Data-Based Filters for Non-Gaussian Dynamic Systems With Unknown Output Noise Covariance

This paper proposes linear and nonlinear filters for a non-Gaussian dynamic system with an unknown nominal covariance of the output noise.The challenge of designing a suitable filter in the presence of an unknown covariance matrix is addressed by focusing on the output data set of the system.Considering that data generated from a Gaussian distribution exhibit ellipsoidal scattering,we first propose the weighted sum of norms(SON)clustering method that prioritizes nearby points,reduces distant point influence,and lowers computational cost.Then,by introducing the weighted maximum likelihood,we propose a semi-definite program(SDP)to detect outliers and reduce their impacts on each cluster.Detecting these weights paves the way to obtain an appropriate covariance of the output noise.Next,two filtering approaches are presented:a cluster-based robust linear filter using the maximum a posterior(MAP)estimation and a clusterbased robust nonlinear filter assuming that output noise distribution stems from some Gaussian noise resources according to the ellipsoidal clusters.At last,simulation results demonstrate the effectiveness of our proposed filtering approaches.

CL2ES-KDBC:A Novel Covariance Embedded Selection Based on Kernel Distributed Bayes Classifier for Detection of Cyber-Attacks in IoT Systems

The Internet of Things(IoT)is a growing technology that allows the sharing of data with other devices across wireless networks.Specifically,IoT systems are vulnerable to cyberattacks due to its opennes The proposed work intends to implement a new security framework for detecting the most specific and harmful intrusions in IoT networks.In this framework,a Covariance Linear Learning Embedding Selection(CL2ES)methodology is used at first to extract the features highly associated with the IoT intrusions.Then,the Kernel Distributed Bayes Classifier(KDBC)is created to forecast attacks based on the probability distribution value precisely.In addition,a unique Mongolian Gazellas Optimization(MGO)algorithm is used to optimize the weight value for the learning of the classifier.The effectiveness of the proposed CL2ES-KDBC framework has been assessed using several IoT cyber-attack datasets,The obtained results are then compared with current classification methods regarding accuracy(97%),precision(96.5%),and other factors.Computational analysis of the CL2ES-KDBC system on IoT intrusion datasets is performed,which provides valuable insight into its performance,efficiency,and suitability for securing IoT networks.

Low-Complexity Reconstruction of Covariance Matrix in Hybrid Uniform Circular Array

Spatial covariance matrix(SCM) is essential in many multi-antenna systems such as massive multiple-input multiple-output(MIMO). For multi-antenna systems operating at millimeter-wave bands, hybrid analog-digital structure has been widely adopted to reduce the cost of radio frequency chains.In this situation, signals received at the antennas are unavailable to the digital receiver, and as a consequence, traditional sample average approach cannot be used for SCM reconstruction in hybrid multi-antenna systems. To address this issue, beam sweeping algorithm(BSA) which can reconstruct the SCM effectively for a hybrid uniform linear array, has been proposed in our previous works. However, direct extension of BSA to a hybrid uniform circular array(UCA)will result in a huge computational burden. To this end, a low-complexity approach is proposed in this paper. By exploiting the symmetry features of SCM for the UCA, the number of unknowns can be reduced significantly and thus the complexity of reconstruction can be saved accordingly. Furthermore, an insightful analysis is also presented in this paper, showing that the reduction of the number of unknowns can also improve the accuracy of the reconstructed SCM. Simulation results are also shown to demonstrate the proposed approach.

Optimal Estimation of High-Dimensional Covariance Matrices with Missing and Noisy Data

The estimation of covariance matrices is very important in many fields, such as statistics. In real applications, data are frequently influenced by high dimensions and noise. However, most relevant studies are based on complete data. This paper studies the optimal estimation of high-dimensional covariance matrices based on missing and noisy sample under the norm. First, the model with sub-Gaussian additive noise is presented. The generalized sample covariance is then modified to define a hard thresholding estimator , and the minimax upper bound is derived. After that, the minimax lower bound is derived, and it is concluded that the estimator presented in this article is rate-optimal. Finally, numerical simulation analysis is performed. The result shows that for missing samples with sub-Gaussian noise, if the true covariance matrix is sparse, the hard thresholding estimator outperforms the traditional estimate method.

