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Robust distributed cooperative RSSbased localization for directed graphs in mixed LoS/NLoS environments
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 19 (2019)
Abstract
The accurate and lowcost localization of sensors using a wireless sensor network is critically required in a wide range of today’s applications. We propose a novel, robust maximum likelihoodtype method for distributed cooperative received signal strengthbased localization in wireless sensor networks. To cope with mixed LoS/NLoS conditions, we model the measurements using a twocomponent Gaussian mixture model. The relevant channel parameters, including the reference path loss, the path loss exponent, and the variance of the measurement error, for both LoS and NLoS conditions, are assumed to be unknown deterministic parameters and are adaptively estimated. Unlike existing algorithms, the proposed method naturally takes into account the (possible) asymmetry of links between nodes. The proposed approach has a communication overhead upperbounded by a quadratic function of the number of nodes and computational complexity scaling linearly with it. The convergence of the proposed method is guaranteed for compatible network graphs, and compatibility can be tested a priori by restating the problem as a graph coloring problem. Simulation results, carried out in comparison to a centralized benchmark algorithm, demonstrate the good overall performance and high robustness in mixed LoS/NLoS environments.
1 Introduction
The wide spread of telecommunication systems has led to the pervasiveness of radiofrequency (RF) signals in almost every environment of daily life. Knowledge of the location of mobile devices is required or beneficial in many applications [1], and numerous localization techniques have been proposed over the years [1–4]. Techniques based on the received signal strength (RSS) are the preferred option when low cost, simplicity, and technology obliviousness are the main requirements. In some standards, e.g., IEEE 802.15.4, an RSS indicator (RSSI) is encoded directly into the protocol stack [5]. In addition, RSS is readily available from any radio interface through a simple energy detector and can be modeled by the wellknown path loss model [6] regardless of the particular communication scheme. Based on that, RSS can be exploited to implement “opportunistic” localization for different wireless technologies, e.g., WiFi [7], FM radio [8], or cellular networks [9]. In the context of wireless sensor networks (WSNs), nodes with known positions (anchors) can be used to localize the nodes with unknown positions (agents). Generally speaking, localization algorithms can be classified according to three important categories.
Centralized vs. distributed. Centralized algorithms, e.g., [10–14], require a data fusion center that carries out the computation after collecting information from the nodes, while distributed algorithms, such as [15, 16], rely on selflocalization and the computation is spread throughout the network. Centralized algorithms are likely to provide more accurate estimates, but they suffer from scalability problems, especially for largescale WSNs, while distributed algorithms have the advantage of being scalable and more robust to node failures [17].
Cooperative vs. noncooperative. In a noncooperative algorithm, e.g., [18], each agent receives information only from anchors. For all agents to obtain sufficient information to perform localization, noncooperative algorithms necessitate either long range (and highpower) anchor transmission or a highdensity of anchors [17]. In cooperative algorithms, such as [19, 20], interagent communication removes the need for all agents to be in range of one (or more) anchors [17].
Bayesian vs. nonBayesian. In nonBayesian algorithms, e.g., expectationmaximization (EM) [21], and its variant, expectationconditional maximization (ECM) [22], the unknown positions are treated as deterministic, while in Bayesian algorithms, e.g., nonparametric belief propagation (NBP) [23] and sumproduct algorithm over wireless networks (SPAWN) [24, 25] and its variant SigmaPoint SPAWN [26], the unknown positions are assumed to be random variables with a known prior distribution.
Many existing works on localization using RSS measures, such as [27] and [28], are based on the assumption that the classical path loss propagation model is perfectly known, mostly via a calibration process. However, this assumption is impractical for two reasons. Firstly, conducting the calibration process requires intensive human assistance, which may be not affordable, or may even be impossible in some inaccessible areas. Secondly, the channel characteristics vary due to multipath (fading), and nonnegligible modifications occur also due to mid to longterm changes in the environment, leading to nonstationary channel parameters [29]. This implies that the calibration must be performed continuously [29, 30] because otherwise the resulting mismatch between design assumptions and actual operating conditions leads to severe performance degradation. These facts highlight the need for algorithms that adaptively estimate the environment and the locations. A further difficulty is due to the existence of nonlineofsight (NLoS) propagation in practical localization environments. Among various works handling the NLoS effect, a majority of them have treated the NLoS meaures as outliers and tried to neglect or mitigate their effect, including the maximum likelihood (ML)based approach [31, 32], the weighted leastsquares (WLS) estimator [32, 33], the constrained localization techniques [34, 35], robust estimators [36, 37], and the method based on virtual stations [38]. In contrast to these works, several approaches, including [21, 22, 39], have proposed specific probabilistic models for the NLoS measures, therewith exploiting the NLoS measures for the localization purpose. In the light of these considerations, our aim is to develop an RSSbased, cooperative localization framework that works in mixed LoS/NLoS environments, requires no knowledge on parameters of the propagation model, and can be realized in a distributed manner.
