Error Covariance Matrix Research Articles

Binary observation‐based identification for finite impulse response systems under denial of service attacks

SummaryThis article considers the parameter identification problem for the finite impulse response system based on binary observations under denial of service attacks. First, in addition to designing the identification algorithm for the unknown parameter, and its convergence is verified, simultaneously obtaining the asymptotic normality. Then, based on the convergence speed of the identification algorithm and the covariance matrix of the estimation error, the optimal attack strategy problem is transformed into an optimization problem with constraints, then the optimal solution is given. Furthermore, a defense strategy with dual time‐scale input design method is proposed and its effectiveness is demonstrated. Finally, numerical simulations are applied to evidence the correctness of the raised method.

Greedy selection of sensors with measurements under correlated noise

We address the sensor selection problem where linear measurements under correlated noise are gathered at the selected nodes to estimate the unknown parameter. Since finding the best subset of sensor nodes that minimizes the estimation error requires a prohibitive computational cost especially for a large number of nodes, we propose a greedy selection algorithm that uses the log-determinant of the inverse estimation error covariance matrix as the metric to be maximized. We further manipulate the metric by employing the QR and LU factorizations to derive a simple analytic rule which enables an efficient selection of one node at each iteration in a greedy manner. We also make a complexity analysis of the proposed algorithm and compare with different selection methods, leading to a competitive complexity of the proposed algorithm. For performance evaluation, we conduct numerical experiments using randomly generated measurements under correlated noise and demonstrate that the proposed algorithm achieves a good estimation accuracy with a reasonable selection complexity as compared with the previous novel selection methods.

Characterization of the Ionospheric Vertical Error Correlation Lengths Based on Global Ionosonde Observations

AbstractData assimilation is one of the most important approaches to monitoring the variations of ionospheric electron densities. The construction of the background error covariance matrix is an important component of ionospheric data assimilations. To construct the background error covariance matrix, the information about the spatial ionospheric correlations is required. We present a statistical analysis on the ionospheric vertical error correlation length (VCL) based on a global network of ionosondes and the Neustrelitz Electron Density Model. We show that the locally derived VCL is well‐defined and the VCL does not show a considerable dependency on the geographical seasons while local time dependencies of the VCL are shown to be present. A novel VCL model is also established based on the ionospheric scale heights. We show that the ionospheric VCL can be characterized by the variance ratio between the ionosphere model and ionospheric measurements. The altitudinal variations of VCLs are controlled by the interactions between the inherent VCLs of the ionosphere model and the measurements. Two experiments are conducted at two different latitudes based on the proposed model. The results show that the proposed model is stable and well‐correlated with the observed VCLs, which implies a potential to be generalized for a global correlation model. The proposed model can be used in the temporal evolution of error covariance matrices in the ionospheric 4D‐Variational (4D‐Var) assimilations, which may overcome the main drawbacks of the static error covariance specifications in the ionospheric 4D‐Var assimilations.

Optimal design and performance analysis of secure estimator for cyber‐physical systems against deception attacks

AbstractCyber‐physical systems (CPSs) are increasingly vulnerable to the threat from malicious adversaries owing to the deep integration of networks and facilities; therefore, CPSs' security against cyber attacks has attracted extensive attention. The deception attack is one of the typical cyber attacks targeting CPSs, and a deception attack mode is established to describe the simultaneous occurrence of actuator and sensor attacks in this paper. Compared with single‐target deception attacks with constant value, the designed attack mode that can describe deception attacks against multiple components and embedded random characteristics has higher applicability in practical engineering processes. To counter such attack behavior, a secure estimator is proposed based on the ideology of augmented state to achieve the optimal design of secure state estimation and deception attack identification. Taking the energy shortage problem of CPSs into account, a novel event‐triggered communication mechanism is introduced on the basis of measurement residuals to improve resource efficiency by reducing low‐value information transmission. The communication scheme and the augmented state estimator are co‐designed by adopting the implicit information of the event‐triggered mechanism. Then, the performance of the co‐designed estimator is analyzed by the upper bound of the estimation error covariance matrix. Besides, the reconciliation between estimation performance and communication energy consumption is achieved by the derivation of Pareto equilibrium. Finally, applications of our approaches to a 3D target tracking system are provided to illustrate the effectiveness of the proposed method.

