Presence Of non-Gaussian Noise Research Articles

Developed square-root cubature Kalman filter-based solution for improving power system state estimation with unknown inputs and non-Gaussian noise

Understanding the ever-changing dynamics of power systems is crucial, and dynamic state estimation (DSE) plays a vital role in achieving this. However, traditional nonlinear Kalman filters (NKFs) face limitations: lack of access to control inputs and presence of non-Gaussian noise in measurements, impacting their accuracy and robustness. This research introduces a novel robust DSE method that tackles these challenges head-on. For the first time in DSE, it leverages the predictive power of Holt-Winters Triple Exponential Smoothing to model the time-varying behavior of control inputs. This innovative approach allows for the simultaneous estimation of dynamic state variables such as the rotor angle and rotor speed changes, as well as transient voltages and control inputs like mechanical input torque and excitation voltage, even in the presence of non-Gaussian noise. Furthermore, the method employs modified projection statistics and a Cauchy function. This unique combination effectively bounds the influence of observation outliers while maintaining high statistical estimation efficiency. This innovative approach utilizes a square cubature Kalman filter (SCKF) for enhanced numerical stability. Extensive simulations under various anomalous conditions demonstrate the method's superior accuracy and efficiency in estimating the state vector. These results highlight its potential to significantly improve power system estimation and pave the way for real-time applications.

Estimation of machinery’s remaining useful life in the presence of non-Gaussian noise by using a robust extended Kalman filter

Estimation of the remaining useful life (RUL) of industrial machinery is essential for condition-based maintenance (CBM). While numerous papers have explored this issues, challenges arise as machinery often works in non-stationary conditions, particularly in harsh environments (like mining machines, wind turbines, helicopters, etc.). The data collected from such environments are affected by non-Gaussian noise, posing difficulties for traditional approaches to non-linear state estimation or prediction. The widely used extended Kalman filter (EKF) suffers from the non-Gaussian noise effect due to its recursive minimum L2-norm filtering. To address these issues, we propose a robust EKF based on the maximum correntropy criterion. This method effectively estimates the RUL of the time-varying degradation process in the presence of non-Gaussian noise, also enabling confidence interval computation for uncertainty management. The efficiency of our approach was confirmed through application to simulated and benchmark data sets, outperforming Kalman filter-based methods for both simulated and real-world scenarios.

Influence of α-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis

Non-negative matrix factorization (NMF) has been used in various applications, including local damage detection in rotating machines. Recent studies highlight the limitations of diagnostic techniques in the presence of non-Gaussian noise. The authors examine the impact of non-Gaussianity levels on the extraction of the signal of interest (SOI). The simple additive model of the signal is proposed: SOI and non-Gaussian noise. As a model of the random component, i.e., noise, a heavy-tailed α-stable distribution with two important parameters (σ and α) was proposed. If SOI is masked by noise (controlled by σ), the influence of non-Gaussianity level (controlled by α) is more critical. We performed an empirical analysis of how these parameters affect SOI extraction effectiveness using NMF. Finally, we applied two NMF algorithms to several (both vibration and acoustic) signals from a machine with faulty bearings at different levels of non-Gaussian disturbances and the obtained results align with the simulations. The main conclusion of this study is that NMF is a very powerful tool for analyzing non-Gaussian data and can provide satisfactory results in a wide range of a non-Gaussian noise levels.

Correntropy-Based Constructive One Hidden Layer Neural Network

One of the main disadvantages of the traditional mean square error (MSE)-based constructive networks is their poor performance in the presence of non-Gaussian noises. In this paper, we propose a new incremental constructive network based on the correntropy objective function (correntropy-based constructive neural network (C2N2)), which is robust to non-Gaussian noises. In the proposed learning method, input and output side optimizations are separated. It is proved theoretically that the new hidden node, which is obtained from the input side optimization problem, is not orthogonal to the residual error function. Regarding this fact, it is proved that the correntropy of the residual error converges to its optimum value. During the training process, the weighted linear least square problem is iteratively applied to update the parameters of the newly added node. Experiments on both synthetic and benchmark datasets demonstrate the robustness of the proposed method in comparison with the MSE-based constructive network, the radial basis function (RBF) network. Moreover, the proposed method outperforms other robust learning methods including the cascade correntropy network (CCOEN), Multi-Layer Perceptron based on the Minimum Error Entropy objective function (MLPMEE), Multi-Layer Perceptron based on the correntropy objective function (MLPMCC) and the Robust Least Square Support Vector Machine (RLS-SVM).

