Rank Of Covariance Matrix Research Articles

Coherent signals direction estimation using high-order quantities in the acoustic field

To meet the demands of underwater remote detection technology, the receiving array must be extended to low frequencies by increasing the array aperture required for detection. Furthermore, the base array aperture must be increased owing to the loss of effective array aperture in coherent targets detection using most decoherent algorithms; however, this prevents its use on small platforms, such as autonomous underwater platforms. Thus, reducing the array aperture while ensuring high resolution is required. In this article, high-order quantities in the acoustic field are utilized to construct a full rank covariance matrix to Multiple Signal Classification Method (MUSIC), which achieves high-resolution results for small aperture arrays with small incident angular intervals in low-frequency coherent targets. The correlated noise of each sensor is considered in the small aperture arrays and the gain of each element in the new matrix is derived. The theoretical analysis, numerical simulations, and experimental verifications show the validity of the method.

Fast and Accurate Approach for DOA Estimation of Coherent Signals

An algorithm for estimation of direction of arrival (DOA) based on QR decomposition is proposed for coherent signals in this paper. When coherent is present, the rank of the signal covariance matrix is generally less than the signal number, making the estimation of the signal or noise subspace inaccurate. Therefore, we need to eliminate the spatial covariance matrix rank loss. According to the idea of matrix reconstruction, we try to construct three different data matrices, one is the signal covariance matrix, the other is the eigenvector reconstruction matrix of the signal covariance matrix, and the other is reconstructed matrix with the addition of spatial smoothing technology. Based on the resulting data matrix, whereafter, QR decomposition with the iteratively weighted least squares (IWLS) as solver is proposed to reduce the computational complexity of DOA estimation. Compared with other existing algorithms, the simulation results show that our method has high accuracy and great increase in the computational efficiency.

On the general conditions of existence for linear MMSE filters: Wiener and Kalman

State estimation is a fundamental task in a plethora of applications. Two important classes of linear minimum mean square error (MMSE) estimators are Wiener filters (WFs) and Kalman filters (KFs), the latter being the recursive form of the former for linear discrete state-space (LDSS) dynamic models. The minimum set of uncorrelation conditions regarding states and measurements which allows to formulate a general KF form has been recently introduced in the literature, but for both WFs and KFs a standard key assumption is that the measurement noise covariance is invertible. In this contribution we provide i) an alternative derivation of the WF in the case of a non-invertible measurement covariance matrix, and ii) the proof that under the minimum set of uncorrelation conditions lately released, the KF can always be formulated irrespective of the measurement covariance matrix rank. Notice that these results also apply for Kalman predictors and smoothers, linearly constrained WFs and minimum variance distortionless response (MVDR) filters, therefore being a remarkable result of broad interest.

Principal component analysis with drop rank covariance matrix

<p style='text-indent:20px;'>This paper considers the principal component analysis when the covariance matrix of the input vectors drops rank. This case sometimes happens when the total number of the input vectors is very limited. First, it is found that the eigen decomposition of the covariance matrix is not uniquely defined. This implies that different transform matrices could be obtained for performing the principal component analysis. Hence, the generalized form of the eigen decomposition of the covariance matrix is given. Also, it is found that the matrix with its columns being the eigenvectors of the covariance matrix is not necessary to be unitary. This implies that the transform for performing the principal component analysis may not be energy preserved. To address this issue, the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived. Moreover, since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem, the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived. In fact, the unitary matrix with its columns being the eigenvectors of the covariance matrix is only a particular case of the condition. Furthermore, the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is derived. It is found that the unitary matrix with its columns being the eigenvectors of the covariance matrix does not satisfy this condition. Computer numerical simulation results are given to valid the results.

