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.
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].
In general, the problem under study concerns the derivation of new robust methods to solve the Bayesian filtering and smoothing problem in challenging applications such as the LOS/NLOS propagation conditions in indoor localization systems. The statespace models (SSM) of interest are expressed as
where \(\mathbf {x}_{k} \in \mathbb {R}^{n_{x}}\) and \(\mathbf {y}_{k} \in \mathbb {R}^{n_{y}}\) are the hidden states of the system and measurements at time k. f _{ k−1}(·) and h _{ k }(·) are known to be the possibly nonlinear functions of the state; and both process and observation noises, u _{ k } and n _{ k }, assumed to be mutually independent. In real–life applications, we may not have a complete knowledge of the system conditions, thus the measurement noise statistics are assumed to be unknown to a certain extent. In contrast, we consider a known process noise covariance Q _{ k }. Regarding the measurement noise, we assume that it is distributed according to a parametric heavytailed skew tdistribution, \(\mathbf {n}_{k} \sim \mathcal {ST}\left (\boldsymbol {\phi }_{k}\right)\), with ϕ _{ k } representing the set of possibly unknown parameters of the nonGaussian distribution. The probability density function (pdf) of the univariate skew t distribution of interest can be written as [19]
with \(\mu \in \mathbb {R}\), \(\sigma ^{2} \in \mathbb {R}^{+}\), \(\lambda \in \mathbb {R}\), and \(\nu \in \mathbb {R}^{+}\), referring to the distribution location, scale, skewness, and degrees of freedom, respectively. \(\mathcal {T}\left (z; \mu,\sigma ^{2}, \nu \right)\) is the pdf of the Student’s t distribution,
with Γ(·) the gamma function. \(\mathrm {T}(\tilde {z}; 0,1,\nu)\) is the cumulative distribution function (CDF) of the Student’s t distribution with ν degrees of freedom and
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.
Stateoftheart
The skew t distribution has been recently shown to provide a reasonable fit to realistic indoor TOA measurements. For instance, characterizing range measures in NLOS conditions in ultrawideband (UWB) localization [7] or in multipath channels when ranging is computed with longterm evolution (LTE) networks [8]. Interestingly, this distribution allows a Gaussian meanscale mixture (GMSM) representation, which implies that the distribution can be reformulated as hierarchically (conditionally) Gaussian [9, 10]. Mathematically, if we have the skew tdistributed random variable \(z \sim \mathcal {ST}(z;\mu,\sigma ^{2},\lambda,\nu)\), then we can write [41]
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.
Contributions
The main contributions of the article, which generalize the preliminary results in [25], are summarized as:

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.
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.
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 recursive solution provides an estimation framework that is optimal in the Bayesian sense, that is, the characterization of the posterior distribution allows us to compute the minimum meansquared error (MMSE), the maximum a posteriori (MAP) or the median of the posterior (minimax) estimators, addressing optimality in many senses. The multidimensional integrals in the prediction and update steps are analytically intractable in the general case. Actually, there are few cases where the optimal Bayesian recursion can be analytically solved. This is the case of linear/Gaussian models, where the KF yields to the optimal solution [27]. In more general models, one must resort to suboptimal algorithms. A plethora of methods can be found in the literature [28]. A popular tool are particle filters (PF) [29–32], a set of simulationbased methods which are applicable in nonlinear/nonGaussian setups. Under the Gaussian assumption of interest, the quadrature KF (QKF) [14, 15, 33] and cubature KF (CKF) [34] are typically the methods of choice. In this case, the marginal predictive and posterior distributions are
In the prediction step, we compute the marginal predictive distribution mean and covariance as ^{1}
In the update step, the mean and covariance of the marginal posterior are given by the KF Equations [35]
where the Kalman gain is \(\mathbf {K}_{k} = \boldsymbol \Sigma _{xy,kk1} \boldsymbol \Sigma _{y,kk1}^{1}.\) The predicted measurement and both innovation and crosscovariance matrices are computed as
The problem reduces to the approximation of these integrals.
