Noise Covariance Matrix Research Articles

Markov chain Monte Carlo based adaptive Rauch–Tung–Striebel smoother

For linear state space model, the covariance matrix setting errors of process and measurement noise deteriorate the estimation performance of Rauch–Tung–Striebel (RTS) smoother. To address this problem, the Markov Chain Monte Carlo is utilized to sample the state vector and noise covariance matrices simultaneously in this study. The Gibbs sampler is adopted and the corresponding adaptive RTS smoother is designed. Simulation results confirm the performance of proposed smoother.

Random Matrix Theory-Based Reduced-Dimension Space-Time Adaptive Processing under Finite Training Samples

Space-time adaptive processing (STAP) is a fundamental topic in airborne radar applications due to its clutter suppression ability. Reduced-dimension (RD)-STAP can release the requirement of the number of training samples and reduce the computational load from traditional STAP, which attracts much attention. However, under the situation that training samples are severely deficient, RD-STAP will become poor like the traditional STAP. To enhance RD-STAP performance in such cases, this paper develops a novel RD-STAP algorithm using random matrix theory (RMT), RMT-RD-STAP. By minimizing the output clutter-plus-noise power, the estimate of the inversion of clutter plus noise covariance matrix (CNCM) can be obtained through optimally manipulating its eigenvalues, thus producing the optimal STAP weight vector. Specifically, the clutter-related eigenvalues are estimated according to the clutter-related sample eigenvalues via RMT, and the noise-related eigenvalue is optimally selected from the noise-related sample eigenvalues. It is found that RMT-RD-STAP significantly outperforms the RD-STAP algorithm when the RMB rule cannot be satisfied. Theoretical analyses and numerical results demonstrate the effectiveness and the performance advantages of the proposed RMT-RD-STAP algorithm.

On the Capacity of Gaussian MIMO Channels With Memory

The operational capacity of Gaussian MIMO channels with memory was obtained by Brandenburg and Wyner (1974) under certain mild assumptions on the channel impulse response and its noise covariance matrix, which essentuially require channel memory to be not too strong. This channel was also considered by Tsybakov (2006) and its information capacity was obtained in some cases. It was further conjectured, based on numerical evidence, that these capacities are the same in all cases. This conjecture is proved here. An explicit closed-form expression for the optimal input power spectral density matrix is also given. The obtained result is further extended to the case of joint constraints, including per-antenna and interference power constraints as well as energy harvesting constraints. These results imply the information-theoretic optimality of OFDM-type transmission systems for such channels with memory.

Estimation of Step Length With Wearable Thigh Sensor Using an Unscented Kalman Filter.

The determination of step length, an important gait parameter, has been a challenging task. Although unobtrusive sensors (inertial measurement units) have been developed recently, they cannot facilitate the automatic estimation of step length. In this article, we use a model-based technique to determine the step length using the Unscented Kalman Filter with angular velocity from a gyroscope inside the thigh pocket. We then propose a novel covariance estimation algorithm based on a screening technique that performs a search for the optimal Process Noise Covariance matrix. Upon implementing the Unscented Kalman Filter, the step length is found using the horizontal position of the foot relative to the hip using a patient-independent robust peak detection algorithm. This research article paves the way for algorithms that are computationally much faster than black box methods, with more scope for the development of better algorithms for covariance estimation using the one proposed in this article as a foundation.

Modified Spatio-Temporal Matched Filtering for Brain Responses Classification

In this article, we apply the method of spatio-temporal filtering (STF) to electroencephalographic (EEG) data processing for brain responses classification. The method operates similarly to linear discriminant analysis (LDA) but contrary to most applied classifiers, it uses the whole recorded EEG signal as a source of information instead of the precisely selected brain responses, only. This way it avoids the limitations of LDA and improves the classification accuracy. We emphasize the significance of the STF learning phase. To preclude the negative influence of super–Gaussian artifacts on accomplishment of this phase, we apply the discrete cosine transform (DCT) based method for their rejection. Later, we estimate the noise covariance matrix using all data available, and we improve the STF template construction. The further modifications are related with the constructed filters operation and consist in the changes of the STF interpretation rules. Consequently, a new tool for evoked potentials (EPs) classification has been developed. Applied to the analysis of signals stored in a publicly available database, prepared for the assessment of modern algorithms aimed in EPs detection (in the frames of the 2019 IFMBE Scientific Challenge), it allowed to achieve the second best result, very close to the best one, and significantly better than the ones achieved by other contestants of the challenge.

