Bayesian Classifier Based on Robust Kernel Density Estimation and Harris Hawks Optimisation

In real-world applications, datasets frequently contain outliers, which can hinder the generalization ability of machine learning models. Bayesian classifiers, a popular supervised learning method, rely on accurate probability density estimation for classifying continuous datasets. However, achieving precise density estimation with datasets containing outliers poses a significant challenge. This paper introduces a Bayesian classifier that utilizes optimized robust kernel density estimation to address this issue. Our proposed method enhances the accuracy of probability density distribution estimation by mitigating the impact of outliers on the training sample’s estimated distribution. Unlike the conventional kernel density estimator, our robust estimator can be seen as a weighted kernel mapping summary for each sample. This kernel mapping performs the inner product in the Hilbert space, allowing the kernel density estimation to be considered the average of the samples’ mapping in the Hilbert space using a reproducing kernel. M-estimation techniques are used to obtain accurate mean values and solve the weights. Meanwhile, complete cross-validation is used as the objective function to search for the optimal bandwidth, which impacts the estimator. The Harris Hawks Optimisation optimizes the objective function to improve the estimation accuracy. The experimental results show that it outperforms other optimization algorithms regarding convergence speed and objective function value during the bandwidth search. The optimal robust kernel density estimator achieves better fitness performance than the traditional kernel density estimator when the training data contains outliers. The Naïve Bayesian with optimal robust kernel density estimation improves the generalization in the classification with outliers.

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Here the estimating problem of a single sinusoidal signal in the additive symmetricα-stable Gaussian(ASαSG)noise is investigated.The ASαSG noise here is expressed as the additive of a Gaussian noise and a symmetricα-stable distributed variable.As the probability density function(PDF)of the ASαSG is complicated,traditional estimators cannot provide optimum estimates.Based on the Metropolis-Hastings(M-H)sampling scheme,a robust frequency estimator is proposed for ASαSG noise.Moreover,to accelerate the convergence rate of the developed algorithm,a new criterion of reconstructing the proposal covar-iance is derived,whose main idea is updating the proposal variance using several previous samples drawn in each iteration.The approximation PDF of the ASαSG noise,which is referred to the weighted sum of a Voigt function and a Gaussian PDF,is also employed to reduce the computational complexity.The computer simulations show that the performance of our method is better than the maximum likelihood and the lp-norm estimators.

Robustifying Biased Estimation in Linear Model

The parameter estimation problem in linear model is considered when multicollinearity and outliers exist simultaneously.A class of new estimators,robust general shrunken estimators,are proposed by grafting the robust estimation techniques philosophy into the biased estimator,and their statistical properties are discussed.By appropriate choices of the shrinking parameter matrix,we obtain many useful and important estimators.A numerical example is used to illustrate that these new estimators can not only effectively overcome difficulty caused by multicollinearity but also resist the influence of outliers.

GNSS receiver autonomous integrity monitoring(RAIM)algorithm based on robust estimation

Integrity is significant for safety-of-life applications. Receiver autonomous integrity monitoring（RAIM） has been developed to provide integrity service for civil aviation. At first,the conventional RAIM algorithm is only suitable for single fault detection, single GNSS constellation. However, multiple satellite failure should be considered when more than one satellite navigation system are adopted. To detect and exclude multi-fault, most current algorithms perform an iteration procedure considering all possible fault model which lead to heavy computation burden. An alternative RAIM is presented in this paper based on multiple satellite constellations（for example, GPS and Bei Dou（BDS） etc.） and robust estimation for multi-fault detection and exclusion, which can not only detect multi-failures,but also control the influences of near failure observation. Besides, the RAIM algorithm based on robust estimation is more efficient than the current RAIM algorithm for multiple constellation and multiple faults. Finally, the algorithm is tested by GPS/Bei Dou data.