Gaussian smoothing
In the previous section, we summarized the general Gaussian Bayesian filtering solution but sometimes it may be interesting to obtain an estimate of the smoothing posterior and not its filtering counterpart. In the problem under study, we consider a forwardbackward smoother formulation [36] to obtain the marginal smoothing posterior, p(x _{ k }y _{1:N }),
where we used the state that is Markovian and then
The forwardbackward smoothing [36] performs two filtering passes, that is, first a standard forward filtering from time k=1 to N, and then, the backward filtering from k=N to 1, backwards in time. Notice that the predictive and filtering distributions may be obtained from the standard Bayesian filtering solution. At time k, if we consider that we know the filtering distribution, \(\mathcal {N}\left (\mathbf {x}_{k} ; \hat {\mathbf {x}}_{kk}, \mathbf {\Sigma }_{x,kk} \right)\), and the predictive distribution, \(\mathcal {N}\left (\mathbf {x}_{k+1} ; \hat {\mathbf {x}}_{k+1k}, \mathbf {\Sigma }_{x,k+1k} \right)\), from the forward filtering, together with the smoothed density at k+1, \(\mathcal {N} \left (\mathbf {x}_{k+1}; \hat {\mathbf {x}}_{k+1N}, \mathbf {\Sigma }_{x,k+1N} \right)\), because the smoother is running backwards, then the analytical solution to the marginal smoothing posterior is obtained as follows: using the Markovian properties of states, we have that p(x _{ k }x _{ k+1},y _{1:N })=p(x _{ k }x _{ k+1},y _{1:k }), and then we can obtain the conditional smoothing distribution of x _{ k } as
with
The joint smoothing distribution p(x _{ k },x _{ k+1}y _{1:N }) is
which can be used to obtain the smoothing distribution by marginalization over x _{ k+1},
Under the Gaussian assumption, the problem is to recursively obtain the mean and covariance of the Gaussian marginal smoothing posterior distribution, which is given by [37, 38]
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.
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.
Filtering
Consider a set of sigmapoints, \(\phantom {\dot {i}\!}\{ \boldsymbol {\xi }_{i}, \omega _{i}\}_{i=1,\ldots,L_{M}}\). Then, construct the transformed set which captures the mean and covariance of the posterior distribution, \(\phantom {\dot {i}\!}\mathbf {x}_{i,k1k1}=\mathbf {S}_{x,k1k1} \boldsymbol {\xi }_{i}+\hat {\mathbf {x}}_{k1k1}\), with \(\phantom {\dot {i}\!}\mathbf {\Sigma }_{x,k1k1} = \mathbf {S}_{x,k1k1}\mathbf {S}_{x,k1k1}^{\top }\). The integrals in the prediction step can be approximated as
In the following update step, first compute the transformed set to capture the mean and covariance of the predictive marginal distribution, \(\phantom {\dot {i}\!}\mathbf {x}_{i,kk1}=\mathbf {S}_{x,kk1} \boldsymbol {\xi }_{i}+\hat {\mathbf {x}}_{kk1}\), with \(\phantom {\dot {i}\!}\mathbf {\Sigma }_{x,kk1} = \mathbf {S}_{x,kk1} \mathbf {S}_{x,kk1}^{\top }\). Then, we approximate the integrals of interest as,
Smoothing
The smoothed state (17) is obtained using the predicted filtering and smoothing states, \(\hat {\mathbf {x}}_{k+1k}\) and \(\hat {\mathbf {x}}_{k+1N}\), respectively. Define again a set of sigmapoints and weights, \(\{ \boldsymbol {\xi }_{i}, \omega _{i}\}_{i=1,\ldots,L_{M}}\phantom {\dot {i}\!}\), and the transformed set which captures the corresponding mean and covariance, \(\mathbf {x}_{i,kk}=\mathbf {S}_{x,kk}\boldsymbol {\xi }_{i}+\hat {\mathbf {x}}_{kk}\phantom {\dot {i}\!}\). Use this transformed sigmapoints to estimate the predicted subspace state, its prediction error covariance, and the crosscovariance as
Finally, estimate the smoothed subspace and covariance as
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.
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.
