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        Bayesian Cooperative Localization Using 
        Received Signal Strength With Unknown Path 
        Loss Exponent: Message Passing Approaches 
        Di Jin, Feng Yin, Carsten Fritsche, Fredrik Gustafsson and Abdelhak M. Zoubir 
        The self-archived postprint version of this journal article is available at Linköping 
        University Institutional Repository (DiVA): 
        http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-165498 
          
          
        N.B.: When citing this work, cite the original publication. 
        Jin, Di, Yin, F., Fritsche, C., Gustafsson, F., Zoubir, A. M., (2020), Bayesian Cooperative Localization 
        Using Received Signal Strength With Unknown Path Loss Exponent: Message Passing Approaches, 
        IEEE Transactions on Signal Processing, 68, 1120-1135. https://doi.org/10.1109/TSP.2020.2969048 
        Original publication available at: 
        https://doi.org/10.1109/TSP.2020.2969048 
        Copyright: Institute of Electrical and Electronics Engineers (IEEE) 
        http://www.ieee.org/index.html 
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        IEEE.  
             
         
                            
                                                                                                                                                                           1
                 Bayesian Cooperative Localization Using Received
               Signal Strength With Unknown Path Loss Exponent:
                                                  Message Passing Approaches
                                      Di Jin, Feng Yin, Carsten Fritsche, Fredrik Gustafsson, and Abdelhak M. Zoubir
                   Abstract—We propose a Bayesian framework for the received-                   approach [7], convex-optimization-based algorithms [13]–[17],
               signal-strength-based cooperative localization problem with un-                  multidimensional scaling (MDS) [18], [19] and expectation-
               known path loss exponent. Our purpose is to infer the marginal                   conditional maximization (ECM) [20]. On the other hand, the
               posterior of each unknown parameter: the position or the path                    Bayesian approaches treat the positions as random variables
               loss exponent. This probabilistic inference problem is solved using              and formulate cooperative localization as a probabilistic in-
               message passing algorithms that update messages and beliefs
               iteratively. For numerical tractability, we combine the variable                 ference problem. These approaches take advantage of prior
               discretization and Monte-Carlo-based numerical approximation                     information of parameters. Most importantly, the posterior
               schemes. To further improve computational efficiency, we develop                  distribution of each position is inferred, which contains much
               an auxiliary importance sampler that updates the beliefs with                    more information than just one deterministic point estimate,
               the help of an auxiliary variable. An important ingredient of the                e.g., the modality of the position and its associated uncertainty.
               proposed auxiliary importance sampler is the ability to sample
               from a normalized likelihood function. To this end, we develop                   Representative probabilistic approaches include the nonpara-
               a stochastic sampling strategy that mathematically interprets                    metric belief propagation (NBP) [21], [22], sum-product algo-
               and corrects an existing heuristic strategy. The proposed mes-                   rithm over a network (SPAWN) [8] and their low-complexity
               sage passing algorithms are analyzed systematically in terms                     variants [23]–[26].