Improved adaptively robust estimation algorithm for GNSS spoofer considering continuous observation error

Once the spoofer has controlled the navigation sys-tem of unmanned aerial vehicle(UAV),it is hard to effectively control the error convergence to meet the threshold condition only by adjusting parameters of estimation if estimation of the spoofer on UAV has continuous observation error.Aiming at this problem,the influence of the spoofer’s state estimation error on spoofing effect and error convergence conditions is theoretically analyzed,and an improved adaptively robust estimation algo-rithm suitable for steady-state linear quadratic estimator is pro-posed.It enables the spoofer’s estimator to reliably estimate UAV status in real time,improves the robustness of the estima-tor in responding to observation errors,and accelerates the con-vergence time of error control.Simulation experiments show that the mean value of normalized innovation squared(NIS)is reduced by 88.5%,and the convergence time of NIS value is reduced by 76.3%,the convergence time of true trajectory error of UAV is reduced by 42.3%,the convergence time of estimated trajectory error of UAV is reduced by 67.4%,the convergence time of estimated trajectory error of the spoofer is reduced by 33.7%,and the convergence time of broadcast trajectory error of the spoofer is reduced by 54.8%when the improved algorithm is used.The improved algorithm can make UAV deviate from pre-set trajectory to spoofing trajectory more effectively and more subtly.

Numerical Robust Stability Estimation in Milling Process

The conventional prediction of milling stability has been extensively studied based on the assumptions that the milling process dynamics is time invariant. However, nominal cutting parameters cannot guarantee the stability of milling process at the shop floor level since there exists many uncertain factors in a practical manufacturing environment. This paper proposes a novel numerical method to estimate the upper and lower bounds of Lobe diagram, which is used to predict the milling stability in a robust way by taking into account the uncertain parameters of milling system. Time finite element method, a milling stability theory is adopted as the conventional deterministic model. The uncertain dynamics parameters are dealt with by the non-probabilistic model in which the parameters with uncertainties are assumed to be bounded and there is no need for probabilistic distribution densities functions. By doing so, interval instead of deterministic stability Lobe is obtained, which guarantees the stability of milling process in an uncertain milling environment, In the simulations, the upper and lower bounds of Lobe diagram obtained by the changes of modal parameters of spindle-tool system and cutting coefficients are given, respectively. The simulation results show that the proposed method is effective and can obtain satisfying bounds of Lobe diagrams. The proposed method is helpful for researchers at shop floor to making decision on machining parameters selection.

Robust Spectral Estimation of Track Irregularity

Because the existing spectral estimation methods for railway track irregularity analysis are very sensitive to outliers, a robust spectral estimation method is presented to process track irregularity signals. The proposed robust method is verified using 100 groups of clean/contaminated data reflecting he vertical profile irregularity taken from Bejing-Guangzhou railway with a sampling frequency of 33 data every ~10 m, and compared with the Auto Regressive (AR) model. The experimental results show that the proposed robust estimation is resistible to noise and insensitive to outliers, and is superior to the AR model in terms of efficiency, stability and reliability.

Exploiting Robust Estimators in Phase Correlation of 3D Point Clouds for 6 DoF Pose Estimation

Point cloud registration is a fundamental task in both remote sensing,photogrammetry,and computer vision,which is to align multiple point clouds to the same coordinate frame.Especially in LiDAR odometry,by conducting the transformation between two adjacent scans,the pose of the platform can be estimated.To be specific,the goal is to recover the relative six-degree-of-freedom(6 DoF)pose between the source point cloud and the target point cloud.In this paper,we explore the use of robust estimators in the phase correlation when registering two point clouds,enabling a 6 DoF pose estimation between point clouds in a sub-voxel accuracy.The estimator is a rule for calculating an estimate of a given quantity based on observed data.A robust estimator is an estimation rule that is insensitive to nonnormality and can estimate parameters of a given objective function from noisy observations.The proposed registration method is theoretically insensitive to noise and outliers than correspondence-based methods.Three core steps are involved in the method:transforming point clouds from the spatial domain to the frequency domain,decoupling of rotations and translations,and using robust estimators to estimate phase shifts.Since the estimation of transformation parameters lies in the calculation of phase shifts,robust estimators play a vital role in shift estimation accuracy.In this paper,we have tested the performance of six different robust estimators and provide comparisons and discussions on the contributions of robust estimators in the 3D phase correlation.Different point clouds from two urban scenarios and one indoor scene are tested.Results validate the proposed method can reach performance that predominant rotation and translation errors reaching less than 0.5°and 0.5 m,respectively.Moreover,the performance of various tested robust estimators is compared and discussed.

