Measurement Noise Variance Research Articles

An improved mean-square weight deviation-proportionate gain algorithm based on error autocorrelation

This paper presents an alternative approach to the gain distribution policy used in the z2-proportionate algorithm. The gain policy of the z2-proportionate uses a rule that combines the mean-square weight deviation-proportionate gain and a uniform one to obtain the whole algorithm gain distribution, leading to very good convergence characteristics. However, such a gain combination law is dependent on the knowledge of the measurement noise variance in the system, which in practice is not always readily available. Here, aiming to circumvent such dependence, a new strategy of gain distribution based on error autocorrelation is introduced. The proposed approach makes the use of the mean-square weight deviation-proportionate gain more attractive for real-world applications. Simulation results show that the proposed algorithm outperforms the z2-proportionate in terms of convergence characteristics for cases in which the measurement noise variance is either unknown or poorly estimated.

Random errors in carbon and water vapor fluxes assessed with Gaussian Processes

The flow of carbon between terrestrial ecosystems and the atmosphere is mainly driven by nonlinear, complex and time-lagged processes. Understanding the associated ecosystem responses is a key challenge regarding climate change questions such as the future development of the terrestrial carbon sink. However, high temporal resolution measurements of ecosystem variables (with the eddy covariance method) are subject to random error, that needs to be accounted for in model-data fusion, multi-site syntheses and up-scaling efforts.Gaussian Processes (GPs), a nonparametric regression method, have recently been shown to capture relationships in high-dimensional, nonlinear and noisy data. Heteroscedastic Gaussian Processes (HGPs) are a specialized GP method for data with inhomogeneous noise variance, such as eddy covariance measurements.Here, it is demonstrated that the HGP model captures measurement noise variances well, outperforming the model residual method and providing reasonable flux predictions at the same time. Based on meteorological drivers and temporal information, uncertainties of annual sums of carbon flux and water vapor flux at six different tower sites in Europe and North America are estimated. Similar noise patterns with different magnitudes were found across sites. Random uncertainties in annual sums of carbon fluxes were between 9.80 and 31.57gCm−2yr−1 (or 4–9% of the annual flux), and were between 2.54 and 8.13mmyr−1 (or 1–2% of the annual flux) for water vapor fluxes. The empirical HGP model offers a general method to estimate random errors at half-hourly resolution based on entire annual records of measurements. It is introduced as a new tool for random uncertainty assessment widely throughout the FLUXNET network.

An improved multi-target tracking algorithm based on CBMeMBer filter and variational Bayesian approximation

Random finite set (RFS) filters have been demonstrating a promising algorithm for tracking an unknown number of targets in real time. However, these methods can only be used in the multi-target tracking systems with known measurement noise variances; otherwise, their tracking performances will decline greatly. To solve this problem, an improved multi-target tracking algorithm is proposed based on the cardinality-balanced multi-target multi-Bernoulli (CBMeMBer) filter and the variational Bayesian (VB) approximation technique to recursively estimate the joint posterior distributions of the multi-target states and the time-varying measurement noise variances. First, the variational calculus method is employed to derive the multi-target estimate recursions, and then the Gaussian and inverse Gamma mixture distributions are introduced to approximate the joint posterior density, and achieve a Gaussian closed-form solution. Simulation results show that the proposed algorithm can effectively estimate the unknown measurement noise variances and has a good performance of multi-target tracking with a strong robustness.

Cubature Kalman Fusion For Bearings-Only Tracking Networks

In this paper, we study the nonlinear data fusion problem for a kind of Bearings-only tracking network system with correlations among measurement noises. As a result, a decentralized cubature Kalman fusion algorithm is proposed by using an information filtering form (CIF) of cubature Kalman filter (CKF) and a noise de-correlation way. Based on the CIF and an augmented measurement, a centralized cubature Kalman fusion algorithm is firstly established and a problem on computational performance is pointed out. In order to improve computational performance of the centralized fusion method, the noise de-correlation technology is used to obtain a functionally equivalent Bearings-only tracking fusion system without the noise correlations, namely the variance of an augmented measurement noise is diagonalized. Accordingly, the proposed decentralized fusion method based on the CIF can be used to achieve a more effective tracking fusion estimate.

Augmented instrumental variable method for position and heading estimation with RDOA measurements

In this paper, we proposed a position and heading estimation algorithm using only range difference of arrival (RDOA) measurements. Based on RDOA measurements, an uncertain linear measurement model is derived and both position and heading are estimated with the instrumental variable (IV) method which can show unbiased estimation results for the uncertainty of the model. In addition, to remove the unknown bias included in the measurement model error, we augment the bias to the state vector of the model. Since the proposition inherits the characteristic of the IV method, it does not need the stochastic information of the RDOA measurement excepting the assumption that the RDOA measurement noise is zero mean and white, and the zero mean error performance can be guaranteed when variances of RDOA measurement noises are identical. Through simulations, the performance of the proposed algorithm is verified at various positions and headings in the sensor network and compared with the robust least squares method which shows a zero mean error performance under the assumption that the stochastic information is known exactly.

