NLOS mitigation in indoor localization by marginalized Monte Carlo Gaussian smoothing
 Jordi VilàValls^{1}Email author and
 Pau Closas^{2}
https://doi.org/10.1186/s1363401704984
© The Author(s) 2017
Received: 24 March 2017
Accepted: 18 August 2017
Published: 29 August 2017
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
1 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 [2–4], 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.
Declarations
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.
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Authors’ Affiliations
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