non-Gaussian Measurement Research Articles

Robust EKF based on the framework of dynamic data reconciliation for state estimation of chemical processes with gaussian/non-gaussian measurement noise

State estimation plays a critical role in modern industry. The extended Kalman filter (EKF) is effective for nonlinear chemical processes with Gaussian noise, but it struggles when gross errors are present. This paper addresses this limitation by reformulating the EKF within the dynamic data reconciliation (DDR) framework, resulting in a robust DDR-based EKF tailored for state estimation in chemical processes with non-Gaussian measurement noise. The combination of random and gross errors is modeled using a contaminated Gaussian distribution. Model predictions are incorporated as prior knowledge, and a fixed-point iterative strategy is employed to update the posterior probability. Additionally, a first-order linearization technique is applied for convergence analysis. The robustness and effectiveness of the DDR-based EKF are demonstrated through both a classic mathematical example and a styrene polymerization reaction. Simulation results show that the DDR-based EKF effectively mitigates gross errors, achieving reliable state estimation.

Event-Driven Maximum Correntropy Filter Based on Cauchy Kernel for Spatial Orientation Using Gyros/Star Sensor Integration.

Gyros/star sensor integration provides a potential method to obtain high-accuracy spatial orientation for turntable structures. However, it is subjected to the problem of accuracy loss when the measurement noises become non-Gaussian due to the complex spatial environment. This paper presents an event-driven maximum correntropy filter based on Cauchy kernel to handle the above problem. In this method, a direct installation mode of gyros/star sensor integration is established and the associated mathematical model is derived to improve the turntable's control stability. Based on this, a Cauchy kernel-based maximum correntropy filter is developed to curb the influence of non-Gaussian measurement noise for enhancing the gyros/star sensor integration's robustness. Subsequently, an event-driven mechanism is constructed based on the filter's innovation information for further reducing the unnecessary computational cost to optimize the real-time performance. The effectiveness of the proposed method has been validated by simulations of the gyros/star sensor integration for spatial orientation. This shows that the proposed filtering methodology not only has strong robustness to deal with the influence of non-Gaussian measurement noise but can also achieve superior real-time spatial applications with a small computational cost, leading to enhanced performance for the turntable's spatial orientation using gyros/star sensor integration.

Maximum correntropy polynomial chaos Kalman filter for underwater navigation

This paper develops an underwater navigation solution that utilizes a strapdown inertial navigation system (SINS) and fuses a set of auxiliary sensors such as an acoustic positioning system, Doppler velocity log, depth meter, and magnetometer to accurately estimate an underwater vessel's position and orientation. The conventional integrated navigation system assumes Gaussian measurement noise, while in reality, the noises are non-Gaussian, particularly contaminated by heavy-tailed impulsive noises. To address this issue, and to fuse the system model with the acquired sensor measurements efficiently, we develop a square root polynomial chaos Kalman filter based on maximum correntropy criteria. The proposed method uses Hermite polynomial chaos expansion to tackle the nonlinearity, and it has the potential to estimate the states in a more accurate way in presence of a non-Gaussian measurement noise. The filter is initialized using acoustic beaconing to accurately locate the initial position of the vehicle. The computational complexity of the proposed filter is calculated in terms of flops count. The proposed method is compared with the existing maximum correntropy sigma point filters in terms of estimation accuracy and computational complexity. It is found from the simulation results that the proposed method is more accurate compared to the conventional deterministic sample point filters and Huber's M-estimator.

Generalized interacting multiple model Kalman filtering algorithm for maneuvering target tracking under non-Gaussian noises

The traditional interacting multiple model Kalman filtering algorithm (IMM-KF) can deal with the maneuvering target problem under Gaussian noise by soft switching among possible motion models. In practice, its performance is likely to degrade when handling non-Gaussian noise. We introduce the Gaussian mixture model (GMM) into the IMM-KF, and the GMM is utilized to model the non-Gaussian measurement noise as a mixture of multiple Gaussian probability densities with a certain probability. Then, a GIMM framework is proposed that enables accurate switching and fusion among multiple possible motion and noise models. And combined with Kalman filtering (KF), a GIMM-KF algorithm is proposed that enables accurate state estimation of maneuvering targets under non-Gaussian noise conditions. Subsequently, the provided simulations and experiments validate that the GIMM-KF algorithm outperforms existing methods in terms of accuracy, stability and robustness.

Exploring the possibility of a complex-valued non-Gaussianity measure for quantum states of light

We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements.

Cameron–Martin type theorem for a class of non-Gaussian measures

In this article, we study the quasi-translation-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron–Martin space and derive the explicit Radon-Nikodym density in terms of the Wiener integral with respect to the fractional Brownian motion. Moreover, we show an integration by parts formula for the derivative operator in the directions of the Cameron–Martin space. As a consequence, we derive the closability of both the derivative and the corresponding gradient operators.

