Eigenvalues Of Covariance Matrix Research Articles

Source Enumeration Method using Eigenvalue Gap Ratio and Performance Comparison in Rayleigh Fading

In electronic warfare, source enumeration and direction-of-arrival estimation are important. The source enumeration method based on eigenvalues of covariance matrix from received is one of the most used methods. However, there are some drawbacks such as accuracy less than 100 % at high SNR, poor performance at low SNR and reduction of maximum number of estimating sources. We suggested new method based on eigenvalues gaps, which is named AREG(Accumulated Ratio of Eigenvalues Gaps). Meanwhile, FGML(Fast Gridless Maximum Likelihood) which reconstructs the covariance matrix was suggested by Wu et al., and it improves performance of the existing source enumeration methods without modification of algorithms. In this paper, first, we combine AREG with FGML to improve the performance. Second, we compare the performance of source enumeration and direction-of-arrival estimation methods in Rayleigh fading. Third, we suggest new method named REG(Ratio of Eigenvalues Gaps) to reduce performance degradation in Rayleigh Fading environment of AREG.

Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory

We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in a fixed basis even though the measurement record is not informationally complete. We then compare this with the information generated and fidelities obtained by application of a different Haar random unitary at each time step. We give an upper bound on the maximal information that can be obtained in tomography and show that a covariance matrix taken from the Wishart-Laguerre ensemble of random matrices and the associated Marchenko-Pastur distribution saturates this bound. We find that physically, this corresponds to an application of a different Haar random unitary at each time step. We show that repeated application of random diagonal unitaries gives a covariance matrix in tomographic estimation that corresponds to a new ensemble of random matrices. We analytically and numerically estimate eigenvalues of this ensemble and show the information gain to be bounded from below by the Porter-Thomas distribution.

Shrinkage estimation of large covariance matrices: Keep it simple, statistician?

Under rotation-equivariant decision theory, sample covariance matrix eigenvalues can be optimally shrunk by recombining sample eigenvectors with a (potentially nonlinear) function of the unobservable population covariance matrix. The optimal shape of this function reflects the loss/risk that is to be minimized. We solve the problem of optimal covariance matrix estimation under a variety of loss functions motivated by statistical precedent, probability theory, and differential geometry. A key ingredient of our nonlinear shrinkage methodology is a new estimator of the angle between sample and population eigenvectors, without making strong assumptions on the population eigenvalues. We also introduce a broad family of covariance matrix estimators that can handle all regular functional transformations of the population covariance matrix under large-dimensional asymptotics. In addition, we compare via Monte Carlo simulations our methodology to two simpler ones from the literature, linear shrinkage and shrinkage based on the spiked covariance model.

Derived vectorcardiogram based automated detection of posterior myocardial infarction using FBSE-EWT technique

The early detection of posterior myocardial infarction (PMI) is an important task as it can cause cardiac failure. Due to the absence of extra posterior leads in the standard 12-lead electrocardiogram (ECG), the PMI detection sensitivity degrades. To improve it, additional posterior leads can be included in standard 12-lead ECG system. Another approach utilizing vectorcardiogram (VCG) has been in practice with specific 7 electrodes with one posterior lead. Although the VCG approach is promising, the arrangement of posterior lead causes patient discomfort. To overcome aforesaid issue, derived VCG (dVCG) obtained through the Dowers inverse transformation has been proposed, wherein, from the standard 12-lead ECG, a three lead VCG is derived, which then can be utilized for PMI detection. Relying on the dVCG approach, in the present paper, we have proposed a novel methodology for PMI detection using Fourier-Bessel series expansion based empirical wavelet transform (FBSE-EWT). The non-stationary behavior of dVCG has been captured using FBSE-EWT, followed by the principal component analysis employing eigenvalues of covariance matrix. The constructed feature space is then fed to decision tree (DT), support vector machine (SVM), and K-nearest neighbor (KNN) classifiers. The KNN classifier with inverse squared city block distance resulted in the overall classification accuracy of 97.92% with sensitivity and specificity of 96.63% and 98.50%, respectively. The experiments were performed over Physikalisch-Technische Bundesanstalt Diagnostic (PTBD) dataset. The proposed method has the potential to be utilized in accurate and robust PMI detection from dVCG signals in the clinical settings.

