Eigenvalues Of Covariance Matrix Research Articles

Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations

For the high-dimensional covariance estimation problem, when limn→∞p/n=c∈(0,1), the orthogonally equivariant estimator of the population covariance matrix proposed by Tsai and Tsai exhibits certain optimal properties. Under some regularity conditions, the authors showed that their novel estimators of eigenvalues are consistent with the eigenvalues of the population covariance matrix. In this paper, under the multinormal setup, we show that they are consistent estimators of the population covariance matrix under a high-dimensional asymptotic setup. We also show that the novel estimator is the MLE of the population covariance matrix when c∈(0,1). The novel estimator is used to establish that the optimal decomposite TT2-test has been retained. A high-dimensional statistical hypothesis testing problem is used to carry out statistical inference for high-dimensional principal component analysis-related problems without the sparsity assumption. In the final section, we discuss the situation in which p>n, especially for high-dimensional low-sample size categorical data models in which p>>n.

An Improved Reduced-Dimension Robust Capon Beamforming Method Using Krylov Subspace Techniques.

A reduced-dimension robust Capon beamforming method using Krylov subspace techniques (RDRCB) is a diagonal loading algorithm with low complexity, fast convergence and strong anti-interference ability. The diagonal loading level of RDRCB is known to become invalid if the initial value of the Newton iteration method is incorrect and the Hessel matrix is non-positive definite. To improve the robustness of RDRCB, an improved RDRCB (IRDRCB) was proposed in this study. We analyzed the variation in the loading factor with the eigenvalues of the reduced-dimensional covariance matrix and derived the upper and lower boundaries of the diagonal loading level; the diagonal loading level of the IRDRCB was kept within the bounds mentioned above. The computer simulation results show that the IRDRCB can effectively solve the problems of a sharp decline in the signal-to-noise ratio gain and an invalid diagonal loading level. The experimental results demonstrate that the interference noise of the IRDRCB is 3~5 dB higher than that of conventional adaptive beamforming, and the computational complexity is reduced by 15% to 20% compared with that of the RCB method.

Eigenvalues of the noise covariance matrix in ocean waveguides.

The eigenvalue (EV) spectra of the theoretical noise covariance matrix (CM) and observed sample CM provide information about the environment, source, and noise generation. This paper investigates these spectra for vertical line arrays (VLAs) and horizontal line arrays (HLAs) in deep and shallow water numerically. Empirically, the spectra are related to the width of the conventional beamforming output in angle space. In deep water, the HLA noise CM tends to be isotropic regardless of the sound speed profile. Thus, the EV spectrum approaches a step function. In contrast, the VLA noise CM is non-isotropic, and the EVs of the CM jump in two steps. The EVs before the first jump are due to sea surface noise, while those between the first and second jump are due to bottom-reflected noise. In shallow water, the VLA noise CM is affected by the environment (sound speed profile and seabed density, sound speed, attenuation, and layers) and array depth, the EVs have a more complicated structure. For Noise09 VLA experimental data, the noise sample CM EVs match the waveguide noise model better than the three-dimensional isotropic noise model.

SPURIOUS FACTORS IN DATA WITH LOCAL-TO-UNIT ROOTS

This paper extends the spurious factor analysis of Onatski and Wang (2021, Spurious factor analysis.Econometrica, 89(2), 591–614.) to high-dimensional data with heterogeneous local-to-unit roots. We find a spurious factor phenomenon similar to that observed in the data with unit roots. Namely, the “factors” estimated by the principal components analysis converge to principal eigenfunctions of a weighted average of the covariance kernels of the demeaned Ornstein–Uhlenbeck processes with different decay rates. Thus, such “factors” reflect the structure of the strong temporal correlation of the data and do not correspond to any cross-sectional commonalities, that genuine factors are usually associated with. Furthermore, the principal eigenvalues of the sample covariance matrix are very large relative to the other eigenvalues, creating an illusion of the “factors”capturing much of the data’s common variation. We conjecture that the spurious factor phenomenon holds, more generally, for data obtained from high frequency sampling of heterogeneous continuous time (or spacial) processes, and provide an illustration.