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.

Optimizing kernel width for new risk-sensitive loss: A generalized algorithmic approach

This paper outlines a novel approach to the development of adaptive filtering algorithms, which minimizes susceptibility to the effects of non-Gaussian measurement noise. Using the generalized Gaussian density (GGD) function as the kernel function, we developed a generalized kernel risk-sensitive loss (GKRSL) criterion by which to derive the generalized minimum kernel risk-sensitive loss (GMKRSL) algorithm. We also developed a variable kernel width algorithm for GMKRSL (VKW-GMKRSL) to address the trade-off between convergence rate and steady-state misalignment resulting from a constant kernel width. In simulations, GMKRSL and VKW-GMKRSL outperformed existing works in system identification performance.

Mesoscopic ultrafast nonlinear optics—the emergence of multimode quantum non-Gaussian physics

Over the last few decades, nonlinear optics has become significantly more nonlinear, traversing nearly a billionfold improvement in energy efficiency, with ultrafast nonlinear nanophotonics in particular emerging as a frontier for combining both spatial and temporal engineering. At present, cutting-edge experiments in nonlinear nanophotonics place us just above the mesoscopic regime, where a few hundred photons suffice to trigger highly nonlinear dynamics. In contrast to classical or deep-quantum optics, the mesoscale is characterized by dynamical interactions between mean-field, Gaussian, and non-Gaussian quantum features, all within a close hierarchy of scales. When combined with the inherent multimode complexity of optical fields, such hybrid quantum-classical dynamics present theoretical, experimental, and engineering challenges to the contemporary framework of quantum optics. In this review, we highlight the unique physics that emerges in multimode nonlinear optics at the mesoscale and outline key principles for exploiting both classical and quantum features to engineer novel functionalities. We briefly survey the experimental landscape and draw attention to outstanding technical challenges in materials, dispersion engineering, and device design for accessing mesoscopic operation. Finally, we speculate on how these capabilities might usher in some new paradigms in quantum photonics, from quantum-augmented information processing to nonclassical-light-driven dynamics and phenomena to all-optical non-Gaussian measurement and sensing. The physics unlocked at the mesoscale present significant challenges and opportunities in theory and experiment alike, and this review is intended to serve as a guide to navigating this new frontier in ultrafast quantum nonlinear optics.

Integrated elastic identification and LOS angular rate extraction for slender rockets: A continuous-discrete maximum correntropy Kalman filter approach

Slender rockets experience significant elastic deformation during flight due to aeroelastic forces, impacting their dynamic behavior and posing challenges for traditional line-of-sight (LOS) angular rate estimation methods. This paper addresses this challenge by proposing a novel and highly accurate line-of-sight angular rate extraction model. This model integrates elastic identification with line-of-sight angular rate extraction, employing estimated generalized coordinates to calculate seeker elastic deformation. After elastic decoupling of the seeker, the slender rocket's line-of-sight angular rate can be estimated accurately. Additionally, the model incorporates a continuous-discrete maximum correntropy Kalman filter (CD-MCKF) to effectively handle the non-Gaussian measurement noise and limitations associated with discretizing the continuous model. Simulations demonstrate that the proposed method achieves significantly higher accuracy compared to existing approaches. The improved accuracy of this method can provide more precise calculations for the design of slender rocket guidance and control systems, ultimately improving the flight stability and control accuracy of rockets, and laying a solid foundation for the integrated guidance and control research of slender rockets.

A robust state estimation method for power systems using generalized correntropy loss function

The accurate estimation of power system states is crucial for effective monitoring and control. However, the performance of conventional state estimators, which assume Gaussian measurement noise and do not account for denial-of-service attacks, can deteriorate significantly in real power systems. To address these issues, this paper proposes a novel robust state estimation method based on the quadratic function (QF) and the generalized correntropy loss function (GCL). The proposed QF-GCL state estimation method can effectively deal with non-Gaussian measurement noise and denial-of-service attacks. To enhance the computational efficiency, an influence function based solving method is developed. To determine the optimal parameters for the proposed QF-GCL state estimation method, a new state estimation error covariance equation is further derived. Simulations are performed on the IEEE 30-bus, 118-bus and 300-bus systems, to demonstrate the accurate and robust performance of the proposed QF-GCL robust state estimation method.

Kalman filtering based on dynamic perception of measurement noise

In this work, we focus on the online state estimation problem for linear systems in non-Gaussian measurement noise. Specifically, the measurement noise is modeled as a Gaussian mixture model (GMM). Then, Kalman innovation is used to approximate the current measurement noise, and dynamically perceive its responsiveness to each sub-model of the GMM. The responsiveness from different Gaussian scales is mapped to a new cost function, while the corresponding Kalman filter algorithm is derived. The theoretical steady-state error and computational complexity analysis of the algorithm are also given. The simulation and real experimental results agree with the theoretical predictions and demonstrate the superior performance of the proposed algorithm.