               of computational complexity, demonstrating the computational                        Among different position-related signal metrics, received
               efficiency of the proposed auxiliary importance sampler. Various
               simulations are conducted to validate the overall good perfor-                   signal strength (RSS) has gained much attention due to its
               mance of the proposed algorithms.                                                ubiquitousness in wireless radio frequency signals [27]. For
                   Index Terms—Belief propagation, cooperative localization,                    instance, an RSS indicator (RSSI) has been encoded in the
               message passing, received signal strength, stochastic sampling.                  IEEE 802.15.4 standards [28]. Despite its comparatively high
                                                                                                uncertainty about position, RSS measurement can be exploited
                                                                                                to enable low-cost, simple and opportunistic localization sys-
                                          I. INTRODUCTION                                       tems, without the need of additional hardware. However, many
                      OCATION awareness plays a key enabling role in the                        existing works on RSS-based localization, such as [14], [29],
               Lexpending location-based services and has a wide range                          are based on the assumption that the classical path loss
               of emerging applications, such as medical services [1], in-                      propagation model is perfectly known. This oversimplified
               telligent transportation systems [2], Internet of Things [2],                    assumption is impractical for two reasons. Firstly, the estima-
               autonomous vehicles [3], crowd sensing [4], [5] and smart                        tion of these model parameters usually relies on a laborious
               environment [6]. Cooperative localization [7]–[10], where all                    calibration phase, where a large amount of training data needs
               internode measurements can be exploited, has many appeal-                        to be collected and processed. However, such a calibration
               ing advantages, among others, expanding the capabilities of                      step is very time consuming and even impossible in many
               locating positions without ambiguity and improving the per-                      scenarios, such as monitoring and surveillance applications
               formance on estimation accuracy. The benefits of cooperation                      in hostile or inaccessible environments [30]. Secondly, these
               among nodes have been theoretically demonstrated in [11],                        model parameters, particularly the path loss exponent (PLE),
               [12]. The existing cooperative localization algorithms can be                    are time varying, due to the changing environment, e.g.,
               categorized into deterministic and probabilistic approaches,                     weather conditions or human behaviors [31], [32]. Without a
               depending on whether the localization problem is formulated                      frequent recalibration, the resulting mismatch will significantly
               in a probabilistic manner, In the first category, the positions                   deteriorate the localization performance. To overcome this
               (and model parameters if any) are assumed to be deterministic                    problem, these model parameters should be assumed unknown
               but unknown, and only a deterministic point estimate is                          and jointly estimated with the positions.
               provided for each unknown parameter. Classical approaches,                          In this paper, we focus on the case with an unknown PLE,
               to mention some, include the maximum likelihood (ML)                             because a slight deviation of PLE may severely deteriorate the
                                                                                                localization performance, as theoretically and algorithmically
                  D. Jin and A. M. Zoubir are with the Signal Processing Group at Technische    demonstrated in [33], [34]. For the case of noncooperative
                         ¨
               Universitat Darmstadt, Darmstadt, Germany. F. Yin (corresponding author) is      localization, there exist several works dealing with unknown
               with the School of Science and Engineering at Chinese University of Hong         PLE. In [30], the target position and the PLE are estimated
               Kong,Shenzhen,China.C.FritscheandF.GustafssonarewiththeDepartment
                                                 ¨                     ¨
               of Electrical Engineering at Linkoping University, Linkoping, Sweden.            jointly by solving an ML problem using the Levenberg-
                                                                                                                                                2
             Marquardt algorithm. In [34], [35], the ML problem is first          in Section VII. Finally, Section VIII concludes the paper.
             relaxed by linearizing the problem and then simplified by              Notation: Throughout this paper, boldface lowercase letter x
             replacing the position variable with a function of the PLE          is reserved for vectors. k·k stands for the Euclidean norm, and
             variable. By doing so, the cost function depends only on            | · | denotes the cardinality of a set. N(µ,σ2) denotes a Gaus-
             the one-dimensional (1D) PLE variable, and the resulting            sian distribution with mean µ and variance σ2; U [a,b) denotes
             optimization problem can be readily solved using grid search.       a uniform distribution with boundaries a and b; logN(µ,σ2)
             In [30], [36], the location is estimated by eliminating the         denotes a log-normally distributed random variable x with µ
             nuisance parameter: the PLE parameter (or several other model       and σ2 being the mean and variance of logx. f(·) and p(·) are
             parameters). The original ML problem in [30] is simplified by        reserved for probability density functions (pdf) and probability
             representing the PLE as a function of the position variable in      mass functions, respectively. f   denotes the pdf of a Gaussian
                                                                                                                N
             [33]. In [37], along with several model parameters, the location    distribution. Γ\i represents a set consisting all elements in the
                                                                                                                   l L
             is estimated based on the expectation and maximization crite-       set Γ excluding the element i. {x }     symbolizes a collection
                                                                                                        	           l=1
             rion. In [38], [39], the location and the PLE are estimated in an   of samples   x1,...,xL .
