Robust time-of-arrival source localization employing error covariance of sample mean and sample median in line-of-sight/non-line-of-sight mixture environments

Park and Chang EURASIP Journal on Advances in Signal Processing (2016) 2016:89 DOI 10.1186/s13634-016-0385-4 RESEARCH EURASIP Journal on Advances in Signal Processing Open Access Robust time-of-arrival source localization employing error covariance of sample mean and sample median in line-of-sight/ non-line-of-sight mixture environments Chee-Hyun Park1 and Joon-Hyuk Chang2* Abstract We propose a line-of-sight (LOS)/non-line-of-sight (NLOS) mixture source localization algorithm that utilizes the weighted least squares (WLS) method in LOS/NLOS mixture environments, where the weight matrix is determined in the algebraic form. Unless the contamination ratio exceeds 50 %, the asymptotic variance of the sample median can be approximately related to that of the sample mean. Based on this observation, we use the error covariance matrix for the sample mean and median to minimize the weighted squared error (WSE) loss function. The WSE loss function based on the sample median is utilized when statistical testing supports the LOS/NLOS state, while the WSE function using the sample mean is employed when statistical testing indicates that the sensor is in the LOS state. To testify the superiority of the proposed methods, the mean square error (MSE) performances are compared via simulation. Keywords: Adaptive selection, Loss function, Sample mean, Sample median, Statistical testing, Error variance 1 Introduction The aim of the source localization system is to find a geometrical point of intersection using the measurements from each receiver, such as the time difference of arrival (TDOA), time of arrival (TOA), or received signal strength (RSS). Localizing a point source in which passive and stationary sensors are used has been a repeated and popular research issue in the areas of radar, sonar, global positioning system, video conferencing, and telecommunication. Even though location estimation problems have been investigated extensively in the existing literature [1–6], there are still some unresolved problems. One of the key challenges of the localization problem is to estimate the position of the source in dense cluttered nonline-of-sight (NLOS) environments [7, 8]. NLOS scenarios occur when there is an obstruction between transmitters and receivers located in indoor environments and outdoor situations such as urban areas. In general, the research *Correspondence: 2 Department of Electronic Engineering, Hanyang University, Seoul 133-791, Republic of Korea Full list of author information is available at the end of the article fields of localization for the LOS/NLOS mixture problem can be categorized into two parts: (1) the constrained least squares (LS) method using optimization such as the semidefinite relaxation and second-order cone relaxation [9–12] and (2) localization using robust statistics. While localization using the optimization method has comparatively high accuracy, the computational load is higher than that of the analytical solution. Therefore, we concentrate on localization using robust statistics. The existing robust LOS/NLOS mixture position estimators are usually based on the concept of the median, e.g., least median squares (LMedS) [13, 14], M-estimator [15, 16], and the HodgesLehmann estimator [17]. The sample mean is an efficient estimator under the normal distribution; thus, it outperforms the accuracy of the sample median in the normal distribution. However, the accuracy of the sample mean is severely degraded when outliers exist in the heavy-tailed distribution, e.g., t distribution or double exponential distribution. On the other hand, the sample median is robust to outliers if the contamination ratio is smaller than 50 % but is inferior to the sample mean when outlier does not exist, i.e., the asymptotic variance of the sample mean approximately amounts to 64 % of variance for the sample © 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Park and Chang EURASIP Journal on Advances in Signal Processing (2016) 2016:89 median when the number of samples is large and the noise distribution follows the normal distribution [18]. Accordingly, the weighted squared error (WSE) loss function based on the sample mean is utilized in the line-of-sight (LOS) condition, where the sensor state is determined by the classical statistical testing [19, 20]. In contrast, when statistical testing indicates that the sensor in the LOS/NLOS state is valid, the WSE loss function using the sample median is adopted. Then, the source position is determined by minimizing the sum of WSE loss functions based on the sample mean of the sensor expected to be in the LOS condition and sample median of the sensor determined to be in the LOS/NLOS state. The motivation of this paper is as follows. The weighted least squares (WLS) estimator utilizes the weight matrix whose diagonal components amount to the inverse of the error variances of independent noisy measurements. However, the algorithm that utilizes the covariance information of the sample median and mean in the LOS/NLOS mixture state has not yet been reported; that is, the WLS in the LOS/NLOS mixture situation, in which the weight matrix is derived in the algebraic form, has not yet been developed. Thus, we employ the WLS algorithm for the LOS/NLOS mixture state, which is extended from the LOS state, and the diagonal elements of the weight matrix amount to the inverse of asymptotic variances for the sample mean and median. The proposed robust position estimation algorithms differ from the existing two-step WLS algorithm for the LOS conditions, because the proposed methods employ the error variance for the sample median, and this error variance has not been applied to the two-step WLS estimator. The proposed robust localization methods are divided into the iteration method and closed-form algorithm. The Taylor series expansion is utilized in the iteration method, and the two-step WLS algorithm is adopted in the closedform method. The proposed methods exhibit the superior mean square error (MSE) performances when compared to that of existing methods. Moreover, the performance of the proposed closed-form LOS/NLOS mixture localization method using the two-step WLS method is similar to that of the Taylor series-based iteration method, with the advantages of low computational complexity and avoidance of the divergence problem of the solution. When solution diverges, it can reach a solution, which is far from the true solution, or sometimes it fails to produce a solution when the initial value is not appropriately chosen. The LOS/NLOS mixture localization method fo (...truncated)