NLOS mitigation in indoor localization by marginalized Monte Carlo Gaussian smoothing
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Abstract
One of the main challenges in indoor timeofarrival (TOA)based wireless localization systems is to mitigate nonlineofsight (NLOS) propagation conditions, which degrade the overall positioning performance. The positive skewed nonGaussian nature of TOA observations under LOS/NLOS conditions can be modeled as a heavytailed skew tdistributed measurement noise. The main goal of this article is to provide a robust Bayesian inference framework to deal with target localization under NLOS conditions. A key point is to take advantage of the conditionally Gaussian formulation of the skew tdistribution, thus being able to use computationally light Gaussian filtering and smoothing methods as the core of the new approach. The unknown nonGaussian noise latent variables are marginalized using Monte Carlo sampling. Numerical results are provided to show the performance improvement of the proposed approach.
Keywords
Robust Bayesian inference Gaussian filtering and smoothing NLOS mitigation Skew tdistributed measurement noise Indoor localization Monte Carlo integration1 Introduction
The knowledge of position is ubiquitous in many applications and services, playing an important role. The widely diffused Global Navigation Satellite System (GNSS) offers a worldwide service coverage due to a network of dedicated satellites [1]. GNSS is recognized to be the de facto system in outdoor environments when it is available. Under the assumption that its reception is not obstructed or jammed [2, 3, 4], there is no doubt that GNSS is the main enabler for locationbased services (LBS). One of such situations is indoor positioning and tracking, where satellite signals are hardly useful (unless extremely large integration times are considered). In indoor scenarios, a plethora of alternative and complementary technologies can be considered [1, 5, 6].
We are interested in a particular propagation phenomena encountered in most positioning technologies (both outdoor and indoor), known as nonlineof sight (NLOS). It is one of the most challenging problems for tracking. Particularly, when considering timeofarrival (TOA) measurements as range estimates, the measured distance can be severely degraded. These ranges are typically positively biased with respect to the true distances, therefore seen as outliers at the receiver. It is of interest to develop NLOS mitigation techniques, providing enhanced robustness to tracking methods based on TOA measurements [6].
Notice that the standard Student t distribution is \(\mathcal {T}\left (z;\mu, \sigma ^{2}, \nu \right)= \mathcal {ST}\left (z;\mu,\sigma ^{2}, \lambda =0, \nu \right)\), the skew normal pdf is \(\mathcal {SN}\left (z;\mu, \sigma ^{2}, \lambda \right)= \mathcal {ST}\left (z;\mu,\sigma ^{2}, \lambda, \nu \rightarrow \infty \right)\), and \(\mathcal {N}\left (z;\mu, \sigma ^{2} \right) = \mathcal {ST}\left (z;\mu,\sigma ^{2}, \lambda = 0, \nu \rightarrow \infty \right)\) the normal distribution.
1.1 Stateoftheart
with \(\tau \sim \mathcal {G}\left (\tau ; \frac {\nu }{2},\frac {\nu }{2}\right)\), \(\gamma \tau \sim \mathcal {N}_{+} \left (\gamma ; 0, \tau ^{1}\right)\), and \(\mathcal {N}_{+}\left (\cdot \right) \) and \(\mathcal {G}(\cdot)\) as the positive truncated normal and gamma distributions. This is a key point in our problem formulation, because under the knowledge of the noise parameters (i.e., μ,σ ^{2},λ,ν,γ and τ in (3)), both the conditional marginal filtering and smoothing posterior distributions of the states, p(x _{ k }y _{1:k }) and p(x _{ k }y _{1:N }), turn to be Gaussian and thus we are able to use computationally light Gaussian smoothing methods to infer the states of the system.
In the literature, some contributions dealing with conditionally Gaussian SSMs corrupted by both heavytailed symmetric and skewed noise distributions were proposed. A particle filter (PF) solution for linear SSMs in symmetric αstable (S α S) noise was presented in [11]. This idea was further explored in [12] for nonlinear systems and generalized to other symmetric distributions in [13]. The key idea was to take advantage of the conditionally Gaussian form and use a sigmapoint Gaussian filter [14, 15] for the nonlinear state estimation. A robust filtering variational Bayesian (VB) approach was considered for linear systems in [16] and further extended to nonlinear SSMs in [17] considering a symmetric Student t measurement noise. But symmetric distributions may not always be appropriate to characterize the system noise. Recently, two interesting approaches to deal with linear SSMs under skewed noise were proposed, the first one uses a marginalized PF [18] and the other considers a VB solution [7, 19]. It is important to point out that (i) these contributions deal with either nonlinear systems corrupted by symmetric distributed noises or linear SSMs under skewed noise and (ii) the core of these methods use standard Bayesian filtering algorithms, then the smoothing problem needs to be further analyzed within this context.
