Heavy-tailed Measurement Noise Research Articles

A variational Bayesian based robust filter for unknown measurement bias and inaccurate noise statistics

In many practical fields, the unknown time-varying measurement biases (additive and multiplicative bias) and heavy-tailed measurement noise caused by some unpredictable anomalous behaviors may degrade the performance of conventional Kalman filter seriously. To solve the state estimation problem of systems with time-varying measurement biases and heavy-tailed measurement noise, this paper proposes a new variational Bayesian (VB) based robust filter. Firstly, the non-Gaussian measurement likelihood probability density function (ML-PDF) with multiplicative and additive measurement bias is built. Then, the conjugate prior distributions for unknown bias and noise scale parameters are selected, and the VB method is utilized to jointly infer the system state, unknown measurement biases and inaccurate measurement noise covariance matrix. Finally, a VB based robust filter is derived and its effectiveness is verified by the numerical simulations.

Robust Kalman filters based on the sub-Gaussian [formula omitted]-stable distribution

Motivated by filtering tasks under a linear system with non-Gaussian heavy-tailed noise, various robust Kalman filters (RKFs) based on different heavy-tailed distributions have been proposed. Although the sub-Gaussian α-stable (SGαS) distribution captures heavy tails well and is applicable in various scenarios, its potential has not yet been explored for RKFs. The main hindrance is that there is no closed-form expression of its mixing density. This paper proposes a novel RKF framework, RKF-SGαS, where the process noise is assumed to be Gaussian and the heavy-tailed measurement noise is modelled by the SGαS distribution. The corresponding joint posterior distribution of the state vector and auxiliary random variables is approximated by the Variational Bayesian approach. Also, four different minimum mean square error (MMSE) estimators of the scale function are presented. The first two methods are based on the Importance Sampling (IS) and Gauss–Laguerre quadrature (GLQ), respectively. In contrast, the last two estimators combine a proposed Gamma series (GS) based method with the IS and GLQ estimators and hence are called GSIS and GSGL. Besides, the RKF-SGαS is compared with the state-of-the-art RKFs under three kinds of heavy-tailed measurement noises, and the simulation results demonstrate its estimation accuracy and efficiency.

An adaptive outlier-robust Kalman filter based on sliding window and Pearson type VII distribution modeling

Due to inaccurate noise modeling and only using current measurement, existing robust filtering algorithms cannot obtain good state estimation results when measurement is interfered by outliers. In this article, a Pearson type VII (PTV) distribution-based adaptive sliding window outlier-robust Kalman filter (PTVSWAKF) is proposed. According to the characteristics of the parameters, the PTV and inverse Wishart distributions are respectively modeled as prior distributions for heavy-tailed measurement noises (HMN) and covariance. Two different Gamma distributions are modeled as priori distributions for the two degree of freedom (DOF) parameters to achieve adaptive updating of them. By the smooth posterior distribution of the sliding window state values, the state, measurement noise covariance, auxiliary random variable and DOF parameters are jointly iterated and updated to solve the outlier interference problem through variational Bayesian. Finally, the simulation experiments verify the proposed PTVSWAKF can obtain more accurate state estimation than existing filters in the environments with varying degrees of HMN and unknown non-stationary HMN.

A Gaussian–Pearson type VII adaptive mixture distribution-based outlier-robust Kalman filter

Considering that existing robust filtering algorithms rely on the selection of initial values of degree of freedom (DOF) parameters in outlier interference environments and cannot effectively cope with unknown non-stationary heavy-tailed measurement noises (HMN), a Gaussian–Pearson type VII (PTV) adaptive mixture distribution-based outlier-robust Kalman filter (GPTVMAKF) is proposed. In order to determine whether the current measurement is a normal value or an outlier, a judgment factor subject to the Beta-Bernoulli distribution is introduced. PTV distribution is used to model HMN caused by outliers, and two Gamma distributions are used to model the two different DOF parameters, which can make the PTV distribution have the adaptive adjustment ability. By introducing the inverse Wishart distribution as the prior distribution of the measurement noise covariance, which is adaptively estimated to cope with the unknown time-varying measurement noises. The state and parameters are jointly estimated by variational Bayesian. Finally, the simulation experiments verify that the proposed GPTVMAKF can obtain more accurate state estimation than existing filters in the environments with varying degrees of HMN and unknown non-stationary HMN.