Algorithms for recursive/semi-recursive bias-compensating least squares system identification within the errors-in-variables framework

Algorithms for the recursive/semi-recursive estimation of the system parameters as well as the measurement noise variances for linear single-input single-output errors-in-variables systems are considered. Approaches based on three offline techniques are presented: namely, the bias eliminating least squares, the Frisch scheme and the extended bias compensating the least squares method. Whilst the underlying equations used within these approaches are identical under certain design choices, the performances of the recursive/semi-recursive algorithms are investigated via simulation, in order to determine the most suitable technique for practical applications.

Self-tuning weighted measurement fusion Wiener filter for autoregressive moving average signals with coloured noise and its convergence analysis

For the multisensor single-channel autoregressive moving average (ARMA) signal with common coloured measurement noise, applying the modern time-series analysis method, based on the ARMA innovation model, the optimal weighted measurement fusion Wiener filter is presented. When the model parameters of coloured measurement noise and partial noise variances are unknown, by applying the recursive instrumental variable, the correlation method and the Gevers–Wouters iterative algorithm with dead band, their local estimates are obtained, then the fused estimates are obtained by taking the average of all corresponding local estimates. Substituting these fused estimates into the optimal weighted measurement fusion Wiener filter, a self-tuning weighted measurement fusion Wiener filter is obtained. By applying the dynamic error system analysis method, it is rigorously proved that the self-tuning weighted measurement fusion Wiener filter converges to the corresponding optimal weighted measurement fusion Wiener filter in a realisation, so that it has asymptotically global optimality. A simulation example shows its effectiveness.

Adaptive Kalman Filtering for Histogram-Based Appearance Learning in Infrared Imagery

Targets of interest in video acquired from imaging infrared sensors often exhibit profound appearance variations due to a variety of factors, including complex target maneuvers, ego-motion of the sensor platform, background clutter, etc., making it difficult to maintain a reliable detection process and track lock over extended time periods. Two key issues in overcoming this problem are how to represent the target and how to learn its appearance online. In this paper, we adopt a recent appearance model that estimates the pixel intensity histograms as well as the distribution of local standard deviations in both the foreground and background regions for robust target representation. Appearance learning is then cast as an adaptive Kalman filtering problem where the process and measurement noise variances are both unknown. We formulate this problem using both covariance matching and, for the first time in a visual tracking application, the recent autocovariance least-squares (ALS) method. Although convergence of the ALS algorithm is guaranteed only for the case of globally wide sense stationary process and measurement noises, we demonstrate for the first time that the technique can often be applied with great effectiveness under the much weaker assumption of piecewise stationarity. The performance advantages of the ALS method relative to the classical covariance matching are illustrated by means of simulated stationary and nonstationary systems. Against real data, our results show that the ALS-based algorithm outperforms the covariance matching as well as the traditional histogram similarity-based methods, achieving sub-pixel tracking accuracy against the well-known AMCOM closure sequences and the recent SENSIAC automatic target recognition dataset.

Asymptotically Efficient Estimation of a Nonlinear Model of the Heteroscedasticity

The paper is devoted to the estimation of a nonlinear parametric model of the heteroscedasticity. The heteroscedasticity occurs in regression when the measurement noise variance is non-constant. Sometimes, the noise variance can be represented as a parameterized function of independent variables, so-called variance function. The maximum likelihood estimation (MLE) of variance function parameters leads to a system of nonlinear equations. The iterative solution of these nonlinear equations is based entirely on a successful choice of initial conditions. Hence, in the practice, the nonlinear MLE is intractable. To overcome this difficulty, another linear quasi-MLE estimator is proposed. It is strongly consistent, asymptotically Gaussian and only slightly less efficient than the Cramer-Rao lower bound. By using this estimator as an initial condition, an asymptotically efficient estimation is obtained by using one-step non-iterative Newton method. This approach has been applied to the GPS navigation in the constrained (urban) environment.