Uniform approximation of common Gaussian process kernels using equispaced Fourier grids

The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels—Matérn and squared exponential—in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Matérn case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.

An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry

All sensing systems have some inherent error. Often, these errors are systematic, and observations taken within a similar region of space and time can have correlated error structure. However, the data from these systems are frequently assumed to have completely independent and uncorrelated error. This work introduces a correlated error model for GNSS reflectometry (GNSS-R) using observations from NASA’s Cyclone Global Navigation Satellite System (CYGNSS). We validate our model against near-simultaneous observations between two CYGNSS satellites and double-difference our results with modeled observables to extract the correlated error structure due to the observing system itself. Our results are useful to catalog for future GNSS-R missions and can be applied to construct an error covariance matrix for weather data assimilation.

A Novel Tracking Algorithm Based on Waveform Selection for Maneuvering Targets in Clutter

The cognition radar utilizes the dynamic waveform configuration to adapt to the environment to improve tracking performance. A novel tracking algorithm based on waveform selection is proposed to address the maneuvering aircraft tracking problem in clutter. Based on the modified current statistical (MCS) model, the modified probabilistic data association filter (MPDAF) is integrated with the square-root cubature Kalman filter (SCKF) to deal with the nonlinear measurement and the clutter. Additionally, the waveform library is established by applying the fractional Fourier transform (FRFT) to a base waveform, and a direct method is proposed to select the optimal waveform to minimize the posterior state error covariance matrix. The simulation results showed that, compared with the filters with the fixed waveform as well as the state-of-the-art algorithms, the proposed algorithm achieved higher estimation precision and lower track loss percentage while maintaining a low computational burden.

Collaborative Parameter Estimation of Multiple Unmanned Surface Vessels: A Robust Distributed Estimator-Based Approach

In this paper, the collaborative parameter estimation of multiple unmanned surface vessels (USVs) with model structure uncertainties is studied. The considered parameter estimation problem is first converted into a distributed state and parameter joint estimation problem. Then, the robust distributed estimator is constructed to handle the inevitable model structure uncertainties, and the upper bounds of prediction and estimation error covariance matrices are derived respectively. Additionally, the upper bound of the estimation error covariance matrix is minimized by designing appropriate estimator gains. Finally, the advantages of the proposed distributed parameter estimation approach from the perspectives of information interaction and consideration of model structure uncertainties are verified via simulations and practical experiments.

Event-Triggered State Filter Estimation for Nonlinear Systems with Packet Dropout and Correlated Noise

This paper begins by exploring the challenge of event-triggered state estimations in nonlinear systems, grappling with packet dropout and correlated noise. A communication mechanism is introduced that determines whether to transmit measurement values based on whether event-triggered conditions are violated, thereby minimizing redundant communication data. In designing the filter, noise decorrelation is initially conducted, followed by the integration of the event-triggered mechanism and the unreliable network transmission system for state estimator development. Subsequently, by combining the three-degree spherical–radial cubature rule, the numerical implementation steps of the proposed state estimation framework are outlined. The performance estimation analysis highlights that by adjusting the event-triggered threshold appropriately, the estimation performance and transmission rate can be effectively balanced. It is established that when there is a lower bound on the packet dropout rate, the covariance matrix of the state estimation error remains bounded, and the stochastic stability of the state estimation error is also confirmed. Ultimately, the algorithm and conclusions that are proposed in this paper are validated through a simulation example of a target tracking system.

How Not to Make the Joint Extended Kalman Filter Fail with Unstructured Mechanistic Models.

The unstructured mechanistic model (UMM) allows for modeling the macro-scale of a phenomenon without known mechanisms. This is extremely useful in biomanufacturing because using the UMM for the joint estimation of states and parameters with an extended Kalman filter (JEKF) can enable the real-time monitoring of bioprocesses with unknown mechanisms. However, the UMM commonly used in biomanufacturing contains ordinary differential equations (ODEs) with unshared parameters, weak variables, and weak terms. When such a UMM is coupled with an initial state error covariance matrix P(t=0) and a process error covariance matrix Q with uncorrelated elements, along with just one measured state variable, the joint extended Kalman filter (JEKF) fails to estimate the unshared parameters and state simultaneously. This is because the Kalman gain corresponding to the unshared parameter remains constant and equal to zero. In this work, we formally describe this failure case, present the proof of JEKF failure, and propose an approach called SANTO to side-step this failure case. The SANTO approach consists of adding a quantity to the state error covariance between the measured state variable and unshared parameter in the initial P(t = 0) of the matrix Ricatti differential equation to compute the predicted error covariance matrix of the state and prevent the Kalman gain from being zero. Our empirical evaluations using synthetic and real datasets reveal significant improvements: SANTO achieved a reduction in root-mean-square percentage error (RMSPE) of up to approximately 17% compared to the classical JEKF, indicating a substantial enhancement in estimation accuracy.