Multiaperture Focusing for Spaceborne Transmitter/Ground-Based Receiver Bistatic SAR

The article proposes a methodology to perform azimuth focusing of spaceborne transmitter-stationary receiver bistatic synthetic aperture radar data across multiple along-track apertures to increase azimuth resolution. The procedure uses as input several azimuth apertures (continuous groups of range compressed pulses) from one or more satellite bursts and comprises the following stages: antenna pattern compensation, slow time resampling, reconstruction of missing azimuth samples between neighboring sets of pulses using an autoregressive (AR) model and back-projection focusing of the resulting multiaperture range image. A novel, highly efficient method is proposed to estimate the optimal order for the AR model. It differs from the traditional approach that uses the Akaike Information Criterion to directly estimate the order because the proposed method estimates the order indirectly by detecting the number of targets using principle component analysis. Spatial smoothing is used to obtain a full rank covariance matrix, whose eigenvalues are then analyzed using minimum description length. The optimal order is an integer multiple of the number of targets, which depends on signal-to-noise ratio. The approach is evaluated with real bistatic data acquired over an area of Bucharest city, Romania.

A spatial rank–based multivariate EWMA chart for monitoring process shape matrices

AbstractIn this article, we propose a nonparametric EWMA control chart for monitoring the shape matrix of a multivariate process based on a spatial rank test and the exponentially weighted moving average scheme. The proposed control chart is essentially developed using an estimated spatial rank covariance matrix to test the shape matrix of the covariance matrix of multivariate distributions with heavy tails. Based on our simulation studies, the proposed control chart outperforms the only existing nonparametric control chart in many practical out‐of‐control scenarios for monitoring the shape matrix of the covariance matrix of many multivariate processes. Further, we point out the weaknesses of both the nonparametric EWMA control charts for monitoring the shape matrix of multivariate processes in real applications and propose one possible method to overcome these weaknesses. We also use an example from a white wine production process to demonstrate the applicability and implementation of the proposed control chart.

Joint Spatial Division and Diversity for Massive MIMO Systems

We propose a downlink beamforming scheme that combines spatial division and orthogonal space-time block coding (OSTBC) in multi-user massive MIMO systems. The beamformer is divided into two parts: a pre-beamforming matrix to separate the users into different beams with no interference between each other, which is designed based on the low rank covariance matrix of the downlink channel, and a linear precoding matrix using partial or even no channel state information (CSI) concatenated by an OSTBC. To construct the pre-beamforming matrix, a simple method that selects columns from DFT matrix is presented. To design the linear precoding matrix with partial CSI of the effective channel after the pre-beamforming, we solve an optimization problem to minimize the pairwise error probability (PEP) of the users under an individual power or sum power constraint, respectively. For the individual power constraint, a semi-definite relaxing (SDR) method with a sufficient condition achieving the globally optimal solution is proposed to provide a performance benchmark. In addition, an efficient iterative successive convex approximation (SCA) method is provided to achieve a suboptimal solution. Furthermore, closed form solutions are derived under some special cases. For the sum power constraint, we consider two different designs, i.e., minimizing the average PEP and minimizing the maximum PEP of all users. We find that both non-convex problems have a similar structure, and proposed a unified SCA-Alternating Direction Method of Multipliers (ADMM) algorithm to handle them. The SCA-ADMM method can be implemented in a parallel manner, and thus is with great efficiency. Simulation results show the efficiency of our proposed JSDD scheme and the optimization method.

Interference-Free Criterion for Interference-Unaware Receive Transform in MIMO Co-Channel Interference

In this letter, we derive the upper bound and the lower bound of the achievable rate for the interference-unaware receive transformation (IURT), which does not require interference knowledge or precoding, to obtain the criterion for making IURT become interference free in multiple-input–multiple-output (MIMO) systems under co-channel interference. We prove that if the receive-antenna number is larger than or equal to the interference-free threshold, which is equal to the summation of the transmit-antenna number and the interference’s covariance-matrix rank, then the derived upper bound reaches the interference-free MIMO channel capacity. Furthermore, we prove that by increasing both the transmit-antenna number and the receive-antenna number the derived lower bound can come closer to the interference-free MIMO channel capacity. Simulation results show that if the receive-antenna number is larger than the interference-free threshold, then the achievable rate of IURT can achieve the interference-free MIMO channel capacity.