State estimation for jump Markov linear systems by variational Bayesian approximation

This article studies the problem of state estimation for jump Markov linear systems with unknown measurement noise variance parameters. By using the concept of conjugate prior distributions for noise statistics, a novel estimator is developed by applying the basic interacting multiple model (IMM) approach and the Kalman filtering theory. The main difficulties encountered are the exponentially growing terms in the interacting stage of the IMM and the coupled state and noise variance in the likelihood functions. They are overcome by employing the merging scheme via matching the first two moments and the variational Bayesian approximation technique, respectively. Simulation results are presented to verify the effectiveness of the proposed filter via a manoeuvring target tracking example.

A Measurement-Based Robust Adaptive Kalman Filtering Algorithm

In the case that the accuracy of standard kalman filter (SKF) declines when the noise statistical characteristics are unknown or changing, a measurement-based adaptive kalman filtering algorithm (MAKF) is presented. Based on the contrastive analysis of measurement characteristics of different measurement systems, MAKF is put forward to estimate adaptively the measurement noise variance R by co-difference measurement sequences. Simulation is performed by applying this algorithm to the GPS/INS integrated navigation system, the results show that MAKF can track the GPS measurement noise in real time on condition that the GPS measurement noise is unknown or changing, and the filtering accuracy and robustness are superior to those of SKF and an improved Sage-Husa adaptive kalman filtering algorithm.

INS/GPS Integration System using Adaptive Filter for Estimating Measurement Noise Variance

The inertial navigation system (INS)/GPS integration system is designed by employing an adaptive filter that can estimate measurement noise variance using the residual of the measurement. To verify the efficiency of the proposed loosely-coupled INS/GPS integration system, simulations were performed by assuming that GPS information has large position errors. The simulation results show that the proposed integration system with the adaptive filter is more effective in estimating the position and attitude errors than a system that uses the extended Kalman filter.

Information Fusion Identification Method for the Multidimension ARMA Signal with Sensor Bias and Common Disturbance Noise

For the multisensor multi-dimension autoregressive moving average(ARMA) signal system with a common disturbance measurement noise and sensor bias, when the model parameters, sensor bias and noise variances are all unknown, their consistent estimates are obtained by the multistage information fusion identification method. Firstly, by multi-dimension recursive extended least squares (RELS) algorithm, the estimates of the autoregressive parameters and sensor bias are obtained. Secondly, applying the correlation method, the estimates of the measurement noise variances are obtained. Finally, the fused estimates of the moving average(MA) parameters and the process noise variances are obtained by the Gevers-Wouters algorithm with a dead band. A simulation example verifies the consistency of unknown parameters estimates.

Design of an Adaptive Passive Collision Warning System for UAVs

Collision warning systems are used for commercial air traffic to provide pilots an extra layer of situational awareness (as well as avoidance actions), and are of increased importance due to the recent interest in shared-airspace deployment of unmanned aerial vehicles (UAVs). The design for consistency of a passive collision warning system (PCWS) that uses low- resolution infrared (IR) cameras mounted to the airframe is presented. The lack of range information, as well as the unknown measurement noise statistics, make tracking and the decision for collision warning difficult. Of the utmost importance are estimation of the measurement noise variance of the sensor and consistency of the resulting tracks. The proposed PCWS adaptively estimates this noise variance. The resulting system was found to provide consistent errors as well as accurate estimates of the angular velocity of detected targets. The results were verified through a number of test flights that included an aircraft approaching and passing over a stationary camera, and also flying across its field of view at a nearly constant range. Subsequently, the filter was modified for use on a light aircraft in conjunction with an inertial navigation system (INS).

Unbiased parameter estimation of continuous-time system based on modulating functions with input and output white noises

An efficient unbiased estimation method is proposed for the direct identification of linear continuous-time system with noisy input and output measurements. Using the Gaussian modulating filters, by numerical integration, an equivalent discrete identification model which is parameterized with continuous-time model parameters is developed, and the parameters can be estimated by the least-squares (LS) algorithm. Even with white noises in input and output measurement data, the LS estimate is biased, and the bias is determined by the variances of noises. According to the asymptotic analysis, the relationship between bias and noise variances is derived. One equation relating to the measurement noise variances is derived through the analysis of the LS errors. Increasing the degree of denominator of the system transfer function by one, an extended model is constructed. By comparing the true value and LS estimates of the parameters between original and extended model, another equation with input and output noise variances is formulated. So, the noise variances are resolved by the set of equations, the LS bias is eliminated and the unbiased estimates of system parameters are obtained. A simulation example by comparing the standard LS with bias eliminating LS algorithm indicates that the proposed algorithm is an efficient method with noisy input and output measurements.