Robust estimation algorithm for multiple-structural data

This paper proposes a robust method of parameter estimation and data classification for multiple-structural data based on the linear error in variable（EIV） model.The traditional EIV model fitting problem is analyzed and a robust growing algorithm is developed to extract the underlying linear structure of the observed data.Under the structural density assumption,the C-step technique borrowed from the Rousseeuw＇s robust MCD estimator is used to keep the algorithm robust and the mean-shift algorithm is adopted to ensure a good initialization.To eliminate the model ambiguities of the multiple-structural data,statistical hypotheses tests are used to refine the data classification and improve the accuracy of the model parameter estimation.Experiments show that the efficiency and robustness of the proposed algorithm.

Integrated Navigation Based on Robust Estimation Outputs of Multi-sensor Measurements and Adaptive Weights of Dynamic Model Information

An integrated navlgation based on the kinematic or dynamic state model and the raw measurements has the advantages of high redundancy, high reliability, as well as high ability of fault tolerance and simplicity in calculation. In order to control the influences of measurements outliers and the kinematic model errors on the integrated navigation results, a robust estimation method and an adaptive data fusion method are applied. An integrated navigation example using simulated data is performed and analyzed.

Robust H_2 estimation and control

This paper is concerned with the H2 estimation and control problems for uncertain discretetime systems with norm-bounded parameter uncertainty. We first present an analysis result on H2 norm bound for a stable uncertain system in terms of linear matrix inequalities (LMIs). A solution to the robust H2 estimation problem is then derived in terms of two LMIs. As compared to the existing results, our result on robust H2 estimation is more general. In addition, explicit search of appropriate scaling parameters is not needed as the optimization is convex in the scaling parameters. The LMI approach is also extended to solve the robust H2 control problem which has been difficult for the traditional Riccati equation approach since no separation principle has been known for uncertain systems. The design approach is demonstrated through a simple example.

Robust and Efficient Reliability Estimation for Exponential Distribution

In modeling reliability data,the exponential distribution is commonly used due to its simplicity.For estimating the parameter of the exponential distribution,classical estimators including maximum likelihood estimator represent the most commonly used method and are well known to be efficient.However,the maximum likelihood estimator is highly sensitive in the presence of contamination or outliers.In this study,a robust and efficient estimator of the exponential distribution parameter was proposed based on the probability integral transform statistic.To examine the robustness of this new estimator,asymptotic variance,breakdown point,and gross error sensitivity were derived.This new estimator offers reasonable protection against outliers besides being simple to compute.Furthermore,a simulation study was conducted to compare the performance of this new estimator with the maximum likelihood estimator,weighted likelihood estimator,and M-scale estimator in the presence of outliers.Finally,a statistical analysis of three reliability data sets was conducted to demonstrate the performance of the proposed estimator.

Data-driven source-load robust optimal scheduling of integrated energy production unit including hydrogen energy coupling

A robust low-carbon economic optimal scheduling method that considers source-load uncertainty and hydrogen energy utilization is developed.The proposed method overcomes the challenge of source-load random fluctuations in integrated energy systems(IESs)in the operation scheduling problem of integrated energy production units(IEPUs).First,to solve the problem of inaccurate prediction of renewable energy output,an improved robust kernel density estimation method is proposed to construct a data-driven uncertainty output set of renewable energy sources statistically and build a typical scenario of load uncertainty using stochastic scenario reduction.Subsequently,to resolve the problem of insufficient utilization of hydrogen energy in existing IEPUs,a robust low-carbon economic optimal scheduling model of the source-load interaction of an IES with a hydrogen energy system is established.The system considers the further utilization of energy using hydrogen energy coupling equipment(such as hydrogen storage devices and fuel cells)and the comprehensive demand response of load-side schedulable resources.The simulation results show that the proposed robust stochastic optimization model driven by data can effectively reduce carbon dioxide emissions,improve the source-load interaction of the IES,realize the efficient use of hydrogen energy,and improve system robustness.