Empirical Likelihood Statistical Inference for Compound Poisson Vector Processes under Infinite Covariance Matrix

The paper discusses the statistical inference problem of the compound Poisson vector process(CPVP)in the domain of attraction of normal law but with infinite covariance matrix.The empirical likelihood(EL)method to construct confidence regions for the mean vector has been proposed.It is a generalization from the finite second-order moments to the infinite second-order moments in the domain of attraction of normal law.The log-empirical likelihood ratio statistic for the average number of the CPVP converges to F distribution in distribution when the population is in the domain of attraction of normal law but has infinite covariance matrix.Some simulation results are proposed to illustrate the method of the paper.

Evaluation of Multiplicative Weight of Covariance Matrix on Hybrid Data Assimilation Schemes

This research develops a comparative study between different multiplicative weights that are assigned to the covariance matrix that represents the background error in two hybrid assimilation schemes: 3DEnVAR and 4DEnVAR. These weights are distributed between the static and time-invariant matrix and the matrix generated from the perturbations of a previous ensemble. The assigned values are 25%, 50%, and 75%, always having as a reference the ensemble matrix. The experiments are applied to the short-range Prediction System (SisPI) that works operationally at the Institute of Meteorology. The impact of Tropical Storm Eta on November 7 and 8, 2020 was selected as a study case. The results suggest that by giving the main weight to the ensemble matrix more realistic solutions are achieved because it shows a better representation of the synoptic flow. On the other hand, it is observed that 3DEnVAR method is more sensitive to multiplicative weight change of the first guess. More realistic results are obtained with 50% and 75% relations with 4DEnVAR method, whereas with 3DEnVAR a weight of 75% for the ensemble matrix is required.

Sensor management of LEO constellation based on covariance control

This paper studies the multi-sensor management problem for low earth orbit(LEO) infrared warning constellation used to track a midcourse missile. A covariance control approach, which selects sensor combinations or subset based on the difference between the desired covariance matrix and the actual covariance of each target, is used for sensor management, including some matrix metrics to measure the differentia between two covariance matrices. Besides, to meet the requirements of the space based warning system, the original covariance control approach is improved. Simulation results demonstrate that the covariance control approach is able to provide a better tracking performance by providing a well-designed desired covariance and balance tracking performance goals with system demands.

Reservoir history matching and inversion using an iterative ensemble Kalman filter with covariance localization

Reservoir inversion by production history matching is an important way to decrease the uncertainty of the reservoir description. Ensemble Kalman filter （EnKF） is a new data assimilation method. There are two problems have to be solved for the standard EnKF. One is the inconsistency between the updated model and the updated dynamical variables for nonlinear problems, another is the filter divergence caused by the small ensemble size. We improved the EnKF to overcome these two problems. We use the half iterative EnKF （HIEnKF） for reservoir inversion by doing history matching. During the H1EnKF process, the prediction data are obtained by rerunning the reservoir simulator using the updated model. This can guarantee that the updated dynamical variables are consistent with the updated model. The updated model can nonlinearly affect the prediction data. It is proved that HIEnKF is similar to the first iteration of the EnRML method. Covariance localization is introduced to alleviate filter divergence and spurious correlations caused by the small ensemble size. By defining the shape and size of the correlation area, spurious correlation between the gridblocks far apart is alleviated. More freedom of the model ensemble is preserved. The results of history matching and inverse problem obtained from the HIEnKF with covariance localization are improved. The results show that the model freedom increases with a decrease in the correlation length. Therefore the production data can be matched better. But too small a correlation length can lose some reservoir information and this would cause big errors in the reservoir model estimation.