A novel maximum likelihood and moving weighted average based adaptive Kalman filter

For the state estimation with inaccurate noise statistics, the existing adaptive Kalman filters (AKFs) usually have substantial computational complexity or are not easy to estimate online. Inspired by the fact, a new computationally efficient AKF based on maximum likelihood and moving weighted average (MMAKF) is proposed. Firstly, to reduce computational complexity, instead of estimating the noise covariance matrixes, the maximum likelihood principle is introduced to directly estimate the prediction error covariance matrix and innovation covariance matrix. Subsequently, a new moving weighted average algorithm is designed to optimize the estimated results. Then, a computationally efficient AKF is derived, and its convergence performance and application are discussed. Simulation results for the target tracking example illustrate that the proposed AKF can effectively reduce error caused by inaccurate noise statistics and basically keep simplicity and elegance of the classical KF.

Adaptive fuzzy neural network-aided progressive Gaussian approximate filter for GPS/INS integration navigation

To address the difficulty of high uncertainty priori information in nonlinear filtering that occurs in tightly coupled navigation and positioning systems, a novel adaptive fuzzy neural network-aided progressive Gaussian filter is proposed in this paper to further improve the robustness and accuracy of the filter. A joint estimation of the step size and measurement noise covariance matrix is performed by a variational Bayesian approach. The estimation accuracy of the progressive Gaussian approximation filter is thereby enhanced. In particular, the measurement noise covariance matrix is adjusted using an adaptive fuzzy algorithm, and the state estimation error covariance matrix is optimized exploiting a neural network model. The proposed filter is experimentally verified to have higher accuracy than the existing state-of-the-art filters.

An Efficient Sparse Bayesian Learning STAP Algorithm with Adaptive Laplace Prior

Space-time adaptive processing (STAP) encounters severe performance degradation with insufficient training samples in inhomogeneous environments. Sparse Bayesian learning (SBL) algorithms have attracted extensive attention because of their robust and self-regularizing nature. In this study, a computationally efficient SBL STAP algorithm with adaptive Laplace prior is developed. Firstly, a hierarchical Bayesian model with adaptive Laplace prior for complex-value space-time snapshots (CALM-SBL) is formulated. Laplace prior enforces the sparsity more heavily than Gaussian, which achieves a better reconstruction of the clutter plus noise covariance matrix (CNCM). However, similar to other SBL-based algorithms, a large degree of freedom will bring a heavy burden to the real-time processing system. To overcome this drawback, an efficient localized reduced-dimension sparse recovery-based space-time adaptive processing (LRDSR-STAP) framework is proposed in this paper. By using a set of deeply weighted Doppler filters and exploiting prior knowledge of the clutter ridge, a novel localized reduced-dimension dictionary is constructed, and the computational load can be considerably reduced. Numerical experiments validate that the proposed method achieves better performance with significantly reduced computational complexity in limited snapshots scenarios. It can be found that the proposed LRDSR-CALM-STAP algorithm has the potential to be implemented in practical real-time processing systems.

Identification of linear systems with multiplicative noise from multiple trajectory data

The paper studies identification of linear systems with multiplicative noise from multiple-trajectory data. An algorithm based on the least-squares method and multiple-trajectory data is proposed for joint estimation of the nominal system matrices and the covariance matrix of the multiplicative noise. The algorithm does not need prior knowledge of the noise or stability of the system, but requires only independent inputs with pre-designed first and second moments and relatively small trajectory length. The study of identifiability of the noise covariance matrix shows that there exists an equivalent class of matrices that generate the same second-moment dynamic of system states. It is demonstrated how to obtain the equivalent class based on estimates of the noise covariance. Asymptotic consistency of the algorithm is verified under sufficiently exciting inputs and system controllability conditions. Non-asymptotic performance of the algorithm is also analyzed under the assumption that the system is bounded. The analysis provides high-probability bounds vanishing as the number of trajectories grows to infinity. The results are illustrated by numerical simulations.

