Maximum Likelihood Localization Research Articles

An efficient estimator for source localization in WSNs using RSSD and TDOA measurements

Range-based localization has received considerable attention in wireless sensor networks due to its ability to efficiently locate the unknown source of a signal. However, the localization accuracy with a single set of measurements may be inadequate, especially in dynamic and noisy environments. To mitigate this problem, received signal strength difference (RSSD) and time difference of arrival (TDOA) measurements are used to develop an efficient estimator to reduce the bias and improve localization accuracy. First, the RSSD/TDOA-based maximum likelihood (ML) localization problem is transformed into a hybrid information nonnegative constrained least squares (HI-NCLS) framework. Then, this framework is used to develop an effective bias-reduction localization approach (BRLA) with a two-step linearization process. The first step employs a linear solving method (LSM) which exploits an active set method to obtain a sub-optimal estimator. The second step uses a bias reduction method (BRM) to mitigate the correlation from linearization and a weighted instrumental variables matrix (IVM) which is weakly correlated with the noise but strongly correlated with the data matrix (DM) is used in place of the DM. Performance results are presented which demonstrate that the proposed BRLA provides better localization performance than state-of-the-art methods in the literature.

Maximum likelihood localization of a network of moving agents from ranges, bearings and velocity measurements

Localization is a fundamental enabler technology for many applications, like vehicular networks, IoT, and even medicine. While Global Navigation Satellite Systems solutions offer great performance, they are unavailable in scenarios like indoor or underwater environments, and, for large networks, the instrumentation cost is prohibitive. We develop a localization algorithm from ranges and bearings, suitable for generic mobile networks. Our algorithm is built on a tight convex relaxation of the Maximum Likelihood position estimator. To serve positioning to mobile agents, a horizon-based version is developed accounting for velocity measurements at each agent. To solve the convex problem, a distributed gradient-based method is provided. This constitutes an advantage over centralized approaches, which usually exhibit high latency for large networks and present a single point of failure. Additionally, the algorithm estimates all required parameters and effectively becomes parameter-free. Our solution to the dynamic network localization problem is theoretically well-founded and still easy to understand. We obtain a parameter-free, outlier-robust and trajectory-agnostic algorithm, with nearly constant positioning error regardless of the trajectories of agents and anchors, achieving better or comparable performance to state-of-the-art methods, as our simulations show. Furthermore, the method is distributed, convex and does not require any particular anchor configuration.

Confidence interval comparison: Precision of maximum likelihood estimates in LLOQ affected data.

When data is derived under a single or multiple lower limits of quantification (LLOQ), estimation of distribution parameters as well as precision of these estimates appear to be challenging, as the way to account for unquantifiable observations due to LLOQs needs particular attention. The aim of this investigation is to characterize the precision of censored sample maximum likelihood estimates of the mean for normal, exponential and Poisson distribution affected by one or two LLOQs using confidence intervals (CI). In a simulation study, asymptotic and bias-corrected accelerated bootstrap CIs for the location parameter mean are compared with respect to coverage proportion and interval width. To enable this examination, we derived analytical expressions of the maximum likelihood location parameter estimate for the assumption of exponentially and Poisson distributed data, where the censored sample method and simple imputation method are used to account for LLOQs. Additionally, we vary the proportion of observations below the LLOQs. When based on the censored sample estimate, the bootstrap CI led to higher coverage proportions and narrower interval width than the asymptotic CI. The results differed by underlying distribution. Under the assumption of normality, the CI's coverage proportion and width suffered most from high proportions of unquantifiable observations. For exponentially and Poisson distributed data, both CI approaches delivered similar results. To derive the CIs, the point estimates from the censored sample method are preferable, because the point estimate of the simple imputation method leads to higher bias for all investigated distributions. This biased simple imputation estimate impairs the coverage proportion of the respective CI. The bootstrap CI surpassed the asymptotic CIs with respect to coverage proportion for the investigated choice of distributional assumptions. The variety of distributions for which the methods are suitable gives the applicant a widely usable tool to handle LLOQ affected data with appropriate approaches.

