Covariance Matrix Estimation Method Research Articles

Robust speckle covariance matrix estimation of sea clutter based on spectral symmetry

Speckle covariance matrix estimation plays an important role in adaptive target detection of maritime radars. Spatial heterogeneity of sea clutter incurs insufficient secondary data in estimation, which degrades detection performance. It is an effective way of estimation improvement to exploit the Doppler spectrum knowledge of sea clutter for structural dimension reduction of the speckle covariance matrix. This paper proposes a speckle covariance matrix estimation method based on the spectral symmetry knowledge of sea clutter, indicating that the spectrum of sea clutter is symmetric with respect to the Doppler offset. First of all, the spectral symmetry of sea clutter is analyzed by using the measured data from several opened databases and the results show that most of the data excluding sea spikes have symmetric Doppler spectra. Secondly, the spectral symmetry constrains the speckle covariance matrix into the Hadamard product of a special rank-one Toeplitz matrix and a real positive definite symmetric matrix. The degrees of freedom are reduced to almost half of that of a Hermitian matrix and thus fewer secondary data are required. Thirdly, a robust iterative maximum likelihood estimator is proposed to estimate the speckle covariance matrix with structural constraint and is free of the clutter texture distribution. The performance of the estimator is verified by the measured data and the results show that it is superior to traditional estimators, particularly in fewer secondary data.

Shrinkage and thresholding approaches for expected utility portfolios: An analysis in terms of predictive ability

In this paper, we estimate Expected Utility Portfolios (EUPs) in high-dimensional, low-sample settings using various covariance matrix estimation methods, including shrinkage and thresholding-based methods. We perform synthetic experiments comparing these methods, using Average Out-of-Sample Variance (AOV) for Global Minimum Variance (GMV) portfolios and Average Out-of-Sample Utility (AOU) for EUPs. Additionally, we propose a practical method for fund managers to select optimal models based on empirical data, relying on AOV and AOU performance measures. The results indicate that shrinkage-based methods outperform thresholding-based ones in high-dimensional settings, with non-linear shrinkage being particularly effective.

Clutter Covariance Matrix Estimation for Radar Adaptive Detection Based on a Complex-Valued Convolutional Neural Network

In this paper, we address the problem of covariance matrix estimation for radar adaptive detection under non-Gaussian clutter. Traditional model-based estimators may suffer from performance loss due to the mismatch between real data and assumed models. Therefore, we resort to a data-driven deep-learning method and propose a covariance matrix estimation method based on a complex-valued convolutional neural network (CV-CNN). Moreover, a real-valued (RV) network with the same framework as the proposed CV network is also constructed to serve as a natural competitor. The obtained clutter covariance matrix estimation based on the network is applied to the adaptive normalized matched filter (ANMF) detector for performance assessment. The detection results via both simulated and real sea clutter illustrate that the estimator based on CV-CNN outperforms other traditional model-based estimators as well as its RV competitor in terms of probability of detection (PD).

Covariance matrix estimation method of FDA-MIMO radar with low snapshot size

Frequency diverse array and multi-input and multi-output (FDA-MIMO) radar have been applied to many fields due to its angle-range-dependent beampattern. A part of the related array processing algorithms can be successfully used on the basis of the known covariance matrix (CM). The estimation accuracy of the CM directly influences the algorithm performance. In particular, the estimation performance of the sample covariance matrix (SCM) will be degraded shapely once the sample size is less than the channel number. Aiming to improve the estimation accuracy of the SCM with FDA-MIMO radar, we propose a novel shrinkage-to-tapering (ST)-based method combined with the block Toeplitz rectification (STT). Firstly, each block matrix in the CM is processed by using the Toeplitz rectification method, and then, three different estimation matrix structures based on the ST method are proposed. Next, by plugging the unbiased estimators into the optimal shrinkage coefficient and normalized mean square error (MSE), their closed forms of their estimators can be given. The numerical simulation results demonstrate that the proposed three different STT approaches are superior to a number of existing methods in terms of CM estimation performance, and the estimation performance is related with the choice of tapering matrix.

A New Statistical Method for Determining the Clutter Covariance Matrix in Spatial-Temporal Adaptive Processing of a Radar Signal.