             alternating manner. More precisely, the position is estimated
             based on an initialized (or estimated) PLE, and afterwards the                      II. PROBLEM FORMULATION
             PLE is estimated based on the updated position estimate. This         Consider a wireless sensor network (WSN) in 2D space
             procedure iterates until certain termination condition is met. In   with two types of nodes: blindfolded nodes (agents) with
             the cooperative case, RSS-based localization with an unknown        unknown locations and reference nodes (anchors) with known
             PLE is even more challenging. To the best of our knowledge,                                       T
                                                                                 locations. Let x    = [x ,y ]    denote the location of each
                                                                                                  i        i  i
             only very limited works exist, including [15], [17], [39], where    node, where i ∈ Su = {1,...,Nu} for an agent and
             the alternating strategy is adopted to handle the unknown           i ∈ S = {N +1,...,N} for an anchor. The index set of all
                                                                                      a       u                                S
             PLE, like in the noncooperative case. In our view, despite its      nodes is denoted by S, and we have S = Su        Sa. Two nodes
             straightforwardness and simplicity, such an alternating strategy    i and j are neighbors if they are allowed to communicate with
             is quite heuristic and lack of theoretical support.                 each other. We denote the index set of node i’s neighbors by
                In contrast to previous works, we treat the PLE as a             Γ .
                                                                                  i
             random variable and formulate the problem in a Bayesian               Using the well-known log-distance path loss propagation
             framework. The reasons are as follows. First, when the PLEs         model, the RSS measurement rij, coming from node i and
             between different propagation links differ, a random variable       received by node j, is given by
             characterizing the averaging behavior of the collection of all                   r   =A −10αlog (d /d )+v ,                      (1)
             PLEs is more suitable than just one deterministic PLE value.                      ij     i           10  ij   0     ij
             Second, characterizing the PLE as a random variable enables         where d0 is a predefined reference distance. Ai denotes the
             us to integrate any prior information, if available, into the       reference power in dBm at d0, and is assumed to be known.
             parameter estimation. Under the Bayesian umbrella, the coop-        α denotes the path loss exponent (PLE) that is assumed to
             erative localization problem with an unknown PLE becomes            be an unknown random variable and d        ,kx −x k is the
                                                                                                                          ij      i     j
             a probabilistic inference problem. In this problem, we derive       Euclidean distance. v    stands for the log-normal shadowing
                                                                                                       ij
             message passing algorithms to infer the marginalized posterior      error that is modeled by v     ∼ N(0,σ2). We assume mea-
                                                                                                             ij           ij
             distribution of each unknown parameter: the position or the         surements to be symmetric, i.e., r   and r . The collection of
                                                                                                                    ij      ji
             PLE. For mathematical tractability, we combine the variable         all RSS measurements is denoted by r , {rij : (i,j) ∈ Γ},
             discretization and Monte-Carlo-based numerical approxima-           where (i,j) represents that nodes i and j are neighbors, and
             tion mechanisms. In addition, to reduce the computational           Γ , {(i,j) : j ∈ Γ and j > i;i ∈ S } denotes the set of all
                                                                                                    i                  u
             complexity, we propose an auxiliary importance sampler for          pairs of neighboring nodes. In line with the majority of the
             belief update that has a complexity order scaling linearly with     existing works, we assume that these shadowing measurement
             the number of samples. Moreover, we develop a novel strategy        errors v  for all (i,j) ∈ Γ are independent. The distribution
                                                                                         ij
             for sampling from a normalized likelihood function, which           of v , denoted by f     (v ), is assumed to be known.