Combined Weighted Envelope Spectrum: An enhanced demodulation framework for extracting characteristic frequency of rotating machinery

Envelope spectrum plays a pivotal role in the surveillance and diagnostics of rotating machinery. Two prevailing techniques for constructing this spectral quantity are narrowband demodulation and cyclostationary analysis. However, the former is susceptible to failure under low Signal-to-Noise Ratio (SNR) conditions or in the presence of non-Gaussian noise, while the latter exhibits sensitivity to cyclostationary noise, such as electromagnetic interferences. Notably, specific tasks determine the characteristic frequencies expected to be detected. More precisely, the determination of target component and noise interference depends on the mechanical part under scrutiny. Consequently, the envelope spectra provided by classical methods often manifest deficiencies when confronted with intricate monitoring signals from rotating machinery. To address this challenge, this paper proposes an enhanced demodulation framework—Combined Weighted Envelope Spectrum (CWES), tailored for extracting specific characteristic frequencies. First, a cyclostationary model for rotating machinery is presented, demarcating the target component as solely the cyclostationary signal caused by the mechanical element of concern. Second, the weighting functions with/without prior knowledge of characteristic frequency are elaborately designed, and the Adaptive Combined Weighted Envelope Spectrum (ACWES) and the Optimal Combined Weighted Envelope Spectrum (OCWES) are constructed correspondingly. Moreover, the indicator Characteristic peak-to-Noise peak Ratio (CNR) is proposed to evaluate the demodulation performance of different envelope spectra. Ultimately, the effectiveness of ACWES and OCWES is validated by simulation signals and experimental data from rolling bearing, centrifugal pump, and Francis turbine, and the comparison with the state-of-the-art demodulation methods verifies the superiority of the proposed enhanced demodulation framework.

Beyond 10log10M array gain: A beamforming method under non-Gaussian noise and multi-sources

Recently, deep learning techniques have been applied in the field of direction of arrival (DOA) to enhance beamforming in various practical applications, such as traffic positioning and hydro-acoustic detection. However, the effectiveness of these algorithms relies heavily on the availability of accurate training data that closely resembles real-world data, which can be challenging to obtain, particularly in hydro-acoustic environments. This is primarily due to the limited array gain (AG) and the constant fluctuations in ambient noise. Additionally, the presence of non-Gaussian noise in the background further complicates the process of identifying the desired signal, and it is worth noting that state-of-the-art deep learning methods have a limitation in detecting up to four sources. To address these challenges, we propose a novel method that employs a diagonal beam spectrum (DBS) feature to suppress non-Gaussian noise and enhance array gain. By incorporating DBS, our method surpasses the theoretical 10logM array gain and achieves an impressive 25logM array gain, while effectively detecting signals as low as -38dB (where M denotes the number of sensors). We also employ SA-U-NET, an image edge detection network based on DBS, to detect more than four sound sources and optimize angular positioning. The experiments focus on accuracy and array gain improvements of algorithms. Our proposed method achieves up to 91.67% correctness and 35dB array gain (i.e., 4 times better than the next best baseline) in experiments compared to other methods, demonstrating the potential of DBS and SA-U-NET.

Filtering in Triplet Markov Chain Model in the Presence of Non-Gaussian Noise with Application to Target Tracking