Heuristic techniques for maximum likelihood localization of radioactive sources via a sensor network

Maximum likelihood estimation (MLE) is an effective method for localizing radioactive sources in a given area. However, it requires an exhaustive search for parameter estimation, which is time-consuming. In this study, heuristic techniques were employed to search for radiation source parameters that provide the maximum likelihood by using a network of sensors. Hence, the time consumption of MLE would be effectively reduced. First, the radiation source was detected using the k-sigma method. Subsequently, the MLE was applied for parameter estimation using the readings and positions of the detectors that have detected the radiation source. A comparative study was performed in which the estimation accuracy and time consumption of the MLE were evaluated for traditional methods and heuristic techniques. The traditional MLE was performed via a grid search method using fixed and multiple resolutions. Additionally, four commonly used heuristic algorithms were applied: the firefly algorithm (FFA), particle swarm optimization (PSO), ant colony optimization (ACO), and artificial bee colony (ABC). The experiment was conducted using real data collected by the Low Scatter Irradiator facility at the Savannah River National Laboratory as part of the Intelligent Radiation Sensing System program. The comparative study showed that the estimation time was 3.27 s using fixed resolution MLE and 0.59 s using multi-resolution MLE. The time consumption for the heuristic-based MLE was 0.75, 0.03, 0.02, and 0.059 s for FFA, PSO, ACO, and ABC, respectively. The location estimation error was approximately 0.4 m using either the grid search-based MLE or the heuristic-based MLE. Hence, heuristic-based MLE can provide comparable estimation accuracy through a less time-consuming process than traditional MLE.

Localization Through Transceivers in Unknown Constant Velocity Trajectories

A distant object is difficult to locate in unique coordinates, either it is too far to resolve the range or the radiated signal is too weak to be seen. This paper explores the possibility of utilizing moving transceivers to improve the geometry and extend the signal propagation range for the localization of a distant object, by time delay measurements. In addition to the object location, the motion parameters and re-transmission time lags of the transceivers are not known. We first investigate the observability of such an approach for localization, when the transceivers are in constant velocity motion. Under certain conditions that are related to the geometry, we show localization by such an approach is feasible and derive the minimum numbers of sensors and transceivers needed for 2-D and 3-D positionings. Both the joint and sequential estimations of the transceiver parameters and the object location are studied next. The use of an inaccurate calibration copy for compensating the transceiver time lags is also examined. Simulations validate the theoretical development and present the Maximum Likelihood localization performance.

Target localization in distributed MIMO radars via improved semidefinite relaxation

Target localization is an important problem in distributed multiple-input multiple-output (MIMO) radar systems. In this paper, a new algorithm using bistatic range measurements is developed for target localization in distributed MIMO radars. Unlike most existing schemes, the proposed algorithm firstly applies semidefinite relaxation to convert the maximum likelihood localization problem into a convex optimization problem. Subsequently, a novel procedure is devised to improve the solution accuracy of the convex optimization problem. Our scheme exhibits evidently better threshold behavior than the state-of-the-art approaches. Moreover, it does not require any initial estimate of the target position. Simulation results verify the superiority of the proposed algorithm over various existing methods.

A Robust Indoor Mobile Localization Algorithm for Wireless Sensor Network in Mixed LOS/NLOS Environments

Wireless sensor network (WSN) is a self-organizing network which is composed of a large number of cheap microsensor nodes deployed in the monitoring area and formed by wireless communication. Since it has the characteristics of rapid deployment and strong resistance to destruction, the WSN positioning technology has a wide application prospect. In WSN positioning, the nonline of sight (NLOS) is a very common phenomenon affecting accuracy. In this paper, we propose a NLOS correction method algorithm base on the time of arrival (TOA) to solve the NLOS problem. We firstly propose a tendency amendment algorithm in order to correct the NLOS error in geometry. Secondly, this paper propose a particle selection strategy to select the standard deviation of the particle swarm as the basis of evolution and combine the genetic evolution algorithm, the particle filter algorithm, and the unscented Kalman filter (UKF) algorithm. At the same time, we apply orthogon theory to the UKF to make it have the ability to deal with the target trajectory mutation. Finally we use maximum likelihood localization (ML) to determine the position of the mobile node (MN). The simulation and experimental results show that the proposed algorithm can perform better than the extend Kalman filter (EKF), Kalman filter (KF), and robust interactive multiple model (RIMM).

