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A reducedcomplexity scheme using message passing for location tracking
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 121 (2012)
Abstract
This article presents a lowcomplexity and highaccuracy algorithm using messagepassing approach to reduce the computational load of the traditional tracking algorithm for location estimation. In the proposed tracking scheme, a state space model for the locationestimation problem can be divided into many mutualinteraction local constraints based on the inherent messagepassing features of factor graphs. During each iteration cycle, the message with reliability information is passed efficiently with an adaptive weighted technique and the error propagation law, and then the messagepassing approach based on predictioncorrection recursion is to simplify the implementation of the Bayesian filtering approach for locationestimation and tracking systems. As compared with a traditional tracking scheme based on Kalman filtering (KF) algorithms derived from Bayesian dynamic model, the analytic result and the numerical simulations show that the proposed forward and onestep backward tracking approach not only can achieve an accurate location very close to the traditional KF tracking scheme, but also has a lower computational complexity.
1. Introduction
With the rapid advance in technologies and infrastructures, wireless communication systems have been getting great momentum due to the convenience that they can provide. Among the various applications in the systems, locationbased services (LBSs) are appearing in our daily life both indoors and outdoors [1]. In other words, various LBSs have been set up in companies, universities, stations, airports, shopping malls, and even households; they are becoming more and more popular, and they are consequently attracting high attentions in industry and academia to investigate their characteristics in every aspect. However, providing customers with tailored LBSs is a fundamental problem [2, 3], and the positioning and tracking mobile terminals (MTs) are considered a key problem in wireless environments. Therefore, the location accuracy and computational complexity are two major challenges in locationestimation systems. Furthermore, for integrating the use of information and communications technology and the use of telecommunications and informatics (telematics) in wireless networks, good location information can help optimize resource allocation and improve cooperation. In other words, accurate positioning and tracking schemes are essential for useful locationestimation systems [4–6], and accurate locations can be improved with location tracking algorithms.
The fundamental phases of tracking systems are based on the prediction (time update) and the correction (measurement update) [7]; the role of location tracking algorithms is to perform recursive state estimation, which is given by state equation and observation equations. For locationestimation techniques using tracking algorithms, Kalman filtering (KF) algorithms are considered optimal for the linear Gaussian model, and they have been introduced to enhance the accurate estimation of the locationestimation system [7–20]. Nevertheless, the high accurate location based on the traditional KF algorithm requires high computational complexity, and direct implementation of the KF equations may be too complex to be applied to practical systems [21]. To improve locationestimation efficiency, it would be useful to develop an algorithm with high location accuracy and low computational complexity. Consequently, some fixed coefficient or degenerate form algorithms were proposed to avoid repeatedly calculating the Kalman gain, and the computational load of these schemes is lower than the traditional KF algorithm [7, 17, 22–25]. However, these schemes are only suitable for tracking the MT in steadystate environments; the coefficients of these algorithms have to be extracted again when these algorithms change its coefficients for different situations. Therefore, to reduce the computational load of the KF tracking algorithm and to get more flexible tracking schemes than some fixed coefficient or degenerate form algorithms are worth developing lowcomplexity techniques for locationestimation and tracking systems. For these reasons, some locationestimation algorithms are based on factor graphs (FGs) [26–30], the errors of these algorithm are expressed in the form of a Gaussian probability density function (PDF). In addition, with computationally attractive technique, location tracking schemes based on a state space model can be derived from the Bayesian dynamic model [31–33], which is a probabilistic framework for state estimation using the Markov assumption. Namely, to effectively simplify the implementation of Bayesian approach, the message of tracking scheme can be represented by the statistical properties of the estimated variables. In other words, for a linear dynamic system, these approaches can be applied for locationestimation systems, and then a location tracking algorithm based on messagepassing propagation can efficiently be cycled between the prediction and correction phases.
In this article, an efficient location tracking scheme based on an adaptive weighted technique with low computational load and good location accuracy is proposed to estimate an MT's location. Specifically, instead of using the traditional KF tracking scheme, the FG with the distributed feature makes the decomposition of the joint distribution to be functions of the variables. Therefore, a forward approach for location estimation and onestep backward approach for speed estimation are proposed to simplify the implementation for the locationestimation system. The analytic and simulation results demonstrate that the proposed forward and onestep backward (FOSB) tracking scheme passing the reliable messages between the prediction and correction phases can achieve an accurate location very close to that of the KF tracking scheme. In addition, the computational load of the proposed scheme is lower than the traditional KF algorithm, and the proposed tracking scheme can easily be implemented.
The remainder of this article is organized as follows. Section 2 presents the background on inference for locationestimation techniques. Section 3 applies the proposed FOSB tracking scheme to overcome the computational complexity of the traditional tracking scheme. Performance evaluation and simulation results are discussed in Section 4. Finally, the conclusion is given in Section 5.
2. Background
2.1. Preliminary
For the system and the measurement of the MT at time k, let us consider the dynamic system described in state space form. The mathematical models based on probability densities can be taken as [31]
where x _{ k }, fun_{ x } (·), and u_{ k } are the state vector, transition function, and process noise with known distribution, respectively; z _{ k }, fun_{ z }(·), and v_{ k } are the observation vector, observation function, and observation noise with known distribution, respectively. f (x_{k + 1}x _{ k }) and h(z _{ k }x _{ k }) are the transition PDF and observation PDF, respectively. By these equations, the hidden states x_{ k } disturbed by u_{ k } and the data z_{ k } disturbed by v_{ k } are assumed to be generated by functions fun_{ x } (·), and fun_{ z }(·), respectively. We assume that the initial state x_{0} is distributed according to a density function f(x_{0}), and that h(z_{0}x_{0}) is the distribution of the initial observation. In addition, the Bayesian approach assumes that the dynamic system is Markovian, and all relevant information is contained in the currentstate variable. As a result, the joint probability density of states and observations based on the probability chain rule can be expressed as
where x_{0:k}≜ (x_{0}, ..., x _{ k }) and z_{0:k}≜ (z_{0}, ..., z _{ k }). The solution of the sequential estimation problem is obtained by the algorithm sequentially estimating the states of a system as a set of observations becomes available. In fact, both directed and undirected graphs allow a global function of several variables to be expressed as a product of factors over subsets of those variables [34]. According to the concept of directed graphs, with Bayesian networks for expressing causal relationships between random variables, a graphical representation of (3) between states and observations can be illustrated in Figure 1[31].
In addition, for the Bayesian approach at time k, assume that p(x_{ k } z_{0:k}) is known, and the objective is to find p(x_{k+1}z_{0:k+1}). According to the Markov structure and Bayes's theorem, the predictioncorrection recursion can be written as follows.