Related to the problem under study, it is worth saying that several contributions deal with the filtering problem in nonlinear/nonGaussian SSM under model uncertainty using sequential Monte Carlo (SMC) methods, for instance, joint state and parameter estimation solutions [20], model selection strategies using interacting parallel PFs [21, 22], or model information fusion within the SMC formulation [23]. The main drawback of SMC methods is their high computational complexity and the curseofdimensionality [24]. That is the reason why we propose to take advantage of the underlying conditional Gaussian nature of the problem and use more efficient methods in this context.
1.2 Contributions

New Bayesian filtering and smoothingbased solutions for SSMs corrupted by parametric heavytailed skewed measurement noise

Marginalization of the unknown nonGaussian noise latent variables by Monte Carlo integration

TOAbased robust target tracking, where the LOS/NLOS propagation is modeled using a skew tdistributed measurement noise. Whereas a Gaussian filter and smoother deals with the nonlinear state estimation problem, the timevarying skew t distribution parameters are marginalized via Monte Carlo sampling
The article is organized as follows: first, we provide a discussion on Gaussian filtering and smoothing in nonlinear/Gaussian systems, together with the sigmapointbased approximation of the multidimensional integrals in the conceptual solution, being computationally more efficient than SMC methods under the Gaussian assumption; then, we provide the conditionally Gaussian formulation of the measurement noise and a method to deal with the unknown nonGaussian noise latent variables, and finally, we propose a NLOS indoor localization solution, based on the Gaussian smoother and the sequential noise latent variables marginalization. Numerical results are provided in realistic scenarios using UWB signals.
2 Gaussian filtering and smoothing
This section reviews the general filtering and smoothing solutions in the case of nonlinear/Gaussian systems, this material corresponds to Sections 2.1 and 2.2, respectively. Then, in Section 2.3, we provide the implementation details when sigmapoints are used to solve the filtering/smoothing equations. Notice that when the system is linear/Gaussian, the optimal solutions are given by the standard Kalman filter (KF) [35] and Kalman smoother (KS) [36], and for general nonlinear/nonGaussian systems, one should consider more sophisticated SMC techniques [29].
For the formal derivation of the Gaussian filter/smoother, we assume that the measurement noise in (2) is zeromean Gaussian with known covariance R _{ k }. Later, in Section 3, we discuss how the method can be used in the context of conditionally Gaussian models.
2.1 Bayesian Gaussian filtering
From a theoretical point of view, all necessary information to infer information of the unknown states resides in the marginal posterior distribution of the states, p(x _{ k }y _{1:k }). Thus, the Bayesian filtering problem is one of evaluating this distribution. It can be recursively computed [26] in two steps: (1) prediction of p(x _{ k }y _{1:k−1}) using the prior information and the previous filtering distribution and (2) update with new measurements y _{ k } to obtain.
The problem reduces to the approximation of these integrals.
2.2 Gaussian smoothing
with \(\mathbf {D}_{k} = \mathbf {\Sigma }_{k,k+1k} \mathbf {\Sigma }^{1}_{x,k+1k}\) and Σ _{ k,k+1k } referring to the crosscovariance between x _{ k } and x _{ k+1}. Note that in practice we do not require the computation of the smoothing estimation error covariance, Σ _{ x,kN }, for the smoother recursion. However, it is useful in order to have a measure of the smoothing uncertainty. The smoother gain D _{ k } can be easily obtained from the standard forward filtering pass, therefore adding very few extra computation.
2.3 Sigmapoint Gaussian filtering and smoothing
An appealing class of filters and smoothers within the nonlinear Gaussian framework are the sigmapoint Gaussian filters (SPGF) [14, 15, 34, 39, 40] and smoothers (SPGS) [37, 38], a family of derivativefree algorithms which are based on a weighted sum of function values at specified (i.e., deterministic) points within the domain of integration, as opposite to the stochastic sampling performed by particle filtering methods. The idea is to use a set of socalled sigmapoints to efficiently characterize the propagation of the normal distribution over the nonlinear system. In the sequel, we detail the formulation of such approximation and how it can be used to perform filtering or smoothing.