The following distinction is made: only algorithms that directly use RSS measures as inputs are considered RSSbased in the strict sense, while algorithms such as NBP [23], SPAWN [24], distributedECM (DECM) [22], and their variants are here called pseudoRSSbased, because they use range estimates as inputs. Generally speaking, among these two options, RSSbased location estimators are preferred for the following reasons. Firstly, inferring the range estimates from the RSS measures usually requires knowledge on the (estimated) propagation model parameters. The assumption of a priori known parameters violates the calibrationfree requirement. Secondly, even with perfectly known model parameters, there exists no efficient estimator for estimating the ranges from the RSS measures, as proven in [40]. Thirdly, dropping the idealistic assumption of known channel parameters and using their estimates introduce an irremovable large bias, as demonstrated in [41]. Based on these considerations, pseudoRSSbased approaches do not meet the requirements in this work. Furthermore, Bayesian approaches, including NBP [23] and SPAWN [24, 25], do not consider mixed LoS/NLoS environments. As one representative RSSbased cooperative localization algorithm, Tomic et al.’s semidefiniteprogramming (SDP) estimator in [20] requires no knowledge on the propagation model, but it does not apply to and cannot be readily extended to a mixed LoS/NLoS environment. To the best of our knowledge, the existing works on RSSbased calibrationfree localization in a mixed LoS/NLoS environment is rather limited. Yin et al. have proposed an EMbased estimator in [21], but only for the singleagent case. In this paper, we consider a multiagent case and aim to develop a location estimator that is RSSbased, cooperative, and calibrationfree and works in a mixed LoS/NLoS environment. To capture the mixed LoS/NLoS propagation conditions, we adopt the modedependent propagation model in [21]. The key difference between this work and [21] lies in whether the localization environment is cooperative or not. More precisely, [21] is concerned with the conventional singleagent localization while this work studies cooperative localization in case of multiagent. Furthermore, we develop a distributed algorithm, where model parameters and positions are updated locally by treating the estimated agents as anchors, inspired by the works in [42–44]. A succinct characterization of the proposed work and its related works is listed in Table 1.
Original contributions: We address the problem of RSSbased cooperative localization in a mixed LoS/NLoS propagation environment, requiring no calibration. To characterize such a mixed LoS/NLoS environment, we assume a modedependent propagation model with unknown parameters. We derive and analyze a robust, calibrationfree, RSSbased distributed cooperative algorithm, based on the ML framework, which is capable of coping with mixed LoS/NLoS conditions. Simulation results, carried out in comparison with a centralized ML algorithm that serves as a benchmark, show that the proposed approach has good overall performance. Moreover, it adaptively estimates the channel parameters, has acceptable communication overhead and computation costs, thus satisfying the major requirements of a practically viable localization algorithm. The convergence analysis of the proposed algorithm is conducted by restating the problem as a graph coloring problem. In particular, we formulate a graph compatibility test and show that for compatible network structures, the convergence is guaranteed. Unlike existing algorithms, the proposed method naturally takes into account the (possible) asymmetry of links between nodes.
The paper is organized as follows. Section 2 formulates the problem and details the algorithms. Section 3 discusses convergence. Section 4 presents the simulations results, while Section 5 concludes the paper. Finally, Appendices A and B contain some analytical derivations which would otherwise burden the reading of the paper.
2 Methods/experimental
2.1 Problem formulation
Consider^{Footnote 1} a directed graph with N_{a} anchor nodes and N_{u} agent nodes, for a total of N=N_{a}+N_{u} nodes. In a twodimensional (2D) scenario, we denote the position of node i by \(\boldsymbol {x}_{i} = [\!x_{i} \ y_{i}]^{\top } \in \mathbb {R}^{2 \times 1}\), where ^{⊤} denotes transpose. Between two distinct nodes i and j, the binary variable o_{j→i} indicates if a measure, onto direction j→i, is observed (o_{j→i}=1) or not (o_{j→i}=0). In the case when i=j, since a node does not selfmeasure, we have o_{i→i}=0. This allows us to define the observation matrix \(\boldsymbol {\mathcal {O}} \in \mathbb {B}^{N \times N}\) with elements \(o_{i,j} \triangleq o_{i \rightarrow j}\) as above. The aforementioned directed graph has connection matrix \(\boldsymbol {\mathcal {O}}\). It is important to remark that, for a directed graph, \(\boldsymbol {\mathcal {O}}\) is not necessarily symmetric; physically, this models possible channel anisotropies, missdetections and, more generally, link failures. Let m_{j→i} be a binary variable, which denotes if the link j→i is LoS (m_{j→i}=1) or NLoS (m_{j→i}=0). Due to physical reasons, m_{j→i}=m_{i→j}. We define the LoS/NLoS matrix^{Footnote 2}\(\mathbf {L} \in \mathbb {B}^{N \times N}\) of elements \(l_{i,j} \triangleq m_{i \rightarrow j}\), and we observe that, since m_{j→i}=m_{i→j}, the matrix is symmetric, i.e., L^{⊤}=L. We stress that this symmetry is preserved regardless of \(\mathbf {\mathcal {O}}\), as it derives from physical reasons only. Let Γ(i) be the (open) neighborhood of node i, i.e., the set of all nodes from which node i receives observables (RSS measures), formally: \(\Gamma (i) \triangleq \{ j \neq i : o_{j \rightarrow i}=1 \}\). We define Γ_{a}(i) as the anchorneighborhood of node i, i.e., the subset of Γ(i) which contains only anchor nodes as neighbors of node i. We also define Γ_{u}(i) as the agentneighborhood of node i, i.e., the subset of Γ(i) which contains only agent nodes as neighbors of node i. In general, Γ(i)=Γ_{a}(i)∪Γ_{u}(i).
2.2 Data model
In the sequel, we will assume that all nodes are stationary and that the observation timewindow is sufficiently short in order to neglect correlation in the shadowing terms. In practice, such a model simplification allows for a more analytical treatment of the localization problem and has also been used, for example, in [11, 12]. Following the path loss model and the data models present in the literature [17, 21] and denoting by K the number of samples collected on each link over a predetermined time window, we model the received power at time index k for anchoragent links as
while, for the agentagent link,
where:

i,u, with u∈Γ_{u}(i), are the indexes for the unknown nodes;

a∈Γ_{a}(i) is an index for anchors;

k=1,…,K is the discrete time index, with K samples for each link;

\(p_{0_{\text {LOS/NLOS}}}\) is the reference power (in dBm) for the LoS or NLoS case;

α_{LOS/NLOS} is the path loss exponent for the LoS or NLoS case;

x_{a} is the known position of anchor a;

x_{u} is the unknown position of agent u (similarly for x_{i});

Γ_{a}(i), Γ_{u}(i) are the anchor and agentneighborhoods of node i, respectively;

The noise terms w_{a→i}(k),v_{a→i}(k),w_{u→i}(k), and v_{u→i}(k) are modeled as serially independent and identically distributed (i.i.d.), zeromean, Gaussian random variables, independent from each other (see below), with variances:
\(\text {Var} [\! w_{a \rightarrow i}(k) ] = \text {Var} [\! w_{u \rightarrow i}(k) ] = \sigma ^{2}_{\text {LOS}}\),
\(\text {Var}[\!v_{a \rightarrow i}(k)] = \text {Var}[\!v_{u \rightarrow i}(k)] = \sigma ^{2}_{\text {NLOS}}\),
and \(\sigma ^{2}_{\text {NLOS}} > \sigma ^{2}_{\text {LOS}} > 0\).
More precisely, letting δ_{i,j} be Kronecker’s delta^{Footnote 3}, the independence assumption is formalized by the following equations
for any k_{1},k_{2},i_{1},i_{2},j_{1}∈Γ(i_{1}),j_{2}∈Γ(i_{2}). The previous equations imply that two different links are always independent, regardless of the considered time instant. In this paper, we call this property link independence. If only one link is considered, i.e., j_{2}=j_{1} and i_{2}=i_{1}, then independence is preserved by choosing different time instants, implying that the sequence \(\left \{ w_{j \rightarrow i} \right \}_{k} \triangleq \left \{ w_{j \rightarrow i}(1), w_{j \rightarrow i}(2), \dots \right \}\) is white. The same reasoning applies to the (similarly defined) sequence {v_{j→i}}_{k}. As a matter of notation, we denote the unknown positions (indexing the agents before the anchors) by \(\boldsymbol {x} \triangleq \left [\boldsymbol {x}^{\top }_{1} \ \cdots \ \boldsymbol {x}^{\top }_{N_{u}}\right ]^{\top } \in \mathbb {R}^{2 N_{u} \times 1}\) and we define η as the collection of all channel parameters, i.e., \(\boldsymbol {\eta } \triangleq \left [\boldsymbol {\eta }^{\top }_{\text {LOS}} \ \boldsymbol {\eta }^{\top }_{\text {NLOS}}\right ]^{\top }\), with \(\boldsymbol {\eta }_{\text {LOS}} \triangleq \left [p_{0_{\text {LOS}}} \ \alpha _{\text {LOS}} \ \sigma ^{2}_{\text {LOS}}\right ]^{\top } \in \mathbb {R}^{3 \times 1}\), \(\boldsymbol {\eta }_{\text {NLOS}} \triangleq \left [p_{0_{\text {NLOS}}} \ \alpha _{\text {NLOS}} \ \sigma ^{2}_{\text {NLOS}}\right ]^{\top } \in \mathbb {R}^{3 \times 1}\).
It is important to stress that, in a more realistic scenario, channel parameters may vary from link to link and also across time. However, such a generalization would produce an underdetermined system of equations, thus giving up uniqueness of the solution and, more generally, analytical tractability of the problem. For the purposes of this paper, the observation model above is sufficiently general to solve the localization task while retaining analytical tractability.
2.3 Timeaveraged RSS measures
Motivated by a result given in Appendix A, we consider the timeaveraged RSS measures, defined as
as our new observables^{Footnote 4}. While it would have been preferable to work with the original data from a theoretical standpoint, several considerations lead to the preference of timeaveraged data, most notably: (1) comparison with other algorithms present in the literature, where the data model assumes only one sample per link, i.e., K=1, which is simply a special case in this paper; (2) reduced computational complexity in the subsequent algorithms; (3) if the RSS measures onto a given link needs to be communicated between two nodes, the communication cost is notably reduced, since only one scalar, instead of K samples, needs to be communicated; (4) formal simplicity of the subsequent equations.
Moreover, from Appendix A, it follows that, assuming known L, the ML estimators of the unknown positions based upon an original data or a timeaveraged data are actually the same. To see this, it suffices to choose \(\boldsymbol {\theta } = (\boldsymbol {x}, p_{0_{\text {LOS}}}, p_{0_{\text {NLOS}}}, \alpha _{\text {LOS}}, \alpha _{\text { NLOS}})\) and
for j∈Γ(i) and splitting the additive noise term as required. For a fixed link, only one of two cases (LoS or NLoS) is verified, thus applying (34) of Appendix A yields
where \(\sigma _{j \rightarrow i}^{2}\) is either \(\sigma ^{2}_{\text {LOS}}\) or \(\sigma ^{2}_{\text {NLOS}}\) and the general result follows from link independence.
We define \(\mathcal {R}_{i}\) as the set of all RSS measures that node i receives from anchor neighbors, i.e., \(\mathcal {R}_{i} \triangleq \left \{ r^{(m)}_{a \rightarrow i}(1), \dots, r^{(m)}_{a \rightarrow i}(K) : a \in \Gamma _{a}(i) \right \}\), \(\mathcal {Z}_{i}\) as the set of all RSS measures that node i receives from agent neighbors, i.e., \(\mathcal {Z}_{i} \triangleq \left \{ r^{(m)}_{j \rightarrow i}(1), \dots, r^{(m)}_{j \rightarrow i}(K) : j \in \Gamma _{u}(i) \right \}\), and \(\mathcal {Y}_{i}\) as the set of all RSS measures locally available to node i, i.e., \(\mathcal {Y}_{i} \triangleq \mathcal {R}_{i} \cup \mathcal {Z}_{i}\). Analogously, for timeaveraged measures, we define \(\bar {\mathcal {R}}_{i} \triangleq \left \{\bar {r}_{a \rightarrow i} : a \in \Gamma _{a}(i)\right \}\), \(\bar {\mathcal {Z}}_{i} \triangleq \left \{\bar {r}_{j \rightarrow i} : j \in \Gamma _{u}(i)\right \}\), and \(\bar {\mathcal {Y}}_{i} = \bar {\mathcal {R}}_{i} \cup \bar {\mathcal {Z}}_{i}\). Finally, we define
which represents the information available to the whole network.
2.4 Singleagent robust maximum likelihood (ML)
We first consider the singleagent case, which we will later use as a building block in the multiagent case. The key idea is that instead of separately treating the LoS and NLoS cases, e.g., by hypothesis testing, we resort to a twocomponent Gaussian mixture model for the timeaveraged RSS measures. More precisely, we assume that the probability density function (pdf), p(·), of the timeaveraged RSS measures, for anchoragent links, is given by
and, for agentagent links,
where:

λ_{i}∈(0,1) is the mixing coefficient for anchoragent links of node i;

