Abstract
Robust techniques critically improve bearing-only target localization when the relevant measurements are being corrupted by impulsive noise. Resistance to isolated gross errors refers to the conventional least absolute residual (LAR) method, and its estimate can be determined by linear programming when pseudolinear equations are set. The LAR approach, however, cannot reduce the bias attributed to the correlation between system matrices and noise vectors. In the present study, perturbations are introduced into the elements of the system matrix and the data vector simultaneously, and the total optimization problem is formulated based on least absolute deviations. Subsequently, an equivalent form of total least absolute residuals (TLAR) is obtained, and an algorithm is developed to calculate the robust estimate by dual ascent algorithms. Moreover, the performance of the proposed method is verified through the numerical simulations by using two types of localization geometries, i.e., random and linear. As revealed from the results, the TLAR algorithm is capable of exhibiting significantly higher localization accuracy as compared with the LAR method.
1. Introduction
Bearing-only target localization (BOTL) by exploiting spatially distributed sensors can have extensive applications in vehicle [1] or gunshot [2] positioning, animal habit monitoring [3], network localization [4], multiagent systems [5, 6], recently massive arrays [7], etc. BOTL algorithms have received broad studies under the measurement noise of Gaussian [8–11]. For numerous applications, however, especially for military applications, the bearing sensors exhibit vulnerability to external interference, enemy attack, or node failure. The bearing measurements may be subject to impulsive noises [12–16], thereby dramatically reducing the localization performance. How to maintain high positioning performance under unreliable measurements should be considered in this paper.
On the whole, the existing robust BOTL methods to process outlier data can fall to the outlier detection [17] and the M-estimate [18]. The outlier detection method aims to first detect suspected outlier data, separate them from the original data set, and then exploit the remaining data to complete the localization task. Picard and Weiss and Picard [17] proposed a sparse representation method to detect the outlier data of time-of-arrival, time-difference-of-arrival, and direction-of-arrival, as solved by linear programming. Xiong et al. [19] developed a robust expectation-maximization algorithm for distance outlier detection. Though the outlier detection method is intuitive and effective, it does not apply to large data sets or complex application scenarios.
The other important aspect refers to M-estimate, capable of estimating robust positions without preprocessing data. M-estimate primarily aims to comply with some other criteria, instead of the least-squares criterion, which is more robust to impulsive noise, as an attempt to improve estimators to be less sensitive to model errors. The robust criteria consist of Huber [20], Bi-square [18], the negative log-likelihood of the Cauchy distribution [21], Wilcoxon [22], Chebyshev norm [23], Lp-norms () [24], etc. Based on energy information for sensor networks, Liu et al. [18] presented a distributed robust localization algorithm. Such algorithm employs Bi-square function as the cost function of M-estimate. Panigrahi et al. [22] proposed a distributed incremental least mean square algorithm based on Wilcoxon norm for parameter estimation of sensor networks. Wu et al. [25] proposed a robust structure total least-squares algorithm for passive localization. The algorithm adopts the optimized Danish weight function to reduce the effect of outlier data on the localization performance. Furthermore, the Lp-norms () are useful for robust estimation since less weight is given to isolated deviations. Under the application of the L1-norm, such M-estimate is termed as the least absolute residuals (LAR) [26] method. For outlier suppression, the L1-norm appears to be markedly superior among the Lp-norms () [27].
From a statistical perspective, the least-squares optimization equals the maximum likelihood (ML) estimation when the measurements are corrupted with independent and identically distributed Gaussian noise. However, the noise distribution is altered with the presence of outliers. The Laplace distribution refers to a probability distribution that accommodates large residuals. The ML estimator of this distribution leads to the LAR solution [28]. Though the ML method is optimal for statistics, its cost function is nonlinear and nonconvex with respect to the target location parameters. The iterative numerical search is inevitable for ML, and the absolute norm should be considered in the respective iteration. The iterative algorithms tend to diverge when poorly initialized and computationally expensive. To remedy the defects of the ML method, [29] proposed a pseudolinear estimator (PLE) by lumping the nonlinearities into the noise term. However, the PLE is subjected to severe bias [30], and the bias remains with the increase in the number of sensors due to the correlations between system matrices and measurement noises. Various algorithms have been proposed to reduce the bias, including instrumental variable (IV) [31, 32] and total least-squares (TLS) [33]. The IV method [31] is capable of reducing biases by setting an instrumental matrix that is asymptotically uncorrelated with the noise vector. Inconsistent with the IV method, the TLS algorithm [33] attempts to reduce biases by minimizing the errors in the system matrix and the measurement vector. However, the IV and TLS estimators fail to improve the bias performance if the measuring angles have gross errors.