In hidden Markov chain (HMC) models, widely used for target tracking, the process noise and measurement noise are in general assumed to be independent and Gaussian for mathematical simplicity. However, the independence and Gaussian assumptions do not always hold in practice. For instance, in a typical radar tracking application, the measurement noise is correlated over time as the sampling frequency of a radar is generally much higher than the bandwidth of the measurement noise. In addition, target maneuvers and measurement outliers imply that the process noise and measurement noise are non-Gaussian. To solve this problem, we resort to triplet Markov chain (TMC) models to describe stochastic systems with correlated noise and derive a new filter under the maximum correntropy criterion to deal with non-Gaussian noise. By stacking the state vector, measurement vector, and auxiliary vector into a triplet state vector, the TMC model can capture the complete dynamics of stochastic systems, which may be subjected to potential parameter uncertainty, non-stationarity, or error sources. Correntropy is used to measure the similarity of two random variables; unlike the commonly used minimum mean square error criterion, which uses only second-order statistics, correntropy uses second-order and higher-order information, and is more suitable for systems in the presence of non-Gaussian noise, particularly some heavy-tailed noise disturbances. Furthermore, to reduce the influence of round-off errors, a square-root implementation of the new filter is provided using QR decomposition. Instead of the full covariance matrices, corresponding Cholesky factors are recursively calculated in the square-root filtering algorithm. This is more numerically stable for ill-conditioned problems compared to the conventional filter. Finally, the effectiveness of the proposed algorithms is illustrated via three numerical examples.

Comparing Apples with Apples: Robust Detection Limits for Exoplanet High-contrast Imaging in the Presence of Non-Gaussian Noise

Over the past decade, hundreds of nights have been spent on the world’s largest telescopes to search for and directly detect new exoplanets using high-contrast imaging (HCI). Thereby, two scientific goals are of central interest: first, to study the characteristics of the underlying planet population and distinguish between different planet formation and evolution theories. Second, to find and characterize planets in our immediate solar neighborhood. Both goals heavily rely on the metric used to quantify planet detections and nondetections. Current standards often rely on several explicit or implicit assumptions about noise. For example, it is often assumed that the residual noise after data postprocessing is Gaussian. While being an inseparable part of the metric, these assumptions are rarely verified. This is problematic as any violation of these assumptions can lead to systematic biases. This makes it hard, if not impossible, to compare results across data sets or instruments with different noise characteristics. We revisit the fundamental question of how to quantify detection limits in HCI. We focus our analysis on the error budget resulting from violated assumptions. To this end, we propose a new metric based on bootstrapping that generalizes current standards to non-Gaussian noise. We apply our method to archival HCI data from the NACO instrument at the Very Large Telescope and derive detection limits for different types of noise. Our analysis shows that current standards tend to give detection limits that are about one magnitude too optimistic in the speckle-dominated regime. That is, HCI surveys may have excluded planets that can still exist.

State of charge estimation for Li-ion batteries based on iterative Kalman filter with adaptive maximum correntropy criterion

Nonlinear filter methods are commonly employed for model-driven based battery state of charge (SOC) estimation because of their ability to suppress Gaussian noise. Due to the uncertainty of complicated operational conditions, the minimum mean square error (MMSE) criterion based Kalman filter (KF) methods may not guarantee sufficient SOC estimation accuracy in the presence of non-Gaussian noise. To enhance the anti-interference performance of the KFs based SOC estimation method, a novel methodology that combines the adaptive kernel width based maximum correntropy criterion (AMCC) with the Levenberg-Marguardt (L-M) principle based adaptive iterative extended KF (AIEKF) is proposed. Firstly, the MMSE criterion of the EKF is replaced by the MCC criterion and an adaptive kernel width strategy is designed to weaken the effect of anomalous data. Then, the AMCC-corrected covariance matrix and state are updated using an L-M optimized multi-step iterative filter during the measurement update phase. Finally, the combination of AMCC and AIEKF is devised to execute SOC estimation. Several sets of data from different temperatures and operational conditions are used to verify the validity of the proposed AMCC-AIEKF method. Compared with the adaptive EKF (AEKF) and the adaptive iterative EKF (AIEKF) method, the AMCC-AIEKF method has the advantage in terms of SOC estimation accuracy and robustness under different operational conditions with noiseless, Gaussian noise and non-Gaussian noise.

Using long-term condition monitoring data with non-Gaussian noise for online diagnostics

The number of timely diagnoses based on condition monitoring data is increasing with the growing usage of monitoring systems. In most of the methods used in these systems, a pre-established fault detection threshold is needed, while there are no specific limit values or thresholds in many cases, especially when the machine is unique. Also, in most actual applications, due to the kind of process and harsh environment, the noise inherent in the observed process exhibits non-Gaussian characteristics, making it a challenging task for diagnostics based on condition monitoring (CM) data. Therefore, this paper introduced a robust methodology based on the switching maximum correntropy Kalman filter (SMCKF) to address the mentioned problems (threshold and online diagnostics in the presence of non-Gaussian noise by using CM data). This approach uses multiple dynamic system models to explain different degradation stages, utilizing robust Bayesian estimation. As this approach is based on dynamic behavior, a threshold for diagnostics is no longer needed. Ultimately, the proposed approach is applied to the online diagnosis of simulated and actual data sets. The results of both simulated and real data sets prove the method’s efficacy.