Addressing challenges in nuclear data adjustment techniques using integral experiments based upon uranium solution thermal systems

A stochastic technique in view of determining neutronic parameters is used along with the JEFF-3.3 nuclear data library, trying to adjust in conjunction with the Asymptotic Generalized Linear Least-Squares (AGLLS) assimilation methodology, a demonstrative limited number of basic ENDF parameters for a few nuclides, which are U-235, U-238 and hydrogen. Correspondingly, the current investigations are primarily focused on low-enriched and intermediate-enrichment uranium solution thermal systems for which the effective multiplication factor, keff, which is the parameter to assimilate, has been measured. Even though the nuclear data seems well known for these nuclides, which is confirmed by the overall good performance of prior, unadjusted data, some deficiencies are highlighted, justifying dedicated assimilation studies.Similarly to fast-spectrum systems, it is confirmed that meaningful, weak adjustments can be obtained on the one hand if the chosen assimilation database does not exhibit too strong analytical cross-correlations among the individual multiplication factors. For keff, which is a global parameter, such strong correlations are anticipated for systems having similar ratios of the overall number of atoms of hydrogen to those of U-235 in the core, largely determining the slowing-down characteristics. Another condition for reasonable adjustments, on the other hand, is that the benchmarks part of the database should be consistent among each other.It is also verified that a sufficiently large number of multiplication factors would be required to reach a converged, reference adjustment. However, without enlarging the range of nuclides and ENDF parameters in addition to the selection of systems, a continuous increase of the number of parameters to assimilate is not possible in a meaningful way without having to extend the database to other integral parameters than keff such as spectral indices, which unfortunately are not available for these thermal systems. Namely, adding new parameters of the same type to databases consisting of three sufficiently decoupled systems, always increases remarkably the analytical cross-correlation of the keffs; in which cases AGLLS either provides stronger, non-physical adjustments or even diverges, as at least one eigenvalue of the analytical covariance matrix of the parameters to assimilate decreases too much, approaching singularity in the worst case.

Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

Note. The best result in each experiment is highlighted in bold.The optimal solutions of many decision problems such as the Markowitz portfolio allocation and the linear discriminant analysis depend on the inverse covariance matrix of a Gaussian random vector. In “Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator,” Nguyen, Kuhn, and Mohajerin Esfahani propose a distributionally robust inverse covariance estimator, obtained by robustifying the Gaussian maximum likelihood problem with a Wasserstein ambiguity set. In the absence of any prior structural information, the estimation problem has an analytical solution that is naturally interpreted as a nonlinear shrinkage estimator. Besides being invertible and well conditioned, the new shrinkage estimator is rotation equivariant and preserves the order of the eigenvalues of the sample covariance matrix. If there are sparsity constraints, which are typically encountered in Gaussian graphical models, the estimation problem can be solved using a sequential quadratic approximation algorithm.

Estimation of the Number of Spiked Eigenvalues in a Covariance Matrix by Bulk Eigenvalue Matching Analysis

ABSTRACT The spiked covariance model has gained increasing popularity in high-dimensional data analysis. A fundamental problem is determination of the number of spiked eigenvalues, K. For estimation of K, most attention has focused on the use of top eigenvalues of sample covariance matrix, and there is little investigation into proper ways of using bulk eigenvalues to estimate K. We propose a principled approach to incorporating bulk eigenvalues in the estimation of K. Our method imposes a working model on the residual covariance matrix, which is assumed to be a diagonal matrix whose entries are drawn from a gamma distribution. Under this model, the bulk eigenvalues are asymptotically close to the quantiles of a fixed parametric distribution. This motivates us to propose a two-step method: the first step uses bulk eigenvalues to estimate parameters of this distribution, and the second step leverages these parameters to assist the estimation of K. The resulting estimator aggregates information in a large number of bulk eigenvalues. We show the consistency of under a standard spiked covariance model. We also propose a confidence interval estimate for K. Our extensive simulation studies show that the proposed method is robust and outperforms the existing methods in a range of scenarios. We apply the proposed method to analysis of a lung cancer microarray dataset and the 1000 Genomes dataset.