An Adaptive Filtering Method for Cooperative Localization in Leader-Follower AUVs.

In the complex and variable marine environment, the navigation and localization of autonomous underwater vehicles (AUVs) are very important and challenging. When the conventional Kalman filter (KF) is applied to the cooperative localization of leader–follower AUVs, the outliers in the sensor observations will have a substantial adverse effect on the localization accuracy of the AUVs. Meanwhile, inaccurate noise covariance matrices may result in significant estimation errors. In this paper, we proposed an improved Sage–Husa adaptive extended Kalman filter (improved SHAEKF) for the cooperative localization of multi-AUVs. Firstly, the measurement anomalies were evaluated by calculating the Chi-square test statistics based on the innovation. The detection threshold was determined according to the confidence level of the Chi-square test, and the Chi-square test statistics exceeding the threshold were regarded as measurement abnormalities. When measurement anomalies occurred, the Sage–Husa adaptive extended Kalman filter algorithm was improved by suboptimal maximum a posterior estimation using weighted exponential fading memory, and the measurement noise covariance matrix was adjusted online. The numerical simulation of leader–follower multi-AUV cooperative localization verified the effectiveness of the improved SHAEKF and demonstrated that the average root mean square and the average standard deviation of the localization errors based on the improved SHAEKF were significantly reduced in the case of the presence of measurement abnormalities.

Adaptive Fading-Memory Receding-Horizon Filters and Smoother for Linear Discrete Time-Varying Systems

In this paper, an adaptive fading-memory receding-horizon (AFMRH) filter is proposed by combining the receding-horizon structure and the adaptive fading-memory method. In the recent finite horizon, state error covariance is adapted with an adaptive fading factor; then the process noise covariance matrix adaption is realized by adjusting the properties of systems. An AFMRH fixed-lag smoother is also proposed by combining the proposed AFMRH filtering algorithm and a Rauch–Tung–Striebel smoothing algorithm to improve the estimation accuracy. Because the proposed AFMRH filter and smoother are reduced to the optimal receding-horizon (RH) filter and smoother when all measurements have the same weight, the proposed adaptive RH estimators could provide a more general solution than the optimal RH filter and smoother. To reduce the complexity and improve the estimation performance of the proposed RH estimators, an adaptive horizon adjustment method and a switching filtering algorithm based on an adaptive fading factor are also proposed. In particular, the proposed adaptive horizon adjustment method is designed to be computationally efficient, which makes it suitable for online and real-time applications. Through computer simulation, the performance and adaptiveness of the proposed approaches were verified by comparing them with existing fading-memory approaches.

Trajectory PHD Filter for Adaptive Measurement Noise Covariance Based on Variational Bayesian Approximation

In order to solve the problem that the measurement noise covariance may be unknown or change with time in actual multi-target tracking, this paper brings the variational Bayesian approximation method into the trajectory probability hypothesis density (TPHD) filter and proposes a variational Bayesian TPHD (VB-TPHD) filter to obtain measurement noise covariance adaptively. By modeling the unknown covariance as the random matrix that obeys the inverse gamma distribution, VB-TPHD filter minimizes the Kullback–Leibler divergence (KLD) and estimates the sequence of multi-trajectory states with noise covariance matrices simultaneously. We propose the Gaussian mixture VB-TPHD (AGM-VB-TPHD) filter under adaptive newborn intensity for linear Gaussian models and also give the extended Kalman (AEK-VB-TPHD) filter and unscented Kalman (AUK-VB-TPHD) filter in nonlinear Gaussian models. The simulation results prove the effectiveness of the idea that the VB-TPHD filter can form robust and stable trajectory filtering while learning adaptive measurement noise statistics. Compared with the tag-VB-PHD filter, the estimated error of the VB-TPHD filter is greatly reduced, and the estimation of the trajectory number is more accurate.