A Novel Weighted Block Sparse DOA Estimation Based on Signal Subspace under Unknown Mutual Coupling

A Novel Weighted Block Sparse DOA Estimation Based on Signal Subspace under Unknown Mutual Coupling

Решение оптимизационной задачи оценивания моделей полносвязной линейной регрессии

<p>This article is devoted to the problem of estimating fully connected linear regression models using the maximum likelihood method. Previously, a special numerical method was developed for this purpose, based on solving a nonlinear system using the method of simple iterations. At the same time, the issues of choosing initial approximations and fulfilling sufficient conditions for convergence were not studied. This article proposes a new method for solving the optimization problem of estimating fully connected regressions, similar to the method of estimating orthogonal regressions. It has been proven that, with equal error variances of interconnected variables, estimates of b-parameters of fully connected regression are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse covariance matrix. And if the ratios of the error variances of the variables are equal to the ratios of the variances of the variables, then the b-parameter estimates are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse correlation matrix, multiplied by the specific ratios of the standard deviations of the variables. A numerical experiment was carried out to confirm the correctness of the developed mathematical apparatus. The proposed method for solving the optimization problem of estimating fully connected regressions can be effectively used when solving problems of constructing multiple fully connected linear regressions.</p>

Vibration characteristics of defective axle box bearings in high-speed trains under track irregularity excitation

Track irregularity is a type of excitation source for the wheel-rail system, and it is also the major cause of vibration and wheel-rail forces on the axle box bearings of vehicles, and it is commonly found on in-service lines. Therefore, this paper mainly focuses on the dynamic characteristics of high-speed train axle box bearings under track irregularity. Firstly, a nonlinear system containing a faulty double-row tapered roller bearings is established, the obtained equations of motion are solved numerically using the Runge–Kutta and comparing it with the varying compliance vibration frequency and fault characteristic frequency obtained by formula, the comparison results verify the effectiveness of the model. Then, the track irregularity is coupled to the above bearing system, the model is also verified by using the rolling and vibrating test rig of single wheelset. Secondly, the eigenvalue of covariance matrix is introduced as an index to analyze the influence trend of different fault sizes, loads, and inner rotational speeds on the axial trajectory. Finally, the influence trend of track irregularity on the stability of vertical acceleration is analyzed by simulating different operating conditions.

An asymptotically exact estimate of the median noise eigenvalue of sample covariance matrices

The Dominant Mode Rejection beamformer [Abraham & Owsley (1990)] averages the noise subspace eigenvalues of the sample covariance matrix (SCM) to estimate the background noise power. This noise power estimate from snapshot deficient SCMs can be unreliable for large arrays in time-varying environments with limited snapshots. Median filtering offers robustness to outliers and subspace mismatch for these challenging problems. Recent numerical experiments [Campos Anchieta & Buck (2022)] identified a simple regression relating the median sample eigenvalue to the true background power for snapshot deficient SCMs. However, the complicated expression of the Marchenko-Pastur (MP) distribution for the noise eigenvalues of the SCM thwarted prior attempts to find a closed form estimator for the MP distribution median. This talk exploits a coordinate transform to obtain a closed-form median of the MP distribution that is exact for the leading term in the power series expansion of the distribution. The new median estimator is more accurate than the previous numerical regression when the ratio of sensors to snapshots is 2 or more. Given the central role of SCM eigenvalues in principal component analysis and constant false alarm rate detectors, the new median expression should find application in other data science algorithms for underwater acoustics. [Work supported by ONR Code 321US.]