TDOA-Based Joint Synchronization and Localization Algorithm for Asynchronous Wireless Sensor Networks

This paper presents a joint synchronization and localization algorithm based on the time-difference-of-arrival (TDOA) for asynchronous Wireless Sensor Networks (WSNs), where the positions of anchors are relatively fixed but unknown. To obtain time synchronization and anchor positions, each anchor broadcasts signals periodically, while other anchors receive the broadcasting signals and stamp times-of-arrival (TOAs). Based on these TOAs, we estimate the internal clock parameters and anchor positions in three steps: least square estimation (LSE) of the relative clock skew based on TDOAs, maximum likelihood localization (MLE) of anchors using biased time of flight (TOF), and LSE estimation of relative clock offsets. Anchor pairs stamp the TDOAs when a tag (a wireless sensor node that requires localization) transmits a signal. The biased TDOAs, due to asynchronous local clocks, are compensated by relative clock offset estimation. The maximum likelihood estimation is used for the tag localization based on the Gaussian noise model. We evaluate the performance of the proposed synchronization and localization algorithm using the Cramer-Rao lower bound (CRLB). Simulations are carried out to verify the validity of the algorithm in this paper.

A Triple-Filter NLOS Localization Algorithm Based on Fuzzy C-means for Wireless Sensor Networks.

With the rapid development of communication technology in recent years, Wireless Sensor Network (WSN) has become a promising research project. WSN is widely applied in a number of fields such as military, environmental monitoring, space exploration and so on. The non-line-of-sight (NLOS) localization is one of the most essential techniques for WSN. However, the NLOS propagation of WSN is largely influenced by many factors. Hence, a triple filters mixed Kalman Filter (KF) and Unscented Kalman Filter (UKF) voting algorithm based on Fuzzy-C-Means (FCM) and residual analysis (TF-FCM) has been proposed to cope with this problem. Firstly, an NLOS identification algorithm based on residual analysis is used to identify NLOS errors. Then, an NLOS correction algorithm based on voting and NLOS errors classification algorithm based on FCM are used to process the NLOS measurements. Hard NLOS measurements and soft NLOS measurements are classified by FCM classification. Secondly, KF and UKF are applied to filter two categories of NLOS measurements. Thirdly, maximum likelihood localization (ML) is employed to estimate the position of mobile nodes. The simulation result confirms that the accuracy and robustness of TF-FCM are better than IMM, UKF and KF. Finally, an experiment is conducted to test and verify our algorithm which obtains higher localization accuracy.

Maxlifd: Joint Maximum Likelihood Localization Fusing Fingerprints and Mutual Distances

Fusing mutual distance information with fingerprints can substantially improve indoor localization accuracy. Such distance information may be spatial (e.g., measurement among users or from installed beaconing devices) or temporal (e.g., via dead-reckoning). Previous approaches on distance-fusion often require deterministic distance measurement, consider fingerprints and distances separately, or are narrowly applicable to some specific sensing technology or scenario. Given the fact that fingerprint and distance measurements are intrinsically random, we propose Maxlifd , an accurate indoor localization framework fusing fingerprints and distances of arbitrary distributions via joint maximum likelihood. Maxlifd is a generic statistical/probabilistic framework applicable to a wide range of sensors (peer-assisted, INS, iBeacon, etc.) and fingerprints (Wi-Fi, RFID, etc.). It achieves low localization errors by a novel optimization formulation jointly considering mutual distances and fingerprint signals. Using generic probabilistic formulation, we further derive the lower bound on localization error for comprehensive performance analysis. We have implemented Maxlifd, and conducted extensive simulation and experimental trials in an international airport and our university campus. Our results show that Maxlifd achieves significantly lower errors than other state-of-the-art schemes (often by more than 30 percent). We experimentally verify that its performance does not depend sensitively on the exact knowledge of the underlying distributions beyond simple Gaussian.