In brief, robust pseudolinear algorithms for BOTL aim to reduce biases attributed to both large residuals and the correlation between system matrix and noise vector. The least absolute residuals (LAR) minimization can be adopted to reduce the bias attributed to outlier data. However, LAR faces a major problem of the correlation bias that remains with the increase in the number of sensors. In the present study, these two types of bias are reduced by conducting total least absolute residuals (TLAR) optimization [34]. We first formulate the problem of BOTL subject to outlier data. Moreover, the pseudolinear measurement model for BOTL is reviewed, and the bias of PLE is analyzed. Subsequently, the TLAR algorithm is developed for BOTL with significantly reduced bias and root mean square error as compared with the LAR method.
The main contributions of the proposed method can be summarized as follows:(i)Development of a new bias reduced estimator based on TLAR for BOTL under bearing gross errors(ii)Development of an algorithm for TLAR optimization using dual ascent algorithms(iii)Demonstration of the performance improvement achieved by the TLAR estimator with respect to the LAR method
The rest of this paper is organized as follows. In Section 2, the measurement model is described. In Section 3, the pseudolinear equations from the BOTL problem are reviewed, the two types of bias for the PLE method are analyzed, and a LAR solution is presented when bearing measurements are subjected to large deviations. In Section 4, the TLAR approach is presented, and an algorithm is developed for TLAR based on dual ascent algorithms in Section 5. In Section 6, numerical examples illustrating the performance of PLE, TLS, LAR, and TLAR are presented. Lastly, conclusions are drawn in Section 7.
2. Model Description
A group of sensors are formed by normal sensors and abnormal sensors with significantly larger than . The total number of sensors reaches . Each node is capable of measuring a bearing in the sensor between the positive horizontal direction and the straight line from the target to the node. It is noteworthy that the normal sensors conduct the effective measurements, while the abnormal sensors collect the wrong observations as impacted by object occlusion, interference or network attack, etc. It is assumed that whether the bearing measurements have outliers cannot be distinguished. Thus, robust localization methods should be designed to prevent the location performance degradation attributed to outlier data. The problem of robust BOTL is to estimate an unknown target position in as accurately as possible by all bearing measurements.
The localization geometry is illustrated in Figure 1, where denotes the target position vector, represents the sensor location vector for the kth measurement, is the true bearing at sensor k, and the angle is positive to the counterclockwise direction, . The relationship between the bearing angle, target position, and sensor location is expressed as the following nonlinear equation:where denotes the four-quadrant arctangent and .
In fact, the observed bearings have errors, and the kth measurement can be described aswhere denotes the measurement error; represents the independent and identically distributed (i.i.d.) zero mean Gaussian noise with variance ; represents a bias term; and is a binary random variable defined as follows:
Since the prior knowledge of is unknown, which measurement is reliable and which is not cannot be distinguished in advance. Accordingly, it is necessary to develop a robust localization algorithm when both reliable and unreliable measurements are used.
3. The Least Absolute Residuals Method
To develop a robust localization method for BOTL, we first review the PLE method by converting nonlinear bearing measurements to pseudolinear equations. We then derive the least absolute residual (LAR) algorithm for BOTL. Lastly, two types of bias for PLE are analyzed.
3.1. Pseudolinear Equation
The measurement equation in (2) is nonlinear with respect to the unknown target location which makes BOTL a nontrivial task. A natural option as we will illustrate in the following would be to model a pseudolinear equation by lumping the nonlinearities into the noise term. For this end, an orthogonal vector sum is first established between the measured angle vector and the true angle vector from Figure 1 given bywhere denotes the true angle vector between and ; represents the measured angle vector starting from and generates the noisy bearing according to the horizontal direction; and indicates the error vector. By defining and as two orthogonal unit trigonometric vectors, and are written in terms of and :
Using the fact that , multiplying (4) with yieldswhere is a nonlinear transformed measurement error. Collecting the pseudolinear equation errors as a vector , we obtainwhere and are the measurement matrix and vector, respectively. The PLE requires that be estimated by minimizing with respect to in the least-squares sense. The position estimate can be obtained by solving
The solution to (8) is given by
It is termed as pseudolinear estimator (PLE).