Low Complexity Radio Frequency Interference Mitigation for Radio Astronomy Using Large Antenna Array

With the ongoing growth in radio communications, there is an increased contamination of radio astronomical source data, which hinders the study of celestial radio sources. In many cases, fast mitigation of strong radio frequency interference (RFI) is valuable for studying short lived radio transients so that the astronomers can perform detailed observations of celestial radio sources. The standard method to manually excise contaminated blocks in time and frequency makes the removed data useless for radio astronomy analyses. This motivates the need for better RFI mitigation techniques for array of size M antennas. Although many solutions for mitigating strong RFI improves the quality of the final celestial source signal, many standard approaches require all the eigenvalues of the spatial covariance matrix ([Formula: see text]) of the received signal, which has [Formula: see text] computation complexity for removing RFI of size d where [Formula: see text]. In this work, we investigate two approaches for RFI mitigation, (1) the computationally efficient Lanczos method based on the Quadratic Mean to Arithmetic Mean (QMAM) approach using information from previously-collected data under similar radio-sky-conditions, and (2) an approach using a celestial source as a reference for RFI mitigation. QMAM uses the Lanczos method for finding the Rayleigh–Ritz values of the covariance matrix [Formula: see text], thus, reducing the computational complexity of the overall approach to [Formula: see text]. Our numerical results, using data from the radio observatory Long Wavelength Array (LWA-1), demonstrate the effectiveness of both proposed approaches to remove strong RFI, with the QMAM-based approach still being computationally efficient.

Rates of Bootstrap Approximation for Eigenvalues in High-Dimensional PCA

In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix $\Sigma$. However, when the data are high-dimensional, there are relatively few theoretical guarantees that quantify the performance of the bootstrap. Our aim in this paper is to analyze how well the bootstrap can approximate the joint distribution of the leading eigenvalues of the sample covariance matrix $\hat\Sigma$, and we establish non-asymptotic rates of approximation with respect to the multivariate Kolmogorov metric. Under certain assumptions, we show that the bootstrap can achieve the dimension-free rate of ${\tt{r}}(\Sigma)/\sqrt n$ up to logarithmic factors, where ${\tt{r}}(\Sigma)$ is the effective rank of $\Sigma$, and $n$ is the sample size. From a methodological standpoint, our work also illustrates that applying a transformation to the eigenvalues of $\hat\Sigma$ before bootstrapping is an important consideration in high-dimensional settings.

A Robust Noise Estimation Algorithm Based on Redundant Prediction and Local Statistics.

Blind noise level estimation is a key issue in image processing applications that helps improve the visualization and perceptual quality of images. In this paper, we propose an improved block-based noise level estimation algorithm. The proposed algorithm first extracts homogenous patches from a single noisy image using local features, obtaining the covariance matrix eigenvalues of the patches, and constructs dynamic thresholds for outlier discrimination. By analyzing the correlations between scene complexity, noise strength, and other parameters, a nonlinear discriminant coefficient regression model is fitted to accurately predict the number of redundant dimensions and calculate the actual noise level according to the statistical properties of the elements in the redundancy dimension. The experimental results show that the accuracy and robustness of the proposed algorithm are better than those of the existing noise estimation algorithms in various scenes under different noise levels. It performs well overall in terms of performance and execution speed.

Advanced Dielectric Anomaly Detection for Non-Destructive Fruit Inspection Using Electromagnetic Microwave Technique

This paper presents a comprehensive investigation into the application of an electromagnetic microwave technique combined with the dielectric anomaly approach algorithm for the non-destructive inspection of fruits. The proposed antenna configuration comprises eight elements arranged in a circular array, enabling the collection of signals scattered by objects, specifically fruits, placed in their path. The collected data undergoes a series of processing steps, including Fast Fourier Transform, covariance matrix estimation, eigenvalue and eigenvector computation, and spatial spectrum construction. The dielectric anomaly algorithm is then applied to detect defects in the fruits. The study covers five different types of fruits, both healthy and defective, and gathers essential dielectric properties. Furthermore, a novel hybrid IQR method is introduced for outlier detection in the dielectric data. The results demonstrate the effectiveness of the proposed methodology in providing valuable insights into the internal structures of fruits and detecting anomalies, contributing to the enhancement of quality assessment in fruit inspection processes.