Research on Dynamic Measurement Method of Drilling Tool Attitude Near Bit Based on Suppression of Heavy-Tailed Measurement Noise

Research on Dynamic Measurement Method of Drilling Tool Attitude Near Bit Based on Suppression of Heavy-Tailed Measurement Noise

Robust IMM Filtering Approach with Adaptive Estimation of Measurement Loss Probability for Surface Target Tracking

A suitable jump Markov system (JMS) filtering approach provides an efficient technique for tracking surface targets. In complex surface target tracking situations, due to the joint influences of lost measurements with an unknown probability and heavy-tailed measurement noise (HTMN), the estimation accuracy of conventional interacting multiple model (IMM) methods may be seriously degraded. Aiming to address the filtering issues in JMSs with HTMNs and random measurement losses, this paper presents an IMM filtering approach with the adaptive estimation of unknown measurement loss probability. In this study, we assumed that the measurement noises obey student’s t-distributions and then proposed Bernoulli random variables (BRVs) to characterize the random measurement loss. Notably, by converting the two likelihood functions from the weighted sum form to exponential multiplication, we established hierarchical Gaussian state space models to directly utilize the variational inference method. The system state vectors, unknown distribution parameters, BRVs, and unknown measurement loss probabilities were estimated simultaneously according to the variational Bayesian inference in the IMM framework. The results of maneuvering target tracking simulations verified that the presented filtering approach demonstrated superior estimation accuracy compared to existing IMM filters.

A Novel Robust IMM Filtering Method for Surface-Maneuvering Target Tracking with Random Measurement Delay

A proper filtering method for jump Markov system (JMS) is an effective approach for tracking a maneuvering target. Since the coexisting of heavy-tailed measurement noises (HTMNs) and one-step random measurement delay (OSRMD) in the complex scenarios of the surface maneuvering target tracking, the effectiveness of typical interacting multiple model (IMM) techniques may decline severely. To solve the state estimation problem in JMSs with HTMN and OSRMD simultaneously, this article designs a novel robust IMM filter utilizing the variational Bayesian (VB) inference framework. This algorithm models the HTMNs as student’s t-distribuitons, and presents a random Bernoulli variable to describe the OSRMD in JMSs. By transforming measurement likelihood function form from weighted summation to exponential product, this paper constructs hierarchical Gaussian state space models. Then, the state vectors, random Bernoulli vairable, and model probability are inferred jointly according to VB inference. The surface maneuvering target tracking simulation example result indicates that the presented IMM filter achieves superior target state estimation accuracy among existing IMM filters.

Design of robust Gaussian approximate filter and smoother with latency probability identification

The Kalman filter is a linear optimal estimator when it can accurately receive measurements and the noise obeys a Gaussian distribution. However, the measurement noise may have heavy-tailed characteristics because of outlier interference. Actually, real-world measurement information may have one-step randomly delayed measurements (ORDMs) with unknown latency probability (LP) due to exogenous disturbances. Moreover, systems are usually not linear. To overcome the influence of the state estimator of nonlinear systems from heavy-tailed measurement noise (HMN) and the unknown LP, this paper focuses on the robust Gaussian approximate (GA) filter and smoother for the above issues. First, the one-step predicted probability density function (PDF) is modeled as a Gaussian distribution, and the likelihood PDF is modeled as a Normal–Gamma–Beta (NGBM) distribution. Furthermore, novel robust Gaussian approximate filter and smoother are proposed based on the variational Bayesian technique. They provide general estimation frameworks for estimating state of nonlinear systems with HMN and ORDMs, and different nonlinear estimators can be implemented using different numerical techniques. In addition, the computational complexity of the proposed algorithms is calculated. Finally, taking a univariate nonstationary growth model, a passive ranging problem, and target tracking as examples, the effectiveness and superiority of the proposed robust algorithms contrasted with the existing algorithms are shown. The outcomes show that the proposed methods have higher estimation precision; however, the computational burden is marginally increased.

Multi-Kernel Maximum Correntropy Kalman Filter

Maximum correntropy criterion (MCC) has been widely used in Kalman filter to cope with heavy-tailed measurement noises. However, its performance on mitigating non-Gaussian process noises and unknown disturbance is rarely explored. In this letter, we extend the definition of correntropy from a single kernel to multiple kernels. Then, we derive a multi-kernel maximum correntropy Kalman filter (MKMCKF) to cope with multivariate non-Gaussian noises and disturbance. Three examples are provided to show the effectiveness of the proposed methods.

A new Gaussian approximate filter with colored non-stationary heavy-tailed measurement noise

The filtering issue of a nonlinear system with colored non-stationary heavy-tailed measurement noise (CNSHMN) is addressed in this study via designing a new Gaussian approximate filter. By utilizing the state expansion method and the measurement difference method, the nonlinear filtering problem with the one-step delayed state and the white non-stationary heavy-tailed measurement noise (NSHMN) after the difference is turned into the traditional nonlinear filtering problem with NSHMN. A Gaussian student t-mixed distribution (GSTM) with Bernoulli random variable is utilized to describe the differenced measurement noise. The state vector, intermediate random variables (IRV), mixed probability and Bernoulli random variable (BRV) are simultaneously inferred by introducing variational Bayesian (VB) technique. Target tracking simulation examples reveal that the proposed filter is superior to the existing methods in the nonlinear filtering issue of CNSHMN.