In this article, a new statistical method for estimating the clutter covariance matrix in space-time adaptive radar signal processing (STAP) is presented and studied. The new method was designed for multiple-input-multiple-output (MIMO) radar with time division multiplexing (TDM). An extensive analysis of statistical and non-statistical methods for estimating the clutter covariance matrix in STAP is presented in this paper. In addition, the STAP algorithm for the standard statistical SMI clutter covariance matrix estimation method, which is based on QR distribution, has been presented. The new method is based on LU distribution with partial pivoting. Simulation results confirm the validity of the presented model and theoretical assumptions. In addition, more accurate object detection results were demonstrated for specific computational examples than for other statistical methods. Considering the current analysis of the literature, it is noted that attention has now been focused worldwide on the study of non-statistical methods for estimating clutter covariance matrices in heterogeneous environments. Hence, it should be emphasized that the posted study fills a gap in current research on STAP.

Rotor Noise-Aware Noise Covariance Matrix Estimation for Unmanned Aerial Vehicle Audition

A noise covariance matrix (NCM) estimation method for unmanned aerial vehicle (UAV) audition is proposed with rotor noise reduction as its primary focus. The proposed NCM estimation method could be incorporated into audio processing algorithms using UAV-mounted microphone array systems. The NCM is formed through accurate estimation of the microphone array input signal's amplitude and phase by using a multi-sensory rotor noise power spectral density (PSD) estimator and a filter formed by exploiting the acoustical relationship between the microphone array and the rotor noise sources, respectively. The estimated NCM aims to be readily incorporable into several source enhancement algorithms to reduce the effects of rotor noise and improve the resultant audio quality. Experiment evaluation using real in-flight UAV rotor noise recordings shows that the estimated NCM significantly improves rotor noise reduction (of up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sim 28$</tex-math></inline-formula> dB) and the quality of the target sound.

Fingerprint-Based Covariance Matrix Estimation for Cell-Free Distributed Massive MIMO Systems

Channel covariance matrix, or large-scale channel state information, is usually assumed to be perfectly known, which is not practical due to the quickly changing environment. This letter proposes a highly accurate channel covariance matrix estimation method based on fingerprint-based localization. In initial pilot assignment stage, we eliminate the impact of inference and noise by introducing random phase shift to estimate channel covariance matrices, which are regarded as fingerprints for estimating locations. Users are clustered based on locations and pilots are reassigned accordingly to improve estimation accuracy. Simulation results show that the proposed method achieves higher covariance matrix estimation accuracy.

A Novel Clutter Covariance Matrix Estimation Method Based on Feature Subspace for Space-Based Early Warning Radar

Accurate estimation of the clutter covariance matrix for the cell under test (CUT) is a committed step in the spatial-temporal adaptive processing (STAP) algorithm. The unique nonstationary characteristic of signal for space-based early warning radar (SBEWR) leads to the spatial variation of training sample and the insufficient number of optional independent identically distributed (i.i.d.) training samples, which brings difficulties to training sample selection and covariance matrix estimation. To improve the estimation accuracy of clutter covariance matrix and the performance of STAP for SBEWR in a heterogeneous environment, a novel training sample selection and clutter covariance matrix estimation method is proposed. The method based on clutter subspace reconstruction and spectrum correction technology can improve the estimation accuracy of clutter covariance matrix in the case of nonstationary signals and heterogeneous environments. The clutter covariance matrix estimated by the proposed method is similar to the clutter covariance matrix of the CUT, and the performance of STAP is improved. The experimental results confirm the performance of the proposed method.

Random matrix improved covariance estimation for a large class of metrics* *This article is an updated version of: Tiomoko M, Couillet R, Bouchard F and Ginolhac G 2019 Random matrix improved covariance estimation for a large class of metrics Proc. Machine Learning Research vol 97 pp 6254–63.

Relying on recent advances in statistical estimation of covariance distances based on random matrix theory, this article proposes an improved covariance and precision matrix estimation method for a wide family of metrics. This method is shown to largely outperform the sample covariance matrix estimate and to compete with state-of-the-art methods, while at the same time being computationally simpler and faster. Applications to linear and quadratic discriminant analyses also show significant gains, therefore suggesting a practical relevance for statistical machine learning.

Multivariate analysis of covariance with potentially singular covariance matrices and non-normal responses