Robust switching Kalman filter for diagnostics of long-term condition monitoring data in the presence of non-Gaussian noise

Machinery condition prognosis system uses long-term historical data to predict remaining useful life (RUL). One of the critical steps to reach this purpose is to segment long-term data into two or several degradation stages (Healthy, Unhealthy, and Critic stage). Finding changing points between regimes may be a crucial preliminary task for further predicting the degradation process. However, finding the accurate partition into two or more regimes is a challenging task in the actual application when the noise inherent in the observed process is non-Gaussian. Therefore, this paper introduced a robust methodology based on switching Kalman filters to address the problems mentioned. This approach uses multiple dynamic system models to explain different degradation stages, utilizing robust Bayesian estimation. Also, based on this fact, this approach works based on dynamic behavior; a threshold for diagnostics is no longer needed. Ultimately, the proposed approach is applied for the online diagnosis of simulated data sets in the presence of Gaussian and non-Gaussian noise. The result of the applied methodology on simulated data sets proves the method’s efficacy.

Diffusion Normalized Maximum Versoria Criterion Robust to Impulsive Noise

In this letter, we propose a new diffusion normalized maximum Versoria criterion (d-NMVC) algorithm, which is based on maximization of the normalized maximum Versoria criterion (MVC) cost function to enhance the performance of the distributed estimation over networks in the presence of non-Gaussian noise. Convergence of the proposed algorithm, in the mean square sense and evolution behavior, under impulsive noise environment is also analyzed. Simulation results show the robustness of the proposed algorithm under impulsive noise environment against various non-Gaussian noise distributions.

A robust diffusion recursive generalized modified Blake-Zisserman algorithm for distributed estimation under an adaptive kernel width

In adaptive networks, the performance of distributed estimation is degraded in the presence of non-Gaussian noises. A number of robust criteria for diffusion approaches, such as generalized maximum correntropy and hyperbolic cosine function have been developed towards non-Gaussian/impulsive background noises. However, these algorithms are affected by high steady-state misadjustment. Therefore, to improve the robustness under non-Gaussian noise and decrease steady-state misadjustment, a generalized modified Blake-Zisserman (GMBZ) robust loss function is represented in this study. Also, we propose a new robust diffusion recursive least squares (RLS) based on GMBZ. Additionally, to enhance tracking ability in non-stationary environments, the proposed method is extended by an adaptive strategy for kernel width selection. The convergence analysis and the steady-state performance of the proposed method are also discussed. Simulation results demonstrate the effectiveness and robustness of the proposed algorithms for system identification scenarios in the presence of α-stable noise in stationary and non-stationary environments.

A Dynamic Self-Tuning Maximum Correntropy Kalman Filter for Wireless Sensors Networks Positioning Systems

To improve the accuracy of the maximum correntropy Kalman filter (MCKF) in wireless sensors networks (WSNs) positioning, a dynamic self-tuning maximum correntropy Kalman filter (DSTMCKF) is proposed, where innovation and the sensors information of the WSNs are used to adjust the noise covariance matrices, and the maximum correntropy criterion is the criterion for the filter’s optimality. By dynamically adjusting the noise covariance matrices, the DSTMCKF ensures that the correntropy distribution is accurate in the presence of non-Gaussian noise (NGN), thus improving its ability to handle the NGN. In simulation and real environment positioning experiments, the DSTMCKF is used to compare with the MCKF, variable kernel width–maximum correntropy Kalman filter (VKW-MCKF) and robust minimum error entropy Kalman filter (R-MEEKF). Among the four filters, the DSTMCKF has the highest accuracy, and the error of the DSTMCKF is reduced by 34.5, 42.9 and 40.0%, respectively, compared with the MCKF, VKW-MCKF and R-MEEKF in the real-world environment positioning experiment. The application of the DSTMCKF in WSNs positioning systems improves the stability of the control systems because of the rising positioning accuracy, which makes WSNs positioning systems more widely used in scenarios requiring high stability, such as automatic parking.