Eigenvalue-based super-resolution DOA algorithm for arbitrary arrays

In order to solve the problem that the traditional DOA estimation algorithm has low angular resolution under low SNR and small snapshot conditions, a super-resolution DOA algorithm is proposed based on Pseudo data reconstruction. Firstly, using the odd function of the array steering vector function, a new matrix is constructed by using original array received covariance matrix and its complex conjugate. Then, the scanning source is introduced into the new matrix to construct a scanning covariance matrix. When the DOA of the scanning source is the same as the DOA of the target or the symmetric DOA of the target, the first noise eigenvalue of the scanning covariance matrix is twice as large as the first eigenvalue of the noise subspace corresponding to the original signal covariance matrix. Otherwise, there is no such relationship. In consequence, this property is used to construct the spatial spectrum for the DOA estimation. Then the interpolation method is used to overcome the shortcomings that the algorithm in this paper cannot be used for arbitrary arrays. Finally, simulation experiments verify the correctness and feasibility of the proposed algorithm.

THE TARGET DETECTION ON THE SEA SURFACE BASED ON STANDARD DEVIATION OF THE MAXIMUM EIGENVALUE OF THE POLARIZATION COVARIANCE MATRIX

This paper proposes a novel method to detect small sea targets, using polarimetric radar. The standard deviation of maximum eigenvalue of the polarization covariance matrix is ultilized for the detection. In order to evaluate the new algorithm in practical applications, the real data collected from polarimetric IPIX radar at MC Master University is tested with the new algorithm, and the result is displayed on the radar screen. The results of this method is also compared with those of the DoP and SP-GLR methods. Initial results show that the novel method significantly improves the detectability of small sea targets on the sea surface.

Sphericity and Identity Test for High-dimensional Covariance Matrix Using Random Matrix Theory

This paper addresses the issue of testing sphericity and identity of high-dimensional population covariance matrix when the data dimension exceeds the sample size. The central limit theorem of the first four moments of eigenvalues of sample covariance matrix is derived using random matrix theory for generally distributed populations. Further, some desirable asymptotic properties of the proposed test statistics are provided under the null hypothesis as data dimension and sample size both tend to infinity. Simulations show that the proposed tests have a greater power than existing methods for the spiked covariance model.

Eigenvalue-based cooperative spectrum sensing using kernel fuzzy c-means clustering

In this paper, novel techniques of eigenvalue-based cooperative spectrum sensing (CSS) using Kernel fuzzy c-means (KFCM) clustering are proposed. Test vectors derived from measured eigenvalues are categorized into channel available and unavailable class by performing clustering in two/three dimensional space. This is in contrast to existing eigenvalue-based spectrum sensing techniques, where sensing decision is made on the basis of a single test statistic in one dimension. Though multiple eigenvalues are used in those techniques, finally a single test statistic is computed and used for spectrum sensing. It is shown that proposed CSS using KFCM clustering in multidimensional space provides improvement in detection performance. Three different spectrum sensing techniques, utilizing different combinations of eigenvalues of the signal covariance matrix, are proposed and studied. Proposed algorithm is then compared to some of the recent techniques for spectrum sensing in literature. Extensive simulation results are given which highlight improvement in detection performance offered by the proposed algorithm.