An ensemble of visual features for Gaussians of local descriptors and non-binary coding for texture descriptors

This paper presents an improved version of a recent state-of-the-art texture descriptor called Gaussians of Local Descriptors (GOLD), which is based on a multivariate Gaussian that models the local feature distribution that describes the original image. The full rank covariance matrix, which lies on a Riemannian manifold, is projected on the tangent Euclidean space and concatenated to the mean vector for representing a given image. In this paper, we test the following features for describing the original image: scale-invariant feature transform (SIFT), histogram of gradients (HOG), and weber's law descriptor (WLD). To improve the baseline version of GOLD, we describe the covariance matrix using a set of visual features that are fed into a set of Support Vector Machines (SVMs). The SVMs are combined by sum rule. The scores obtained by an SVM trained using the original GOLD approach and the SVMs trained with visual features are then combined by sum rule. Experiments show that our proposed variant outperforms the original GOLD approach. The superior performance of the proposed system is validated across a large set of datasets. Particularly interesting is the performance obtained in two widely used person re-identification datasets, CAVIAR4REID and IAS, where the proposed GOLD variant is coupled with a state-of-the-art ensemble to obtain an improvement of performance on these two datasets. Moreover, we performed further tests that combine GOLD with non-binary features (local ternary/quinary patterns) and deep transfer learning. The fusion among SVMs trained with deep features and the SVMs trained using the ternary/quinary coding ensemble is demonstrated to obtain a very high performance across datasets. The MATLAB code for the ensemble of classifiers and for the extraction of the features will be publicly available11https://www.dei.unipd.it/node/2357 (+Pattern Recognition and Ensemble Classifiers) to other researchers for future comparisons.

Identifying Broad and Narrow Financial Risk Factors with Convex Optimization

Factor analysis of security returns aims to decompose a return covariance matrix into systematic and specific risk components. To date, most commercially successful factor analysis has been based on fundamental models, although there is a large academic literature on statistical models. While successful in many respects, traditional statistical approaches like principal component analysis and maximum likelihood suffer from several drawbacks. These include a lack of robustness, strict assumptions on the underlying model of returns, and insensitivity to narrow factors such as industries and currencies, which affect only a small number of securities, but in an important way.We apply convex optimization methods to decompose a security return covariance matrix into low rank and sparse parts. The low rank component includes the market and other broad factors that affect most securities. The sparse component includes narrow factors and security specific effects.We measure the variance forecasting accuracy of a low rank plus sparse covariance matrix estimator on an equally weighted portfolio of 100 securities simulated from a model with two broad factors and 21 narrow factors. We find that the low rank plus sparse estimators are more accurate than estimates made with classical principal component analysis, in particular, at forecasting risk due to narrow factors. Finally, we illustrate a low rank plus sparse decomposition of an empirical covariance matrix of 100 equities drawn from 21 countries.

SINR maximization in colocated MIMO radars using transmit covariance matrix

The main waveform design features in multiple-input multiple-output (MIMO) radars include (a) signal transmission with full rank covariance matrix in order to use the maximum waveform diversity and to suppress more number of interferers, (b) constant envelope in order to have simplicity in deployment and to reduce the destructive effect of nonlinear amplifiers and (c) small side lobe level (SLL) in order to reduce the effect of interferers with unknown location. Therefore, in order to maximize the signal-to-interference-plus-noise ratio (SINR) and to exploit the advantages of MIMO radars, in this paper we have proposed two full rank transmit covariance matrices and maximum achievable SINR is calculated analytically for both. We have shown that the two proposed covariance matrices can be used to generate BPSK waveforms which satisfy constant modulus constraint. Simulation results show that when the angle location of interferences is known, the first proposed matrix achieves a higher level of SINR compared to the second one, while the second proposed matrix has a lower SLL compared with the first one. Also we have shown that the two proposed covariance matrices can handle more interferences compared to phased-array and the recently proposed methods in MIMO radars.