Prediction step (time update)
Correction step (measurement update)
where p(z_{k+1}z_{0:k}) is the predictive distribution of z_{k+1}given the past observations z_{0:k}. The predictioncorrection recursion relations in (4) and (5) form the sequential scheme for the Bayesian approach.
2.2. State and measurement models
In this article, the useful notation of the tracking algorithm can be taken as follows. The value of vector x(t) at a discrete time instant t = t_{ k } is denoted by x _{ k }. A best estimate of x(t) at time t = t_{ k } given the observations up to time t = t_{ j } is denoted by a doublesubscript notation with x_{kj}. Moreover, for location tracking, there are three useful cases denoted as follows: the onestep fixedlag smoothing is ${\mathbf{x}}_{kk+1}={\stackrel{\u2323}{\mathbf{x}}}_{k}$ when k < j = k + 1: the best estimation is ${\mathbf{x}}_{kk}={\widehat{\mathbf{x}}}_{k}$ when k = j; the best onestep prediction is ${\mathbf{x}}_{k+1k}={\stackrel{\u0303}{\mathbf{x}}}_{k+1}$ when k +1 > j = k. Therefore, for the linear Gaussian model, the mathematical models of the linear dynamic system and of the measurement can be denoted as a state space model by
State equation
Observation equation
where x _{ k }, Φ_{ k } , u _{ k }, and Q_{ k } are the state matrix, state transition matrix, model noise matrix, and model noise covariance matrix, respectively; z _{ k }, H _{ k }, v _{ k }, and R_{ k } are the actual measurement matrix, measurement transition matrix, measurement noise matrix, and measurement noise covariance matrix, respectively; u_{ k } and v_{ k } are zeromean independent Gaussian vectors with covariance matrices Q_{ k } and R _{ k }, respectively. According to the prediction and correction steps in (4) and (5), the mathematical equations and phases of the KF algorithm derived from Bayesian dynamic model are described in Appendix 1.
2.3. Operations of predictioncorrection recursion
According to the inherent distributed feature of the FG and the distributive law of multiplications to make the decomposition of the joint distribution to be functions of the variables [34], the messages of the reliable information can be processed and passed among variable nodes and factor nodes. The fundamental concepts about the works of FG algorithm with messagepassing structure are illustrated in Appendix 2. Therefore, the useful operations of predictioncorrection recursion based on message passing are introduced as follows.
2.3.1. The correction phase (measurement update)
Assume that each message is a Gaussian PDF, with the notation as
The product of any K Gaussian PDF is also Gaussian and is expressed as [26, 34]
where ${\sigma}_{k}^{2}$ are the variance, and m_{ k } are the mean; $\frac{1}{{\widehat{\sigma}}^{2}}=\sum _{k=1}^{K}\frac{1}{{\sigma}_{k}^{2}}$ and $\widehat{m}={\widehat{\sigma}}^{2}\sum _{k=1}^{K}\frac{{m}_{k}}{{\sigma}_{k}^{2}}$. Consider a correction phase with two incoming Gaussian messages illustrated as the left part of Figure 2. According to (5) and (9), the mean (m_{3}) is the estimate result based on data reliability passing from the prediction and observation for location tracking, which is taken as
where (10) is defined as oper_{1} in Table 1, i.e., a message from a variable node to a factor node is the product of incoming messages. The mean of oper_{1} will be dominated by the messages with higher reliability. Namely, the message from a variable node to a factor node can be taken as the correction phase for location tracking.
2.3.2. The prediction phase (time update)
For two continuous variables x and y, the marginal density function of y is obtained by integrating the joint distribution over variable x. The integration of the product of any two Gaussian PDFs is Gaussian and can be calculated by [26]
where (11) is defined as oper_{2} in Table 1. In addition, consider a prediction phase with two incoming Gaussian messages illustrated as the right part of Figure 2. According to (9) and (11), a message from a factor node to a variable node is the integration of the product of two incoming messages enforced on the factor node. In other words, the integration of the product of any three Gaussian PDFs is Gaussian and can be calculated as
where (12) is defined as oper_{3} in Table 1, and the message from a factor node to a variable node can be taken as the prediction phase for location tracking.
2.4. Relationship with received signal strength for location estimation
For the approaches of incorporated measurement uncertainties in terms of received signal strengths (RSSs) or beacons in twodimensional (2D) coordinate system, locationestimation approaches extracting X and Y locations were proposed in [18, 20, 25, 35, 36]; the concepts of FG approaches based on measuring distance and RSSs were proposed by Chen et al. [28] and Huang et al. [30]; the concept of distributing approach based on the measuring distance in terms of RSSs was proposed by Wang et al. [10]. In addition, for estimating X and Y locations, one of the popular commercial ZigBee positioning systems using the TI CC2431 location engine in terms of RSS indicators, a measurement of the power present in a received radio signal, gains X and Y locations independently [36]. However, location tracking approaches with recursive state estimation can be used to improve location accuracy. Consequently, this article only focuses on location tracking approaches in terms of X and Y groups independently. In addition, for a fair comparison between the algorithms, the input observations of the KFbased tracking scheme is based on the location information extracted from positioning approaches, and the performance of the KFbased approach is also as a comparing bound for location tracking schemes. In short, the proposed approach focuses on distributed approach based on a weighting concept with the prediction and correction phases for location tracking in decoupled coordinates.
3. Methodology
In this article, the proposed adaptive locationtracking approach is formulated as a filtering problem in terms of the state space model, where the prediction phase projects the current state estimate ahead of the sampling time, and the correction phase adjusts the projected estimate by the actual measurement at that time.
3.1. Problem formulation
Although the state and measurement models are only based on a 2D model (cf. [8, 20, 35, 36]) in this article, the extension of the proposed scheme to a threedimensional model is straightforward. Consequently, for the motion model of the MT based on speed noise [17, 18, 20], by adding a random component to the MT, the 2D model describing the motion and observing the location of the MT is taken as
where the vector ${\mathbf{x}}_{k}={\left[{x}_{1,k}\phantom{\rule{0.3em}{0ex}}{x}_{2,k}\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0307}{x}}_{1,k}\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0307}{x}}_{2,k}\right]}^{T}$ denotes the state of the MT at time k. For the 2D model with an X and Y coordinate system, x_{1, k}and x_{2, k}are the locations; ${\stackrel{\u0307}{x}}_{1,k}={s}_{1,k}$ and ${\stackrel{\u0307}{x}}_{2,k}={s}_{2,k}$ are the speeds. Δ _{ k } is the measurement interval between k and k + 1. As compared (6) and (7) with (13) and (14), u_{ k } = [u_{1, k}u_{2, k}u_{3, k}u_{4, k}] ^{T} , and z_{ k } = [z_{1, k}z_{2, k}] ^{T} , and v_{ k } = [v_{1, k}v_{2, k}] ^{T} are the process noise, observed location, and measurement noise corresponding to the MT at time k, respectively. In addition, for the motion model of the MT based on acceleration noise [17, 19], by adding a random component to the MT, the 2D model describing the motion is taken as
where E {Δ_{ n }η_{ n }(Δ_{ k }η _{ k } )^{T} } = Q_{ k } ; η_{ k } = [η_{1, k}η_{2, k}] ^{T} is the process noise for an MT motion.
