satellite orbit data in the simulations.The standard deviation of the measurement noise in the absence of MP signals is set to σ s;1 ¼ 10 m, where s ∈ 1; 2; 3; 4 f g.In this simulation, the second satellite PR measurement (s ¼ 2) is affected by MP interferences, resulting in μ 2;2 ¼ 0 and σ 2;2 ¼ 30 m, while the fourth satellite PR measurement (s ¼ 4) is simultaneously affected by the mean value and variance changes, resulting in μ 4;2 ¼ 10 m and σ 4;2 ¼ 20 m.Accordingly, the noise GMM associated with all in-view satellite measurements is defined as where μ 1 ¼ diag 0; 0; 0; 0 ½ �, Σ 1 ¼ diag 10 2 ; 10 2 ; 10 2 ; 10 2 ½ � and μ 2 ¼ diag 0; 0; 0; 10 ½ �, Σ 2 ¼ diag 10 2 ; ½ 30 2 ; 10 2 ; 20 2 �, where diag ρ 1 ; …; ρ N s � � denotes a diagonal matrix with diagonal elements ρ 1 ; …; ρ N s .In addition, the mixture coefficients are set to π 1 ¼ 0:7 and π 2 ¼ 0:3, respectively.

| Validation of the proposed algorithm
In order to validate the capability of the proposed approach for adaptively determining the number of components, the number of GMM components is set to K ¼ 3. The transition probability matrix Π used for computing the predicted mode probabilities is defined as [39] Π ¼ 0:98 0:01 0:01 0:01 0:98 0:01 0:01 0:01 0:98 0 @ 1 A : Accordingly, the initial values for each Gaussian component are set to π k;0 ¼ 1=K, α k;0 ¼ 1, m k;0 ¼ 0; 0; 0 ð Þ T , β k;0 ¼ 1, W k;0 ¼ diag 0:1 2 ; 0:1 2 ; 0:1 2  ½ � and ν k;0 ¼ N s with k ∈ 1; 2; 3 f g.The simulation time and the number of particles are set to T ¼ 600s and N ¼ 2000. 1 The different algorithms are run for N m ¼ 100 Monte Carlo (MC) runs.The MAP estimates of r t and the root mean square errors (RMSEs) of the states for the different MC runs are defined as follows: and RMSE ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi where b u k;t m ð Þ and b x t m ð Þ are the mth estimates (m ¼ 1; …; N m ) of the measurement mode probability and the state at the tth time instant.All algorithms have been coded 1 The number of particles has been chosen to meet a compromise between the estimation accuracy of the proposed approach and its computational load.A larger number of particles might be considered for other applications.CHENG ET AL.
-9 using MATLAB and run on a laptop with Intel i-5 and 8 GB RAM.
Figure 1 displays the error rates ϵ 2 for the MAP estimates of r t for the 100 MC simulations.The corresponding averaged value of the error rates is 4:47%, demonstrating that the measurement mode can be well-identified by using the proposed MPF.In addition, the means of the mixing coefficients computed using 100 MC simulations are depicted in Figure 2. It is clear that the means of b π 1;t m ð Þ and b π 2;t m ð Þ gradually converge to the true values (i.e.0.7 and 0.3), whereas the mean of b π 3;t m ð Þ approaches zero when time increases (m ¼ 1; …; N m ).These results show that the number of mixture components and the values of the mixing coefficients can be adaptively determined by the proposed approach.
The RMSEs of the mean and standard deviation of each Gaussian component (i.e.b μ s;1 , b σ s;1 , and b μ s;2 , b σ s;2 for s ¼ 1; 2; 3; 4 f g) obtained with the proposed approach are depicted in Figures 3 and 4. The RMSEs associated with the first (s ¼ 1) and third (s ¼ 3) PR measurements are obviously smaller than those associated with the second (s ¼ 2) and fourth (s ¼ 4) PR measurements.This is due to the fact that the first and third PR measurements were not affected by MP signals, contrary to the other two PR measurements.Moreover, the estimation results obtained for the second and fourth PR measurements show that the false identification of the measurement mode impairs the estimation accuracy of the GMM parameters, especially for the fourth PR measurement, where the noise mean and variance change simultaneously, due to the presence of MP.