GOLD: Gaussians of Local Descriptors for image representation

The Bag of Words paradigm has been the baseline from which several successful image classification solutions were developed in the last decade. These represent images by quantizing local descriptors and summarizing their distribution. The quantization step introduces a dependency on the dataset, that even if in some contexts significantly boosts the performance, severely limits its generalization capabilities. Differently, in this paper, we propose to model the local features distribution with a multivariate Gaussian, without any quantization. The full rank covariance matrix, which lies on a Riemannian manifold, is projected on the tangent Euclidean space and concatenated to the mean vector. The resulting representation, a Gaussian of Local Descriptors (GOLD), allows to use the dot product to closely approximate a distance between distributions without the need for expensive kernel computations. We describe an image by an improved spatial pyramid, which avoids boundary effects with soft assignment: local descriptors contribute to neighboring Gaussians, forming a weighted spatial pyramid of GOLD descriptors. In addition, we extend the model leveraging dataset characteristics in a mixture of Gaussian formulation further improving the classification accuracy. To deal with large scale datasets and high dimensional feature spaces the Stochastic Gradient Descent solver is adopted. Experimental results on several publicly available datasets show that the proposed method obtains state-of-the-art performance.

STAP Covariance Matrix Rank Analysis for Airborne MIMO Radar in the Presence of Jammers

该文分析了正侧阵机载MIMO雷达杂波加有源干扰协方差矩阵的结构，得到杂波加有源干扰协方差矩阵秩的上界为杂波的秩与有源干扰的秩之和减去有源干扰个数。并由此得到MIMO雷达杂波加有源干扰协方差矩阵非满秩而SIMO雷达杂波加有源干扰协方差矩阵满秩时有源干扰个数的范围。当有源干扰的数目在此范围时，SIMO雷达的理论性能严重下降，而MIMO雷达在理论上仍然有足够的自由度来抑制杂波和有源干扰，从而保证有较好的性能。通过仿真实验验证了上述结论。

Monitoring the covariance matrix with fewer observations than variables

Multivariate control charts are essential tools in multivariate statistical process control. In real applications, when a multivariate process shifts, it occurs in either location or scale. Several methods have been proposed recently to monitor the covariance matrix. Most of these methods deal with a full rank covariance matrix, i.e., in a situation where the number of rational subgroups is larger than the number of variables. When the number of features is nearly as large as, or larger than, the number of observations, existing Shewhart-type charts do not provide a satisfactory solution because the estimated covariance matrix is singular. A new Shewhart-type chart for monitoring changes in the covariance matrix of a multivariate process when the number of observations available is less than the number of variables is proposed. This chart can be used to monitor the covariance matrix with only one observation. The new control chart is based on using the graphical LASSO estimator of the covariance matrix instead of the traditional sample covariance matrix. The LASSO estimator is used here because of desirable properties such as being non-singular and positive definite even when the number of observations is less than the number of variables. The performance of this new chart is compared to that of several Shewhart control charts for monitoring the covariance matrix.

Waveguide invariant minimum variance scatterer depth classification for active sonar.

Active sonar systems are plagued by false alarms due to confusion between returns from water-column targets and backscatter from the bottom. Both feature-based and physics-based classifiers are notoriously susceptible to mismatch of the environment used for training and/or modeling active sonar returns. In this paper, in order to achieve more robust classification, uniformly sub-sampled DFT-coefficients from a single snapshot of the wideband active sonar return are used to define a waveguide-invariant spectral density matrix (WI-SDM). The WI-SDM facilitates adaptive matched-filtering based approaches for target depth estimation, where the waveguide invariant property is exploited to obtain uncorrelated snapshots without inflating covariance matrix rank. Depth classification is then performed by designing a waveguide-invariant minimum variance filter (WI-MVF) with adaptive weights which minimize ambiguous depth sidelobes. Simulation and real data results in a shallow-water Mediterranean environment are presented to illustrate the approach. [Work sponsored by ONR.]

The rank of the covariance matrix of an evanescent field

Evanescent random fields arise as a component of the 2D Wold decomposition of homogeneous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space–time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem.