3.2. Bias Analysis
The bias of obtained from (9) is defined by
The bias of includes two parts. The first part is attributed to the correlation between and . The second part is formed by large residuals. Let and be the pseudomeasurement matrix and vector obtained from normal sensors. Without considering the abnormal measurements, the PLE solution becomes
If the number of normal sensors is significantly bigger than that of abnormal sensors, the first type of bias can be approximated aswhere denotes the pseudomeasurement noise with its kth entry given by . Based on the Slutsky theorem [33], the first type of bias can be asymptotically computed by
As goes to infinity, (13) becomes an equality. For finite , obtained by (13) is a good approximation to (12) [35]. After is calculated, the second type of bias is . An example of PLE bias is depicted in Figure 2(b), Section 6. The second type of bias appears to dominate if the standard deviation of is small enough. Thus, the performance of PLE will degrade dramatically under impulsive noise since L2-norm optimization can be severely affected by pseudolinear errors with large residuals. To ensure such items have less influence, we could instead minimize a cost function that gives less weight to large deviations.
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3.3. Least Absolute Residual
A common choice to alleviate the effect of gross errors is the absolute value metric denoted by , where represents L1-norm. As such, the LAR optimization can be achieved by
The derivative of function is bounded for all by the value , demonstrating that the cost function is robust for all deviations. For the PLE criterion, the derivative is not bounded, and it increases linearly with .
In the literature, numerous algorithms have been developed to solve the minimization problem of (e.g., iteratively reweighted least-squares (IRWLS) [36], expectation-maximization (EM) procedure [37], and linear programming [38]). A weight matrix is defined as
The WLS algorithm is explicitly given by
The problem of (16) is that the weights become extraordinarily large for or numerically indeterminate for . Benefiting from convex optimization, LAR problem can be relaxed to identify the minimum bound of the absolute value:where denotes the upper bound of ; is a column vector of ones. To be specific, two nonnegative vectors are denoted, i.e., and . The equivalent form of (17) is
Note that (23) refers to a standard linear programming problem, which can be solved by using the existing CVX software [39]. When the pseudolinear errors follow i.i.d. Laplacian distribution with zero mean, (14) is equivalent to the ML estimator. However, the LAR method implicitly assumes that only is subjected to errors. This is not the case since the system matrix is corrupted with measurement noises as well. The correlation between and causes the LAR estimator to be inconsistent. Though the LAR estimator has bias, it gives a reasonable estimate and provides an initial guess for other robust estimators.
4. Total LAR Optimization
The LAR algorithm expressed in (10) implicitly indicates that only has errors and the gross errors are reduced by using the weights all restricted in . In fact, matrix is also subject to measurement errors. When both and are disturbed with noise, the LAR solution of (9) will inevitably cause large bias as and are statistically dependent. To increase the accuracy of the LAR estimator, the idea of total least absolute residual (TLAR) can be exploited to reduce the errors in both and .
The concept of TLAR is that the disturbance vector is adopted to correct the data vector , while the disturbance matrix is employed to disturb the data matrix . In other words, we use the following equations in the TLAR problem:where denotes a augmented matrix; represents a perturbation matrix; is a perturbation matrix of ; is a perturbation vector of ; is a vector; and is a scaling factor. Notably, both and are corrupted with noise. Next, the error statistics for and are examined. The error matrix is defined aswhere the entries of are given by the difference between and the noiseless augmented matrixwhere , . If is sufficiently small, it yields , . Thus, and become
The mean and second-order moments of the error items are written as
Note that from (23), the mean value of errors in is nonzero as impacted by the outlier data. In addition, (24) and (25) are nonzero even if equals zero. It is therefore suggested that the error terms in the identical row of are correlated with each other. To more effectively reduce the bias of LAR, TLAR metric is adopted to minimize the disturbance matrix and vector simultaneously, and the TLAR problem for BOTL is formulated as
If the L1-norm in (26) is replaced by the Frobenius matrix norm, (26) becomes the well-known TLS problem. Both TLAR and TLS fall to the domain of “total approximation problem,” known as the total least pth problem for . To solve the TLAR problem effectively, an equivalent form of (26) is explored. The following result holds.