                                                                                     ij               vij  ij
             plays an important role in the auxiliary importance sampler           It is important to stress that the propagation environment
             andmathematically interprets and corrects an existing heuristic     implied by the measurement model (1) is homogeneous, i.e.,
             sampling strategy. The proposed sampling strategy will benefit       all links share a common PLE α. On the other hand, an
             many existing works, such as [8], [22], since this task is an       inhomogeneous environment is defined for links with different
             embedded step in many message-passing-based cooperative             PLEs. The same definitions can be found in [40] as well.
             localization algorithms.                                            It is true that each propagation link, e.g., the connection
                This paper is organized as follows: In Section II, we            betweennodeiandnodej,shouldhaveanindividualPLEα .
                                                                                                                                              ij
             formulate the RSS-based cooperative localization problem            However, this will lead to an under-determined problem, since
             with an unknown PLE mathematically. Fundamental concepts            besides the unknown locations, there will be one unknown
             in message passing algorithms are given in Section III. We          parameter α     associated with each measurement r . As a
                                                                                              ij                                        ij
             discuss how to approximate the messages in Section IV and           compromise between model accuracy and feasibility, a ho-
             demonstrate how to update the beliefs approximately in Sec-         mogeneous propagation environment is considered throughout
             tion V. Some important issues are discussed in Section VI. The      this work. It is noticeable that even for a homogeneous prop-
             proposed algorithms are evaluated using extensive simulations       agation environment, the problem of RSS-based cooperative
                                                                                                                                                      3
              localization with an unknown PLE is readily challenging and                               fj                       f
                                                                                      f , t ∈ Γ \i                                i    f , s ∈ Γ \j
              rarely studied.                                                          tj        j                                      si        i
                From a Bayesian perspective, we treat the PLE α and each                                                m (x )
                                                                                                                          ij   i
              position xi,i ∈ S, as random variables, whose prior distribu-                  .                                                .
                                                                                             .          x           f            x            .
              tions are denoted by f(α) and f(xi),i ∈ S, respectively. All                   .            j           ij          i           .
              positions and the PLE variable are assumed to be mutually
              independent, i.e., f(α,x1,...,xN) = f(α)·f(x1)···f(xN).               Fig. 1: An illustration of factor graph and belief propagation. For
              Our purpose is to infer the marginalized posterior distribution       clarity, the overlap between two rectangle blocks are omitted. Here,
              (marginal posterior) of each unknown parameter, which is              f is the short notation for f(x ) and f   for the likelihood function
              f(α|r) or f(x |r), i ∈ S , from the measurements r and the             i                             i       ij
                                                                                    f(r |x ,x ).
                             i           u                                              ij  i  j
              prior information about all parameters.
                III. FUNDAMENTALS ON COOPERATIVE LOCALIZATION                       (m˜ (xj)) on position xj, taking into account the likelihood
                                                                                                    
                                                                                                    
                                  VIA MESSAGE PASSING                               function f(rij xi,xj). Collecting all messages from factors
                To infer the marginal posterior of the PLE variable α and           that are connected to position variable xi, belief at xi, denoted
              that of each position x ,i ∈ S , we start with the joint poste-       by Bn(xi), is obtained as
                                      i       u                                                                       Y
              rior distribution f(x1,...,xN,α|r). Under the assumptions                            Bn(xi) ∝ f(xi)         mn        (xi).
                                                                                                                            f →x
              made in the preceding section, it has the form of                                                              ij    i
                                                                                                                     j∈Γ
                                                                                                                       i
                                             N
              f(x1,...,xN,α|r)∝ f(α)Y f(xi) Y f(rij|xi,xj,α) .                      The belief Bn(xi) can be interpreted as collecting the sta-
                                             i=1      j∈Γ ,j>i                      tistical information on xi coming from all its neighbors
                                                          i                            n
                                                                              (2)   (m         (x ), ∀j ∈ Γ ) and its own prior (f(x )). Such
                                                                                       f →x ,     i             i                            i
                                                                                        ij   i
              Intuitively, the marginal posterior, e.g., f(xi |r), can be cal-      message passing procedure iterates until certain termination
              culated as follows:                                                   condition is fulfilled. Upon convergence, the belief B(xi) is
                               Z      Z                                                                                                  
                                                                                                                                         
                                                                                    an approximation of the marginal posterior f(xi r).
                  f(xi|r) =       · · ·  f(x1,...,xN,α|r) dx1:N\idα.