Estimation of time-varying noise parameters for unscented Kalman filter

The unscented Kalman filter (UKF) is a promising method for system state and structural parameters estimation. However, its performance depends on the process noise and measurement noise covariance matrices, which are usually unknown in practice. Arbitrary selection of these covariance matrices may lead to unreliable or even diverging estimation results. To resolve this critical problem, we propose a Bayesian probabilistic algorithm for the estimation of the noise covariance matrices based on the response measurement. The proposed Noise-Parameters-Identified Unscented Kalman Filter (NPI-UKF) has the following salient features: (1) the divergence problem is resolved; (2) reliable estimation results including uncertainty quantification can be obtained; and (3) NPI-UKF is applicable to nonstationary situations. These salient features are illustrated through the numerical applications to a bridge structure and a laboratory experiment to a shear building model. The efficacy and robustness of NPI-UKF will be validated.

Indirect and direct estimation of pharmacokinetic parameters in dynamic diffuse fluorescence tomography by adaptive extended Kalman filtering.

Pharmacokinetic parameter estimation with the support of dynamic diffuse fluorescence tomography (DFT) can provide helpful diagnostic information for tumor differentiation and monitoring. Adaptive extended Kalman filtering (AEKF) as a nonlinear filter method has the merits of high quantitativeness, noise robustness, and initialization independence. In this paper, indirect and direct AEKF schemes combining with a commonly used two-compartment model were studied to estimate the pharmacokinetic parameters based on our self-designed dynamic DFT system. To comprehensively compare the performances of both schemes, the selection of optimal noise covariance matrices affecting estimation results was first studied, then a series of numerical simulations with the metabolic time ranged from 4.16 min to 38 min was carried out and quantitatively evaluated. The comparison results show that the direct AEKF outperforms the indirect EKF in estimation accuracy at different metabolic velocity and demonstrates stronger stability at the large metabolic velocity. Furtherly, the in vivo experiment was conducted to achieve the indocyanine green pharmacokinetic-rate images in the mouse liver. The experimental results confirmed the capability of both schemes to estimate the pharmacokinetic-rate images and were in agreement with the theory predictions and the numerical simulation results.

Quantifying the impact of temporal analysis of products reactor initial state uncertainties on kinetic parameters

AbstractThe temporal analysis of products (TAP) reactor provides a route to extract intrinsic kinetics from transient measurements. Current TAP uncertainty quantification only considers the experimental noise present in the outlet flow signal. Additional sources of uncertainty such as initial surface coverages, catalyst zone location, inert void fraction, gas pulse intensity, and pulse delay, are not included. For this reason, a framework for quantifying initial state uncertainties present in TAP experiments is presented and applied to a carbon monoxide oxidation case study. Two methods for quantifying these sources of uncertainty are introduced. The first utilizes initial state sensitivities to approximate the parameter variances and provide insights into the structural certainty of the model. The second generates parameter confidence distributions through an ensemble‐based sampling algorithm. The initial state covariance matrix can ultimately be merged with the experimental noise covariance matrix, providing a unified description of the parameter uncertainties for a TAP experiment.

Nonnegative block-sparse Bayesian learning algorithm for EEG brain source localization

Localizing electrical sources on the cortex surface from scalp recorded electroencephalogram (EEG) is challenging, due to ill-posed problem, noises/artifacts contamination, etc. Applying sparse Bayesian learning (SBL) in this field can automatically give sparse solution to ill-posed problem, while most of the SBL-based algorithms require precise or estimated version of noise statistics information. As EEG signals are more likely to stem from locally synchronized neural masses, modeling source block-sparsity on the cortex surface would bring benefits. In this paper, we develop an EEG brain source localization algorithm in SBL framework, with innovative modeling at sensor level as well as source level. For sensor-level modeling, the distribution of sample covariance matrix of multi-electrode measurements is considered, to circumvent the requirement of noise covariance matrix information. The innovation of source-level modeling is that, with block-sparsity prior used, the block-sparse signal reconstruction problem is transformed to an atom-sparse one, in which variance parameters of brain regions are to be estimated. As these parameters are nonnegative, their priors are modeled by nonnegative Gaussian, which was neglected by previous studies, and ultimately a nonnegative block-SBL (NNBSBL) algorithm is proposed in expectation–maximization (EM) approach. Simulations demonstrate that the proposed NNBSBL algorithm has excellent performance in variations of source number, source locations, homoscedastic/heteroscedastic noises, signal-to-noise ratio (SNR), and number of samples, compared to benchmark and state-of-the-art algorithms. The performance of the proposed algorithm is also evaluated through real P300 EEG data, which is proved consistent with the conclusions of P300 source locations in literature.