A new robust Kalman filter with measurement loss based on mixing distribution

A new robust Kalman filter (KF) based on mixing distribution is presented to address the filtering issue for a linear system with measurement loss (ML) and heavy-tailed measurement noise (HTMN) in this paper. A new Student’s t-inverse-Wishart-Gamma mixing distribution is derived to more rationally model the HTMN. By employing a discrete Bernoulli random variable (DBRV), the form of measurement likelihood function of double mixing distributions is converted from a weighted sum to an exponential product, and a hierarchical Gaussian state-space model (HGSSM) is therefore established. Finally, the system state, the intermediate random variables (IRVs) of the new STIWG distribution, and the DBRV are simultaneously estimated by utilizing the variational Bayesian (VB) method. Numerical example simulation experiment indicates that the proposed filter in this paper has superior performance than current algorithms in processing ML and HTMN.

A New Variational Bayesian-Based Kalman Filter with Unknown Time-Varying Measurement Loss Probability and Non-Stationary Heavy-Tailed Measurement Noise.

In this paper, a new variational Bayesian-based Kalman filter (KF) is presented to solve the filtering problem for a linear system with unknown time-varying measurement loss probability (UTVMLP) and non-stationary heavy-tailed measurement noise (NSHTMN). Firstly, the NSHTMN was modelled as a Gaussian-Student’s t-mixture distribution via employing a Bernoulli random variable (BM). Secondly, by utilizing another Bernoulli random variable (BL), the form of the likelihood function consisting of two mixture distributions was converted from a weight sum to an exponential product and a new hierarchical Gaussian state-space model was therefore established. Finally, the system state vector, BM, BL, the intermediate random variables, the mixing probability, and the UTVMLP were jointly inferred by employing the variational Bayesian technique. Simulation results revealed that in the scenario of NSHTMN, the proposed filter had a better performance than current algorithms and further improved the estimation accuracy of UTVMLP.

Application of state-space model with skew-t measurement noise to blood test value prediction

State-space models (SSMs) have been widely used for analyzing time-series data in the fields of economics and bioinformatics to express the dynamic behavior of data. Recently, filtering and smoothing algorithms applied to linear discrete SSMs with skewed and heavy-tailed measurement noise have been proposed for a more appropriate model because measurement noise does not often follow a Gaussian distribution. In this paper, we propose a linear SSM with skew-t measurement noise for predicting blood test values, along with a method for estimating their parameter values to ensure consistency with the data when using a generalized expectation-maximization (EM) algorithm. To validate the effectiveness of the proposed model and method, we analyze time-series blood test data using both Gaussian and skew-t measurement noise and compared their prediction accuracy for future values. Then, we predicted future blood test values of the unhealthy participant under his current and improved lifestyles. By comparing these predicted results under different lifestyles, we demonstrate that he will overcome lifestyle-related diseases with the improved lifestyle.

A novel robust Kalman filter with adaptive estimation of the unknown time-varying latency probability

To better model one-step randomly delayed measurements (ORDM) with unknown time-varying latency probability (UTLP) in linear systems with heavy-tailed measurement noise (HMN), a novel Normal-Gamma-Beta mixture (NGBM) distribution is presented. By introducing a Bernoulli random variable, the probability density function of the proposed NGBM distribution can be reformulated as a Gaussian hierarchical form. Based on this, a novel robust Kalman filter is designed using the variational Bayesian technique. A target tracking simulation verifies the potential of the proposed robust filter, which has higher filtering accuracy than existing cutting-edge filters and can adaptively estimate the UTLP. Furthermore, it is concluded that when HMN and ORDM concurrently exist, the HMN has more influence on the accuracy of the filter than the ORDM.

A Novel Robust Nonlinear Kalman Filter Based on Multivariate Laplace Distribution

In this brief, we investigate the robust state estimation of nonlinear systems with heavy-tailed noise. Based on the fact that the multivariate Laplace (ML) distribution has heavier tails than Gaussian distribution but is only determined by its mean and covariance, we utilize the ML distribution to model the heavy-tailed measurement noise. Based on the Gaussian mixture form of ML distribution, the system state and the covariance of ML distribution are inferred by variational Bayesian method, and then a closed recursive robust nonlinear Kalman filter is obtained with the nonlinear integrations calculated by numerical integration methods. Simulation results demonstrate that the proposed algorithm is insensitive to its free parameters and can obtain better estimation accuracy than related robust algorithms.

A Robust SMC-PHD Filter for Multi-Target Tracking with Unknown Heavy-Tailed Measurement Noise.