In applied research, it is often sensible to account for one or several covariates when testing for differences between multivariate means of several groups. However, the “classical” parametric multivariate analysis of covariance (MANCOVA) tests (e.g., Wilks’ Lambda) are based on quite restrictive assumptions (homoscedasticity and normality of the errors), which might be difficult to justify. Furthermore, existing potential remedies (e.g., heteroskedasticity-robust approaches) become inappropriate in cases where the covariance matrices are singular or close to singular. Nevertheless, such scenarios are frequently encountered in the life sciences and other fields, when, for example, in the context of standardized assessments, a summary performance measure as well as its corresponding subscales are analyzed. Moreover, computational issues may also lead to singular covariance structures. In the present manuscript, we consider a general MANCOVA model, allowing for potentially heteroskedastic and even singular covariance matrices as well as non-normal errors. We combine heteroskedasticity-consistent covariance matrix estimation methods with our proposed modified MANCOVA ANOVA-type statistic (MANCATS) and apply two different bootstrap approaches. We provide the proofs of the asymptotic validity of the respective testing procedures as well as the results from an extensive simulation study, which indicate that especially the parametric bootstrap version of the MANCATS outperforms its competitors in most small-sample scenarios, both in terms of type I error rates and power. These considerations are further illustrated and substantiated by examining real-life data from standardized achievement tests. Yet, for large sample sizes, Wald-type approaches are an attractive alternative option. Moreover, the choice between Wald-type and MANCATS-based tests might depend on the particular setting (e.g., the covariance structure, or the alternative under consideration).

Comparison of NMC and Ensemble-Based Climatological Background-Error Covariances in an Operational Limited-Area Data Assimilation System

The evaluation of several climatological background-error covariance matrix (defined as the B matrix) estimation methods was performed using the ALADIN limited-area modeling data-assimilation system at a 4 km horizontal grid spacing. The B matrices compared were derived using the standard National Meteorological Center (NMC) and ensemble-based estimation methods. To test the influence of lateral boundary condition (LBC) perturbations on the characteristics of ensemble-based B matrix, two ensemble prediction systems were established: one used unperturbed lateral boundary conditions (ENS) and another used perturbed lateral boundary conditions (ENSLBC). The characteristics of the three B matrices were compared through a diagnostic comparison, while the influence of the different B matrices on the analysis and quality of the forecast were evaluated for the ENSLBC and NMC matrices. The results showed that the lateral boundary condition perturbations affected all the control variables, while the smallest influence was found for the specific humidity. The diagnostic comparison showed that the ensemble-based estimation method shifted the correlations toward the smaller spatial scales, while the LBC perturbations gave rise to larger spatial scales. The influence on the analysis showed a smaller spatial correlation for the ensemble B matrix compared to that of the NMC, with the most pronounced differences for the specific humidity. The verification of the forecast showed modest improvement for the experiment with the ensemble B matrix. Among the methods tested, the results suggest that the ensemble-based data-assimilation method is the favorable approach for background-error covariance calculation in high-resolution limited-area data assimilation systems.

A permutation-based Bayesian approach for inverse covariance estimation

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.

Spectral efficiency of massive MIMO networks with pilot contamination and channel covariance matrix estimation

This study addresses spectral efficiencies (SEs) in a massive multiple-input multiple-output cellular network with pilot contamination. Both uplink and downlink SEs are investigated with practical channel state information. A downlink precoder minimising signal detection error and power leakage with considerations on spatial correlation has been derived. Also, a low-complexity covariance matrix estimation method has been proposed and compared with the existing estimation method in terms of normalised mean-squared error. It has been demonstrated via simulations that both uplink and downlink SEs using the proposed covariance matrix estimation method can grow with the number of antennas even with a relatively small number of pilot symbols.

An Outlier Detection Method Based on Mahalanobis Distance for Source Localization.

This paper addresses the problem of localization accuracy degradation caused by outliers of the angle of arrival (AOA). The problem of outlier detection of the AOA is converted into the detection of the estimated source position sets, which are obtained by the proposed division and greedy replacement method. The Mahalanobis distance based on robust mean and covariance matrix estimation method is then introduced to identify the outliers from the position sets. Finally, the weighted least squares method based on the reliable probabilities and distances is proposed for source localization. The simulation and experimental results show that the proposed method outperforms representative methods when unreliable AOAs are present.

Covariance Matrix Estimation in Massive MIMO

Interference during the uplink training phase significantly deteriorates the performance of a massive MIMO system. The impact of the interference can be reduced by exploiting second order statistics of the channel vectors, e.g., to obtain minimum mean squared error estimates of the channel. In practice, the channel covariance matrices have to be estimated. The estimation of the covariance matrices is also impeded by the interference during the training phase. However, the coherence interval of the covariance matrices is larger than that of the channel vectors. This allows us to derive methods for accurate covariance matrix estimation by appropriate assignment of pilot sequences to users in consecutive channel coherence intervals.

Information Geometry for Covariance Estimation in Heterogeneous Clutter with Total Bregman Divergence.