Theorem 1. Suppose that , where denotes the augment matrix with and defined in (7), is the error matrix with and defined in (19), and is a vector. Subsequently, the estimation of from the minimization of can be achieved by
Proof. From , it yields . Let us denote and . Then, can be computed asNote that , for . Accordingly, it yields . If the infinite norm of satisfies , the above inequality becomes . Lastly, we get the conclusion that the minimization of subject to can be obtained from (27).
Theorem 1 presents a practicable solution for BOTL since . However, (27) is not a convex problem since the feasible region does not fall to . Thus, the global minimum is not ensured. Indeed, its local minima can be computed by adopting the Lagrange multiplier formulation of (27).
5. Algorithm Development
In the present section, an algorithm is derived to solve (27). Since the optimization of (27) is not convex, a stationary point satisfying the first-order necessary conditions is calculated. First, (27) is transformed into an unconstrained minimization problem by leveraging the Lagrange multiplier method. The Lagrange objective function is defined as
And the dual problem of (27) is given bywhere denotes the Lagrange multiplier. Set as the optimal value of (30). By substituting into (29), the minimization of generates the primal optimal point . Such dual problem can be solved by a dual ascent (DS) algorithm [40] as expressed below.(1) is assumed as the optimal solution of the dual problem (30) at the jth step.(2)The primal optimal point can be determined from by minimizing :(3)The dual variable is updated by where is a step size. The -update is realized by using gradient ascent. With proper choice of , increases for each step, i.e., .(4) Next, we investigate the problem of minimizing . Based on Taylor’s series expansion of around up to the first order, it yields(5) Note that , where is the angle between and . Thus, the steepest decent direction is . Let denote the subdifferential of , where(6) is a subgradient vector and its kth element is given by(7)The subdifferential of at , denoted by , is defined by where conv denotes the convex hull and is a vector whose ith element is 1 and all other elements are 0. Based on (34) and (36), is written as
6. Experimental Results
In the present section, numerical examples are studied to compare the localization performance of the proposed TLAR algorithm with PLE, TLS, and LAR under abnormal sensor and nonabnormal sensor. In addition, two typical localization geometries are applied in the simulations. The first one is randomly distributed sensors, and the other one is linearly distributed sensors. In the respective scenario, four cases are considered. To be specific, (1) the number of abnormal sensors varies and the total number of sensors is fixed, (2) both the numbers of normal and abnormal sensors vary, (3) the numbers of normal and abnormal sensors are fixed, and (4) all sensors are normal. The bearing data achieved from abnormal sensors exhibit uniform distribution . For normal sensors, the bearing measurement errors are assumed as i.i.d. zero mean Gaussian with standard deviation σ. Simulation comparisons are drawn based on Mc = 1000 Monte-Carlo simulation runs. This study employs bias and root mean square error (RMSE) for localization performance comparison, which are written aswhere and denote the estimates of target location parameters for the mth Monte-Carlo run of the simulation.
6.1. Randomly Distributed Sensors
In this section, all sensors are randomly placed in a 100 × 100 m2 region centered at (50, 50) m (Figure 2(a)). The unknown target is placed at (100, 100) m. In the first example, the number of abnormal sensors is elevated from 2 to 6, and that of normal sensors decreases from 18 to 14. The total number of sensors is fixed at 20. σ is set to (3°). Figure 2(a) presents the RMSE results with the increase in the number of abnormal sensors. The LAR and TLAR methods are capable of reducing the outliers, as indicated in the two plots of Figure 2(a), the blue line with “+” and the red line with “square.” The RMSE of LAR is 0.618 m above that of TLAR when the number of outliers is set to 2. Such value increases to 1.551 m when the number of outliers is kept at 6. The estimation bias is presented in Figure 2(b). As the number of outliers is elevated, the value of bias turns more significant.
In the next example, the number of outliers is fixed at 3. The sensors have a total number of twenty. σ ranges from to (1° to 5°). The number of abnormal sensors (red circles) is set to three, and the rest are normal sensors (blue circles), as illustrated in Figure 3(a). The simulated biases of PLE are plotted in Figure 3(b). Under small σ, the first type of bias attributed to the correlation between and keeps at low level, and the second type of bias formed by large residuals dominates the theoretical bias of PLE. With the increase in the measurement noise variance, the effect of the second type of bias turns out to be less significant. Figures 4(a) and 4(b) illustrate the RMSE and bias curves of various methods in the presence of abnormal sensors. The PLE method and the TLS estimator fail to give accurate target location estimates since they are not robust to outlier data. The blue line with “+” in Figure 4(a) represents the RMSE value determined by using the LAR method, and the red line with “square” represents the RMSE curve for the TLAR algorithm. They are well separated and the gap between these two lines increases as σ becomes large, suggesting the reduction of the first type of bias. Furthermore, this phenomenon is verified in Figure 4(b), where the bias of TLAR is significantly lower than that of the LAR method.