ζ_{i}∈(0,1) is the mixing coefficient for agentagent links of node i.
Empirically, we can intuitively interpret λ_{i} as the fraction of anchoragent links in LoS (for node i), while ζ_{i} as the fraction of agentagent links in LoS (for node i). As in [21], the Markov chain induced by our model is regular and timehomogeneous. From this, it follows that the Markov chain will converge to a twocomponent Gaussian mixture, giving a theoretical justification to the proposed approach.
Assume that there is a single agent, say node i, with a minimum of three anchors^{Footnote 5} in its neighborhood (Γ_{a}(i)≥3), in a mixed LoS/NLoS scenario. Our goal is to obtain the maximum likelihood estimator (MLE) of the position of node i. Let \(\boldsymbol {\bar {r}}_{i} = \left [\bar {r}_{1 \rightarrow i} \ \cdots \ \bar {r}_{ \Gamma _{a}(i) \rightarrow i}\right ]^{\top } \in \mathbb {R}^{ \Gamma _{a}(i) \times 1}\) be the collection of all the timeaveraged RSS measures available to node i. Using the previous assumptions and the independency between the links, the joint likelihood function^{Footnote 6}\(p(\boldsymbol {\bar {r}}_{i} ; \boldsymbol {\theta })\) is given by
where θ=(x_{i},λ_{i},η). Thus, denoting with \(L(\boldsymbol {\theta }; \boldsymbol {\bar {r}}_{i})\) the loglikelihood, we have
The MLE of θ is given by
where the maximization is subject to several constraints: λ_{i}∈(0,1), α_{LOS} > 0, α_{NLOS}>0, \(\sigma ^{2}_{\text {LOS}} > 0\), and \(\sigma ^{2}_{\text {NLOS}} > 0\). In general, the previous maximization admits no closedform solution, so we must resort to numerical procedures.
2.5 Multiagent robust MLbased scheme
In principle, our goal would be to have a ML estimate of all the N_{u} unknown positions, denoted by x. Let \(\boldsymbol {\lambda } \triangleq \left [\lambda _{1} \ \cdots \ \lambda _{N_{u}}\right ]^{\top }\), \(\boldsymbol {\zeta } \triangleq \left [\zeta _{1} \ \cdots \ \zeta _{N_{u}}\right ]^{\top }\) be the collections of the mixing coefficients. Defining θ=(x,λ,ζ,η), the ML joint estimator
is, in general, computationally unfeasible and naturally centralized. In order to obtain a practical algorithm, we now resort to a suboptimal but computationally feasible and distributed approach. The intuition is as follows. Assume, for a moment, that a specific node i knows η, λ, ζ and also all the true positions of its neighbors (which we denote by \(\mathcal {X}_{i}\)). Then, the ML joint estimation problem is notably reduced, in fact,
We now make the suboptimal approximation of avoiding nonlocal information in order to obtain a distributed algorithm, thus resorting to
where we made explicit the functional dependence on all the other parameters (which, for now, are assumed known). Due to the i.i.d. hypothesis, the “local” likelihood function has the form
where the marginal likelihoods are Gaussianmixtures and we underline the (formal and conceptual) separation between anchoragent links and agentagent links. By taking the natural logarithm, we have
The maximization problem in (15) then reads
We can now relax the initial assumptions: instead of assuming known neighbors positions \(\mathcal {X}_{i}\), we will substitute them with their estimates, \(\hat {\mathcal {X}}_{i}\). Moreover, since the robust MLbased selfcalibration can be done without knowing the channel parameters η, we also maximize over them. Lastly, we maximize with respect to the mixing coefficients λ_{i},ζ_{i}. Thus, our final approach is
where \(\hat {\Gamma }_{u}(i)\) is the set of all agent neighbors of node i for which estimated positions exist. We can iteratively construct (and update) the set \(\hat {\Gamma }_{u}(i)\), in order to obtain a fully distributed algorithm, as summarized in Algorithm 1.
A few remarks are now in order. First, this algorithm imposes some restrictions on the arbitrariness of the network topology, since the information spreads starting from the agents which were able to selflocalize during initialization; in practice, this requires the network to be sufficiently connected. Second, convergence of the algorithm is actually a matter of compatibility: if the network is sufficiently connected (compatible), convergence is guaranteed. Given a directed graph, compatibility can be tested a priori and necessary and sufficient conditions can be found (see Section 4). Third, unlike many algorithms present in the literature, symmetrical links are not necessary, nor do we resort to symmetrization (like NBP): this algorithm naturally takes into account the (possible) asymmetrical links of directed graphs.
2.6 Distributed maximum likelihood (DML)
As a natural competitor of the proposed RDML algorithm, we derive here the distributed maximum likelihood (DML) algorithm, which assumes that all links are of the same type. As its name suggests, this is the nonrobust version of the previously derived RDML. As usual, we start with the singleagent case as a building block for the multiagent case. Using the assumption that all links are the same and the i.i.d. hypothesis, the joint pdf of the timeaveraged RSS measures, received by agent i, is given by
We can now proceed by estimating, with the ML criterion, first p_{0} as a function of the remaining parameters, followed by α as a function of x_{i} and finally x_{i}. We have
Defining \(s_{a,i} \triangleq 10 \log _{10} \ \boldsymbol {x_{i}}  \boldsymbol {x}_{a} \\) as the logdistance, \(\boldsymbol {s}_{i} \triangleq \left [s_{1,i} \ s_{2,i} \ \cdots \ s_{\Gamma _{a}(i),i}\right ]^{\top } \in \mathbb {R}^{\Gamma _{a}(i) \times 1}\) the columnvector collecting them and \(\boldsymbol {1}_{n} =\ [\!1 \cdots 1]^{\top } \in \mathbb {R}^{n \times 1}\) an allones vector of dimension n, the previous equation can be written as
which is a leastsquares (LS) problem and its solution is
By using this expression, the problem of estimating α as a function of x_{i} is
where, given a fullrank matrix \(\boldsymbol {A} \in \mathbb {R}^{m \times n}\), with m≥n, \(\boldsymbol {P}^{{}^{\perp }}_{\boldsymbol {A}}\) is the orthogonal projection matrix onto the orthogonal complement of the space spanned by the columns of A. It can be computed via \(\boldsymbol {P}^{{}^{\perp }}_{\boldsymbol {A}} = \boldsymbol {I}_{m}  \boldsymbol {P_{A}}\), where P_{A}=A(A^{⊤}A)^{−1}A^{⊤} is an orthogonal projection matrix and I_{m} is the identity matrix of order m. The solution to problem (24) is given by
where \(\boldsymbol {\tilde {r}}_{i} = \boldsymbol {P}^{{}^{\boldsymbol {\perp }}}_{\boldsymbol {1}_{\Gamma _{a}(i)}} \boldsymbol {\bar {r}}_{i}\) and \(\boldsymbol {\tilde {s}}_{i} = \boldsymbol {P}^{\boldsymbol {\perp }}_{\boldsymbol {1}_{\Gamma _{a}(i)}} \boldsymbol {s}_{i}\). By using the previous expression, we can finally write
which, in general, does not admit a closedform solution, but can be solved numerically. After obtaining \(\hat {\boldsymbol {x}}_{i}\), node i can estimate p_{0} and α using (23) and (25).
The multiagent case follows an almost identical reasoning of the RDML. Approximating the true (centralized) MLE by avoiding nonlocal information and assuming to already have an initial estimate of p_{0} and α, it is possible to arrive at
where (again) an initialization phase is required and the set of estimated agentsneighbors \(\hat {\Gamma }_{u}(i)\) is iteratively updated. The key difference with RDML is that, due to the assumption of the links being all of the same type, the estimates of p_{0} and α are broadcasted and a common consensus is reached by averaging. This increases the communication overhead, but lowers the computational complexity, operating a tradeoff. The DML algorithm is summarized in Algorithm 2.
Similar remarks as for the RDML can be made for the DML. Again, the network’s topology cannot be completely arbitrary, as the information must spread throughout the network starting from the agents which selflocalized, implying that the graph must be sufficiently connected. Necessary and sufficient conditions to answer the compatibility question are the same as RDML. Secondly, the (strong) hypothesis behind the DML derivation (i.e., all links of the same type) allows for a more analytical derivation, up to position estimation, which is a nonlinear leastsquares problem. However, it is also its weakness since, as will be shown later, it is not a good choice for mixed LoS/NLoS scenarios.
2.7 Centralized MLE with known nuisance parameters (CMLE)
The centralized MLE of x with known nuisance parameters, i.e., assuming known L and η, is chosen here as a benchmark for both RDML and DML. In the following, this algorithm will be denoted by CMLE. Its derivation is simple (see Appendix B) and results in
where \(\left (p_{0_{j \rightarrow i}}, \alpha _{j \rightarrow i}, \sigma ^{2}_{j \rightarrow i}\right)\) are either LoS or NLoS depending on the considered link. It is important to observe that, if all links are of the same type, the dependence from \(\sigma ^{2}_{j \rightarrow i}\) in (28) disappears. From standard ML theory [45], CMLE is asymptotically (K→+∞) optimal. The optimization problem (28) is computationally challenging, as it requires a minimization in a 2N_{u}dimensional space, but still feasible for small values of N_{u}.
3 Convergence analysis
The convergence test of our proposed algorithm (and also of DML) can be restated as a graph coloring problem: if all the graph can be colored, then it is compatible and convergence is guaranteed. As it is common in the literature on graph theory, let G=(V,E) be a directed graph, with V denoting the set of nodes and E the set of directed edges. The set of nodes is such that V=V_{a}∪V_{u}, where V_{a} is the (nonempty) set of anchor nodes and V_{u} is the (nonempty) set of agent nodes.
Definition 1
(RDMLinitializable) A directed graph G is said to be RDMLinitializable if and only if there exists at least one agent node, say x, such that Γ_{a}(x)≥3.
As can be easily checked, the previous statement is a necessary condition: if a graph is not RDMLinitializable, then it is incompatible with RDML. To give a necessary and sufficient condition, we introduce the notion of “color.” A node can be either black or white; all anchors are black and all agent nodes start as white, but may become black if some condition is satisfied. The RDML can be rewritten as a graph coloring problem. In order to do this, we define the set \(\hat {\Gamma }_{u}(i) \triangleq \{ j \in \Gamma _{u}(i) : \text {agent} \ j \ \text {is black} \}\), i.e., the subset of agent neighbors of node i which contains only black agent nodes. In general, \(\hat {\Gamma }_{u}(i) \subseteq \Gamma _{u}(i)\). We also define the set \(B_{u} \triangleq \{ i \in V_{u}: \text {agent } {i} \text { is black} \}\), i.e., the set of black agents. In general, B_{u}⊆V_{u}. Given a graph G, we can perform a preliminary test by running the following RDMLcoloring algorithm (a better test will be derived later):