A Kalman filter-based approach to reduce the effects of geometric errors and the measurement noise in the inverse ECG problem

In this article, we aimed to reduce the effects of geometric errors and measurement noise on the inverse problem of Electrocardiography (ECG) solutions. We used the Kalman filter to solve the inverse problem in terms of epicardial potential distributions. The geometric errors were introduced into the problem via wrong determination of the size and location of the heart in simulations. An error model, which is called the enhanced error model (EEM), was modified to be used in inverse problem of ECG to compensate for the geometric errors. In this model, the geometric errors are modeled as additive Gaussian noise and their noise variance is added to the measurement noise variance. The Kalman filter method includes a process noise component, whose variance should also be estimated along with the measurement noise. To estimate these two noise variances, two different algorithms were used: (1) an algorithm based on residuals, (2) expectation maximization algorithm. The results showed that it is important to use the correct noise variances to obtain accurate results. The geometric errors, if ignored in the inverse solution procedure, yielded incorrect epicardial potential distributions. However, even with a noise model as simple as the EEM, the solutions could be significantly improved.

Noise Power Gain for Discrete-Time FIR Estimators

The noise power gain (NPG) matrix is specialized in state space for transversal finite impulse response (FIR) estimators intended for filtering, prediction, and smoothing of discrete time-variant K-state models with M states measured. A computationally efficient iterative algorithm for NPG associated with unbiased estimation is provided along. Based on a numerical example, we show that the estimates are well bounded with the error bound (EB) specified in the three-sigma sense by the main components of the NPG matrix and measurement noise variance. In turn, the cross-components in the NPG matrix represent interactions in the estimator channels. It is concluded that EB can serve as an efficient measure of errors in optimal and suboptimal FIR and Kalman structures.

Implementation of the Rauch-Tung-Striebel Smoother for Sensor Compatibility Correction of a Fixed-Wing Unmanned Air Vehicle

This paper presents a complete procedure for sensor compatibility correction of a fixed-wing Unmanned Air Vehicle (UAV). The sensors consist of a differential air pressure transducer for airspeed measurement, two airdata vanes installed on an airdata probe for angle of attack (AoA) and angle of sideslip (AoS) measurement, and an Attitude and Heading Reference System (AHRS) that provides attitude angles, angular rates, and acceleration. The procedure is mainly based on a two pass algorithm called the Rauch-Tung-Striebel (RTS) smoother, which consists of a forward pass Extended Kalman Filter (EKF) and a backward recursion smoother. On top of that, this paper proposes the implementation of the Wiener Type Filter prior to the RTS in order to avoid the complicated process noise covariance matrix estimation. Furthermore, an easy to implement airdata measurement noise variance estimation method is introduced. The method estimates the airdata and subsequently the noise variances using the ground speed and ascent rate provided by the Global Positioning System (GPS). It incorporates the idea of data regionality by assuming that some sort of statistical relation exists between nearby data points. Root mean square deviation (RMSD) is being employed to justify the sensor compatibility. The result shows that the presented procedure is easy to implement and it improves the UAV sensor data compatibility significantly.

Self-tuning distributed measurement fusion Kalman estimator for the multi-channel ARMA signal

For the multisensor multi-channel autoregressive moving average (ARMA) signal with white measurement noises and a common disturbance measurement white noise, when the model parameters and the noise variances are all unknown, a multi-stage information fusion identification method is presented, where the consistent fused estimates of the model parameters and noise variances are obtained by the multi-dimension recursive instrumental variable (RIV) algorithm, correlation method and Gevers–Wouters algorithm with a dead band. Substituting these estimates into the optimal distributed measurement fusion Kalman signal estimator, a self-tuning distributed measurement fusion Kalman signal estimator is presented. Its convergence is proved by the dynamic error system analysis (DESA) method, so that it has asymptotical global optimality. In order to reduce computational load, a fast recursive inversion algorithm for a high-dimension matrix is presented by the inversion formula of partitioned matrix. Especially, when the process and measurement noise variance matrices are all diagonal matrices, the inversion formula of a high-dimension matrix is presented, which extends the formula of the inverse of Pei–Radman matrix. Applying the proposed inversion algorithm, the computation of the fused measurement and fused noise variance is simplified and their computational burden is reduced. A simulation example shows effectiveness of the proposed method.

Weight over-estimation problem in GMP-PHD filter

The Gaussian mixture particle probability hypothesis density (GMP-PHD) filter is a promising nonlinear multi-target tracking algorithm. However, when the variance of measurement noise is small, and if there are some particles nearby clutters, the average weight of the particles will be much greater than the clutter density, because the peak value of the likelihood function is much greater than the number of particles. Therefore, the weights of Gaussian components updated by the clutter will be greater than the actual values. The present authors call this phenomenon the weight over-estimation problem, which can be solved by some modifications of the weight updating formula. Simulation results show that the proposed algorithm has better performance than the GMP-PHD filter, implying good application prospects.