A novel time-domain polarization filter based on a correlation matrix analysis

Polarization filters (PFs) are widely used for denoising seismic data. These filters are applied in the fields of seismology, microseismic monitoring, vertical seismic profiling, and subsurface imaging. They are primarily useful to suppress ground roll in seismic reflection data and enhance P- and S-wave arrivals. Traditional implementations of PFs involved the analysis of the covariance matrix or the singular value decomposition (SVD) of a 3C seismogram matrix. The linear polarization level, known as rectilinearity, is expressed as a function of the covariance matrix eigenvalues or by the data matrix singular values. Wavefield records that are linearly polarized are amplified, whereas others are attenuated. Besides the described implementations, we have developed a new time-domain PF based on the analysis of the seismic data correlation matrix. The principal idea is to extend the notion of the correlation coefficient in 3D space. This new filter avoids the need for diagonalization of the covariance matrix or SVD of data matrix, which are often time-consuming. The new implementation facilitates the choice of the rectilinearity threshold: We determine that linear polarization in three dimensions can be represented as three classic 2D correlations. A good linear polarization is detected when a high linear correlation between the three seismogram components is mutually observed. The tuning parameters of the new filter are the length of the time window, the filter order, and the rectilinearity threshold. Tests using synthetic seismograms indicate that optimal results are reached with a filter order that spans between 2 and 4, a rectilinearity threshold between 0.3 and 0.7, and a window length that is equivalent to one to three times the period of the signal wavelet. Compared with covariance-based filters, the new filter can enhance the signal-to-noise ratio by 6–20 dB and reduces computational costs by 25%.

Order Estimation via Matrix Completion for Multi-Switch Antenna Selection

This letter addresses the problem of order estimation for uniform linear arrays (ULAs) with multi-switch antenna selection in the small-sample regime. Multi-switch antenna selection results in a data matrix with missing entries, a scenario for which existing order estimation methods that build on the eigenvalues of the sample covariance matrix do not perform well. A direct application of the Davis-Kahan theorem allows us to show that the signal subspace is quite robust in the presence of missing entries. Based on this finding, this letter proposes a matrix completion (MC) subspace-based order estimation criterion that exploits the shift-invariance property of ULAs. A recently proposed shift-invariant matrix completion (SIMC) method is used for reconstructing the data matrix, and the proposed order estimation criterion is based on the chordal subspace distance between two submatrices extracted from the reconstructed matrix for increasing values of the dimension of the signal subspace. Our simulation results show that the method provides accurate order estimates with percentages of missing entries higher than 50 $\%$ .

Generalized likelihood ratio test for optical subpixel objects’ detection with hypothesis-dependent background covariance matrix

Much interest has arisen in the problem of detecting weak optical subpixel objects in a sequence of images immersed in a heavy homogeneous Gaussian clutter background. In optical systems, the presence of the objects changes the background plus the channel noise covariance matrix. Hence, this matrix may be different under null and alternative hypotheses. Because the maximum likelihood estimate of the background covariance matrix is not available when using the primary data set, we use an approximate estimate of a certain number of eigenvalues of the background covariance matrix that are available for estimation. We derived the general likelihood ratio test for the problem at hand and obtained the modified spectrum matching detector (MSpMD). Using the simulation results, we contrast it with well-known spectrum matching detectors and the hypothesis-independent matched subspace detectors. We also present experimental results of small subpixel objects detection on an agitated sea surface, using digital optical experiments that demonstrate the validation of the simulating results. We show the advantage of the MSpMD that allows to detect small subpixel floating objects on an agitated sea surface even with the same means and variances of the reflected signals from the sea surface and the object.

A Novel Goodness of Fit Test Spectrum Sensing Using Extreme Eigenvalues

The existing Goodness of fit (GoF) test based spectrum sensing algorithms mostly use samples or energies as observations to make decisions, which can hardly achieve satisfactory performance especially when the Primary user (PU) signals are highly correlated. Meanwhile, the eigenvalue of covariance matrix can reflect signal correlations well. Motivated by this, we study the distribution of eigenvalue and propose an eigenvalue based GoF spectrum sensing algorithm. In the proposed scheme, we use the ratios of maximum to minimum eigenvalue as observations and thus it can bring performance improvements through capturing correlation of PU signals. We also provide the related theoretical analysis for the proposed method. Simulation results show that the proposed method overcomes the problem of noise uncertainty and achieves performance improvement over the classical samples-based GoF test.

A Novel Direction-of-Arrival Estimation Algorithm Without Knowing the Source Number

As the eigenvalue ranking problem is analyzed, a direction-of-arrival (DOA) estimation method without knowing the number of sources is proposed. The proposed method is utilized through ranking the eigenvalue of specific covariance matrix to get the smallest eigenvalue and construct a new spatial spectrum. By deducing the novel algorithm, the mapping between the eigenvalues and power of signals is presented. Using the mapping, the minimum detection threshold for this algorithm is derived. Besides, the reason for the formation of pseudo-peaks is interpreted, and an approach to parameter selection is provided. The validity of the related theory of the algorithm is further investigated by simulations.