Robust Adaptive Beamforming for Interference Suppression Based on SNR

Robust adaptive beamforming (RAB) can be used to suppress interference signals while retaining the desired signals received by a sensor array. However, desired signal self-cancellation and model mismatch can affect RAB performance. In this paper, a novel interference-plus-noise covariance matrix (INCM) reconstruction method is proposed for RAB to solve these problems. The proposed method divides the desired signal into two ranges according to the input signal-to-noise ratio (SNR), namely low SNR and high SNR. In the low SNR range, INCM reconstruction directly uses the same sample covariance matrix as the sample matrix inversion (SMI) beamformer to retain the advantages of the traditional SMI algorithm. In the high SNR range, the eigenvalues of the sample covariance matrix are used to estimate the interference power and noise power. The optimized interference steering vector (SV) is obtained by solving a quadratic convex optimization problem in an interference subspace. The INCM is reconstructed from the interference SVs, interference power, and noise power. The reconstructed INCM is then used to correct the desired signal SV via maximizing the beamformer output power. This is achieved by solving a quadratically constrained quadratic programming (QCQP) problem. Analysis and simulation results are presented which demonstrate that the proposed method performs well under a variety of mismatch conditions.

CORRELATION MATRIX OF EQUI-CORRELATED NORMAL POPULATION: FLUCTUATION OF THE LARGEST EIGENVALUE, SCALING OF THE BULK EIGENVALUES, AND STOCK MARKET

Given an N-dimensional sample of size [Formula: see text] form a sample correlation matrix [Formula: see text]. Suppose that N and T tend to infinity with [Formula: see text] converging to a fixed finite constant [Formula: see text]. If the population is a factor model, then the eigenvalue distribution of [Formula: see text] almost surely converges weakly to Marčenko–Pastur distribution such that the index is Q and the scale parameter is the limiting ratio of the specific variance to the ith variable [Formula: see text]. For an N-dimensional normal population with equi-correlation coefficient [Formula: see text], which is a one-factor model, for the largest eigenvalue [Formula: see text] of [Formula: see text], we prove that [Formula: see text] converges to the equi-correlation coefficient [Formula: see text] almost surely. These results suggest an important role of an equi-correlated normal population and a factor model in Laloux et al. [(2000) Random matrix theory and financial correlations, International Journal of Theoretical and Applied Finance3 (3), 391–397]: the histogram of the eigenvalue of sample correlation matrix of the returns of stock prices fits the density of Marčenko–Pastur distribution of index [Formula: see text] and scale parameter [Formula: see text]. Moreover, we provide the limiting distribution of the largest eigenvalue of a sample covariance matrix of an equi-correlated normal population. We discuss the phase transition as to the decay rate of the equi-correlation coefficient in N.

P‐15.3: Color Speckle Evaluation Using Eigenvalues Method

In the current international standard (IEC 62906‐5‐4), the variance and covariance of color coordinates u’ and v’, as well as photometric speckle contrast are used to evaluate color speckle. There are some shortcomings in the covariance method that it is difficult for users to match the speckle distribution with covariance intuitively. In this paper, the eigenvalues of the sample covariance matrix generated by u’ and v’ are proposed to evaluate color speckle, which can be more intuitive and more suitable as an evaluation metric in measurement.