Eddy Covariance Tilt Corrections over a Coastal Mountain Area in South-east China:Significance for Near-Surface Turbulence Characteristics

Turbulence characteristics of an atmospheric surface layer over a coastal mountain area were investigated under different coordinate frames. Performances of three methods of coordinate rotation： double rotation （DR）, triple rotation （TR）, and classic planar-fit rotation （PF） were examined in terms of correction of eddy covariance flux. Using the commonly used DR and TR methods, unreasonable rotation angles are encountered at low wind speeds and cause significant run-to-run errors of some turbulence characteristics. The PF method rotates the coordinate system to an ensemble-averaged plane, and shows large tilt error due to an inaccurate fit plane over variable terrain slopes. In this paper, we propose another coordinate rotation scheme. The observational data were separated into two groups according to wind direction. The PF method was adapted to find an ensemble-averaged streamline plane for each group of hourly runs with wind speed exceeding 1.0 m s-1. Then, the coordinate systems were rotated to their respective best- fit planes for all available hourly observations. We call this the PF10 method. The implications of tilt corrections for the turbulence characteristics are discussed with a focus on integral turbulence characteristics, the spectra of wind-velocity components, and sensible heat and momentum fluxes under various atmospheric stabilities. Our results show that the adapted application of PF provides greatly improved estimates of integral turbulence characteristics in complex terrain and maintains data quality. The comparisons of the sensible heat fluxes for four coordinate rotation methods to fluxes before correction indicate that the PF10 scheme is the best to preserve consistency between fluxes.

ON THE OPTIMAL BACKGROUND ERROR COVARIANCES: DIFFERENT SCALE ERRORS' CONTRIBUTION

The large-scale and small-scale errors could affect background error covariances for a regional numerical model with the specified grid resolution.Based on the different background error covariances influenced by different scale errors,this study tries to construct a so-called"optimal background error covariances"to consider the interactions among different scale errors.For this purpose,a linear combination of the forecast differences influenced by information of errors at different scales is used to construct the new forecast differences for estimating optimal background error covariances.By adjusting the relative weight of the forecast differences influenced by information of smaller-scale errors,the relative influence of different scale errors on optimal background error covariances can be changed.For a heavy rainfall case,the corresponding optimal background error covariances can be estimated through choosing proper weighting factor for forecast differences influenced by information of smaller-scale errors.The data assimilation and forecast with these optimal covariances show that,the corresponding analyses and forecasts can lead to superior quality,compared with those using covariances that just introduce influences of larger-or smallerscale errors.Due to the interactions among different scale errors included in optimal background error covariances,relevant analysis increments can properly describe weather systems(processes)at different scales,such as dynamic lifting,thermodynamic instability and advection of moisture at large scale,high-level and low-level jet at synoptic scale,and convective systems at mesoscale and small scale,as well as their interactions.As a result,the corresponding forecasts can be improved.

Eddy covariance measurements of water vapor and energy flux over a lake in the Badain Jaran Desert,China

Exploring the surface energy exchange between atmosphere and water bodies is essential to gain a quantitative understanding of regional climate change, especially for the lakes in the desert. In this study, measurements of energy flux and water vapor were performed over a lake in the Badain Jaran Desert, China from March 2012 to March 2013. The studied lake had about a 2-month frozen period （December and January） and a 10-month open-water period （February-November）. Latent heat flux （LE） and sensible heat flux （Hs） acquired using the eddy covariance technique were argued by measurements of long＇wave and shortwave radiation. Both fluxes of longwave and shortwave radiation showed seasonal dynamics and daily fluctuations during the study period. The reflected solar radiation was much higher in winter than in other seasons. LE exhibited diurnal and seasonal variations. On a daily scale, LE was low in the morning and peaked in the afternoon. From spring （April） to winter （January）, the diurnal amplitude of LE decreased slowly. LE was the dominant heat flux throughout the year and consumed most of the energy from the lake. Generally speaking, LE was mostly affected by changes in the ambient wind speed, while Hs was primarily affected by the product of water-air temperature difference and wind speed. The diurnal LE and Hs were negatively correlated in the open-water period. The variations in Hs and LE over the lake were differed from those on the nearby land surface. The mean evaporation rate on the lake was about 4.0 mm/d over the entire year, and the cumulative annual evaporation rate was 1445 mm/a. The cumulative annual evaporation was 10 times larger than the cumulative annual precipitation. Furthermore, the average evaporation rates over the frozen period and open-water period were approximately 0.6 and 5.0 mm/d, respectively. These results can be used to analyze the water balance and quantify the source of lake water in the Badain Jaran Desert.