2.3.1 Filtering
2.3.2 Smoothing
with \(\mathbf {D}_{k} = \mathbf {\Sigma }_{k,k+1k} \mathbf {\Sigma }_{x,k+1k}^{1}\). Notice that the smoother gain can be embedded into the prediction step of the forward filtering, then only the last step is performed in the backward recursion. At time k=N, both filtering and smoothing estimates are the same, then the backward pass runs from time N−1 to 1. Compared to the filtering process, implementation of the smoothing solution only impacts in having additional steps ?? and ?? in Algorithm 1, where we use the notation \(\mathbf {\Sigma }_{x} = \mathbf {S}_{x}\mathbf {S}_{x}^{\top }\) for the factorized covariances.
3 Hierarchically Gaussian measurement noise formulation
In the previous Section 2, we assumed a Gaussian measurement noise with known covariance matrix. But in challenging applications such as the NLOS propagation conditions of interest here, the Gaussian assumption does not hold and noise parameters may be unknown to a certain extent. In such scenarios, one may have outliers or impulsive behaviors that produce biased estimates, for instance, under NLOS conditions the receiver is likely to estimate distances to the anchors larger than the true ones [6]; therefore, we must account for more accurate observation models.
with \(\mathcal {ST}\left (n_{k,i}; \mu, \sigma ^{2}, \lambda, \nu \right)\) defined in Section 1.
While τ _{ k,i } controls the heavytailed behavior, γ _{ k,i } controls the skewness of the distribution.
We can define the vector with 2×n _{ y } noise distribution latent variables, \(\boldsymbol {\phi }_{k} = \left.\{ \gamma _{k,i}, \tau _{k,i} \}\right _{i=1,\ldots,n_{y}}\), where we omit the dependence with respect to the hyperparameters (i.e., μ,σ ^{2},λ,ν) for the sake of clarity.
where [m _{ k }(ϕ _{ k })]_{ i }=μ+λ γ _{ k,i } and \([\mathbf {R}_{k}(\boldsymbol {\phi }_{k})]_{i,i} = \tau _{k,i}^{1} \sigma ^{2}\).
The distribution hyperparameters are application dependent and typically assumed a priori known. The standard Gaussian filter/smoother in charge of the state estimation assumes a zeromean Gaussian measurement noise with known parameters. In the skew tdistributed case, at every time step, the filter requires an estimate of the corresponding mean and covariance, m _{ k }(ϕ _{ k }) and R _{ k }(ϕ _{ k }), respectively. In the following, we consider the marginalization of the noise latent variables in the general filter/smoother formulation.
4 Noise latent variables marginalization
from basic conjugate analysis results. Interestingly, the posteriors at k in (38) and (39) can be used as the priors in k+1 instead of (36) and (37). In this way, the algorithm is learning the environment as it progresses over time. However, given the assumed model, it is more meaningful to reset the prior at each time instant instead of sequentially using the latest posterior. The reason is that measurements are assumed independent, so there is no benefit in carrying out information from one time instant to the other. Instead, under these conditions, we suggest to use the values γ _{0}, κ _{0}, α _{0}, and β _{0} at k−1 before updating the distribution with \(\tilde {y}_{k,i}\). Sequential use of the posterior will be interesting when the generative model is known to have some memory.
A further improvement of standard SPGF/S schemes comes from the fact that the filter should preserve the properties of a covariance matrix, namely, its symmetry and positivedefiniteness. In practice, however, due to lack of arithmetic precision, numerical errors may lead to a loss of these properties. To circumvent this problem, a squareroot filter can be considered to propagate the square root of the covariance matrix instead of the covariance itself [33, 34]. We propose to use squareroot cubature and quadrature Kalman filters/smoothers (named SCKF/S and SQKF/S, respectively) [38, 43] as the core implementation of the new squareroot MSPGF/S. These methods resort to cubature [34] and GaussHermite quadrature rules [15] to approximate the integrals in the optimal solution. While the SCKF/S uses L _{ c }=2n _{ x } sigmapoints, in the SQKF/S we have \(L_{q} = \alpha ^{n_{x}}\phantom {\dot {i}\!}\), where α determines the number of sigmapoints per dimension, which is typically set to α=3. A straightforward solution to avoid the exponential computational complexity increase of the standard QKF in highdimensional systems is the use of sparsegrid quadrature rules, which reduce the computational complexity with negligible penalty in numerical accuracy [44, 45].
5 Application to indoor localization
5.1 SSM for the TOAbased localization problem
The Gaussian process noise \(\mathbf {u}_{k} \sim \mathcal {N}\left (\mathbf {u}_{k};\mathbf {0},\sigma ^{2}_{u} \mathbf {I}_{2}\right)\) models an acceleration of σ _{ u } m/s^{2}.