Efficient Matrix Polynomial Expansion Detector for Large-Scale MIMO: An Inverse-Transform-Sampling Approach

Matrix polynomial expansion (MPE) based detector incurs either complicated computation of polynomial coefficients or slow convergence in uplink large-scale multiple-input and multiple-output (LS-MIMO) systems. To solve these issues, an improved MPE (IMPE) detector is proposed, which can speed up the convergence significantly with uncomplicated polynomial coefficients. However, a challenging problem of performing IMPE is needed to compute all the eigenvalues of channel covariance matrix in real time. Unfortunately, directly calculating the eigenvalues of the channel covariance matrix requires complexity, which is as costly as the matrix inverse. To this end, an inverse-transform-sampling based IMPE (ITS-IMPE) detector is proposed to enhance the convergence rate and accuracy in a simple way. First, the closed-form expression of the eigenvalue spectral cumulative distribution function of the channel covariance matrix is deduced analytically, which is a critical factor that influence the eigenvalues estimation. Second, the improved polynomial coefficients of ITS-IMPE are then introduced by a fast online ITS-based eigenvalues estimation algorithm and a least-squares fitting procedure, which achieve a well trade-off between precision and computation. Simulation results exhibit that ITS-IMPE detector is able to achieve a significant enhancement performance with much lower complexity compared with many reported detectors under Rayleigh fading channel and low spatial correlated channel.

Revisiting Polarity Indeterminacy of ICA-Decomposed ERPs and Scalp Topographies.

We propose an alternative method to align the polarities of independent components (ICs) for group-level IC cluster analysis. Current methods are presently limited in how indeterminacy of IC polarities is handled, as when multiplying a weight matrix to a time-series IC activation, the result from 1 × 1 and - 1 × - 1 are indistinguishable. We first clarify the EEGLAB's default solution and define it as the iterative correlation maximization as it maximizes the within-cluster correlations of the IC scalp topographies to the cluster mean. We then propose the covariance maximization method, which determines the polarity of ICs based on the sign of the largest eigenvalue of covariance matrix. We compared the two methods on datasets from a published visual event-related potential (ERP) study. The results were similar when both methods were applied to the IC scalp topographies. However, when the proposed method was applied to IC ERPs, the number of clusters that showed significant ERP amplitudes increased from 5 to 9 out of 9 due to minimization of within-cluster ERP amplitude cancellation. Our study confirm covariance maximization provides an alternative solution to post-ICA group-level analysis that can maximize sensitivity of IC ERPs.

Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices

We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.

Eigenvalue Distribution of a High-Dimensional Distance Covariance Matrix With Application

We introduce a new random matrix model called distance covariance matrix in this paper, whose normalized trace is equivalent to the distance covariance. We first derive a deterministic limit for the eigenvalue distribution of the distance covariance matrix when the dimensions of the vectors and the sample size tend to infinity simultaneously. This limit is valid when the vectors are independent or weakly dependent through a finite-rank perturbation. It is also universal and independent of the details of the distributions of the vectors. Furthermore, the top eigenvalues of this distance covariance matrix are shown to obey an exact phase transition when the dependence of the vectors is of finite rank. This finding enables the construction of a new detector for such weak dependence where classical methods based on large sample covariance matrices or sample canonical correlations may fail in the considered high-dimensional framework.

How to Determine an Optimal Noise Subspace?

The multiple signal classification (MUSIC) algorithm based on the orthogonality between the signal subspace and noise subspace is one of the most frequently used methods in the estimation of direction of arrival (DOA), and its performance of DOA estimation mainly depends on the accuracy of the noise subspace. In the most existing researches, the noise subspace is formed by (defined as) the eigenvectors corresponding to all small eigenvalues of the array output covariance matrix. However, we found that the estimation of DOA through the noise subspace in the traditional formation is not optimal in almost all cases, and using a partial noise subspace can always obtain optimal estimation results. In other words, the subspace spanned by the eigenvectors corresponding to a part of the small eigenvalues is more representative of the noise subspace. We demonstrate this conclusion through a number of experiments. Thus, it seems that which and how many eigenvectors should be selected to form the partial noise subspace would be an interesting issue. In addition, this research poses a much general problem: how to select eigenvectors to determine an optimal noise subspace?