A novel adaptive H‐infinity filtering method for the accurate SOC estimation of lithium‐ion batteries based on optimal forgetting factor selection

AbstractAccurate estimation of the state of charge (SOC) of lithium‐ion batteries is quite crucial to battery safety monitoring and efficient use of energy; to improve the accuracy of lithium‐ion battery SOC estimation under complicated working conditions, the research object of this study is the ternary lithium‐ion battery; the forgetting factor recursive least square (FFRLS) method optimized by particle swarm optimization (PSO) and adaptive H‐infinity filter (HIF) algorithm are adopted to estimate battery SOC. The PSO algorithm is improved with dynamic inertia weight to optimize the forgetting factor to solve the contradiction between FFRLS convergence speed and anti‐noise ability. The noise covariance matrixes of the HIF are improved to realize adaptive correction function and improve the accuracy of SOC estimation. To verify the rationality of the joint algorithm, a second‐order Thevenin model is established to estimate the SOC under three complex operating conditions. The experimental results show that the absolute value of the maximum estimation error of the improved algorithm under the three working conditions is 0.0192, 0.0131, and 0.0111, respectively, which proves that the improved algorithm has high accuracy and offers a theoretical basis for the safe and efficient operation of the battery management system.

Adaptive Kalman Filtering for Recursive Both Additive Noise and Multiplicative Noise

This paper focuses on adaptive Kalman filtering problem for linear systems with unknown covariances of both dynamic multiplicative noise (multiplicative measurement noise) and additive noises (additive process and measurement noises). A recursive-noise adaptive Kalman filter is proposed to estimate both states and covariances of noises by using the varaitional Bayesian (VB) inference and an indirect method. First, we characterize inverse Wishart priors for both measurement noise covariance and process noise covariance and employ the Student's t-distribution to represent the likelihood function, which is non-Gaussian and affected by mixing multiplicative noise and additive measurement noise. Then, an adaptive Kalman filtering for recursive both noise covariance matrices and dynamic state, is proposed following VB inference. Performance analysis for VB procedures and the proposed filter is provided to ensure the convergence and stability. A target tracking example is provided to validate the effectiveness of the proposed filtering algorithm.

Robust underwater direction-of-arrival tracking with uncertain environmental disturbances using a uniform circular hydrophone array.

Underwater direction-of-arrival (DOA) tracking using a hydrophone array is an important research subject in passive sonar signal processing. In this study, considering that an unknown underwater environment results in uncertain disturbances to the measurements, robust underwater DOA tracking with regard to uncertain environmental disturbances was studied. Because the uniform circular array (UCA) is free from the port and starboard ambiguity problem, a UCA was used to obtain the measurements for a long-time tracking scenario. First, a kinematic model of an underwater target and a measurement model based on the received signal of the UCA were established. Then, a DOA tracking algorithm was derived based on the extended Kalman filter (EKF), whose performance is significantly affected by the accuracy of the measurement noise covariance matrix (MNCM). Finally, considering that uncertain disturbances carry out unstable measurement noise, the modified Sage-Husa algorithm was used to obtain accurate MNCMs during the process of the derived EKF-based DOA tracking algorithm. Thus, a robust DOA tracking method with uncertain environmental disturbances using a UCA was proposed. The accuracy and reliability of the suggested method was verified via Monte Carlo simulations of a DOA tracking scenario and an experiment in the South China Sea in July 2021.

Simple Direction-of-Arrival Estimation under Nonuniform Noise Scenarios

The uniform white noise assumption is widely applied for most of the existing direction-of-arrival (DOA) estimation methods, which however may degrade the estimation performance. In practical scenarios, the nonuniform white noise model is more adequate. In this paper, we propose a simple DOA estimation method in the presence of spatially nonuniform white noise with unknown noise covariance matrix. The estimate of the unknown noise covariance matrix is firstly obtained by utilizing the shift property of identical subarrays. Then, the received signal covariance matrix is prewhitened by the estimated noise covariance, which can overcome the biased estimate of signal covariance. Finally, the performance improvement is verified by simulations in terms of resolution ability and estimation accuracy.