Tests and estimates of shape based on spatial signs and ranks

Nonparametric procedures for testing and estimation of the shape matrix in the case of multivariate elliptic distribution are considered. Testing for sphericity is an important special case. The tests and estimates are based on the spatial sign and rank covariance matrices. The estimates based on the spatial sign covariance matrix and symmetrized spatial sign covariance matrix are Tyler's [A distribution-free M-estimator of multivariate scatter, Ann. Statist. 15 (1987), pp. 234–251] shape matrix and and Dümbgen's [On Tyler's M-functional of scatter in high dimension, Ann. Inst. Statist. Math. 50 (1998), pp. 471–491] shape matrix, respectively. The test based on the spatial sign covariance matrix is the sign test statistic in the class of nonparametric tests proposed by Hallin and Paindaveine [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]. New tests and estimates based on the spatial rank covariance matrix are proposed. The shape estimates introduced in the paper play an important role in the inner standardisation of the spatial sign and rank tests for multivariate location. Limiting distributions of the tests and estimates are reviewed and derived, and asymptotic efficiencies as well as finite-sample efficiencies of the proposed tests are compared with those of the classical modified John's [Some optimal multivariate tests, Biometrika 58 (1971), pp. 123–127; The distribution of a statistic used for testing sphericity of normal distributions, Biometrika 59 (1972), pp. 169–173] test and the van der Waerden test (Hallin and Paindaveine, [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]). The symmetrised spatial sign- and rank-based estimates and tests seem to have a very high efficiency in the multivariate normal case, and they are much better than the classical estimate (shape matrix based on the regular covariance matrix) and test (John's test) for distributions with heavy tails.

Rank covariance matrix estimation of a partially known covariance matrix

Classical multivariate methods are often based on the sample covariance matrix, which is very sensitive to outlying observations. One alternative to the covariance matrix is the affine equivariant rank covariance matrix (RCM) that has been studied in Visuri et al. [2003. Affine equivariant multivariate rank methods. J. Statist. Plann. Inference 114, 161–185]. In this article we assume that the covariance matrix is partially known and study how to estimate the corresponding RCM. We use the properties that the RCM is affine equivariant and that the RCM is proportional to the inverse of the regular covariance matrix, and hence reduce the problem of estimating the original RCM to estimating marginal rank covariance matrices. This is a great computational advantage when the dimension of the original data vector is large.

Waveguide invariant focusing for broadband beamforming in an oceanic waveguide

The performance of broadband sonar array processing can degrade significantly in shallow-water environments when interference becomes angularly spread due to multipath propagation. Particularly for towed line arrays near endfire, elevation angle spreading of multipath interference often results in masking of weaker sources of interest. While adaptive beamforming in a series of narrow frequency bands can suppress coherent multipath interference, this approach requires long observation times to estimate the required narrowband covariance matrices. To form wideband covariance matrices which can be estimated with less observation time, plane-wave focusing methods have been used to avoid interference covariance matrix rank inflation. This paper extends wideband focusing to the case of coherent multipath interference. The approach presented here, called waveguide invariant focusing (WIF), exploits a robust relationship for the frequency dependence of horizontal wave number differences. Unlike matched-field methods, WIF does not model multipath wave fronts but rather makes the interference appear to occupy the same rank-one subspace across frequency. This permits formation of wideband covariance matrices without interference rank inflation. Simulation experiments in a realistic ocean environment indicate that adaptive beamforming using WIF covariance matrices can provide a significant array gain improvement over conventional adaptive methods with limited observation time.

Joint space-time interpolation for distorted linear and bistatic array geometries

This paper presents a new joint space-time interpolation technique (STINT) to improve the small sample support performance of space-time adaptive processing (STAP) with distorted linear monostatic arrays and linear bistatic array configurations. Brennan's rule for the space-time clutter covariance matrix rank is extended to monostatic linear arrays with arbitrary intersensor spacing, distorted linear arrays and bistatic geometries. It is shown that both distortion in the array geometry and bistatic operation increase the clutter rank and cause the space-time clutter covariance matrix to become range dependent. This results in lower output signal-to-interference-plus-noise ratio (SINR) for the same number of adaptive degrees of freedom and reduced available sample support. This motivates the development of the STINT technique aimed at compensating for the clutter rank inflation, while also making the clutter statistics appear more stationary across range. More specifically, a linear transformation is designed that maps the received clutter across space and time to that which would be received by a "virtual" monostatic side-looking ULA. By mapping the data to form a reduced rank clutter covariance matrix, fewer snapshots are needed for a statistically stable matrix inversion as required in STAP, thereby improving the short observation time performance. Simulation results for a typical airborne radar scenario indicate up to 10-dB SINR improvement can be obtained using STINT with limited sample support.