| Comparison with the state-of-the art approaches
The position and velocity estimation accuracies obtained with the proposed method are compared to those obtained using the marginalised adaptive particle filter (MAPF) of Ref [24] and the robust particle filter (PF) of Ref [25].Figures 5 and 6 display the RMSEs of position and velocity estimations obtained with the different methods.As shown in Figure 5, since the estimates for the parameters of the noise GMM are inaccurate at the beginning of the simulation, the corresponding position estimation accuracy is reduced for the proposed approach when compared to those obtained with the MAPF and the robust PF.However, it is clear that the proposed approach provides a better positioning estimation accuracy when time increases.Figure 6 shows that the velocity estimation obtained with the proposed method is very competitive with respect to the state-of-the-art approaches.Overall, these results show the interest of using the proposed approach for mitigating the impact of MP signals on GNSSbased positioning in urban canyons.
The computational complexities of the MAPF, the robust PR and the proposed approach are O NT ð Þ, O κ max NT ð Þ, 3 and O KNT ð Þ.Table 1 shows the execution time of a single MC run for the three approaches with different numbers of particles.One can conclude that the increased estimation accuracy is obtained at the price of a moderate computational cost, which will be acceptable in many practical applications.

| CONCLUSION
This work studied a new marginalised particle filter with variational inference for non-linear SSMs with a Gaussian mixture model (GMM) for the observation noise.A latent variable was introduced for indicating the measurement mode of the SSM, corresponding to a specific component of the GMM.The joint posterior distribution of the state, the latent variable and the parameters of the GMM was then derived and sampled using a new marginalised particle filter combined with variational Bayesian inference.The samples generated sequentially by this particle filter were finally used to estimate the unknown parameters of the SSM.A simulation study was conducted for a realistic GNSS-based positioning problem in urban canyons.The proposed approach was compared to the state-of-the-art approaches, more precisely to the marginalised adaptive particle filter and the robust particle filter, providing accurate estimates of the state and noise parameters at the price of a moderate computational cost.In particular, the number of mixture components and the GMM parameters (including the mixing coefficients and the parameters of each Gaussian distribution) were adaptively estimated by the proposed approach in a recursive way.Our future work will be devoted to implementing the proposed approach for SSMs with other mixture noise model, such as the Gaussian-uniform mixture noise [35].In addition, testing of the proposed algorithm in the context of radar target tracking is also an interesting prospect.
−1 and b ν i k;tjt−1 are the predicted hyperparameters of the normal and Wishart distributions for the ith particle at the tth time instant.In order to maintain the conjugacy for the distribution of the noise parameters, b m k;tjt−1 , b β k;tjt−1 , c W k;tjt−1 and b ν k;tjt−1 can be propagated as follows: 1 and b ν i k;t−1 are the estimated hyperparameters of the normal and Wishart distributions for the ith particle at the ðt − 1Þth time instant.The a priori quantities b m k;tjt−1 , b β k;tjt−1 , c W k;tjt−1 and b ν k;tjt−1 are then used as the initial values for VB inference (detailed in the Section 3.2), leading to the a posteriori quantities b

1 F I G U R E 2 3 4 F I G U R 4
Error rate for the maximum a posteriori estimates of r t over 100 MC simulations (proposed approach) Means of the mixing coefficients estimates over 100 MC simulations (proposed approach) Root mean square errors of μ s;1 and σ s;1 for s ∈ 1; 2Root mean square errors of μ s;2 and σ s;2 for s ∈ 1; 2; 3; 4 f g CHENG ET AL.

5 6
Root mean square errors of position estimates with the different approaches Root mean square errors of velocity estimates with the different approaches is an indicator for the kth component of the noise GMM (equal to 1 when r t ¼ k and 0 otherwise) and v s;t ; ð51Þ where y s;t (s ¼ 1; …; N s , N s being the number of in-view satellites) is the PR measurement associated with the sth inview satellite, ∥ ⋅ ∥ is the Euclidean norm, p s;t ¼ Execution times for the three approaches with different numbers of particles NF NT � 100%, where N F and N T denote the number of false MAP estimates and the total number of samples for r t in the time interval T .
T A B L E 1 2 ϵ ¼