In multi-target tracking, the sequential Monte Carlo probability hypothesis density (SMC-PHD) filter is a practical algorithm. Influenced by outliers under unknown heavy-tailed measurement noise, the SMC-PHD filter suffers severe performance degradation. In this paper, a robust SMC-PHD (RSMC-PHD) filter is proposed. In the proposed filter, Student-t distribution is introduced to describe the unknown heavy-tailed measurement noise where the degrees of freedom (DOF) and the scale matrix of the Student-t distribution are respectively modeled as a Gamma distribution and an inverse Wishart distribution. Furthermore, the variational Bayesian (VB) technique is employed to infer the unknown DOF and scale matrix parameters while the recursion estimation framework of the RSMC-PHD filter is derived. In addition, considering that the introduced Student- t distribution might lead to an overestimation of the target number, a strategy is applied to modify the updated weight of each particle. Simulation results demonstrate that the proposed filter is effective with unknown heavy-tailed measurement noise.

Robust Generalized Labeled Multi-Bernoulli Filter for Multitarget Tracking With Unknown Non- Stationary Heavy-Tailed Measurement Noise

A robust generalized labeled multi-Bernoulli (GLMB) filter is presented to perform multitarget tracking (MTT) with unknown non-stationary heavy-tailed measurement noise (HTMN). The HTMN is modeled as a multivariate Student's t-distribution with unknown and time-varying mean. The proposed filter relaxes the restrictive assumption that the mean of HTMN is zero, and can effectively deal with MTT under the condition that the mean of HTMN is unknown and time-varying. The variational Bayesian (VB) approximation is applied in the GLMB filtering framework with the augmented state. The marginal likelihood function is obtained via minimizing the Kullback-Leibler divergence by the variational lower bound. The simulation results demonstrate that the proposed filter can effectively track multiple targets in both linear and nonlinear scenarios when the mean of HTMN is unknown and time-varying.

An Adaptive Filter for Nonlinear Multi-Sensor Systems with Heavy-Tailed Noise.

Aiming towards state estimation and information fusion for nonlinear systems with heavy-tailed measurement noise, a variational Bayesian Student’s t-based cubature information filter (VBST-CIF) is designed. Furthermore, a multi-sensor variational Bayesian Student’s t-based cubature information feedback fusion (VBST-CIFF) algorithm is also derived. In the proposed VBST-CIF, the spherical-radial cubature (SRC) rule is embedded into the variational Bayes (VB) method for a joint estimation of states and scale matrix, degree-of-freedom (DOF) parameter, as well as an auxiliary parameter in the nonlinear system with heavy-tailed noise. The designed VBST-CIF facilitates multi-sensor fusion, allowing to derive a VBST-CIFF algorithm based on multi-sensor information feedback fusion. The performance of the proposed algorithms is assessed in target tracking scenarios. Simulation results demonstrate that the proposed VBST-CIF/VBST-CIFF outperform the conventional cubature information filter (CIF) and cubature information feedback fusion (CIFF) algorithms.

A Novel Robust Gaussian Approximate Smoother Based on EM for Cooperative Localization With Sensor Fault and Outliers

In this article, a novel robust Gaussian approximation smoother based on expectation–maximization (EM) algorithm is proposed for cooperative localization (CL) with faulty Doppler velocity log (DVL) and heavy-tailed measurement noise. In our model, the autonomous underwater vehicle (AUV) velocity information that is not available due to DVL failure and the bias in the underwater acoustic modem are considered as unknown inputs. Then, the Student’s t distribution is used to model the heavy-tailed measurement noise. An EM algorithm is also developed for the state-space model with heavy-tailed measurement noise. The state, noise covariance matrices, and auxiliary random variables are regarded as hidden variables to obtain the maximum likelihood estimation of unknown inputs. Gaussian smoother where modified process and measurement noise covariance are inferred by variational Bayesian (VB) approach is applied to estimate the state. The experimental results illustrate that given the heavy-tailed noise, the proposed method estimates the unknown input, and the state with a high level of accuracy. The proposed algorithm can be used as a backup algorithm for CL in cases where DVL is not available due to failure.

Robust particle filter algorithms for one‐step randomly delayed measurements and heavy‐tailed noise

The development of particle filters for emerging data transmission applications is essential to address network-related issues. In this Letter, a robust particle filter algorithm for non-linear systems with one-step randomly delayed measurements and outlier corrupted measurement noise is investigated. To address the issue of measurement outliers, a Student's-t distribution is utilised to model the measurement noise. For the random delay uncertainties, a Bernoulli random variable is employed, based on which the likelihood function is transformed into an exponential multiplication form. The filter state estimation is obtained by Bayes's theorem and using variational Bayesian approach, the latent random variables and Bernoulli random variable are estimated together. The superior performance of the proposed method is verified through a typical example by comparison with existing particle filters that consider random measurement delay or heavy-tailed measurement noise.