Combing dynamic parameter identification and singular value decomposition adaptive unscented Kalman filter for predictive state of charge of lithium-ion batteries

The development of a secure battery management system (BMS) for electric vehicles depends heavily on the correct assessment of the online state-of-charge (SOC) of Li-ion batteries. The ternary lithium battery is used as the research object in this paper, and a second-order RC equivalent circuit model is developed to characterize the dynamic operating characteristics of the battery. In order to solve the problem that the adaptive unscented Kalman filter (AUKF) algorithm is easy to fail SOC estimation because the error covariance matrix is not positively definite due to the incomplete accuracy of the equivalent circuit model, a corresponding solution is proposed. Considering the poor real-time battery SOC estimate caused by the battery model’s fixed parameters, therefore we propose the Variable Forgetting Factor Recursive Least Squares (VFFRLS) algorithm for joint estimation of Li-battery SOC and the Singular Value Decomposition-AUKF (SVD-AUKF) algorithm. The SVD-AUKF algorithm can accurately estimate the SOC of the battery when the error covariance is negative. The algorithm can be adaptively adjusted in both the parameter identification and SOC estimation stages, which can effectively solve the problem of poor estimation accuracy caused by fixed parameters. According to experiments, under two separate dynamic operating situations, the joint estimation algorithm’s error is less than 2%, and its stability has also been greatly enhanced. At the same time, when the initial SOC value is set incorrectly, the convergence time of the algorithm proposed in this paper can reach within 2.1 seconds for BBDST and DST conditions, which can be well adapted to complex working conditions.

Optimal reconciliation of hierarchical wind energy forecasts utilizing temporal correlation

Independent wind energy forecasts of a wind farm at different time horizons have limited accuracy, and they show disagreement despite relating to the same wind farm. The limited forecast accuracy is attributable to the insufficient information at a particular time horizon of the wind energy time series, whereas applying distinct forecasting methods to several time series of non-identical patterns at different time scales causes disagreement among forecasts. Mutual disagreement among less accurate forecasts negatively impacts the decision-making capabilities in associated power systems activities at distinct time scales. The configuration of time series expressing different time horizons at different levels of a non-overlapped hierarchically aggregated framework manifests a temporal hierarchy. Forecast combination through reconciliation of time series forecasts drawn at different hierarchical levels of temporal hierarchy using any state-of-the-art method facilitates the sharing of diverse information across the hierarchy; consequently, accuracy and mutual agreement of forecasts improve. Such benefits may be further enhanced by embedding intra- and inter-level forecast error correlations in the forecast reconciliation process. However, the forecast error covariance matrix of temporal hierarchy becomes a complex high-dimensional structure while accommodating intra- and inter-level correlations. Estimating such a matrix is challenging since the high-dimensional structure severely impedes the identifiability of model parameters. Besides, in the hierarchical forecast reconciliation process, the number of predictor variables is generally higher than the number of samples. This condition gives rise to a singular covariance matrix, making it non-invertible, and thus obstructs its parameter estimation. This work employs the MinT(shrinkage) covariance matrix estimator that considers all correlations and shrinks the non-diagonal components of the matrix toward zero to avert the complexity and, therefore, the non-identifiability. Additionally, the shrinkage parameter λ of MinT(shrinkage) conveniently obtains the invertible matrix. The case study validates that while incorporating the intra- and inter-level forecast error correlations, MinT(shrinkage) provides competitively accurate and mutually agreed forecasts over other reconciliation methods.

Joint regression analysis of multiple traits based on genetic relationships.