Ocean Data Assimilation with Background Error Covariance Derived from OGCM Outputs

The background error covariance plays an important role in modern data assimilation and analysis systems by determining the spatial spreading of information in the data. A novel method based on model output is proposed to estimate background error covariance for use in Optimum Interpolation. At every model level, anisotropic correlation scales are obtained that give a more detailed description of the spatial correlation structure. Furthermore, the impact of the background field itself is included in the background error covariance. The methodology of the estimation is presented and the structure of the covariance is examined. The results of 20-year assimilation experiments are compared with observations from TOGA-TAO (The Tropical Ocean-Global Atmosphere-Tropical Atmosphere Ocean) array and other analysis data.

Using Analysis State to Construct a Forecast Error Covariance Matrix in Ensemble Kalman Filter Assimilation

Correctly estimating the forecast error covariance matrix is a key step in any data assimilation scheme. If it is not correctly estimated, the assimilated states could be far from the true states. A popular method to address this problem is error covariance matrix inflation. That is, to multiply the forecast error covariance matrix by an appropriate factor. In this paper, analysis states are used to construct the forecast error covariance matrix and an adaptive estimation procedure associated with the error covariance matrix inflation technique is developed. The proposed assimilation scheme was tested on the Lorenz-96 model and 2D Shallow Water Equation model, both of which are associated with spatially correlated observational systems. The experiments showed that by introducing the proposed structure of the forecast error eovariance matrix and applying its adaptive estimation procedure, the assimilation results were further improved.

Representations of Inverse Covariances by Differential Operators

In the cost function of three- or four-dimensional variational dataassimilation, each term is weighted by the inverse of its associated error covariance matrix and thebackground error covariance matrix is usually much larger than the other covariance matrices.Although the background error covariances are traditionally normalized and parameterized by simplesmooth homogeneous correlation functions, the covariance matrices constructed from these correlationfunctions are often too large to be inverted or even manipulated. It is thus desirable to finddirect representations of the inverses of background error correlations. This problem is studied inthis paper. In particular, it is shown that the background term can be written into ∫ dx∣Dυ(x)∣~2, that is, a squared 1/2 norm of a vector differential operator D, called theD-operator, applied to the field of analysis increment υ(x). For autoregressive correlationfunctions, the D-operators are of finite orders. For Gaussian correlation functions, the D-operatorsare of infinite order. For practical applications, the Gaussian D-operators must be truncated tofinite orders. The truncation errors are found to be small even when the Gaussian D-operators aretruncated to low orders. With a truncated D-operator, the background term can be easily constructedwith neither inversion nor direct calculation of the covariance matrix. D-operators are also derivedfor non-Gaussian correlations and transformed into non-isotropic forms.

Fast and accurate covariance matrix reconstruction for adaptive beamforming using Gauss-Legendre quadrature

Most of the reconstruction-based robust adaptive beamforming(RAB)algorithms require the covariance matrix reconstruction(CMR)by high-complexity integral computation.A Gauss-Legendre quadrature(GLQ)method with the highest algebraic precision in the interpolation-type quadrature is proposed to reduce the complexity.The interference angular sector in RAB is regarded as the GLQ integral range,and the zeros of the threeorder Legendre orthogonal polynomial is selected as the GLQ nodes.Consequently,the CMR can be efficiently obtained by simple summation with respect to the three GLQ nodes without integral.The new method has significantly reduced the complexity as compared to most state-of-the-art reconstruction-based RAB techniques,and it is able to provide the similar performance close to the optimal.These advantages are verified by numerical simulations.

Underdetermined DOA estimation via multiple time-delay covariance matrices and deep residual network

Higher-order statistics based approaches and signal sparseness based approaches have emerged in recent decades to resolve the underdetermined direction-of-arrival(DOA)estimation problem.These model-based methods face great challenges in practical applications due to high computational complexity and dependence on ideal assumptions.This paper presents an effective DOA estimation approach based on a deep residual network(DRN)for the underdetermined case.We first extract an input feature from a new matrix calculated by stacking several covariance matrices corresponding to different time delays.We then provide the input feature to the trained DRN to construct the super resolution spectrum.The DRN learns the mapping relationship between the input feature and the spatial spectrum by training.The proposed approach is superior to existing model-based estimation methods in terms of calculation efficiency,independence of source sparseness and adaptive capacity to non-ideal conditions(e.g.,low signal to noise ratio,short bit sequence).Simulations demonstrate the validity and strong performance of the proposed algorithm on both overdetermined and underdetermined cases.