Conditionality of the principal component method in the problem of calculating the composite index of a system from a series of observations

The construction of the integral characteristic of the system based on a number of observations containing errors can be considered as the problem of extracting useful signal against a background of noise. Principal component analysis (PCA) is widely used to find the most characteristic features in the data and to reveal their structure. However, in practice, PCA is rarely used to determine the weights of composite indices. This is partly because the weights determined using PCA often contradict intuitive ideas about the properties of the analyzed system. One of the possible reasons for the occurrence of such situations may be the poor conditioning of the PCA - the influence of irremovable data errors on the determination of the eigenvalues and eigenvectors in PCA. The covariance matrices for different observations can be interpreted as a perturbation of the true unknown covariance matrix that determines the structure of the system. The error in calculating the eigenvalues of an unknown covariance matrix can be estimated a posteriori from the available values of the observed covariance matrices. The obtained estimates should be used when choosing computationally significant principal components that satisfy the criterion of information content, which works correctly in the case of noisy data.

Minimum Eigenvector Collaborative Representation Discriminant Projection for Feature Extraction

High-dimensional signals, such as image signals and audio signals, usually have a sparse or low-dimensional manifold structure, which can be projected into a low-dimensional subspace to improve the efficiency and effectiveness of data processing. In this paper, we propose a linear dimensionality reduction method—minimum eigenvector collaborative representation discriminant projection—to address high-dimensional feature extraction problems. On the one hand, unlike the existing collaborative representation method, we use the eigenvector corresponding to the smallest non-zero eigenvalue of the sample covariance matrix to reduce the error of collaborative representation. On the other hand, we maintain the collaborative representation relationship of samples in the projection subspace to enhance the discriminability of the extracted features. Also, the between-class scatter of the reconstructed samples is used to improve the robustness of the projection space. The experimental results on the COIL-20 image object database, ORL, and FERET face databases, as well as Isolet database demonstrate the effectiveness of the proposed method, especially in low dimensions and small training sample size.

Detection, Classification, and Location of Seismovolcanic Signals with Multicomponent Seismic Data: Example from the Piton de La Fournaise Volcano (La Réunion, France)

AbstractWe apply three different methods based on the analysis of the multicomponent seismic data to detect seismovolcanic tremors and other seismovolcanic signals, to propose an approach to classify them, and to locate their sources. We use continuous seismograms recorded during 1 year by 21 stations at the Piton de la Fournaise volcano (La Réunion, France). The first method allows the detection of seismovolcanic signals based on stability in time of the intercomponent cross‐correlation function. Two other methods based on the simultaneous analysis of the whole network can be used to detect seismovolcanic signals and to locate their sources. In the first network‐based method, the seismic wavefield is analyzed by calculating the width of the network covariance matrix eigenvalue distribution.The second network‐based method consists in performing the 3‐D backprojection of the interstation cross correlations in order to calculate the network response function. Simultaneous analysis of the parameters measured by the three different methods can be used to classify different types of seismovolcanic tremors. Our results demonstrate that all three methods efficiently detect seismovolcanic tremors accompanying the 2010 eruptions and the preceding pre‐eruptive seismic swarms. Furthermore, Methods 2 and 3 based on simultaneous analysis of the whole network detect a large number of volcanic earthquakes. Our location results show that each seismovolcanic tremor is located in a distinct regionof the volcano, close to the eruptive site at a shallow depth, and the preceding seismic crisis is located deeper at about the sea level under the summit crater.

A method for source number estimation with density-based clustering algorithm

Almost all high-performance spatial spectrum estimation algorithms for direction of arrival (DOA) estimation are based on accurate source number estimation. Applied with features of covariance matrix eigenvalues of received data and compared with commonly-used K-means algorithm, this article proposes to use a density-based clustering algorithm, which is ordering points to identify the clustering structure (OPTICS) and its detailed steps, to estimate the accurate source number. The experimental results applying with OPTICS algorithm demonstrate better performance on source number estimation over other algorithms under the same conditions.