5.2 Numerical results
 1.
SQKF/S operating under the Gaussian assumption without accounting for the nonGaussian nature of the measurement noise (SQKF/SG).
 2.
SQKF/S using point estimates of the noise latent variables ψ _{ k,i } as proposed in [25] (SQKF/SP).
 3.
New squareroot SPGF/Sbased solution with marginalized noise latent variables ϕ _{ k } within the filter/smoother formulation via Monte Carlo sampling (MSPGF/S).
 4.
A clairvoyant SQKF/S that knows exactly the realization of the latent variables ϕ _{ k } at each instant k and thus can use m _{ k }(ϕ _{ k }) and R _{ k }(ϕ _{ k }). This is the performance benchmark for the new methodology (SQKF/SK).
We also considered a sampling importance resampling PF with 81 particles (i.e., equivalent to the number of sigmapoints in the SQKF/S), but as already shown in [46], the filter is in general not able to correctly localize the target (i.e., the filter diverges). Moreover, to obtain the same performance than the clairvoyant SQKF/S, we must consider a much larger number of particles. This is the reason why these results are not shown in the figures, since for the fair comparison in terms of number of particles the PF does not provide convergent result.
The proposed MSPGF/S can be implemented using cubature [34] and GaussHermite approximations [15], then using respectively L _{ c }=2n _{ x }=8 and \(L_{q} = \alpha ^{n_{x}}=81\phantom {\dot {i}\!}\) deterministic samples to approximate the integrals of the general solution. In the proposed indoor localization scenario, we tested both cubature and quadrature approximations, and the performance obtained was found strictly equivalent. In practice, the method of choice is the cubaturebased solution, which has the lowest computational complexity.
Notice that all the methods consider known distribution hyperparameters, which are application dependent. We consider an UWB TOAbased indoor localization realistic scenario, with hyperparameters given in [7] and adjusted to match real data: μ=−0.1 m, σ=0.3 m, and λ=0.6 m and ν=4. The corresponding Gaussian approximation is given by μ _{ G }=1.3 and σ _{ G }=1.6.
Mean RMSE of position and velocity versus Monte Carlo sample size L in the MSPGS
Smoother  RMSE position  RMSE velocity 

SQKSK (known noise)  0.0653  0.0362 
MSPGS L=1000  0.1017  0.0421 
MSPGS L=500  0.1020  0.0421 
MSPGS L=100  0.1033  0.0421 
MSPGS L=50  0.1033  0.0421 
SQKSP  0.1338  0.0452 
SQKSG  0.4202  0.0617 
6 Conclusions
This article presented a new Bayesian filtering and smoothing framework to deal with nonlinear systems corrupted by parametric heavytailed skew tdistributed measurement noise. The new method takes advantage of the conditionally Gaussian form of the skew tdistribution, which allows to use a computationally light Gaussian filter and smoother to deal with the state estimation. The unknown nonGaussian noise latent variables are marginalized from the general filtering/smoothing solution via Monte Carlo sampling. The performance of the new solution was evaluated in a representative TOAbased localization scenario, where the positive skewed behavior of NLOS propagation conditions is typically modeled using such nonGaussian distributions.
7 Endnote
^{1} We write (x)^{2}, (y)^{2}, f ^{2}(·), and h ^{2}(·) as the shorthand for x x ^{⊤}, y y ^{⊤}, f(·)f ^{⊤}(·), and h(·)h ^{⊤}(·), respectively. We omitted the dependence with time of f _{ k−1}(·) and h _{ k }(·) for the sake of clarity.