Polygenic scores (PGSs) are widely available and employed in genomic data analyses for predicting and understanding genetic architectures. Existing approaches either require information on SNP level, do not infer clusters of traits sharing genetic characteristic, or do not have any immediate predictive properties. Here, we present geneJAM, which is a novel clustering and estimation method using PGSs for inferring a genetic relationship among multiple, simultaneously measured and potentially correlated traits in a multivariate GWAS.Using graphical lasso, we estimate a sparse covariance matrix of the PGSs and obtain clusters of traits sharing genetic characteristics. We use the clusters to specify the structure of the error covariance matrix of a generalized least squares (GLS) model and use the feasible GLS estimator for estimating a linear regression model with a certain unknown degree of correlation between the residuals.The method suits many biology studies well with traits embedded in some genetic functioning groups and facilitates development of the PGS research. We compare the method with fully parametric techniques on simulated data and illustrate the utility of the methods by examining a heterogeneous stock mouse data set from the Wellcome Trust Centre for Human Genetics. We demonstrate that the method successfully identifies clusters of traits and increases precision, power, and computational efficiency. GeneJAM is implemented in R and available at: https://github.com/abuchardt/geneJAM.

Estimation of road adhesion coefficient and sideslip angle for electric vehicles with slip-aware constraints using strong tracking unscented Kalman filter

Accurate estimation of road adhesion coefficient and sideslip angle are crucial for safer driving and ride quality of electric vehicles (EVs). To that end, an estimation algorithm of road adhesion coefficient and sideslip angle for EVs with slip-aware constraints and strong tracking unscented Kalman filter (SSTUKF) algorithm is designed in this paper. First, a vehicle system model is established as the basis for deriving the desirable estimation algorithm. Then, a strong tracking unscented Kalman filter (STUKF) algorithm with fading factor is presented to strengthen the correction effect of error covariance matrix used in unscented Kalman filter (UKF) algorithm. Meanwhile, slip-aware constraints are considered to calculate the adaptive compensated steering angle, which is obtained by repeated iteration of the fitness function. Additionally, a data fusion estimator is established based on the extremum theory to solve the problem of inaccurate estimation of sideslip angle under complex driving conditions. Finally, the CarSim-MATLAB/Simulink simulation platform is used to simulate under different driving conditions, and the simulation results reveal that the presented SSTUKF algorithm can enhance the estimation performance.

Including the Horizontal Observation Error Correlation in the Ensemble Kalman Filter: Idealized Experiments with NICAM-LETKF

Abstract Densely observed remote sensing data such as radars and satellites generally contain significant spatial error correlations. In data assimilation, the observation error covariance matrix is usually assumed to be diagonal, and the dense data are thinned or spatially averaged to compensate for neglecting the spatial observation error correlation. However, in theory, including the spatial observation error correlation in data assimilation can make better use of the dense data. This study performs perfect model observing system simulation experiments (OSSEs) using the nonhydrostatic icosahedral atmospheric model (NICAM) and the local ensemble transform Kalman filter (LETKF) to assess the impact of assimilating horizontally dense and error-correlated observations. The condition number of the observation error covariance matrix, defined as the ratio of the largest to smallest eigenvalues, is important for the numerical stability of the LETKF computation. A large condition number makes it difficult to compute the ensemble transform matrix correctly. Reducing the condition number by reconditioning is found effective for stable computation. The results show that including the horizontal observation error correlation with reconditioning makes the analysis more accurate but requires 6 times more computations than the case with the diagonal observation error covariance matrix. Significance Statement It is important to effectively utilize observations in data assimilation for more accurate weather prediction. Spatially dense observations are known to have an error correlation that is ignored in the data assimilation. This study explores assimilating dense observations by explicitly including observation error correlations with an idealized experiment. The results shows that the analysis is improved by including the observation error correlations. Also, the condition number of the observation error covariance matrix is essential for stable computations.

A Variational Bayesian Truncated Adaptive Filter for Uncertain Systems with Inequality Constraints

In this paper, a variational Bayesian (VB) truncated adaptive filter for uncertain systems with inequality constraints is proposed. By choosing the skew‐t and inverse Wishart distributions as the prior information of the measurement noise and predicted error covariance matrix, the state vector, the predicted error covariance matrix, and noise parameters are inferred and approximated by using the VB method. To achieve the inequality‐constrained estimation, the constrained state is computed by truncating the probability density function (PDF) of the estimated state after the variational update stage; the mean and covariance of the constrained state are the first and second moments of the truncated PDF. Considering the model uncertainties where the system dynamics are unpredictable, a multiple model VB truncated adaptive filter is proposed in the interacting multiple model framework. The performances of the proposed algorithms are evaluated via the target tracking simulations and the robot positioning experiments. Results show that the proposed algorithms improve estimation accuracy compared with the existing adaptive filters when the states suffer inequality constraints.