In general, these nonGaussian behaviors can be effectively characterized by parametric heavytailed and positiveskewed noise distributions. It has been recently shown experimentally that TOAbased positioning under NLOS conditions [7] and multipath ranging error distributions in LTE networks [8] can be well approximated by a skew tdistribution [9]. Taking into account the problem at hand, we are interested in measurement models with independent observation components and measurement noise models where the noise components are independently univariate skew tdistributed
with \(\mathcal {ST}\left (n_{k,i}; \mu, \sigma ^{2}, \lambda, \nu \right)\) defined in Section 1.
A key point on the problem formulation is to take advantage of the hierarchically (conditionally) Gaussian formulation of the measurement noise distribution. The hierarchical Gaussian representation of the skew tdistribution is written as [41]
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.
The measurement noise in (24) can be written as
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.
Noise latent variables marginalization
In the problem of interest, the measurement noise is conditionally Gaussian with unknown noise latent variables. Therefore, the filtering/smoothing formulation in Section 2 must be modified to take such uncertainty into account. We assume known measurement noise distribution hyperparameters, and thus, we want to marginalize the state estimation with respect to the noise latent variables, γ _{ k,i } and τ _{ k,i }. We can write the marginalized posterior distribution as
Notice that the measurement noise parameters only affect the computation of the innovation in the measurement update of the filter/smoother. Within this context, the predicted measurement is reformulated as
where the first term corresponds to the standard Gaussian case, and the second is a marginalization of the noise term over the last available distribution (i.e., we write n _{ k }(ϕ _{ k }) to explicitly emphasize the dependency on the noise latent variables, ϕ _{ k }). Using a similar formulation, we can rewrite the innovation covariance as
For the marginalization of the noise latent variables, a key point is to obtain the posterior distributions of γ _{ k,i } and τ _{ k,i }. The joint posterior is given by
As the observation components are assumed to be independent, the likelihood function is
We can define a normalized observation
and \(\tilde {\gamma }_{k,i} \triangleq \lambda \gamma _{k,i}/\sigma \). Then, we have the normalized likelihood is given by
Using the conjugate nature of the prior distributions [42], it is possible to obtain the analytical solution for the posterior of γ _{ k,i } and τ _{ k,i }. In this case, we have from (28) that the a priori distributions are
with γ _{0}=0, κ _{0}=σ ^{2}/λ ^{2}, and α _{0}=β _{0}=ν/2. We are interested in updating with the new measurements to get the posterior distributions
with
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.
In [25], we proposed to use a point estimate for γ _{ k,i } and τ _{ k,i } from a single observation using their posterior marginals. The corresponding modes of these distributions were used as point estimates \(\hat {\gamma }_{k,i} = \frac { \tilde {y}_{k,i}  }{2} \frac {\sigma }{\lambda }\) and \(\hat {\tau }_{k,i} = \frac {\alpha  1}{\beta }\). where we took into account that \(\hat {\gamma }_{k,i} \in \mathbb {R}^{+}\) by construction. In this contribution, instead of using a point estimate, we consider a Monte Carlobased marginalization drawing L samples from the joint posterior given by (38) and (39). Using these distributions, we propose to compute the two integrals of interest as
with \(\boldsymbol {\phi }^{(j)}_{k}\) being random samples drawn from the joint posterior distribution of the noise latent variables, ϕ _{ k }. In practice, this can be easily implemented by first drawing a sample from (39) and then, using that sample, draw from (38). These expressions can be further expanded as follows
where \({\gamma }^{(j)}_{k,i}\) are random samples drawn from the posterior of \({\gamma }^{(j)}_{k,i}\), and \(\tilde {\mathbf {R}}_{k}\) is approximated by a diagonal matrix, where the pth element of the diagonal is
Finally, we have that the marginalized Monte Carlo sigmapoint Gaussian filter and smoother (MSPGF/S) proposed in this contribution is given by Algorithm 1 with step ?? modified as
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].