According to (13)(15) and Figure 3 (Appendix 1), the calculations involved in the KF algorithm are matrix computations, which include multiplication, addition, and inverse operations. However, for an n × n matrix, the complexity of an inversion operation is $\mathcal{O}\left({n}^{3}\right)$ with GaussJordan elimination algorithm. Moreover, as shown in Table 2, many elements in the matrices are zeros for the KF tracking algorithm. In other words, the computational load of the traditional KF techniques must be considered and should be reduced for practical realtime applications.
To reduce the computational complexity of the tracking algorithm, this article proposes a locationtracking technique based on the forward message passing approach for location estimation and the onestep backward message passing approach for speed estimation. Moreover, the idea of decoupling X and Y dimensions for different tracking groups is used to reduce the computational complexity. For the 2D problem simplified with two onedimensional (1D) problems, the 2D problem can be represented by two independent main groups, X and Ycoordinate groups (cf. [10, 18, 20, 25, 35, 36]). As a result, the approach of 1D problem based on the proposed forward and onestep backward (FOSBbased) tracking scheme is illustrated in the following sections.
3.2. Adaptive weighted scheme for location tracking
For tracking speed and location of the Xcoordinate group, the joint probability density of states and observations, p(s_{0:K}, x_{0:K}, z_{0:K}) = p (s_{0}, ..., s_{ i } , ..., s_{ K } , x_{0}, ..., x_{ i } , ..., x_{ K } , z_{0}, ..., z_{ i } , ..., z_{ K } ), can be based on the probability chain rule, where i = 1, ..., K; s_{ i } , x_{ i } , and z_{ i } , are the estimated speed, estimated location, and observed location, corresponding to the MT at time i, respectively. As a result, the states of speed and location developed in the probability model and based on the Markov structure are written as
In addition, as the conditional probability density of the observation z_{ k } depends only on x_{ k } , given the location states x_{0:k}, speed states s_{0:k}, and past observations z_{0:k1}, the probability is written as
Therefore, this Markov structure given the observed data z_{0:K}can be used to factor the conditional joint PDF of the state variables s_{0:K}and x_{0:K}as
For the continuous variables, the marginal density function is obtained by integrating the joint distribution over all variables except s_{ k } and x_{ k } as
where \{s_{ k } , x_{ k } } denotes the set of variables with variables s_{ k } and x_{ k } omitted; f(s_{ k } , s_{k1}) ∝ f(s_{ k } s_{k1})≜f_{ k } and g(x_{ k } , x_{k1}, s_{k1})∝g(x_{ k }  x_{k1}, s_{k1})≜g_{ k } are the local speed and the local location transition PDFs, respectively; h(z_{ k } , x_{ k } )∝h(z_{ k } x_{ k } )≜h_{ k } is the local location observation PDF, and the results of X and Y estimated locations given from positioning approaches (cf. [10, 18, 20, 25, 30, 35, 36]) are used as the input information (observation) of h for location tracking in realistic measurements. According to the concept of undirected graphs suited for expressing soft constraints between random variables, the graph representation (cf. [34], see Section "Factor graphs" in Appendix 2) related to tracking speed and location of the Xcoordinate group that describes (20) is illustrated as the black color diagram in Figure 4, where a correction step (variable node) is illustrated by a circle; a prediction step (factor node) is illustrated by a square. Furthermore, some notations defined in the proposed tracking scheme are described in Table 3. Based on a similar method, the mathematical model of the Ycoordinate group also can be implemented.
3.3. Message passing based on error propagation
As the motion of the MT with adding a random component to the MT in constant speed, the mutually interactive constraint rule with an undirected graph representation for all nodes is illustrated in Figure 4. Moreover, according to the error propagation law [37], the PDFs of (13) and (14) for Xcoordinate group are given by
Where x_{1, k}≜x_{ k } , s_{1, k}≜s_{ k } , and z_{1, k}≜z_{ k } . However, in (14), the observed information is only based on the location observations. If the speed iteration cycle is performed without a speed observation, it may cause error propagation in speed and thus reduce the location accuracy. In fact, another speed observation can be refined with the two location estimations, the known measurement interval (Δ), and the error propagation law, which is called the onestep backward algorithm (or onestep fixedlag smoothing), as follow.
where l(s_{ k } , x_{k+1}, x_{ k } )∝l(s_{ k }  x_{k+1}, x_{ k } )≜l_{ k } is the local refined speed observation PDF. That is, for speed estimation, the proposed onestep backward algorithm is to delay the calculation of the estimate until one future observation location is obtained. Consequently, for the proposed FOSBbased tracking scheme, the prediction and correction flows of location messages and of speed messages are illustrated with solid and dashed lines in Figure 4, respectively. Similarly, as the motion of the MT is with a random component based on acceleration noise, the PDFs of (15) for Xcoordinate group are given by
For the model with acceleration noise, another speed observation can be refined as follows.
In brief, the KF approach provides a state estimate based on the present observations for realtime operation. For applications in realistic environment, it is possible to delay the computation of the estimate until future data obtained [38]. Therefore, the proposed onestep backward approach is to establish a postprocessing environment, which is an approach of onestep fixedlag smoothing using one point of future observation for location tracking. That is, a forward approach provides a state estimated based on the present observations as realtime operation in (22) and (26) for location estimation; the onestep backward algorithm is to establish a postprocessing environment through a displacement during a time interval in (25) and (28) for speed estimation. In short, the forward location and the backward speed estimations are incorporated in the proposed FOSBbased location tracking scheme.
Figure 4 illustrates the proposed FOSB tracking flows of the locationestimation system whose unnormalized joint distribution is given by (20). As indicated by rectangles in Figure 4, Steps 1 through 3 give the initial values for location tracking. The initial states x_{0} and s_{0} are distributed according to a density function f(x_{0}); initial observation h (z_{0}x_{0}) is the distribution of the initial observation. In general, the initial location and speed information are given initial guess values; the parameters also can be given by positioning systems, and then the training session of location tracking scheme is performed by comparing its new state and new observation information for the operations of predictioncorrection recursion. In addition, the messages of a single iteration of the tracking scheme can be generated in seven steps (Steps 5 through 11). The equations represent the propagation of messages according to the proposed FOSB approach along the chain, and the detail messagepassing flows are illustrated as follows.