Application of Machine Learning Tools for Long-Term Diagnostic Feature Data Segmentation

In this paper, a novel method for long-term data segmentation in the context of machine health prognosis is presented. The purpose of the method is to find borders between three data segments. It is assumed that each segment contains the data that represent different statistical properties, that is, a different model. It is proposed to use a moving window approach, statistical parametrization of the data in the window, and simple clustering techniques. Moreover, it is found that features are highly correlated, so principal component analysis is exploited. We find that the probability density function of the first principal component may be sufficient to find borders between classes. We consider two cases of data distributions, Gaussian and α-stable, belonging to the class of non-Gaussian heavy-tailed distributions. It is shown that for random components with Gaussian distribution, the proposed methodology is very effective, while for the non-Gaussian case, both features and the concept of moving window should be re-considered. Finally, the procedure is tested for real data sets. The results provided here may be helpful in understanding some specific cases of machine health prognosis in the presence of non-Gaussian noise. The proposed approach is model free, and thus it is universal. The methodology can be applied for any long-term data where segmentation is crucial for the data processing.

Denoising the hob vibration signal using improved complete ensemble empirical mode decomposition with adaptive noise and noise quantization strategies

A novel denoising method combining complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and noise quantization strategies is proposed to solve the problem of the noise of the hob vibration signals disturbing the condition monitoring and feature extraction. The vibration signal is decomposed into several intrinsic mode functions (IMFs) and a residual based on CEEMDAN first. Considering that statistical indicators such as correlation coefficient and kurtosis are not effective in the presence of non-Gaussian noises and modulation because they primarily focus on the signal statistical distribution while ignoring the characteristics of the mechanism, a novel index based on the autocorrelation function analysis called periodic modulation for noise assessment (PMNA) is proposed to quantify the noise of IMFs. Further, IMFs are rearranged in the decreasing order of PMNA. A novel threshold joint with IMFs noise assessment (TJINA) varying with the combination of PMNA and the rearranged IMF retrieval is designed, which has advantages in the local smoothness and small fluctuation. On that basis, IMFs are divided into noise domain and signal domain, IMFs in the noise domain are denoised with TJINA and soft threshold function strategies. The proposed method is applied to the simulated signals with different input signal to noise ratios (SNRin) and two measured gear hobbing vibration signals. The comparison with some state-of-the-art approaches and the ablation experiment reveals that the proposed method performs better in enhancing the effective components and eliminating noise.

Using machine learning to autotune chi-squared tests for gravitational wave searches

The sensitivity of gravitational wave searches is reduced by the presence of non-Gaussian noise in the detector data. These non-Gaussianities often match well with the template waveforms used in matched filter searches, and require signal-consistency tests to distinguish them from astrophysical signals. However, empirically tuning these tests for maximum efficacy is time consuming and limits the complexity of these tests. In this work we demonstrate a framework to use machine-learning techniques to automatically tune signal-consistency tests. We implement a new $\chi^2$ signal-consistency test targeting the large population of noise found in searches for intermediate mass black hole binaries, training the new test using the framework set out in this paper. We find that this method effectively trains a complex model to down-weight the noise, while leaving the signal population relatively unaffected. This improves the sensitivity of the search by $\sim 11\%$ for signals with masses $> 300 M_\odot$. In the future this framework could be used to implement new tests in any of the commonly used matched-filter search algorithms, further improving the sensitivity of our searches.

Generalized Correntropy Sparse Gauss–Hermite Quadrature Filter for Epidemic Tracking on Complex Networks

The dynamic estimation of epidemic spreading on networks is essential for controlling morbidity. Nonlinear Kalman filters, which are capable of estimating the hidden state of a nonlinear system, can be utilized for dynamic state estimation of epidemic spreading. Traditional nonlinear Kalman filters perform optimization using the minimum mean square error (MMSE) criterion. Since observable measurements are generally corrupted by non-Gaussian noise, maximum correntropy criterion (MCC)-based nonlinear Kalman filters have been used for improving robustness against non-Gaussian noise. In comparison with the MCC, generalized correntropy is more robust and flexible in the presence of non-Gaussian noise. In this article, epidemic spreading is described by a compartmental model, i.e., susceptible-infected-recovered-susceptible model. Based on the compartmental model, the generalized correntropy-based sparse Gauss-Hermite quadrature filter (GCSGHQF) is proposed for epidemic tracking on homogeneous networks. The GCSGHQF approximates the Gaussian-weighted integrals by utilizing the sparse Gauss-Hermite quadrature rule. In addition, the generalized correntropy is applied to enhancing robustness in the presence of non-Gaussian noise. Simulation results show that the GCSGHQF exhibits a higher filtering accuracy and improves robustness compared to traditional nonlinear Kalman filters.