Estimation-free Prediction Algorithms

For Time-varying, Time-invariant, and steady-state systems, Kalman Filter can be implemented as a prediction algorithm, since it produces the state prediction and the corresponding prediction error covariance matrix via the state estimation and the corresponding estimation error covariance matrix. Lainiotis Filter is equivalent to Kalman Filter and can be used to compute the prediction. In this paper, for Time-varying, Time-invariant and steady state systems, estimation-free Prediction Algorithms are derived via Kalman and Lainiotis filters; they are equivalent and compute iteratively the prediction and the corresponding prediction error covariance matrix. The estimation and the corresponding estimation error covariance matrix are not needed and are not computed. The proposed estimation-free prediction algorithms are faster than the Kalman filter.

Binary observation‐based FIR system identification under sequence denial of service attacks

AbstractDue to the close interaction between computing, sensing, communication, and driving of the cyber‐physical system (CPS), which makes it highly vulnerable to cyber attacks, the security of CPS has attracted wide attention. In the article, we discuss the system identification problem of the finite impulse response system under sequence denial of service attacks. The identification algorithm of the unknown parameter is designed under binary observations, and its convergence performance is analyzed. The optimal attack strategy problem is modeled as a optimized problem with constraints based on the covariance matrix of estimation error, the solution method and implementation algorithm are given. From the perspective of the defender, an encryption‐type defense algorithm is designed, and then the optimal defense strategy is presented. The numerical simulation is used to verify the validity of the proposed method.

Comparison of Classical and Robust Factor Analyses Methods

Factor analysis is a multivariate statistical analysis technique that has become very popular in recent years. In the factor analysis model, the error covariance matrix is assumed to be the multivariate normal distribution, and outliers are likely to be accounted for. Various estimation methods were compared with Monte Carlo simulation for the factor analysis model. The performances of the estimation methods were evaluated based on the ratio of the total variance explained and the criterion fit values. Considering the MLE, PCA, WLS, and GLS methods for classical factor analysis and the MCD, M, and S methods for robust factor analysis, the ratio of total variance explained, and fit values decreased as the sample size increased. When the number of variables increases, the ratio of total variance explained, and fit values increase at different sample sizes. It can be said that the WLS and GLS methods are better than others for classical factor analysis and the MCD and M methods are better than others for robust factor analysis.

A Space-Time Variational Method for Retrieving Upper-Level Vortex Winds from GOES-16 Rapid Scans over Hurricanes

A space-time variational method is developed for retrieving upper-level vortex winds from geostationary satellite rapid infrared scans over hurricanes. In this method, new vortex-flow-dependent correlation functions are formulated for the radial and tangential components of the vortex wind. These correlation functions are used to construct the background error covariance matrix and its square root matrix. The resulting square root matrix is then employed to precondition the cost function, constrained by an advection equation formulated for rapidly scanned infrared image movements. This newly formulated and preconditioned cost function is more suitable for deriving upper-level vortex winds from GOES-16 rapid infrared scans over hurricanes than the cost function in the recently adopted optical flow technique. The new method was applied to band-13 (10.3 µm) brightness temperature images scanned every min from GOES-16 over Hurricanes Laura on 27 August 2020 and Hurricanes Ida on 29 August 2021. The retrieved vortex winds were shown to not only be much denser than operationally produced atmospheric motion vectors (AMVs) but also more rotational and better organized around the eyewall than the super-high-resolution AMVs derived from optical-flow technique. By comparing their component velocities (projected along radar beams) with limited radar velocity observations available near the cloud top, the vortex winds retrieved using the new method were also shown to be more accurate than the super-high-resolution AMVs derived from the optical-flow technique. The new method is computationally efficient for real-time applications and potentially useful for hurricane wind nowcasts. Furthermore, the combined use of VF-dependent covariance functions and imagery advection equation is not only novel but was also found to be critically important for the improved performance of the method. This finding implies that similar combined approaches can be developed with improved performance for retrieving vortex flows rapidly scanned using other types of remote sensing on different scales, such as tornadic mesocyclones rapidly scanned by phased-array radars.