(1)
A priori estimate:
Step 1: ${\widehat{G}}_{k1}\triangleq {\mu}_{{x}_{k1}\to {g}_{k}}\left({x}_{k1}\right)\triangleq \mathcal{N}\left({x}_{k1};{\widehat{M}}_{{x}_{k1}},{\widehat{V}}_{{x}_{k1}}\right)$
Step 2: $\widehat{F}{g}_{k1}\triangleq {\mu}_{{s}_{k1}\to {g}_{k}}\left({s}_{k1}\right)\phantom{\rule{0.3em}{0ex}}\triangleq \mathcal{N}\left({s}_{k1};\widehat{M}{g}_{{s}_{k1}},\widehat{V}{g}_{{s}_{k1}}\right)$
Step 3: ${\widehat{F}}_{k1}\triangleq {\mu}_{{s}_{k1}\to {f}_{k}}\left({s}_{k1}\right)\phantom{\rule{0.3em}{0ex}}\triangleq \mathcal{N}\left({s}_{k1};{\widehat{M}}_{{s}_{k1}},{\widehat{V}}_{{s}_{k1}}\right)$
Step 4: $\begin{array}{cc}\hfill {\stackrel{\u0303}{G}}_{k}\triangleq {\mu}_{{g}_{k}\to {x}_{k}}\left({x}_{k}\right)& \stackrel{{\mathsf{\text{oper}}}_{3}}{\propto}\mathcal{N}\left({x}_{k};{\widehat{M}}_{{x}_{k1}}+{\Delta}_{k}\cdot \widehat{M}{g}_{{s}_{k1}},{\widehat{V}}_{{x}_{k1}}+{\Delta}_{k1}^{2}\cdot \widehat{V}{g}_{{s}_{k1}}+{Q}_{11,k1}\right)\hfill \\ \phantom{\rule{2.77695pt}{0ex}}\triangleq \mathcal{N}\left({x}_{k};{\stackrel{\u0303}{M}}_{{x}_{k}},{\stackrel{\u0303}{V}}_{{x}_{k}}\right)\hfill \end{array}$

(2)
Prediction state (time update phase):
Step 5: ${\stackrel{\u0303}{F}}_{k}\triangleq {\mu}_{{f}_{k}\to {s}_{k}}\left({s}_{k}\right)\stackrel{{\mathsf{\text{oper}}}_{2}}{\propto}\mathcal{N}\left({s}_{k};{\widehat{M}}_{{s}_{k1}}\phantom{\rule{2.77695pt}{0ex}},\phantom{\rule{2.77695pt}{0ex}}{\widehat{V}}_{{s}_{k1}}+{Q}_{33,k1}\right)\triangleq \mathcal{N}\left({s}_{k};{\stackrel{\u0303}{M}}_{{s}_{k}}\phantom{\rule{2.77695pt}{0ex}},\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0303}{V}}_{{s}_{k}}\right)$
Step 8: $\begin{array}{cc}\hfill {\stackrel{\u0303}{G}}_{k+1}\triangleq {\mu}_{{g}_{k+1}\to {x}_{k+1}}\left({x}_{k+1}\right)& \stackrel{{\mathsf{\text{oper}}}_{3}}{\propto}\mathcal{N}\left({x}_{k+1};{\widehat{M}}_{{x}_{k}}+{\Delta}_{k}\cdot \widehat{M}{g}_{{s}_{k}},{\widehat{V}}_{{x}_{k}}+{\Delta}_{k}^{2}\cdot \widehat{V}{g}_{{s}_{k}}+{Q}_{11,k}\right)\hfill \\ \phantom{\rule{2.77695pt}{0ex}}\triangleq \mathcal{N}\left({x}_{k+1};{\stackrel{\u0303}{M}}_{{x}_{k+1}},{\stackrel{\u0303}{V}}_{{x}_{k+1}}\right)\hfill \end{array}$

(3)
Correction state (measurement update phase):
Step 6: $\widehat{F}{g}_{k}\triangleq {\mu}_{{s}_{k}\to {g}_{k+1}}\left({s}_{k}\right)\phantom{\rule{0.3em}{0ex}}\triangleq \phantom{\rule{0.3em}{0ex}}{\stackrel{\u0303}{F}}_{k}\phantom{\rule{0.3em}{0ex}}\triangleq \phantom{\rule{0.3em}{0ex}}\mathcal{N}\left({s}_{k};\widehat{M}{g}_{{s}_{k}},\widehat{V}{g}_{{s}_{k}}\right)$
Step 7: ${\widehat{G}}_{k}\triangleq {\mu}_{{x}_{k}\to {g}_{k+1}}\left({x}_{k}\right)\stackrel{{\mathsf{\text{oper}}}_{1}}{\propto}\mathcal{N}\left({x}_{k};\frac{{R}_{11,k}\cdot {\stackrel{\u0303}{M}}_{{x}_{k}}+{z}_{k}\cdot {\stackrel{\u0303}{V}}_{{x}_{k}},}{{R}_{11k}+{\stackrel{\u0303}{V}}_{{x}_{k}}},\frac{{R}_{11,k}\cdot {\stackrel{\u0303}{V}}_{{x}_{k}}}{{R}_{11,k}+{\stackrel{\u0303}{V}}_{{x}_{k}}}\right)\phantom{\rule{0.3em}{0ex}}\triangleq \mathcal{N}\left({x}_{k};{\widehat{M}}_{{x}_{k}},{\widehat{V}}_{{x}_{k}}\right)$

(4)
Correction state with the improved speed message:
To avoid the error propagation caused by a shortage of the speed observation in recursive operations, the onestep backward algorithm is proposed for speed estimation. The correction phases with improved messages for the speed estimation are described by the following steps.