ASSESSMENT OF LOCAL INFLUENCE IN A GROWTH CURVE MODEL WITH RAO'S SIMPLE COVARIANCE STRUCTURE

The present paper deals with the problem of assessing the local influence in a growth curve model with Rao's simple covariance structure. Based on the likelihood displacement,the curvature measure is employed to evaluate the effects of some minor perturbations on the statistical inference, thus leading to the large curvature direction, which is the most critical diagnostic statistic in the context of the local influence analysis. As an application, the common covariance-weighted perturbation scheme is thoroughly considered.

Geometric Covariance Modeling for Surface Variation of Compliant Parts Based on Hybrid Polynomial Approximation and Spectrum Analysis

Part variation characterization is essential to analyze the variation propagation in flexible assemblies. Aiming at two governing types of surface variation,warping and waviness,a comprehensive approach of geometric covariance modeling based on hybrid polynomial approximation and spectrum analysis is proposed,which can formulate the level and the correlation of surface variations accurately. Firstly,the form error data of compliant part is acquired by CMM. Thereafter,a Fourier-Legendre polynomial decomposition is conducted and the error data are approximated by a Legendre polynomial series. The weighting coefficient of each component is decided by least square method for extracting the warping from the surface variation. Consequently,a geometrical covariance expression for warping deformation is established. Secondly,a Fourier-sinusoidal decomposition is utilized to approximate the waviness from the residual error data. The spectrum is analyzed is to identify the frequency and the amplitude of error data. Thus,a geometrical covariance expression for the waviness is deduced. Thirdly,a comprehensive geometric covariance model for surface variation is developed by the combination the Legendre polynomials with the sinusoidal polynomials. Finally,a group of L-shape sheet metals is measured along a specific contour,and the covariance of the profile errors is modeled by the proposed method. Thereafter,the result is compared with the covariance from two other methods and the real data. The result shows that the proposed covariance model can match the real surface error effectively and represents a tighter approximation error compared with the referred methods.

Model Selection in Estimation of Covariance Functions for Growth of Angora Goats

Covariance functions have been proposed as an alternative to model longitudinal data in animal breeding because of their various merits in comparison to the classical analytical methods.In practical estimation,different models and polynomial orders fitted can influence the estimates of covariance functions and thus genetic parameters.The objective of this study was to select model for estimation of covariance functions for body weights of Angora goats at 7 time points.Covariance functions were estimated by fitting 6 random regression models with birth year,birth month,sex,age of dam,birth type,and relative birth date as fixed effects.Random effects involved were direct and maternal additive genetic,and animal and maternal permanent environmental effects with different orders of fit.Selection of model and orders of fit were carried out by likelihood ratio test and 4 types of information criteria.The results showed that model with 6 orders of polynomial fit for direct additive genetic and animal permanent environmental effects and 4 and 5 orders for maternal genetic and permanent environmental effects,respectively,were preferable for estimation of covariance functions.Models with and without maternal effects influenced the estimates of covariance functions greatly.Maternal permanent environmental effect does not explain the variation of all permanent environments,well suggesting different sources of permanent environmental effects also has large influence on covariance function estimates.

Uncertainty Principle for the Quaternion Linear Canonical Transform in Terms of Covariance

An uncertainty principle(UP),which offers information about a signal and its Fourier transform in the time-frequency plane,is particularly powerful in mathematics,physics and signal processing community.Under the polar coordinate form of quaternion-valued signals,the UP of the two-sided quaternion linear canonical transform(QLCT)is strengthened in terms of covariance.The condition giving rise to the equal relation of the derived result is obtained as well.The novel UP with covariance can be regarded as one in a tighter form related to the QLCT.It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.