RSS-Based Localization in WSNs Using Gaussian Mixture Model via Semidefinite Relaxation

Energy-based source localization methods are normally developed according to the channel path-loss models in which the noise is generally assumed to follow Gaussian distributions. In this letter, we represent the practical additive noise by the Gaussian mixture model, and develop a localization algorithm based on the received signal strength to achieve a maximum likelihood location estimator. By using Jensen’s inequality and semidefinite relaxation, the initially proposed nonlinear and nonconvex estimator is relaxed into a convex optimization problem, which can be efficiently solved to obtain the globally optimal solution. Besides, the corresponding Cramer–Rao lower bound is derived for performance comparison. Simulation and experimental results show a substantial performance gain achieved by our proposed localization algorithm in wireless sensor networks.

Robustness of statistical algorithms for location of microseismic sources based on surface array data

Here, we present a statistical simulation study of several array data processing techniques used for the location of sources of elastic waves. The source location problem arises in such engineering applications as radio-communications, acoustics, sonar, meteorology, astrophysics, seismology, etcetera. Recently, this problem has become important for applied geophysics in connection with the modern technology of hydrocarbon production based on the hydraulic fracturing of the surface layers of the Earth’s environment. These techniques are required for the monitoring of hydraulic fracturing through the location of sources of numerous microearthquakes that occur during cracking of crustal rocks. We have established theoretical connections among several location algorithms. In particular, two different theoretical interpretations were proposed for the phase alignment location algorithm, recently proposed for the seismic array data processing. The robustness properties of the phase alignment location algorithm and the adaptive maximum likelihood location algorithm were demonstrated through Monte Carlo simulation in regard to variations of those noise statistical features, which are typical under monitoring of microseismicity at the hydrocarbon production sites. In the framework of the Monte Carlo modeling, we also compared the two mentioned location algorithms with the so-called emission tomography algorithm that is developed for the location of microseismic sources during the monitoring of hydraulic fracturing.

Navigation by inertial device and signals of opportunity

Inertial navigation systems are known to yield rather accurate measurements over short time intervals, while their error variance tends to increase with time. In order to keep the error within specification most systems use GPS signals. In the absence of GPS data, due to jamming or spoofing, it is desirable to use signals of opportunity instead. We examine the use of time of arrival measurements of signals of opportunity that have known structure. We propose a low complexity semi-definite relaxation algorithm by converting the maximum likelihood location estimator to a convex optimization problem. Simulation results demonstrate that the proposed algorithms converge to the Cramér–Rao lower bound under some geometrical and noise limitations.

Maximum likelihood localization: When does it fail?

Abstract Maximum likelihood is a criterion often used to derive localization algorithms. In particular, in this paper we focus on a distance-based algorithm for the localization of nodes in static wireless networks. Assuming that Ultra Wide Band (UWB) signals are used for inter-node communications, we investigate the ill-conditioning of the Two-Stage Maximum-Likelihood (TSML) Time of Arrival (ToA) localization algorithm as the Anchor Nodes (ANs) positions change. We analytically derive novel lower and upper bounds for the localization error and we evaluate them in some localization scenarios as functions of the ANs’ positions. We show that particular ANs’ configurations intrinsically lead to ill-conditioning of the localization problem, making the TSML-ToA inapplicable. For comparison purposes, we also show, through some examples, that a Particle Swarm Optimization (PSO)-based algorithm guarantees accurate positioning also when the localization problem embedded in the TSML-ToA algorithm is ill-conditioned.

A Distributed and Maximum-Likelihood Sensor Network Localization Algorithm Based Upon a Nonconvex Problem Formulation

We propose a distributed algorithm for sensor network localization, which is based upon a decomposition of the nonlinear nonconvex maximum likelihood (ML) localization problem. Decomposition and coordination are obtained by applying the alternating direction method of multipliers (ADMM), to provide a distributed, synchronous, and nonsequential algorithm. When penalty coefficients are locally increased under specific conditions, the algorithm is proved to converge irrespective of the chosen starting point. It is also shown to be fast and accurate, providing a performance, which is equivalent to that of a centralized solver. In the comparison with existing literature on distributed sensor network localization, the proposed method involves a much lighter local processing effort, an improved robustness in non-Gaussian scenarios, and a better adherence to the original problem.