Step 9: $\begin{array}{cc}\hfill \widehat{G}{b}_{k+1}\triangleq {\mu}_{{x}_{k+1}\to {l}_{k+1}}\left({x}_{k+1}\right)& \propto \mathcal{N}\left({x}_{k+1};E\left[{x}_{k+1}\right],Var\left[{x}_{k+1}\right]\right)=\mathcal{N}\left({x}_{k+1};{z}_{k+1},{R}_{11,k+1}\right)\hfill \\ \phantom{\rule{2.77695pt}{0ex}}\triangleq \mathcal{N}\left({x}_{k+1};\widehat{M}{b}_{{x}_{k}+1},\widehat{V}{b}_{{x}_{k+1}}\right)\hfill \end{array}$
Step 10: $\begin{array}{cc}\hfill \stackrel{\u0303}{G}f{b}_{k}\triangleq {\mu}_{{l}_{k+1}\to {s}_{k}}\left({s}_{k}\right)& \stackrel{{\mathsf{\text{oper}}}_{3}}{\propto}\mathcal{N}\left({s}_{k};\frac{\widehat{M}{b}_{k+1}{\widehat{M}}_{k}}{{\Delta}_{k}},\frac{1}{{\Delta}_{k}^{2}}\cdot \left(\widehat{V}{b}_{{x}_{k+1}}+{\widehat{V}}_{{x}_{k}}+{Q}_{11,k}\right)\right)\hfill \\ \triangleq \mathcal{N}\left({s}_{k};\stackrel{\u0303}{M}{b}_{{s}_{k}},\stackrel{\u0303}{V}{b}_{{s}_{k}}\right)\hfill \end{array}$
Step 11: ${\widehat{F}}_{k}\triangleq {\mu}_{{s}_{k}\to {f}_{k+1}}\left({s}_{k}\right)\stackrel{{}^{{\mathsf{\text{oper}}}_{1}}}{\propto}\mathcal{N}\left({s}_{k};\frac{{\stackrel{\u0303}{M}}_{{s}_{k}}\cdot \widehat{V}{b}_{{s}_{k}}+\widehat{M}{b}_{{s}_{k}}\cdot {\stackrel{\u0303}{V}}_{{s}_{k}}}{\widehat{V}{b}_{{s}_{k}}+{\stackrel{\u0303}{V}}_{{s}_{k}}},\frac{\widehat{V}{b}_{{s}_{k}}\cdot {\stackrel{\u0303}{V}}_{{s}_{k}}}{\widehat{V}{b}_{{s}_{k}}+{\stackrel{\u0303}{V}}_{{s}_{k}}}\right)\triangleq \mathcal{N}\left({s}_{k};{\widehat{M}}_{{s}_{k}},{\widehat{V}}_{{s}_{k}}\right),$
where ${\stackrel{\u0303}{M}}_{x}$, ${\stackrel{\u0303}{V}}_{x}$, ${\widehat{M}}_{x}$, and ${\widehat{V}}_{x}$ are the mean of location prediction, variance of location prediction, mean of location estimation, and variance of location estimation, respectively; ${\stackrel{\u0303}{M}}_{s}$, ${\stackrel{\u0303}{V}}_{s}$, ${\widehat{M}}_{s}$, and ${\widehat{V}}_{s}$ are the mean of speed prediction, variance of speed prediction, mean of speed estimation, and variance of speed estimation, respectively; $\widehat{M}{g}_{s}$ and $\widehat{V}{g}_{s}$ are the mean of forward speed estimation and variance of forward speed estimation, respectively; $\widehat{M}{b}_{x}$, $\widehat{V}{b}_{x}$, $\stackrel{\u0303}{M}{b}_{s}$ and $\stackrel{\u0303}{V}{b}_{s}$ are the mean of backward location estimation, variance of backward location estimation, mean of backward speed prediction, and variance of backward speed prediction, respectively. For the correction phase in (10), the location estimation is based on weighted reliable information of location prediction and location observation in Step 7; the speed estimation is based on weighted reliable information of speed prediction and speed observation in Step 11. Therefore, it is the important feature of the proposed algorithm to take into consideration the exchange of the reliable information of locations and speeds in the fusion process. In addition, for the FOSBbased proposed locationtracking scheme, there are two useful estimations denoted as follows: the mean of ${\mu}_{{x}_{k}\to {g}_{k+1}}\left({x}_{k}\right)$ and the mean of ${\mu}_{{s}_{k}\to {f}_{k+1}}\left({s}_{k}\right)$ are the location estimation and the speed estimation of the MT of Xcoordinate at the k th state estimation, respectively. Therefore, the means and uncertainties of location estimation and speed estimation are illustrated in (29)(32).
The proposed locationtracking approach is based on both future and past information. The 2D problem is reduced to two independent main groups, X and Ycoordinate groups. According to similar procedures, the Ycoordinate group can be modeled and implemented similarly, too.
As described previously, not only are there many zero elements in the matrixes, but also the calculations based on the KF algorithm (KFbased) are matrix computations [8]. However, for the proposed predictioncorrection recursion, only simple scalar addition, subtraction, multiplication, and division operations are required in the FOSBbased tracking scheme. According to (13)(15), there are four states and two inputs to the system. For this 2D model, a comparison of the computational complexity of the KFbased and the proposed FOSBbased tracking schemes is presented in Table 4. It shows that the proposed tracking scheme can dramatically reduce the computational load of a traditional tracking scheme without decoupling approach. In addition, the results of X and Y estimated locations are decoupled as individual input (observation) of h for location tracking in Figure 4. The computational complexity of the KF approach, a meansquare error (MSE) sense estimator, with decoupling X and Y dimensions is also illustrated in Table 4, where the count of multiplication involved either a 1 or a 0 is eliminated; the count of division involved a 1 is eliminated; the count of additions and subtraction involved a 0 is eliminated. As compared with computational complexity in Table 4, the proposed algorithm is more practical for implementation.
4. Performance evaluation and numerical simulations
The proposed approach focuses on location tracking with distributed approach in terms of a weighting concept for X and Y groups filtered independently. In this section, simulations are conducted to demonstrate the efficiency and accuracy of location estimation. To verify the performance of location results introduced by the proposed scheme, it is assumed that the location parameters are based on indoor wireless local area networks. For the KFbased tracking approach used in this article, the equations of the KF algorithm are illustrated in (36)(45), and the process cycle of KF algorithm is illustrated in Figure 3. To provide fair comparisons in this article, the KFbased tracking scheme and the FOSBbased tracking scheme use the same parameters to analyze and carry out the simulation. Namely, the KFbased tracing scheme is based on the prediction and correction phases for location estimation; the proposed FOSBbased tracking scheme is to distribute and pass the reliable messages between the prediction and correction phases for location estimation.
In this article, three test paths based on the 2D model environment are examined. For the first path, the equations are used to generate and analyze the motion of the MT as follows. By adding a random component ${\sigma}_{k}^{u}=0.4$ to the MT, the 2D model describing the motion of the MT with speed noise is generated by (33); by adding a random component ${\sigma}_{k}^{v}=4$ to the MT, the model describing the observation location of the MT is taken in (14); the model describing the analytic steady motion of the MT with speed noise is based on (13).