Initialization (k=0)

1
All anchors are colored black and all agents white;

2
Every agent i with Γ_{a}(i)≥3 is colored black;

1

Iterative coloring: Start with k=1

1
Every agent with \(\left  \Gamma _{a}(i) + \hat {\Gamma }^{(k1)}_{u}(i) \right  \geq 3\), where \(\hat {\Gamma }^{(k)}_{u}(i)\) is the set \(\hat {\Gamma }_{u}(i)\) at step k, is colored black;

2
Every agent j updates is own \(\hat {\Gamma }^{(k)}_{u}(j)\) with the new colored nodes;

3
The set \(B^{(k)}_{u}\) is updated, where \(B^{(k)}_{u}\) is the set B_{u} at step k;

4
Set k←k+1 and repeat the previous steps until V_{u} contains only black nodes.

1
Suppose that the previous algorithm can color the entire graph black in a finite amount of steps, say n. Then, n is called RDMLlifetime.
Definition 2
(RDMLlifetime) A directed graph G is said to have RDMLlifetime equal to n if and only if the RDMLcoloring algorithm colors black the set V_{u} in exactly n steps. If no such integer exists, by convention, n=+∞.
This allows us to formally define compatibility:
Definition 3
(RDMLcompatibility) A directed graph G is said to be RDMLcompatible if and only if:

1
G is RDMLinitializable;

2
the RDMLlifetime of G is finite.
Otherwise, G is said to be RDMLincompatible.
In practice, there are only two ways for which a graph is RDMLincompatible: either G cannot be initialized, or the RDMLlifetime of G is infinite. Testing the first condition is trivial; the interesting result is that testing the second condition is made simple thanks to the following.
Theorem 1
An RDMLinitializable graph G is RDMLincompatible if and only if there exist an integer h such that
that is, if there is a step h in which no more agents can be colored black and at least one agent is still left white.
Proof
(⇐) First, observe that, by construction, \(B^{(k1)}_{u} \subseteq B^{(k)}_{u}\), as black nodes can only be added. Let \(C_{G} : \mathbb {N} \rightarrow \mathbb {N}\) be the following function
Since \(B^{(k1)}_{u} \subseteq B^{(k)}_{u}\), C_{G} is nondecreasing. An RDMLinitializable graph is RDMLcompatible if and only if it has finite RDMLlifetime, i.e., there must exist n such that C_{G}(n)=V_{u}. But condition (29) implies that there exists h such that C_{G}(h)=C_{G}(h−1)<V_{u}. At step h+1 and all successive steps, C_{G} cannot increase since the set \(B^{(k)}_{u}\) cannot change. To show this, the key observation is that, as the graph G at step h−1 did not satisfy the conditions for \(B^{(h1)}_{u}\) to grow (by hypothesis), the set was equal to itself at step h, i.e., \(B^{(h1)}_{u} = B^{(h)}_{u}\). But since no color change happened in V_{u} at step h, the graph G still does not satisfy the conditions for \(B^{(k)}_{u}\) to grow for k≥h. Thus, C_{G}(k) becomes a constant function for k≥h and can never reach the value V_{u}.
(⇒) Since G is an RDMLincompatible graph by hypothesis, at least one agent must be white, so \(\left B_{u}^{(h)}\right  < V_{u}\) is true for any h. Since C_{G}(k) is nondecreasing, it must become constant for some h>k, but this implies that, for some h, \(\left B^{(h)}_{u}\right  = \left B^{(h1)}_{u}\right \). This implies that, since \(B^{(k1)}_{u} \subseteq B^{(k)}_{u}\) by construction, \(B^{(h)}_{u} = B^{(h1)}_{u}\) for some h. □
Definition 4
(RDMLdepth) Let G be a directed graph. Then,
is called the RDMLdepth of G.
A complete graph has h_{G}=0, as all agents are colored black during the initialization phase of the RDMLcoloring algorithm.
Corollary 1
Let G be a directed graph. Then, h_{G}≤n, where n is the RDMLlifetime of G.
Proof
If G is not RDMLinitializable, h_{G}=0 as \(B^{(0)}_{u} = \emptyset \). If G is RDMLinitializable, there are two cases: either n is finite or not. In the latter, n=+∞ and h_{G} is finite by previous theorem. If n is finite, h_{G}=n since \(\left B^{(n)}_{u}\right  = V_{u}\) by definition of RDMLlifetime. □
The previous corollary proves that h_{G} is always finite, regardless of G. This allows us to write the graph compatibility test, shown in Algorithm 3. Thanks to the previous results, this algorithm always converges and can be used to test a priori if a graph is RDMLcompatible or not.
Remark Algorithm 3 can be intuitively explained via a physical metaphor, where, in a metal grid (representing the graph), “heat” (information) spreads out starting from some initial “hot spots” (nodes that are colored black in the first iteration). This spreading is continued, reaching more and more locations on the grid, until the event occurs that further spreading of “heat” does not change the “heat map.” If, at this point, there are cold spots (nodes that have not been colored black), the graph is RDMLincompatible. By contrast, if heat spreads throughout the grid, the graph is RDMLinitializable.
Figures 1 and 2 show two examples of RDMLinitializable graphs. Figure 1 illustrates the case of a small (N_{a}=4,N_{u}=4) but highly connected network. In contrast, Fig. 2 displays a larger (N_{a}=15,N_{u}=20), but weakly connected network (graph depth = 7). In particular, only 90 out of 680 possible directed links are connected, where anchors have three to four links (out of 20). For both cases, graph compatibility can be shown using Algorithm 3.
4 Results and discussion
In this section, we use standard Monte Carlo (MC) simulation to evaluate the performance of the proposed algorithm and its competitors. As a performance metric, we will show the ECDF (empirical cumulative distribution function) of the localization error, defined as \(e_{i} \triangleq \left \ \hat {\boldsymbol {x}}_{i}  \boldsymbol {x}_{i} \right \\) for agent i, i.e., an estimate of the probability \(\mathcal {P}\left \{ \left \ \hat {\boldsymbol {x}}_{i}  \boldsymbol {x}_{i} \right \ \leq \gamma \right \}\). The ECDFs are obtained by stacking all the localization errors for every agent in a single vector, in order to give a global picture of the algorithm performances. The simulated scenario is as follows. In a square coverage area of 100×100 m^{2}, N_{a}=11 stationary anchors are deployed, as depicted in Fig. 3. The channel parameters are generated as follows: \(p_{0_{\text {LOS}}} \sim \mathcal {U}[30, 0]\), \(\alpha _{\text {LOS}} \sim \mathcal {U}[2,4]\), and σ_{LOS}=6, while, for the NLoS case, \(p_{0_{\text {NLOS}}} \sim \mathcal {N}(0, 25)\), \(\alpha _{\text {NLOS}} \sim \mathcal {U}[3,6]\), and σ_{NLOS}=12. Similar settings on the reference power and the path loss exponent can be found in [46, 47], respectively. At each MC trial, N_{u}=10 agents are randomly (uniformly) generated in the coverage area. Unless stated otherwise, K=40 samples per link are used. Finally, each simulation consists of 100 MC trials.
The optimization problems (27) and (19) have been solved as follows. For DML, a 2D grid search has been used, while, for RDML and CMLE, the optimization has been performed with the MATLAB solver fmincon.
4.1 Gaussian noise
Here, we validate our novel approach by considering fully connected networks in three different scenarios. In Fig. 4, all links are assumed to be LoS. As can be seen from the ECDFs, the robust approach has similar performances to the DML algorithm, which was designed assuming all links to be of the same type. In Fig. 5, all links are assumed to be NLoS and again we see a similar result. The key difference is in Fig. 6, where a mixed LoS/NLoS scenario is considered and the fraction of NLoS links is random (numerically generated as \(\mathcal {U}[0,1]\) and different for each MC trial). Here, the robust approach outperforms DML, validating our strategy. Moreover, it has a remarkably close performance to the centralized algorithm CMLE. As a result, the robust approach is the preferred option in all scenarios.
4.2 NonGaussian noise for NLoS
In order to evaluate robustness, we consider a model mismatch on the noise distribution by choosing, for the NLoS links, a Studentt distribution with ν=5 degrees of freedom (as usual, serially i.i.d. and independent from the LoS noise sequence). For brevity, we consider only the case of a fully connected network in a mixed LoS/NLoS scenario. As shown by Fig. 7, our proposed approach is able to cope with nonGaussianity without a significant performance loss, thanks to the Gaussian mixture model.
4.3 Cooperative gain