              However, this is intractable due to the high dimensionality of        B. Proposed Message Passing Algorithms
              the problem. A well-know local message passing algorithm,                Despite the fact that several works, e.g., [21], [42], exist for
              called belief propagation (BP), enables approximate marginal-         cooperative localization via BP, it is impossible to apply them
              ization for loopy graphs in an efficient fashion [41].                 directly to our problem. The reason is that unlike a pairwise
                                                                                    potential function in the existing works, the likelihood function
              A. Fundamentals on Belief Propagation                                 f(rij |xi,xj,α) in our problem is of order three. This makes
                To facilitate the derivation of the proposed message passing        the BP algorithm for our problem not straightforward, and,
              algorithms, we start with the fundamentals on belief prop-            hence, we will derive it explicitly in what follows. We first
              agation. For that, a graphical model, factor graph, will be           represent the joint posterior distribution f(x1,...,xN,α|r)
              introduced first. For simplicity, we ignore the PLE variable α         using the factor graph, see Fig. 2. The PLE variable is
              in this subsection. There are two distinctive types of nodes in       connected to all likelihood functions as it is related to all
              the factor graph, the position variables in circles and the factors   measurements.
              in squares. An example is depicted in Fig. 1. The factors stand          The key idea of the BP is to update a set of messages
              for functions, such as the likelihood functions and the prior         iteratively, which contribute to calculating the marginal pos-
              distribution functions. Two position variables are connected          teriors. With a slight notation abuse, we use fij as a short-
              via a factor if there is a measurement between them. The basic        handnotation for the likelihood function f(rij |xi,xj,α) from
                                                                                    now on. We denote the message from factor f          to variable α
              operation in BP is the message passing procedure, which can                                                              ij
                                                                                    by m        (α) and that from f      to x by m           (x ). The
              be interpreted as exchanging statistical information on adjacent            f →α                         ij     i       f →x      i
                                                                                           ij                                          ij   i
                                                                                    messages m          (α) and m         (x ) are updated according
              nodes of the factor graph. According to the message passing                        f →α              f →x      i
                                                                                                  ij                ij   i
              rules in BP [41], the message passed from factor fij to variable      to the following rules:
              node xi is obtained as                                                   n            ZZ                             Y n−1
                               Z                        Y                           m        (α) ∝       f(rij |xi,xj,α) f(xi)          m        (xi)
                                                                                      f →α                                               f →x
                n                                            n−1                       ij                                                 si   i
              m        (x ) ∝     f(r   x ,x )f(x )         m         (x ) dx .                                                  s∈Γ \j
                f →x      i          ij   i   j     j         f →x      j      j                                                     i
                 ij   i                                        tj    j                                     Y
                                                      t∈Γ \i                                                       n−1
                                                          j                                      · f(x )        m         (x ) dx dx ,             (3a)
                                                |           {z           }                             j           f →x      j     i    j
                                                                                                                    tj   j
                                                          m˜ (xj)                                         t∈Γ \i
                                                                                                    ZZ       j
              Here, the superscript n is the iteration index, and Γ \i denotes       n                                             Y n−1
                                                                     j             m        (x ) ∝       f(r |x ,x ,α) f(x )            m         (x )
                                                                                     f →x      i             ij   i  j          j         f   →x     j
              the set of all neighbors of node j excluding node i. Roughly            ij   i                                               tj    j
                                                                                                                                  t∈Γ \i
                                         n                                                                Y                          j
              speaking, the message m           (xi) can be interpreted as the
                                         fij→xi                                                                   n−1
                                                                                                 · f(α)        m         (α) dxj dα.               (3b)
              statistical information on position x coming from its neighbor                                      f  →α
                                                    i                                                              uz
              j. This is obtained by transforming the statistical information                         (u,z)∈Γ\(i,j)
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