For the second path, the equations are used to generate and analyze the motion of the MT as follows. By adding a random component ${\sigma}_{k}^{\eta}=0.4$ to the MT, the 2D model describing the motion of the MT with acceleration noise is generated by (34); by adding a random component ${\sigma}_{k}^{v}=4$ to the MT, the model describing the observation location of the MT is taken in (14); the model describing the analytic steady motion of the MT with acceleration noise is based on (15).
where the speed s is set to 2 m/s, and Δ _{ k } , the measurement intervals between k and k + 1, are set to 0.1~4 s for the first and the second paths; u_{ k } and η_{ k } are assumed to be normal random variables. In (14), the measurement noise, v_{ k } , has zero mean and a variance of ${\left({\sigma}_{k}^{v}\right)}^{2}$, where v_{k} is assumed to be an normal random variable. For the third path, the equations are used to generate and analyze the motion of the MT as follows. By adding a random component ${\sigma}_{k}^{\eta}$ to the MT, the 2D model describing the motion of the MT with acceleration noise is generated by (35); by adding a random component ${\sigma}_{k}^{v}$ to the MT, the model describing the observation location of the MT is taken in (14); the model describing the analytic motion of the MT with acceleration noise is based on (15).
where speed s is set to 1.36 m/s (81.6 m/min) of human walking [39, 40]; f = 1/Δ _{ k } is the sampling frequency; ϕ is the moving angle, which is uniformly distributed between 0 and 2π. For indoor environments, the signal strength has a variance of 4.53 dB in an office environment over long time periods [41]. In addition, it is assumed that the measurement variations have zero means and variances of ${\left({\sigma}_{k,x}^{v}\right)}^{2}=4.5$ and ${\left({\sigma}_{k,y}^{v}\right)}^{2}=4.5$ to accommodate positioning errors, and that small values, ${\sigma}_{k,x}^{\eta}=0.4$ and ${\sigma}_{k,y}^{\eta}=0.4$ are chosen. Furthermore, the simulation environment modeled as a person walking indoors is set in a 50 × 50 (m^{2}) square area. The indoor environment is similar to the table of billiard game, and the movement of the MT (person) is similar to a moving billiard ball. Therefore, when the MT moves to boundaries, the MT will change direction and move around the square area. For the simulations of the three paths, 10,000 simulation trials were performed to obtain a confident performance of the location estimates as follows.
For the first and the second paths in terms of the cumulative distribution function (CDF) of the error distance, the simulation results are given in Figures 5 and 6. In addition, in terms of the influence of different measurement intervals (sampling times), the comparisons of tracking schemes are given in these figures. In Figure 5, for the linear trajectory in the first path, the FOSBbased tracking scheme is to delay the calculation of the speed estimate until one future observed location is obtained. As compared with the KFbased tracking scheme, the speed estimate of the FOSBbased tracking scheme is based on the additional information contained in the future data. Therefore, the location accuracy of the FOSBbased tracking scheme with a feature of smoothing is slightly better than the traditional KFbased tracking schemes in stable environments [38]. Nevertheless, if the forward FG tracking scheme does not apply to the onestep fixedlag smoothing technique for sensing additional information, the location accuracy of the FG tracking scheme is slightly worse than the traditional KF tracking scheme. In addition, when the values of sampling times of the tracking algorithms are decreased from 4 to 0.01 s, not only is the accuracy of the estimated location increased, but also the performance of the KFbased tracking scheme is becoming more close to that of the FOSBbased tracking scheme. Namely, for the FOSBbased tracking scheme, decreasing the sampling time will reduce the effect of the future observed information; it will reduce the influence with smoothing out estimated fluctuations. Consequently, the sampling time has a considerable effect on the performance of the tracking schemes.
For the motion of the MT with acceleration noise in the second path, the simulation results of the proposed algorithm are given in Figures 6 and 7. In Figure 7, the result shows that the sampling time too short will affect the initial convergence rate and diminish the accuracy of the FOSBbased tracking scheme. As compared with the KF algorithm, the location accuracy of the proposed tracking scheme is not always better than the traditional KFbased tracking schemes. That is, for the moving trajectory of the second path, the sampling time will influence the covariance scale in (26)(28), and (34); the shorter sampling time will result in smaller covariance, cause an initialvalue problem, and induce larger estimation errors. As shown in Figure 8 (Appendix 1), the effect of the standard deviation of the state equation should be considered when the model describing the analytic motion of the MT is based on acceleration noise. Furthermore, the shortage of the speed observation in recursive operations and noncausal nature can lead to a convergence bias if the proposed FOSBbased tracking scheme is used with an improper initial speed value. However, for short sampling time in Figure 6, as the initialization is finished, the results show that the FOSBbased approach is slightly better than the KFbased approach. In terms of Figure 7, as the tracking scheme is based on a fixedlag smoothing concept with faster sampling frequency, the phenomenon of the worse initial convergence can be reduced with selecting a more closer initial value or can be overcome with the times of the process cycle based on the inherent messagepassing features to exchange information.
In Figures 5 and 6, as the speed s set to 2 m/s and sampling time set to 1 s, the results demonstrate that more than 90 (60) percent of the nontracking method had error distances of less than 8.67 (5.40) m for the first and second paths; the estimates of KFbased tracking scheme had error distances of less than 4.80 (2.98) m for the first path and less than 4.72 (2.94) m for the second path. The estimates of FOSBbased tracking scheme had error distances of less than 4.72 (2.92) m for the first path and less than 4.61 (2.87) m for the second path. According to the results, the location accuracy of the KFbased and FOSBbased tracking schemes is better than the nontracking schemes. However, for the linear Gaussian model, the KFbased tracking scheme is an optimal algorithm, and it gives the linear estimator in an MSE sense [27]. Therefore, the location accuracy of the KF tracking results can be considered a reference CDF bound for location accuracy.