Investigation of the numerics of point spread function integration in single molecule localization.

The computation of point spread functions, which are typically used to model the image profile of a single molecule, represents a central task in the analysis of single molecule microscopy data. To determine how the accuracy of the computation affects how well a single molecule can be localized, we investigate how the fineness with which the point spread function is integrated over an image pixel impacts the performance of the maximum likelihood location estimator. We consider both the Airy and the two-dimensional Gaussian point spread functions. Our results show that the point spread function needs to be adequately integrated over a pixel to ensure that the estimator closely recovers the true location of the single molecule with an accuracy that is comparable to the best possible accuracy as determined using the Fisher information formalism. Importantly, if integration with an insufficiently fine step size is carried out, the resulting estimates can be significantly different from the true location, particularly when the image data is acquired at relatively low magnifications. We also present a methodology for determining an adequate step size for integrating the point spread function.

A Cognitive Algorithm for Received Signal Strength Based Localization

In this paper, we propose a method to localize a wireless and stationary blind node based on received signal strength measurements; the positioning scheme relies on the statistical path loss model. The operating environment is either non-homogeneous, i.e., the attenuation factors of the various links are different, or homogeneous, i.e., all links share one and the same attenuation factor. We introduce a cognitive procedure to circumvent the a priori uncertainty on the actual scenario being in force: the first stage is a properly designed hypothesis test, fed by measurements between anchors, moving along known trajectories, to decide whether or not the environment is homogeneous. Based upon the output of the test, a model-dependent maximum-likelihood localization algorithm is adopted using the measurements made by the blind node and the parameters estimated by the anchors at previous stage. The performance assessment shows that the proposed approach could be a viable means to handle localization in uncertain scenarios.

A divide and conquer strategy for the maximum likelihood localization of low intensity objects

In cell biology and other fields the automatic accurate localization of sub-resolution objects in images is an important tool. The signal is often corrupted by multiple forms of noise, including excess noise resulting from the amplification by an electron multiplying charge-coupled device (EMCCD). Here we present our novel Nested Maximum Likelihood Algorithm (NMLA), which solves the problem of localizing multiple overlapping emitters in a setting affected by excess noise, by repeatedly solving the task of independent localization for single emitters in an excess noise-free system. NMLA dramatically improves scalability and robustness, when compared to a general purpose optimization technique. Our method was successfully applied for in vivo localization of fluorescent proteins.

Mobility-Aided Wireless Sensor Network Localization via Semidefinite Programming

In this paper, considering a mobile wireless sensor network, we study the problem of exploiting sensor mobility information in the process of sensor localization under two range measurement models, namely the time-of-arrival (TOA) model and the received signal strength (RSS) model. To do so, for each model, we first derive the maximum likelihood (ML) location estimator for the case of error-free velocity measurements. As the corresponding optimization problems are non-convex, we resort to semi-definite relaxation (SDR) techniques to find approximate solutions to each problem using semi-definite programming (SDP). We then extend our results to the cases where the velocity measurements are subject to measurement errors. Our simulation results show that exploiting the mobility information in the localization process can significantly improve the performance of the sensor localization. Moreover, mobility-aided localization has the potential to address some of typical positioning problems, such as sensitivity to the ranging measurement errors and the requirement on the number of the anchors needed to uniquely localize the sensor nodes.

Fast Maximum Likelihood Localization of Near-Field Sources

Conventional methods for locating near-field sources generally suffer performance degradation when the assumption of uniform spatial Gaussian noise does not hold. In this paper study the scenario of non-uniform spatial Gaussian noise. First we construct the near-field signal model based on planar sensor array and derive the maximum likelihood method for obtaining the azimuth and distance of sound sources, then we proposed two fast algorithms-stepwise-concentrated maximum likelihood method(SML) and approximate maximum likelihood method(AML) to reduce the high computational complexity of maximum likelihood localization method. Simulation results show that the two proposed methods outperform conventional maximum likelihood method, with lower computational complexity and less mean squared error of both azimuth estimation and distance estimation.