A Novel Hybrid Model Combining Improved VMD and ELM with Extended Maximum Correntropy Criterion for Prediction of Dissolved Gas in Power Transformer Oil

The prediction of dissolved gas change trends in power transformer oil is very important for the diagnosis of transformer faults and ensuring its safe operation. Considering the time series and nonlinear features of the gas change trend, this paper proposes a novel robust extreme learning machine (ELM) model combining an improved data decomposition method for gas content forecasting. Firstly, the original data with nonlinear and sudden change properties will make the forecasting model unstable, and thus an improved variational modal decomposition (IPVMD) method is developed to decompose the original data to obtain the multiple modal dataset, in which the marine predators algorithm (MPA) optimization method is utilized to optimize the free parameters of the VMD. Second, the ELM as an efficient and easily implemented tool is used as the basic model for dissolved gas forecasting. However, the traditional ELM with mean square error (MSE) criterion is sensitive to the non-Gaussian measurement noise (or outliers). In addition, considering the nonlinear non-Gaussian properties of the dissolved gas, a new learning criterion, called extended maximum correntropy criterion (ExMCC), is defined by using an extended kernel function in the correntropy framework, and the ExMCC as a learning criterion is introduced into the ELM to develop a novel robust regression model (called ExMCC-ELM) to improve the ability of ELM to process mutational data. Third, a gas-in-oil prediction scheme is proposed by using the ExMCC-ELM performed on each modal obtained by the proposed IPVMD. Finally, we conducted several simulation studies on the measured data, and the results show that the proposed method has good predictive performance.

Impact of the non-Gaussian measurement noise on the performance of state-of-the-art state estimators for distribution systems

This paper aims to investigate the impact of non-Gaussian measurement noise on state estimation (SE) results in distribution systems. To this end, the measurement noise is assumed to be distributed according to Gaussian or one of the following non-Gaussian probability distribution functions: Uniform, Laplace, Weibull and Gaussian mixture of two Gaussian components. The influence is investigated on three different state-of-the-art SE methods: weighted least squares (WLS) based static SE method, and two Kalman filter based forecasting-aided SE methods, namely extended Kalman filter (EKF) and unscented Kalman filter (UKF). Analyses are conducted on modified IEEE 37-bus system under different operating conditions, including quasi-steady state, sudden state changes and bad data. Performance of the methods in the presence of non-Gaussian measurement noise is compared against their performance when measurement noise is Gaussian distributed. The main conclusions were drawn, summarizing the impacts non-Gaussian measurement noise has on SE and proposing the solutions for overcoming some of the negative impacts.

A Distributed Robust System-Wide State Estimation Method for Power Systems Based on Maximum Correntropy

The distribution of measurement noise applied in practical power system state estimation (PSSE) can deviate from the assumed Gaussian model and the performance of an estimator becomes bad if the Gaussian model is still used. This paper proposes a new distributed robust PSSE method for multi-area power systems. The non-Gaussian model is utilized to fit the measurement noise distribution to reach high model accuracy. The proposed distributed method is derived based on the maximum correntropy criterion to further reduce the impact of non-Gaussian measurement noise and outliers. The influence function combined with the finite-time average consensus algorithm is used to implement the proposed distributed robust method in a fully distributed manner. Simulations conducted on the IEEE 30-bus, 118-bus and 300-bus systems and the Polish 2383-bus system demonstrate the robustness and effectiveness of the proposed distributed method. Each local area can get the system-wide robust state estimation solution by only using local information and small amounts of data from neighboring areas. Our proposed distributed robust method has at least twelve percent improvement in reducing the mean squared error under Gaussian-Uniform noise. It is verified the communication network applied for the proposed method is fairly flexible while the existing distributed approaches use a fixed communication network when the power systems are partitioned. The simulation results also verify the robustness of the proposed method to bad data and communication failure.

Cooperative positioning method of underwater autonomous vehicle formation based on extended Kalman filter

Autonomous Underwater Vehicles (UAVs) have shown significant potential application value in marine environmental surveillance, development, and utilization of resources. However, due to the complex underwater acoustic channel environment and the relative motion between AUVs, this may lead to heavy-tailed non-Gaussian process noise and measurement noise, leading to increased error and the Extended Kalman Filter (EKF) may fail. In prior studies, we propose a multi-AUV formation hierarchical target location algorithm based on the EKF, which can realize multi-AUV self-localization and stationary target localization. The AUV formation consists of one high-precision piloting AUV and several low-precision following AUVs. The following AUVs are divided into two levels, the reference AUVs and the AUVs to be tested. According to the designed positioning period, the reference AUV receives the position parameters from a high-precision piloting AUV and transmits its own position parameters to the AUV to be measured. Then use the EKF to complete the cooperative position correction of the AUV cluster. In this paper, we will study the influence of heavy-tailed noise on this method and study the EKF based on Huber estimation to improve the antijamming capability.