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Specific to the third example, the number of sensors is elevated from ten to thirty, five at a time, in which one is abnormal, and the other four are normal. σ is set to (3°). Figures 5(a) and 5(b) draw a comparison of the RMSE and bias of various methods. With the increase in the number of sensors, the RMSE and bias are reduced gradually since the numbers of abnormal and normal sensors increase simultaneously and are proportional. It is again proved that the TLAR method exhibits the optimal RMSE and bias performance.
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Subsequently, the RMSE and bias performance are determined according to different bearing noise standard deviations when all sensors are normal sensors, as presented in Figures 6(a) and 6(b). In this scenario, the TLS method exhibits the optimal RMSE performance. The RMSE curve of TLAR is slightly higher than that of TLS since the bearing measurement errors overall comply with the Gaussian distribution. In such scenario, the L2-norm criterion is optimal. Unlike the RMSE performance, the bias of TLAR is comparable with that of TLS and much lower than that of LAR or PLE. When all sensors are normal, the LAR and TLAR methods are capable of achieving effective results, compared with PLE and TLS, thereby demonstrating the robustness of the proposed algorithm.
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6.2. Linearly Distributed Sensors
In the present section, all sensors are linearly distributed with the equal spacing from (5, 26) m to (155, −34) m (Figure 7(a)). The target is placed at (200, 40) m. In the first example, the number of abnormal sensors ranges from 2 to 6, and that of sensors is set to 31. σ is set at (3°). Figures 8(a) and 8(b) illustrate the RMSE and bias performance when the number of abnormal sensors is altered. It is therefore demonstrated that the TLAR algorithm exhibits better RMSE and bias performance than the LAR method. As the number of outliers is elevated, the performance is more significantly improved.
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In the next example, all sensors are working properly except for the 8th to 10th sensors. Consistent with Section 6.1, the theoretical bias of PLE is plotted in Figure 7(b). The results comply with those presented in Figure 3(b).
The RMSE and bias results for such example are given in Figures 9(a) and 9(b), respectively. The RMSE value of TLAR is lower than that of the LAR method for the case of linearly distributed sensors, even though only a 0.502 m reduction for σ = 1° and a 4.56 m reduction for σ = 5°. With the increase in the bearing measurement noise variance, the effect of large residuals turns out to be less significant. The amount of bias reduction is 5.04 m for σ = 5° if TLAR is used.
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Specific to the third example, the RMSE and bias results are shown in Figures 10(a) and 10(b) when the number of sensors ranges from 16 to 31, five at a time, one of which is abnormal, and the other four are normal. σ is set to (3°). Figures 10(a) and 10(b) verify that the solution of TLAR has better RMSE and bias performance than that of PLE, TLS, and LAR. Based on more than 21 sensors, the RMSE and bias of LAR suddenly decrease. This is not unexpected since the sensor observation angles are more significantly discriminated.
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In addition, the RMSE and bias curves of PLE, LAR, TLS, and TLAR without abnormal sensors are plotted. Figure 11(a) gives the RMSE results as the bearing standard deviation increases, and Figure 11(b) compares the bias of various solutions. As clearly demonstrated by the RMSE and bias results, TLAR outperforms LAR. The amounts of reduction in terms of RMSE and bias are 1.85 m and 2.845 m, respectively.
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7. Conclusions
The present study presents a TLAR algorithm to solve the BOTL problem in the presence of outlier data. Though the conventional LAR is robust to significant deviations, it has bias formed by the correlation between the system matrix and the noise vector and such bias remains with the increase in the number of sensors. To increase the accuracy of the LAR estimator, the TLAR is proposed by adding minimal perturbations to both system matrix and data vector in least absolute residuals sense, so the perturbed matrix is consistent. As revealed from the experimental results, the TLAR algorithm outperforms the LAR method. When the bearing noise power becomes more significant, the performance of RMSE and bias reduction is more obviously improved.
Data Availability
The simulation data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Science Foundation for Young Scientists of China under Grant 61703129 and in part by the National Key Laboratory of Science and Technology on Blind Signal Processing under Grant 61424131807.