Notes
Acknowledgements
This work has been supported by the Spanish Ministry of Economy and Competitiveness through project TEC201569868C22R (ADVENTURE) and by the Government of Catalonia under Grant 2014–SGR–1567.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.D Dardari, E Falletti, M Luise, Satellite and terrestrial radio positioning techniques: a signal processing perspective (Elsevier Academic Press, Oxford, 2011).Google Scholar
 2.MG Amin, P Closas, A Broumandan, JL Volakis, Vulnerabilities, threats, and authentication in satellitebased navigation systems [scanning the issue]. Proc. IEEE. 104(6), 1169–1173 (2016).CrossRefGoogle Scholar
 3.JT Curran, M Navarro, M Anghileri, P Closas, S Pfletschinger, Coding aspects of secure GNSS receivers. Proc. IEEE. 104(6), 1271–1287 (2016).CrossRefGoogle Scholar
 4.R Ioannides, T Pany, G Gibbons, Known vulnerabilities of global navigation satellite systems, status, and potential mitigation techniques. Proc. IEEE. 104(6), 1174–1194 (2016).CrossRefGoogle Scholar
 5.H Liu, H Darabi, P Banerjee, J Liu, Survey of wireless indoor positioning techniques and systems. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev.37(6), 1067–1080 (2007).CrossRefGoogle Scholar
 6.D Dardari, P Closas, PM Djuric, Indoor tracking: theory, methods, and technologies. IEEE Trans. Veh. Technol.64(4), 1263–1278 (2015).CrossRefGoogle Scholar
 7.H Nurminen, T Ardeshiri, Piche, Ŕ, F Gustafsson, in Proc. of the 2015 International Conference on Indoor Positioning and Indoor Navigation (IPIN). A NLOSrobust TOA positioning filter based on a skewt measurement noise model (IEEE, Banff, AB, Canada, 2015).Google Scholar
 8.P Müller, et al, Statistical trilateration with skewt distributed errors in LTE networks. IEEE Trans. Wirel. Commun.15(10), 7114–7127 (2016).CrossRefGoogle Scholar
 9.TI Lin, JC Lee, WJ Hsieh, Robust mixture modeling using the skew t distribution. Stat. Comput.17:, 81–92 (2007).MathSciNetCrossRefGoogle Scholar
 10.S Zozor, C Vignat, Some results on the denoising problem in the elliptically distributed context. IEEE Trans. Sig. Process. 58(1), 134–150 (2010).MathSciNetCrossRefGoogle Scholar
 11.MJ Lombardi, SJ Godsill, On–line Bayesian estimation of signals in symmetric alphastable noise. IEEE Trans. Sig. Process. 54(2), 775–779 (2006).CrossRefGoogle Scholar
 12.J VilàValls, C FernándezPrades, P Closas, JA FernándezRubio, in Proc. of the European Signal Processing Conference, Eusipco 2011. Bayesian filtering for nonlinear state–space models in symmetric α–stable measurement noise (IEEE, Barcelona, 2011), pp. 674–678.Google Scholar
 13.J VilàValls, P Closas, C FernándezPrades, JA FernándezRubio, in Proc. of the European Signal Processing Conference (Eusipco). Nonlinear Bayesian filtering in the Gaussian scale mixture context (IEEE, Bucharest, 2012), pp. 529–533.Google Scholar
 14.K Ito, K Xiong, Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control.45(5), 910–927 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
 15.I Arasaratnam, S Haykin, RJ Elliot, Discretetime nonlinear filtering algorithms using GaussHermite quadrature. Proc. IEEE. 95(5), 953–977 (2007).CrossRefGoogle Scholar
 16.G Agamennoni, JI Nieto, EM Nebot, Approximate inference in statespace models with heavytailed noise. IEEE Trans. Sig. Process. 60(10), 5024–5037 (2012).MathSciNetCrossRefGoogle Scholar
 17.R Piché, S Särkkä, J Hartikainen, in Proc. of the IEEE International Workshop on Machine Learning for Signal Processing (MLSP). Recursive outlierrobust filtering and smoothing for nonlinear systems using the multivariate Studentt distribution (IEEE, Bucharest, 2012).Google Scholar
 18.S Saha, Noise robust online inference for linear dynamic systems (2015). http://arxiv.org/abs/1504.05723. Accessed 24 Aug 2017.Google Scholar
 19.H Nurminen, T Ardeshiri, R Piché, F Gustafsson, Robust inference for statespace models with skewed measurement noise. IEEE Sig. Process. Lett. 22(11), 1898–1902 (2015).CrossRefGoogle Scholar
 20.N Chopin, PE Jacob, O Papaspiliopoulos, SMC2: an efficient algorithm for sequential analysis of statespace models. J. R. Statist. Soc. B. 75(3), 397–426 (2013).CrossRefGoogle Scholar