Adaptive identification under the maximum correntropy criterion with variable center

The problem of identifying the parameters of a linear object in the presence of non-Gaussian noise is considered. The identification algorithm is a gradient procedure for maximizing the functional, which is a correntropy. This functionality allows you to get estimates that have robust properties. In contrast to the commonly used Gaussian kernels, the centers of which are at zero and effective for distributions with zero mean, the paper considers a modification of the criterion suitable for distributions with nonzero mean. The modification is to use correntropy with a variable center The use of Gaussian kernels with a variable center will allow us to estimate unknown parameters under Gaussian and non-Gaussian noises with zero and non-zero mean distributions and provide an opportunity to develop new technologies for data analysis and processing. It is important to develop a robust identification algorithm based on correntropy with variable center. Their properties in the identification of stationary and non-stationary objects are the subject of research. The goal is to develop a robust identification algorithm that maximizes the criterion of correntropy with a variable center using center configuration procedures and kernel width and to study its convergence in stationary and non-stationary cases under non-Gaussian noise. Expressions for steady-state value of the estimation error are obtained, which depend on the type of noise distribution and the degree of non-stationarity of the estimated parameters The following tasks are solved: to investigate the convergence of the algorithm and determine the conditions for the stability of the established identification process. Methods of estimation theory (identification) and probability theory are used. The following results were obtained: 1) the developed algorithm provides robust estimates in the presence of noises having a distribution with zero and non-zero mean; 2) its convergence was studied in stationary and non-stationary cases under conditions of Gaussian and non-Gaussian noise; 3) simulation of the algorithm was carried out. 1) the developed algorithm consists in the development of a robust identification algorithm that maximizes the criterion of correntropy with a variable center; 2) its convergence in stationary and non-stationary cases in the conditions of Gaussian and non-Gaussian noises is investigated; 3) simulation of the algorithm is performed. Conclusions: The results of the current study will improve existing data processing technologies based on robust estimates and accelerate the development of new computing programs in real time.

Infogram performance analysis and its enhancement for bearings diagnostics in presence of non-Gaussian noise

Local damage detection in rotating machines is a well-established field of study, however, some new challenges have recently appeared. They are associated with the presence of strongly non-Gaussian noise related to harsh environments or technological processes. Searching for cyclic impulses in the presence of noncyclic, high amplitude impulses becomes difficult. Cyclostationary analysis may be the answer to such a problem. However, it does not recognize the impulsive nature of the signal. Thus, the concept of infogram, a tool that investigates diagnostic information both in the time and frequency domain, seems ideally suited for such an application. It uses the negentropy to measure the impulsiveness of the squared envelope (SE infogram) and the periodicity in the spectral domain (SES infogram) to finally get the combined information on impulsiveness and periodicity as the average of the SE and SES infograms, called average infogram or just infogram. The purpose of this study is to examine the performance of the infogram for several benchmark signals i.e. Gaussian noise, signals with cyclic impulses and non-cyclic impulses and the mixture of all of them. Finally, the enhancement of the classical infogram is proposed. It is shown that replacing simple averaging of partial infograms with the logarithmic mean, geometric mean, or normalizing of partial infograms, improves the robustness of the classical infogram in the presence of strongly non-Gaussian noise. As an illustration, two real signals are considered. The first one isthe vibration signal from the ore crusher used in the raw materials’ industry, for mechanical fragmentation of rock oversized pieces. The second one is an acoustic signal from bearing installed in an idler in a belt conveyor. In both cases the impulses occurring in this signal (from the crushing process or interaction between a moving belt joint and coating of idler) make the classical diagnosis very difficult. The proposed modified infogram can deal with such a problem successfully.