Optimizing state-discrimination receivers for continuous-variable quantum key distribution over a wiretap channel

We address a continuous-variable quantum key distribution protocol employing quaternary phase-shift-keying of coherent states and a non-Gaussian measurement inspired by quantum receivers minimizing the error probability in a quantum-state-discrimination scenario. We consider a pure-loss quantum wiretap channel, in which a possible eavesdropper is limited to collect the sole channel losses. We perform a characterization of state-discrimination receivers and design an optimized receiver maximizing the asymptotic secure key rate (SKR), namely the key-rate optimized receiver (KOR), comparing its performance with respect to the pretty good measurement and the heterodyne-based protocol. We show that the KOR increases the SKR for metropolitan-network distances. Finally, we also investigate the implementations of feasible schemes, such as the displacement feed-forward receiver, obtaining an increase in the SKR in particular regimes.

Enhanced square root CKF with mixture correntropy loss for robust state of charge estimation of lithium-ion battery

The measurement data of lithium-ion battery collected in complex operating conditions may be contaminated by non-Gaussian noises (or outliers), a novel robust state of charge (SOC) estimation approach that enhances the robustness of square-root cubature Kalman filter (SRCKF) by incorporating a mixture correntropy loss (MCL) is proposed to overcome the issue of non-Gaussian measurement noise interference. Although the SRCKF can perform non-destructive state estimation for nonlinear system, it is sensitive to the non-Gaussian noises (or outliers) due to the use of mean square error (MSE) loss. To overcome this limitation, the MCL is used as a cost to replace the MSE in CKF framework, which is constructed by mixture of two Gaussian kernel functions with different kernel widths and has outstanding robustness. Moreover, we establish an effective nonlinear regression model that incorporates noise and state error information into the MCL function, and then a fixed-point iteration method is utilized to update the posterior state estimation. Additionally, to further suppress the influence of abnormal noise on the posterior state estimation, we introduce the exponential function of the innovation vector to adjust the measurement covariance matrix in real-time. Accordingly, an enhanced MCL-SRCKF is developed for SOC estimation considering the measurement noises with non-Gaussian distribution. Results from various simulation environments demonstrate that the proposed method achieves high estimation accuracy in different cases.

Measures of non-Gaussianity in axion-string-induced CMB birefringence

The presence of axion strings in the Universe after recombination can leave an imprint on the polarization pattern of the cosmic microwave background radiation through the phenomenon of axion-string-induced birefringence via the hyperlight axion-like particle's coupling to electromagnetism. Across the sky, the polarization rotation angle is expected to display a patchwork of uniform regions with sharp boundaries that arise as the `shadow' of axion string loops. The statistics of such a birefringence sky map are therefore necessarily non-Gaussian. In this article we quantify the non-Gaussianity in axion-string-induced birefringence using two techniques, kurtosis and bispectrum, which correspond to 4- and 3-point correlation functions. If anisotropic birefringence were detected in the future, a measurement of its non-Gaussian properties would facilitate a discrimination across different new physics sources generally, and in the context of axion strings specifically, it would help to break degeneracies between the axion-photon coupling and properties of the string network.

Robust Data-Driven Sparse Estimation of Distribution Factors Considering PMU Data Quality and Renewable Energy Uncertainty - Part I: Theory

Data-driven sparse estimation of distribution factors (DFs) facilitates online power flow sensitivity analysis for secure system operation. However, existing methods are vulnerable to time-varying non-Gaussian PMU measurement noise, bad data, and uncertain renewable energy sources (RESs). Moreover, they lack scalability to large-scale systems. This two-part paper proposes a robust and scalable sparse DF estimation framework considering PMU data quality and RES uncertainty. In this Part I, a novel Adaptive <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -Lasso estimator with theoretically guaranteed robustness is proposed. It mitigates the impacts of measurement and RES uncertainties to yield accurate dominant DF estimates while promoting sparsity. The key idea is to integrate the robust statistics theory with sparse representation techniques, in particular the Huber loss function, adaptively-weighted <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> regularization, concomitant scale estimate, and pseudo-residuals. Two important robustness properties of this estimator are theoretically proven, i.e., the bounded influence function and the asymptotic consistency of dominant DF estimates given limited samples. The breakdown points of this estimator to measurement and RES uncertainties are derived. Test results validate that the proposed estimator allows accurate estimation without relying on power flow models or massive operating data. It achieves significantly superior robustness over existing methods in multiple scenarios.