Application to indoor localization
SSM for the TOAbased localization problem
To show the performance of the proposed approach, we consider a TOAbased localization problem, where a set of N anchor nodes at known locations, x _{ k,i }=[x _{ k,i },y _{ k,i }]^{⊤}, provide range information. If we define the state to be inferred as position and velocity components of the target, \(\mathbf {p}_{k} \triangleq (x_{k}, y_{k})^{\top }\) and \(\mathbf {v}_{k} \triangleq (\dot {x}_{k}, \dot {y}_{k})^{\top }\), respectively, the observed range from each node i to the target, is modeled as \(\hat {\rho }_{k,i} = \rho _{i}(\mathbf {x}_{k}) + n_{k,i}\), i∈{1,…,N}, with n _{ k,i } denoting the ranging error and \(\rho _{i}(\mathbf {x}_{k}) \triangleq \rho _{k,i} = \\mathbf {x}_{k}  \mathbf {x}_{k,i}\\) the true distance from the ith node to the target node at time k. The complete measurement equation is given by
In standard localization applications, the measurement noise is nominally distributed according to \(\mathbf {n}_{t} \sim \mathcal {N}\left (\mathbf {n}_{t} ; \mathbf {0}, \sigma ^{2} \cdot \mathbf {I}_{N}\right)\), where σ depends on the technology used to obtain the ranging estimates. In the case of UWB devices, this is typically on the order of 0.1 to 1 meter. But the Gaussian distribution does not capture the NLOS propagation conditions [7]; thus, we must account for more accurate measurement models such as the skew tdistribution introduced in the previous sections. Using the noise n _{ k } defined in (29), the measurement equation is defined as
Considering the state \(\mathbf {x}_{k} = [x_{k},y_{k},\dot {x}_{k},\dot {y}_{k}]^{\top }\), the process equation is defined as a linear constant acceleration model
with
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}.
Numerical results
For the numerical evaluation of the proposed method, the root mean square error (RMSE) of position is used as the measure of performance, which is obtained from 1000 Monte Carlo runs. The new method was validated in a realistic scenario composed of N=6 anchor nodes, circularly deployed in a 40×40 m^{2} area, and considering σ _{ u }=0.03 m/s^{2}. We compare the tracking performance obtained with four methods:

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.
Figures 1 and 2 show the filtering and smoothing RMSE of position, respectively, for the different methods and considering L=1000 Monte Carlo samples for the MSPGF/S. In both cases, we obtained similar results. Although the clairvoyant filter/smoother (SQKF/SK) with fully known measurement noise parameters outperforms the rest, we have a small performance loss with the proposed methodology considering unknown noise latent variables. The new marginalized approach is more robust and outperforms the SQKF/SP using point estimates first proposed in the filtering context in [25]. The SQKF/SG operating under the full Gaussian assumption, even if the parameters of the underlying Gaussian noise (i.e.,μ _{ G } and σ _{ G }) are correctly obtained to fit the real data, shows the worst performance. This is because this filter/smoother does not take into account the NLOSinduced outliers in the measurement noise. For the sake of completeness, we assess the impact of the Monte Carlo sample size in the MSPGF/S performance. The mean RMSE of position and velocity, for the different methods and several representative values of L, are given in Table 1. The performance of the proposed approach is not degraded when using a sample size as low as L=50 samples, therefore being possible to keep a low overall computational complexity.
Notice that the parameter ν of the skew t distribution controls the tails of the distribution. Lower ν implies heavier tails, thus more outliers and impulsive behaviors. To fully characterize the new method, the performance obtained in the UWB TOAbased localization scenario but now with ν=2, to induce more outliers in the measurements, is shown in Figs. 3 and 4. The proposed method correctly deals with the nonGaussian noise and approaches the optimal solution. In this case, the performance given by the filter/smoother under the Gaussian assumption (SQKF/SG) is really poor. This is because the underlying noise distribution is more heavytailed, then the Gaussian approximation is no longer valid.
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.
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.
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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.
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VilàValls, J., Closas, P. NLOS mitigation in indoor localization by marginalized Monte Carlo Gaussian smoothing. EURASIP J. Adv. Signal Process. 2017, 62 (2017). https://doi.org/10.1186/s1363401704984
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Keywords
 Robust Bayesian inference
 Gaussian filtering and smoothing
 NLOS mitigation
 Skew tdistributed measurement noise
 Indoor localization
 Monte Carlo integration