 21.L Martino, J Read, V Elvira, F Louzada, Cooperative parallel particle filters for online model selection and applications to urban mobility. Digit. Sig. Process.60:, 172–185 (2017).CrossRefGoogle Scholar
 22.CC Drovandi, J McGree, AN Pettitt, A sequential Monte Carlo algorithm to incorporate model uncertainty in Bayesian sequential design. J. Comput. Graph. Stat. 23(1), 3–24 (2014).MathSciNetCrossRefGoogle Scholar
 23.I Urteaga, MF Bugallo, PM Djurić, in Proc. of the IEEE Statistical Signal Processing Workshop (SSP). Sequential Monte Carlo methods under model uncertainty (IEEE, Palma de Mallorca, 2016).Google Scholar
 24.F Daum, J Huang, in Proc. of IEEE Aerospace Conference. Curse of dimensionality and particle filters (IEEE, Big Sky, 2003).Google Scholar
 25.P Closas, J VilàValls, in Proc. of the IEEE Statistical Signal Processing Workshop (SSP). NLOS mitigation in TOAbased indoor localization by nonlinear filtering under skew tdistributed measurement noise (IEEE, Palma de Mallorca, 2016).Google Scholar
 26.H Sorenson, in Recursive estimation for nonlinear dynamic systems, ed. by JC Spall (Marcel Dekker, New York, 1988).Google Scholar
 27.RE Kalman, A new approach to linear filtering and prediction problems. J. Basic Eng. Trans. ASME. 82(1), 35–45 (1960).CrossRefGoogle Scholar
 28.Z Chen, Bayesian filtering: from Kalman filters to particle filters, and beyond (2003). Tech. Rep. Adaptive Syst. Lab. McMaster University. Ontario.Google Scholar
 29.A Doucet, N de Freitas, N Gordon (eds.), Sequential Monte Carlo Methods in Practice (Springer, New York, 2001).Google Scholar
 30.S Arulampalam, S Maskell, N Gordon, T Clapp, A tutorial on particle filters for online nonlinear/nonGaussian Bayesian tracking. IEEE Transactions on Signal Processing. 50(2), 174–188 (2002).CrossRefGoogle Scholar
 31.PM Djurić, JH Kotecha, J Zhang, Y Huang, T Ghirmai, MF Bugallo, J Míguez, Particle filtering. IEEE Signal Processing Magazine. 20(5), 19–38 (2003).CrossRefGoogle Scholar
 32.B Ristic, S Arulampalam, N Gordon (eds.), Beyond the Kalman Filter: Particle Filters for Tracking Applications (Artech House, Boston, 2004).Google Scholar
 33.I Arasaratnam, S Haykin, Squareroot quadrature Kalman filtering. IEEE Trans. Sig. Process.56(6), 2589–2593 (2008).MathSciNetCrossRefGoogle Scholar
 34.I Arasaratnam, S Haykin, Cubature Kalman filters. IEEE Trans. Autom. Control. 54(6), 1254–1269 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
 35.B Anderson, JB Moore, Optimal Filtering (PrenticeHall, Englewood Cliffs, 1979).zbMATHGoogle Scholar
 36.HE Rauch, F Tung, CT Striebel, Maximum likelihood estimates of linear dynamic systems. AIAA J. 3(8), 1445–1450 (1965).MathSciNetCrossRefGoogle Scholar
 37.S Särkkä, Unscented RauchTungStriebel smoother. IEEE Trans. Autom. Control. 53(3), 845–849 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
 38.I Arasaratnam, S Haykin, Cubature Kalman smoothers. Automatica. 47:, 2245–2250 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
 39.SJ Julier, JK Ulhmann, HF DurrantWhyte, A new method for nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Control. 45(3), 472–482 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
 40.M Norgaard, NK Poulsen, O Ravn, New developments in state estimation of nonlinear systems. Automatica. 36:, 1627–1638 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
 41.TI Lin, Robust mixture modeling using multivariate skew t distributions. Stat. Comput. 20:, 343–356 (2010).MathSciNetCrossRefGoogle Scholar
 42.JM Bernardo, AFM Smith, Bayesian Theory, vol. 405 (Wiley, New York, 2009).Google Scholar
 43.S Särkkä, Bayesian Filtering and Smoothing (Cambridge University Press, Cambridge, 2013).CrossRefzbMATHGoogle Scholar
 44.P Closas, J VilàValls, C FernándezPrades, in Proc. of the CAMSAP’15. Computational complexity reduction techniques for quadrature Kalman filters (IEEE, Cancun, 2015).Google Scholar
 45.B Jia, M Xin, Y Cheng, Sparsegrid quadrature nonlinear filtering. Automatica. 48(2), 327–341 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
 46.J VilàValls, P Closas, AF GarcíaFernández, Uncertainty exchange through multiple quadrature Kalman filtering. IEEE Sig. Process. Lett. 23(12), 1825–1829 (2016).CrossRefGoogle Scholar
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