The proposed algorithm exhibits the so called “cooperative gain,” i.e., a performance advantage (according to some performance metric, typically localization accuracy) with respect to a noncooperative approach, where each node tries to localize itself only by selflocalization (12). In our case, the cooperative gain is twofold: first, it allows to improve localization accuracy; second, it allows to localize otherwise nonlocalizable agents. To show the first point, we consider a mixed LoS/NLoS environment (for simplicity, with white Gaussian noise) and a network is generated at each Monte Carlo trial, assuming a communication radius R=70 m, with an ideal probability of detection model [17]. Moreover, the network is generated in a way as to allow all agents to selflocalize and, as a consequence, it is RDMLcompatible. In Fig. 8, the ECDFs of the localization error are shown. To show the second point, we consider the toy network depicted in Fig. 9. In this example, agent X is not able to selflocalize, since no anchors are in its neighborhood, Γ_{a}(X)=∅. Thus, in a noncooperative approach, the position of agent X cannot be uniquely determined^{Footnote 7}, while, in a cooperative approach, the position of agent X can actually be obtained by exchanging information, e.g., the estimated positions of its neighbors.
4.4 Variable K and NLoS fraction
We next analyze the RDML algorithm by varying the number of samples per link, K, and the fraction of NLoS links. In both cases, we consider (for brevity) fully connected networks with white Gaussian noise; for variable K, the fraction of NLoS is randomized, while, for variable NLoS fraction, K is fixed. In Fig. 10, we observe that the performance increases as K increases, as expected. In Fig. 11, where each point represents 100 MC trials, the median error is chosen as a performance metric^{Footnote 8} and RDML shows good performances and is not significantly affected by the actual NLoS fraction, which is evidence for its robustness, while DML is clearly inferior and suffers from model mismatch.
4.5 Communication and computational costs
The communication overhead is evaluated by computing the number of elementary messages sent throughout the network, where an elementary message is simply defined as a single scalar. Sending a ddimensional message, e.g., (x,y) coordinates, is equivalent to sending d elementary messages. In Fig. 12, the communication overhead of RDML and DML is plotted with respect to the number of agents, assuming a complete graph: for N_{u} agents, 2N_{u}(N_{u}−1) scalars, representing estimated 2D positions, are needed for RDML, while 4N_{u}(N_{u}−1) are needed for DML, as estimates of (p_{0},α) are also necessary. Viewing (2D) position has the fundamental information of a message (d=2), in a complete graph, RDML achieves the theoretical minimum cost in order for all nodes to have complete information, while DML uses auxiliary information. For a general graph, the communication overhead depends on the specific graph topology, but is never greater than the aforementioned value. Thus, for a general graph, the communication cost is upperbounded by a quadratic function of N_{u}.
Regarding computational complexity, both RDML and DML scale linearly with N_{u} and they benefit from parallelization, as the main optimization task can be executed independently for each involved node. As already mentioned, DML operates a tradeoff between communication and computational complexity; in fact, the DML optimization problem (27) is easier to solve than the RDML optimization problem (19). Both problems are nonconvex and may have local minima/maxima, so care must be taken in the optimization procedure.
5 Conclusions
We have developed a novel, robust MLbased scheme for RSSbased distributed cooperative localization in mixed LoS/NLoS scenarios, which, while not being optimal, has good overall accuracy, is adaptive to the environment changes, is robust to NLoS propagation, including nonGaussian noise, and has communication overhead upperbounded by a quadratic function of the number of agents and computational complexity scaling linearly with the number of agents, also benefiting from parallelization. The main original contributions are that, unlike many algorithms present in the literature, the proposed approach (a) does not require calibration and (b) does not require symmetrical links (nor does it resort to symmetrization), thus naturally accounting for the topology of directed graphs with asymmetrical links as a result of missdetections and channel anisotropies. To the best of the authors’ knowledge, this is the first distributed cooperative RSSbased algorithm for directed graphs. The main disadvantage is imposing some restrictions on the arbitrariness of networks’ topology, but these restrictions disappear for sufficiently connected networks. We also derive a compatibility test based on graph coloring, which allows to determine whether the given network is compatible. If it is compatible, convergence of the algorithm is guaranteed. Future work may include the consideration of possible approximations, in order to extend this approach to more general networks, and of alternative models, overcoming the limitations of the standard path loss model.
6 Appendix A: On using the timeaveraged sample mean
Let \(\Theta \subseteq \mathbb {R}^{d}\) be the (nonempty) parameter space, θ∈Θ unknown deterministic parameters and \(s : \Theta \rightarrow \mathbb {R}\) a function. Consider the following model
where k=1,…,K and the noise sequence w(k) is zeromean, i.i.d. Gaussian, with deterministic unknown variance σ^{2}>0. Let
be the sample mean and let p(y(1),…,y(k);θ,σ^{2}) denote the joint likelihood function. Then,
Proof
Since the observations are independent,
We can now focus on the term in the exponential. By adding and subtracting \(\bar {y}\) from the quadratic term and expanding it, we get
Observing now that
we are left with
Thus, the joint pdf can be factorized as follows
As a byproduct, by invoking the NeymanFisher factorization theorem [45] and assuming known σ^{2}, \(\bar {y}\) is a sufficient statistic for θ. Resuming our proof, we can now observe that
since no term in the summation depends on k. Thus,
On the other hand,
which completes the proof. □
7 Appendix B: CMLE derivation
The MLE of x is given by
where p(Υ;x) is the joint likelihood function. Since all other parameters are assumed known, the set of all timeaveraged measures \(\bar {\Upsilon } = \cup _{i=1}^{N_{u}} \bar {\mathcal {Y}_{i}}\) is a sufficient statistic for x (see Appendix A), from which it follows that
By link independence,
where \(p(\bar {r}_{j \rightarrow i} ; \boldsymbol {x})\) is the marginal likelihood function. Thus, we have
Taking the natural logarithm and neglecting constants,
which completes the derivation.
Notes
Throughout the paper, vectors and matrices will be denoted in bold, ∥v∥ denotes the Euclidean norm of vector v, \( \mathcal {A} \) denotes the cardinality of set \(\mathcal {A}\). We denote by \(\mathbb {E}[X]\) and Var[X] the statistical expectation and variance, respectively, of random variable X. Finally, \(\mathbb {B} = \{0,1\}\) is the Boolean set.
The values on the main diagonal are arbitrary. Here we choose m_{i→i}=1.
δ_{i,j}=1 if and only if i=j, zero otherwise.
For better readability, the notation ^{(m)} has not been carried over, as it implicit in the formalism.
The reason for this is that localizing a node in 2D requires at least three anchors.
Hereafter, we omit the conditioning on the set {o_{n→i}} of actually observed RSS measures (received by node i) in the joint likelihood function, since it is implicit in the neighborhood formalism.
This easily follows by observing that the deterministic (noiseless) version of all the relevant equations for agent X admit infinite solutions.
This is due to the fact that RSME (Root Mean Square Error) is not a suitable metric when the Error ECDFs are very longtailed, as in our case.
Abbreviations
 BP:

Belief propagation
 CMLE:

Centralized maximum likelihood estimator
 DECM:

Distributed expectationconditional maximization
 DML:

Distributed maximum likelihood
 ECDF:

Empirical cumulative distribution function
 ECM:

Expectationconditional maximization
 EM:

Expectationmaximization
 LoS:

Lineofsight
 LS:

Least squares
 MC:

Monte Carlo
 ML:

Maximum likelihood
 MLE:

Maximum likelihood estimator
 NBP:

Nonparametric belief propagation
 NLoS:

Nonlineofsight
 RDML:

Robust distributed maximum likelihood
 RF:

Radiofrequency
 RMSE:

Root mean square error
 RSS:

Received signal strength
 RSSI:

Received signal strength indicator
 SDP:

Semidefiniteprogramming
 SPAWN:

Sumproduct algorithm over wireless networks
 WLS:

Weighted least squares
 WSN:

Wireless sensor network
 2D:

2dimensional
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Acknowledgements
The authors would like to thank prof. F. Bandiera from University of Salento for having started the collaboration which lead to this work.
Funding
The work of L. Carlino was supported by the “Erasmus+ Traineeship” programme of University of Salento. The work of M. Muma was supported by the “Athene Young Investigator Programme” of Technische Universität Darmstadt.
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The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
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LC has contributed towards the development of the proposed algorithms and the performance analysis. DJ has contributed towards the introduction, the related work, and with other minor revisions throughout the paper. MM has contributed towards the example networks regarding graph connectivity, the overall organization of the paper, and with other minor revisions throughout the paper. As the supervisor, AMZ has proofread the paper several times and provided guidance throughout the whole preparation of the manuscript. All authors read and approved the final manuscript.
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LC received his B.Sc. degree with full marks in Information Engineering at University of Salento, Lecce, Italy, in 2012. He received his M.Sc. degree in Telecommunications Engineering (summa cum laude) at University of Salento, in 2014. He started his Ph.D. studies at University of Salento in 2014 and finished his Ph.D. in 2018. His main area of research is signal processing, while his research topic is applying signal processing techniques to the problem of localization, with a specific focus on localization based upon received signal strength. His other research interests include radar applications, target tracking, data mining and information theory.
DJ received the B.Sc. degree in information and communication engineering from Zhejiang University, Hangzhou, China, in 2011, and the M.Sc. degree in electrical engineering and information technology from Technische Universität Darmstadt, Darmstadt, Germany, in 2014. She is currently working towards the Ph.D. degree in the Signal Processing Group at Technische Universität Darmstadt. Her research interests include localization and tracking, distributed and cooperative inference in wireless networks.
MM received the Dipl.Ing (2009) and the Dr.Ing. degree (summa cum laude) in Electrical Engineering and Information Technology (2014), both from Technische Universität Darmstadt, Darmstadt, Germany. He completed his diploma thesis with the Contact Lens and Visual Optics Laboratory, School of Optometry, Brisbane, Australia, on the role of cardiopulmonary signals in the dynamics of the eye’s wavefront aberrations. Currently, he is a Postdoctoral Fellow at the Signal Processing Group, Institute of Telecommunications and has recently been awarded Athene Young Investigator of Technische Universität Darmstadt. His research is on robust statistics for signal processing with applications in biomedical signal processing, wireless sensor networks, and array signal processing. MM was the supervisor of the Technische Universität Darmstadt student team who won the international IEEE Signal Processing Cup 2015. MM coorganized the 2016 Joint IEEE SPS and EURASIP Summer School on Robust Signal Processing. In 2017, together with his coauthors, MM received the IEEE Signal Processing Magazine Best Paper Award for the paper entitled “Robust Estimation in Signal Processing: A tutorialstyle treatment of fundamental concepts". In 2017, he was elected to the European Association for Signal Processing (EURASIP) Special Area Team in Theoretical and Methodological Trends in Signal Processing (SATTMSP).
AZ is a Fellow of the IEEE and IEEE Distinguished Lecturer (Class 2010 2011). He received his Dr.Ing. from RuhrUniversität Bochum, Germany in 1992. He was with Queensland University of Technology, Australia from 1992–1998 where he was Associate Professor. In 1999, he joined Curtin University of Technology, Australia as a Professor of Telecommunications. In 2003, he moved to Technische Universität Darmstadt, Germany as Professor of Signal Processing and Head of the Signal Processing Group. His research interest lies in statistical methods for signal processing with emphasis on bootstrap techniques, robust detection and estimation and array processing applied to telecommunications, radar, sonar, automotive monitoring and safety, and biomedicine. He published over 400 journal and conference papers on the above areas. AZ served as General Chair and Technical Chair of numerous international conferences and workshops; more recently he was the Technical CoChair of ICASSP14 held in Florence, Italy. He also served on publication boards of various journals, notably as EditorInChief of the IEEE Signal Processing Magazine (2012–2014). AZ was the Chair (2010–2011) of the IEEE Signal Processing Society (SPS) Technical Committee Signal Processing Theory and Methods (SPTM). He served on the Board of Governors of the IEEE SPS (2015–2017) and is the president of the European Association of Signal Processing (EURASIP) (2017–2018).
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Carlino, L., Jin, D., Muma, M. et al. Robust distributed cooperative RSSbased localization for directed graphs in mixed LoS/NLoS environments. J Wireless Com Network 2019, 19 (2019). https://doi.org/10.1186/s1363801813357
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DOI: https://doi.org/10.1186/s1363801813357