For the third path, Figures 9 and 10 present the comparisons among the nontracking, the KFbased tracking, and the FOSBbased tracking schemes. The results in terms of the CDF of the error distance of 10,000 sample trials and in terms of the average rootmeansquare error (RMSE) of the first 500 sample trials are given in Figure 9a, b, respectively. As the sampling time set to 1 s, the results demonstrate that more than 90 (60) percent of the nontracking, KFbased tracking, and FOSBbased tracking schemes have error distances of less than 4.63 (2.87), 3.04 (1.90), and 3.03 (1.90) m, respectively. In Figure 9a, as the sampling time set to 1 or 2 s, the results show that the location accuracy of the proposed tracking scheme is almost the same as the KFbased tracking scheme. For the proposed FOSBbased tracking scheme, despite the location accuracy, adopting the distributive approach using the predictioncorrection recursion to implement the location estimator significantly reduces the computational complexity, as shown in Table 4. However, the tracking approach will result in the corner effect generally [8], which is caused by the values of sampling time or the standard deviation parameter for target acceleration. In Figure 9b, the result shows that the errors of proposed tracking scheme based on onestep fixed lag smoothing are slightly smoother in the moving trajectory of an MT. In addition, the corner effect (the peaks) of the proposed tracking scheme is slightly larger than the KFbased tracking scheme. That is, the speed of proposed algorithm is reformulated by using forward and backward steps, and it further smooth the location state; it results in larger estimation errors around the corners. Furthermore, in terms of (35), the MT's speed and direction of the simulations are based on random acceleration noise and random moving angle. Figure 10a illustrated the simulation result of the samples from 3500 to 4500 within 10,000 trials, the different moving lengths and the random moving angles (or directions) are assumed to simulate the trajectory and direction of an MT moving around hallways in an indoor environment, where the 50 × 50 (m^{2}) square area almost be completely filled with MT's trajectory after 8,000 simulation trials. Figure 10b is the observation trajectory simulated to be extracted from the measurements of RSSs of WLAN in an indoor environment (cf. [8, 14, 41]). In addition, in terms of the first 500 samples in 10,000 simulation trials, Figure 11al are the original trajectory and speeds, the observed trajectory and speeds, the KFbased tracking results, and FOSBbased tracking results, respectively. Figure 11 shows that both the KFbased and the proposed FOSBbased tracking schemes can provide a high degree of accuracy for predicting nextstep location and speed in tracking paths. In addition, the proposed approach is assumed that the speed observations cannot be obtained, and then the speed is defined and obtained by an MT moving through a displacement during a time interval. According to the simulation results in Figure 11e, f, the observed speeds extracted from location observations cause a large variation and lead to instability with lower reliability. However, for the proposed FOSBbased tracking scheme, the speed estimation is based on weighted reliable information of speed prediction and speed observation in Step 11. In terms of the correction phase in (10) and oper_{1} in Table 1, the estimation speed will be dominated by the messages with higher reliability. As compared with Figure 11ai, Figure 11jl shows that the proposed FOSBbased tracking scheme can achieve an accurate performance very close to the traditional KF tracking scheme, and the tracking results are also very close to nondisturbance data. Consequently, when the messagepassing rules used for prediction and correction phases with improved messages, the oscillation observed in Figure 7 may be caused by the proposed approach, where the performance of the proposed approach is close to an MSE estimator.
In brief, according to the proposed approach and Appendix 1, not only is the FOSBbased tracking scheme derived from Bayesian approach close to MSE estimator, but also the KFbased tracking scheme can be derived from Bayesian point of view based on MSE sense. Namely, the proposed FOSBbased tracking scheme passing the reliable messages between the prediction and correction phases can achieve an accurate location very close to the KFbased tracking scheme. Furthermore, the sampling time will influence the state space model and imply the changes of the simulated track. In Figures 5, 6, and 9a, the different values of Δ _{ k } are used to generate and analysis in the different curves. These results show that the smaller Δ _{ k } results in smaller location tracking errors. In terms of (15), (16), (34), and (35), the smaller Δ _{ k } results in smaller random component added to the motion model of state equation. Therefore, for fixed random components ${\sigma}_{k}^{\eta}$ and ${\sigma}_{k}^{v}$, the estimation accuracy will be dominated by the message of the reliable motion model as the state equation is with smaller Δ _{ k } ; the estimation accuracy will be dominated by the message of the reliable observation as the state equation is with larger Δ_{ k }. In other words, when the value of variances or sampling times becomes high, it diminishes the location accuracy of the MT. As compared with the traditional KFbased tracking scheme, the proposed FOSBbased tracking scheme can achieve acceptable location performance and can reduce the computational burden with a slightly large corner effect. Therefore, the approach of the proposed tracking algorithm can work well under the large computational loading conditions.
5. Conclusion
This article presents a forward scheme for location estimation and a onestep backward scheme for speed estimation with messagepassing algorithm to implement the Bayesian approach. According to the proposed algorithm, not only is a new adaptive weighted scheme used to reduce the computational complexity of traditional tracking schemes, but also the idea of decoupling X and Y dimensions is used to simplify the implementation for location estimation. In addition, to deal with a shortage of the speed observation in recursive operations, a fixedlag smoothing concept based on the past and future location information is implemented to avoid the error propagation and then to improve the location accuracy. Namely, the location accuracy of the proposed FOSBbased tracking scheme is weighted and dominated by the incoming messages having higher reliability information of the location and speed between the prediction and correction phases. In comparison, the proposed FOSBbased with the traditional KFbased location tracking schemes, the major differences between them are the computational load, where the advantage of the former is based on the distributive approach and the decoupling idea for location estimation. For the purpose of increasing the execution speed, the lower computational complexity of the proposed FOSBbased tracking scheme is indeed a valuable improvement for estimating the location of an MT. The simulation results demonstrate that the proposed FOSBbased tracking scheme, in comparison with the traditional KFbased tracking scheme, achieves close location accuracy. Consequently, with the inherent feature of the Bayesian approach, the distributive feature of proposed predictioncorrection recursion makes it suitable for practical implementation. To sum up, the proposed FOSBbased tracking scheme, with features of both good location accuracy and low computational complexity, is attractive to use in location estimation for the application of LBSs in our daily life both indoors and outdoors.
Appendix 1
Kalman filtering
In general, the basic ideas of derivation processes for the KF algorithm are based on three distinctive features of the innovation approach which are linear, unbiased, and minimumvariance [38, 42–44]. However, according to the prediction and correction steps in (4) and (5), if the state space model is linear and Gaussian, a KF algorithm can also be derived and applied to the recursive Bayesian estimation [31, 38]. Let the vector x = [x_{1}, ..., x_{ n } ] ^{T} consist of independent components i = 1, ..., n. The PDF of x is the production of the individual PDF's of x_{1}, ..., x_{ n } ; $\mathcal{N}\left(\mathbf{x};\mathbf{m},\mathbf{P}\right)$ is defined as Gaussian density for n dimensions; the ndimension Gaussian density function is defined by
where x, m, and P are the argument, mean, and covariance, respectively. According to (1), (2), (6), and (7), the KF algorithm can be applied as follows.