This paper presents a covariance matrix estimation method based on information geometry in a heterogeneous clutter. In particular, the problem of covariance estimation is reformulated as the computation of geometric median for covariance matrices estimated by the secondary data set. A new class of total Bregman divergence is presented on the Riemanian manifold of Hermitian positive-definite (HPD) matrix, which is the foundation of information geometry. On the basis of this divergence, total Bregman divergence medians are derived instead of the sample covariance matrix (SCM) of the secondary data. Unlike the SCM, resorting to the knowledge of statistical characteristics of the sample data, the geometric structure of matrix space is considered in our proposed estimators, and then the performance can be improved in a heterogeneous clutter. At the analysis stage, numerical results are given to validate the detection performance of an adaptive normalized matched filter with our estimator compared with existing alternatives.

Efficient Clustering-based Noise Covariance Estimation for Maximum Noise Fraction.

Most hyperspectral images (HSI) have important spectral features in specific combination of wave numbers or channels. Noise in these specific channels or bands can easily overwhelm these relevant spectral features. Maximum Noise Fraction (MNF) by Green et al. [1] has been extensively studied for noise removal in HSI data. The MNF transform maximizes the Signal to Noise Ratio (SNR) in feature space, thereby explicitly requiring an estimation of the HSI noise. We present two simple and efficient Noise Covariance Matrix (NCM) estimation methods as required for the MNF transform. Our NCM estimations improve the performance of HSI classification, even when ground objects are mixed. Both techniques rely on a superpixel based clustering of HSI data in the spatial domain. The novelty of our NCM's comes from their reduced sensitivity to HSI noise distributions and interference patterns. Experiments with both simulated and real HSI data show that our methods significantly outperforms the NCM estimation in the classical MNF transform, as well as against more recent state of the art NCM estimation methods. We quantify this improvement in terms of HSI classification accuracy and superior recovery of spectral features.

Improved Global Minimum Variance Portfolio via Tail Eigenvalues Amplification

We identify a few sample eigenvalues adjustment patterns that lead to an improvement in the out-of-sample portfolio Sharpe ratio when the population covariance matrix admits a high-dimensional factor model. These patterns unveil the key to portfolio performance improvement and shed light on the effectiveness of a few well-known covariance matrix estimation methods which were not designed to improve the out-of-sample portfolio performance.

Sparse Steinian Covariance Estimation

ABSTRACTWe consider a new method for sparse covariance matrix estimation which is motivated by previous results for the so-called Stein-type estimators. Stein proposed a method for regularizing the sample covariance matrix by shrinking together the eigenvalues; the amount of shrinkage is chosen to minimize an unbiased estimate of the risk (UBEOR) under the entropy loss function. The resulting estimator has been shown in simulations to yield significant risk reductions over the maximum likelihood estimator. Our method extends the UBEOR minimization problem by adding an ℓ1 penalty on the entries of the estimated covariance matrix, which encourages a sparse estimate. For a multivariate Gaussian distribution, zeros in the covariance matrix correspond to marginal independences between variables. Unlike the ℓ1-penalized Gaussian likelihood function, our penalized UBEOR objective is convex and can be minimized via a simple block coordinate descent procedure. We demonstrate via numerical simulations and an analysis of microarray data from breast cancer patients that our proposed method generally outperforms other methods for sparse covariance matrix estimation and can be computed efficiently even in high dimensions.

Knowledge-Aided Covariance Matrix Estimation in Spiky Radar Clutter Environments

Space-time adaptive processing (STAP) is an important airborne radar technique used to improve target detection in clutter-limited environments. Effective STAP implementation is dependent on accurate space-time covariance matrix estimation. Heterogeneous clutter, including spiky, spatial clutter variation, violates underlying STAP training assumptions and can significantly degrade corresponding detection performance. This paper develops a spiky, space-time clutter model based on the K-distribution, assesses the resulting impact on STAP performance using traditional methods, and then proposes and evaluates the utility of the knowledge-aided parametric covariance matrix estimation (KAPE) method, a model-based scheme that rapidly converges to better represent spatial variation in clutter properties. Via numerical simulation of an airborne radar scenario operating in a spiky clutter environment, we find substantial improvement in probability of detection ( P D ) for a fixed probability of false alarm ( P F A ) for the KAPE method. For example, in the spiky clutter environment considered herein, results indicate a P D of 32% for traditional STAP and in excess of 90% for KAPE at a P F A of 1E-4, with a corresponding difference of 11.5 dB in threshold observed from exceedance analysis. The proposed K-distributed spiky clutter model, and application and assessment of KAPE as an ameliorating STAP technique, contribute to an improved understanding of radar detection in complex clutter environments.