A method of Robust low-angle target height and compound reflection coefficient joint estimation

It is always a challenging issue for radar systems to estimate the height of a low-angle target in the multipath propagation environment.The highly deterministic maximum likelihood estimator has a high accuracy,but the errors of the ground reflection coefficient and the reflecting surface height have serious influence on the method.In this paper,a robust es-timation method with less computation burden is proposed based on the compound reflection coefficient multipath model for low-angle targets.The compound reflection coefficient is es-timated from the received data of the array and then a one-di-mension generalized steering vector is constructed to estimate the target height.The algorithm is robust to the reflecting sur-face height error and the ground reflection coefficient error.Fi-nally,the experiment and simulation results demonstrate the validity of the proposed method.

l_(1)-norm Based GWLP for Robust Frequency Estimation

In this work,we address the frequency estimation problem of a complex single-tone embedded in the heavy-tailed noise.With the use of the linear prediction(LP)property and l_(1)-norm minimization,a robust frequency estimator is developed.Since the proposed method employs the weighted l_(1)-norm on the LP errors,it can be regarded as an extension of the l_(1)-generalized weighted linear predictor.Computer simulations are conducted in the environment of α-stable noise,indicating the superiority of the proposed algorithm,in terms of its robust to outliers and nearly optimal estimation performance.

Robust Variance Components Estimation in the PERG Mixed Distributions of Empirical Variances—PEROBVC Method

A mixed distribution of empirical variances, composed of two distributions the basic and contaminating ones, and referred to as PERG mixed distribution of empirical variances, is considered. In the paper a robust inverse problem solution is given, namely a (new) robust method for estimation of variances of both distributions—PEROBVC Method, as well as the estimates for the numbers of observations for both distributions and, in this way also the estimate of contamination degree.

Robust Estimators for Poisson Regression

The present paper proposes a new robust estimator for Poisson regression models. We used the weighted maximum likelihood estimators which are regarded as Mallows-type estimators. We perform a Monte Carlo simulation study to assess the performance of a suggested estimator compared to the maximum likelihood estimator and some robust methods. The result shows that, in general, all robust methods in this paper perform better than the classical maximum likelihood estimators when the model contains outliers. The proposed estimators showed the best performance compared to other robust estimators.

An Embedded Consensus ADMM Distribution Algorithm Based on Outer Approximation for Improved Robust State Estimation of Networked Microgrids

Networked microgrids(NMGs)are critical in theaccommodation of distributed renewable energy.However,theexisting centralized state estimation(SE)cannot meet the demandsof NMGs in distributed energy management.The currentestimator is also not robust against bad data.This study introducesthe concepts of relative error to construct an improvedrobust SE(IRSE)optimization model with mixed-integer nonlinearprogramming(MINLP)that overcomes the disadvantage ofinaccurate results derived from different measurements whenthe same tolerance range is considered in the robust SE(RSE).To improve the computation efficiency of the IRSE optimizationmodel,the number of binary variables is reduced based on theprojection statistics and normalized residual methods,which effectivelyavoid the problem of slow convergence or divergenceof the algorithm caused by too many integer variables.Finally,an embedded consensus alternating direction of multiplier method(ADMM)distribution algorithm based on outer approximation(OA)is proposed to solve the IRSE optimization model.This algorithm can accurately detect bad data and obtain SE resultsthat communicate only the boundary coupling informationwith neighbors.Numerical tests show that the proposed algorithmeffectively detects bad data,obtains more accurate SE results,and ensures the protection of private information in all microgrids.

THEORETIC RESEARCH ON ROBUSTIFIED LEAST SQUARES ESTIMATOR BASED ON EQUIVALENT VARIANCE-COVARIANCE

Depending on analyzing the abuse of equivalent weights,a set of self-contained theory system on robust estimation based on equivalent variance-covariance is established,which includes ρ function, φ function,equivalent variance-covariance function,influence function and breakdown point.And an example is given to verify that the robust models proposed in this paper are reliable and correct.

A New Method to Solve Robust Data Reconciliation in Nonlinear Process

Data reconciliation is an effective technique for providing accurate and consistent value for chemical process. However, the presence of gross errors can severely bias the reconciled results. Robust estimators can significantly reduce the effect of gross errors and yield less-biased results. In this article, a new method is proposed to solve the robust data reconciliation problem of nonlinear chemical process. By using several technologies including linearization method, penalty function, virtual observation equation, and equivalent weights method, the robust data reconciliation problem can be transformed into least squares estimator problem which leads to the convenience in computation. Simulation results in a nonlinear chemical process demonstrate the efficiency of the proposed method.