It is assumed that the initial state is Gaussian distribution as $p\left({\mathbf{x}}_{0}\right)=\mathcal{N}\left({\mathbf{x}}_{0};{\mathbf{m}}_{0}={\widehat{\mathbf{x}}}_{0},{\mathbf{P}}_{0}={\hat{\mathbf{P}}}_{0}\right)$; the distribution can start from p (x_{0}z_{0}) = p (z_{0}x_{0}) p (x_{0})/p(z_{0}), and then the distribution p(x _{ k }z_{0:k}) is obtained at time k. Therefore, the correction step can be denoted by
According to (4) and (5), the predictioncorrection recursion can be used to get the prediction step as
Equation (6) shows how to update the distribution in one step. Therefore, x_{k+1}is just a summation of two independent Gaussian random vectors, Φ_{ k }x_{ k } and u _{ k }. Under the standard rules for summation of independent Gaussian random vectors, the conditional probability density of the predictive distribution x_{k+1}depends on z_{0:k}, and the probability is written as
where ${\stackrel{\u0303}{\mathbf{x}}}_{k+1}={\mathbf{\Phi}}_{k}{\widehat{\mathbf{x}}}_{k}$ and ${\stackrel{\u0303}{\mathbf{P}}}_{k+1}={\mathbf{\Phi}}_{k}{\widehat{\mathbf{P}}}_{k}{\mathbf{\Phi}}_{k}^{T}+{\mathbf{Q}}_{k}$. In addition, the new data point z_{ k } is incorporated through Bayes' theorem. According to (5), (38), and (41), the correction step (measurement update) of the filtering recursion is as follows.
where ${\widehat{\mathbf{x}}}_{k+1}={\hat{\mathbf{P}}}_{k+1}^{T}\left[{\mathbf{H}}_{k+1}^{T}{\mathbf{R}}_{k+1}^{1}{\mathbf{z}}_{k+1}+{\stackrel{\u0303}{\mathbf{P}}}_{k+1}^{1}{\stackrel{\u0303}{\mathbf{x}}}_{k+1}\right]$, and ${\hat{\mathbf{P}}}_{k+1}^{T}={\left[{\mathbf{H}}_{k+1}^{T}{\mathbf{R}}_{k+1}^{1}{\mathbf{H}}_{k+1}+{\stackrel{\u0303}{\mathbf{P}}}_{k+1}^{T}\right]}^{1}$ The expression can be rearranged as follows [31, 38].
where ${\widehat{\mathbf{x}}}_{k}$ and ${\stackrel{\u0303}{\mathbf{x}}}_{k}$ are the state estimate matrix and state prediction matrix, respectively; ${\widehat{\mathbf{P}}}_{k}$, ${\stackrel{\u0303}{\mathbf{P}}}_{k}$ and K_{ k } are the estimation error covariance matrix, prediction error covariance matrix, and Kalman Gain, respectively; e _{ kj }, ${\widehat{\mathbf{e}}}_{k}$, and ${\stackrel{\u0303}{\mathbf{e}}}_{k}$ are the state error matrix, estimation error matrix, and prediction error matrix, respectively. Therefore, the mathematical equations and phases of the KF algorithm can be summarized as a process cycle of KF given in Figure 3[8].
In addition, to verify the influence of the standard deviation of the state equation in (6) about the MT acceleration for a KF algorithm, the results of two different standard deviation, ${\sigma}_{\mathsf{\text{small}}}^{u}$ and ${\sigma}_{\mathsf{\text{large}}}^{u}$, are selected. In this simulation, it is assumed that the measurement variation has zero mean and a variance of (σ^{v} )^{2} = 9 to accommodate location errors. To simulate the nonconstant speed trajectory, the sampling time is set to 0.1 s to obtain the location estimates in the simulation. Figure 8ac shows the true location, speed, and acceleration of the MT, respectively. Figure 8df shows the estimate errors in KF algorithm when the standard deviation of the MT acceleration is with the small σ value $\left({\sigma}_{\mathsf{\text{small}}}^{u}=0.1\right)$. Figure 8gi shows the same quantities if a large σ value, $\left({\sigma}_{\mathsf{\text{large}}}^{u}=1\right)$ is used. In Figure 8, as compared with the larger σ value, the smaller σ value results in larger estimation errors; the errors of the smaller σ value are smoother. In addition, the results show that the tracking errors are significantly less than the measurement errors (without tracking) even though the state model in (6) does not match the real acceleration of the MT very well.
Appendix 2
Factor graph and sumproduct algorithm
Factor graph
The joint distribution over a set of variables can be expressed as product of factors over subsets by
where x_{ s } denotes a subset of the variables, and factor f_{ s } is a function of a corresponding set of variables x _{ s }. For convenience, the individual variables are denoted by x_{ i }. Therefore, the marginal density function is obtained by summing the joint distribution over all variables except x as follows [26, 34].
where x\x denotes the set of variables in x with variable x omitted. For an FG, there are two types of nodes, and all links go between nodes of opposite type, where every variable node is illustrated by a circle in the distribution; each factor f_{ s } (x _{ s }) node is illustrated by a square in the joint distribution. According to FGs, a function of several variables can be factored into product of smaller functions. For example, let us consider a distribution expressed in terms of the factorization in (48). The distribution can be expressed by the FG shown in Figure 12.
Sumproduct algorithm
In the operation of the sumproduct algorithm [26, 34], μ_{ x→f } (x) denotes the message sent from variable node x to factor node f; μ_{ f→x } (x) denotes the message sent from factor node f to variable node x; ne(x) denotes the set of factor nodes neighboring on a given variable node x; ne(f) denotes the set of variable nodes neighboring on a given factor node f. Therefore, two kinds of messages are denoted by the sumproduct algorithm. One is the message passed from factor nodes to variable nodes, which can be calculated by
where f_{ l } is a function of a corresponding set of variables x _{ l },, \x_{ m } denotes that the summation takes all the variables except x_{ m } , ne(f_{ l } )\x_{ m } denotes the set of variable nodes that are neighbors of the factor node f_{ l } except for the variable node x_{ m } removed. It is important to note that a factor node can send a message to a variable node after the factor node has received incoming messages from all other neighboring variable nodes. As a result, to evaluate the message sent by a factor node to a variable node along the link connecting them, one can take the product of the incoming messages along all other links coming into the factor node, multiply by the factor associated with that node, and then marginalize over all of the variables associated with the incoming messages. A message from a factor node to variable nodes is the sum of factor contributions and is illustrated in the middle and left parts of Figure 13.
The other is the message passed from variable nodes to factor nodes, which can be calculated as
where ne(x_{ m } )\f_{ s } denotes the set of factor nodes that are neighbors of the variable node x_{ m } except for the factor node f_{ s } removed. Therefore, to evaluate the message sent by a variable node to an adjacent factor node along the connecting link, the product of the incoming messages along all of the other links is taken simply. In addition, a variable node can send a message to a factor node after it has received incoming messages from all other neighboring factor nodes. A message from a variable node to a factor node is the product of incoming messages and is illustrated in the middle and right parts of Figure 13.
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Acknowledgements
The editor coordinating the review of this article was P. Oonincx. The authors are very grateful to the anonymous reviewers for their useful comments. This study was supported in part by the National Science Council of the Republic of China under grant NSC 982221E008097MY2.
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Chiou, YS., Tsai, F., Wang, CL. et al. A reducedcomplexity scheme using message passing for location tracking. EURASIP J. Adv. Signal Process. 2012, 121 (2012). https://doi.org/10.1186/168761802012121
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Keywords
 Bayesian approach
 data reliability
 factor graphs
 location tracking
 onestep backward
 wireless communication