Eigenvalues Of Covariance Matrix Research Articles

Maximum Eigenvalue-Based Goodness-of-Fit Detection for Spectrum Sensing in Cognitive Radio

The state-of-the-art goodness-of-fit (GoF) test algorithms for spectrum sensing directly make decisions based on temporal samples or energies of the observations, which perform well under the condition that the received primary user (PU) signals are independent. However, when the received PU signals are highly correlated, these methods cannot achieve satisfactory performance. In this case, the correlation feature is the main character of the signals and the maximum eigenvalue of the sample covariance matrix is a versatile statistic reflecting the correlation feature. Motivated by this, we make full use of the correlation feature involved in the eigenvalues to improve the GoF detection performance. Specifically, we first study the GoF test in eigenvalue domain and design a semi-blind maximum eigenvalue-based GoF detection scheme using the ratio of the maximum eigenvalue to the noise power. Considering that the accurate knowledge of the noise power is not always available in practice, we next design a totally blind maximum eigenvalue-based GoF detection method, which only uses the ratio of the maximum eigenvalue to the trace. Utilizing the recent results of random matrix theory, we provide a theoretical analysis of the proposed methods, including the probabilities of false alarm, detection thresholds, and probabilities of detection. Finally, simulation results show that the proposed algorithms outperform the related GoF detection methods in terms of the detection performance.

Tight Analytical and Asymptotic Upper Bound for the BER and FER of Linear Codes Over Exponentially Correlated Generalized-Fading Channels

In this paper, the upper bound and asymptotical approximation for the bit error rate (BER) as well as for the frame error rate of linear block codes under maximum-likelihood decoding operating over exponentially correlated (EC) generalized-fading channels are proposed. Particularly, our analysis takes the convolutional codes as an example and three different fading scenarios are considered, namely Rice, Nakagami- $m$ , and Weibull fading. Such scenarios have a common property that is spherically invariant random process (SIRP). Our analysis relies on the derivation of new lower bounds per each eigenvalue since the BER as well as the FER expressions are written in terms of the pairwise error probability, which by its turn depends on the eigenvalues of the channel covariance matrix. Based on that, the upper bounds and asymptotic expressions of the considered metrics subject to the three above fading models are derived, which show to be tighter than the previous results published elsewhere in the literature, especially for the high signal-to-noise ratio regions.

Robust Noise Suppression Technique for a LADAR System via Eigenvalue-Based Adaptive Filtering.

The laser detection and ranging system (LADAR) is widely used in various fields that require 3D measurement, detection, and modeling. In order to improve the system stability and ranging accuracy, it is necessary to obtain the complete waveform of pulses that contain target information. Due to the inevitable noise, there are distinct deviations between the actual and expected waveforms, so noise suppression is essential. To achieve the best effect, the filters’ parameters that are usually set as empirical values should be adaptively adjusted according to the different noise levels. Therefore, we propose a novel noise suppression method for the LADAR system via eigenvalue-based adaptive filtering. Firstly, an efficient noise level estimation method is developed. The distributions of the eigenvalues of the sample covariance matrix are analyzed statistically after one-dimensional echo data are transformed into matrix format. Based on the boundedness and asymptotic properties of the noise eigenvalue spectrum, an estimation method for noise variances in high dimensional settings is proposed. Secondly, based on the estimated noise level, an adaptive guided filtering algorithm is designed within the gradient domain. The optimized parameters of the guided filtering are set according to an estimated noise level. Through simulation analysis and testing experiments on echo waves, it is proven that our algorithm can suppress the noise reliably and has advantages over the existing relevant methods.

Study of Subjective Quality and Objective Blind Quality Prediction of Stereoscopic Videos.

We present a new subjective and objective study on full high-definition (HD) stereoscopic (3D or S3D) video quality. In subjective study, we constructed an S3D video dataset with 12 pristine and 288 test videos, and the test videos are generated by applying the H.264 and H.265 compression, blur and frame freeze artifacts. We also propose a no reference (NR) objective video quality assessment (QA) algorithm that relies on measurements of the statistical dependencies between the motion and disparity subband coefficients of S3D videos. Inspired by the Generalized Gaussian Distribution (GGD) approach in liu2011statistical, we model the joint statistical dependencies between the motion and disparity components as following a Bivariate Generalized Gaussian Distribution (BGGD). We estimate the BGGD model parameters (α,β) and the coherence measure (Ψ) from the eigenvalues of the sample covariance matrix (M) of the BGGD. In turn, we model the BGGD parameters of pristine S3D videos using a Multivariate Gaussian (MVG) distribution. The likelihood of a test video's MVG model parameters coming from the pristine MVG model is computed and shown to play a key role in the overall quality estimation. We also estimate the global motion content of each video by averaging the SSIM scores between pairs of successive video frames. To estimate the test S3D video's spatial quality, we apply the popular 2D NR unsupervised NIQE image QA model on a frame-by-frame basis on both views. The overall quality of a test S3D video is finally computed by pooling the test S3D video's likelihood estimates, global motion strength and spatial quality scores. The proposed algorithm, which is 'completely blind' (requiring no reference videos or training on subjective scores) is called the Motion and Disparity based 3D video quality evaluator (MoDi3D). We show that MoDi3D delivers competitive performance over a wide variety of datasets including the IRCCYN dataset, the WaterlooIVC Phase I dataset, the LFOVIA dataset and our proposed LFOVIAS3DPh2 S3D video dataset.

High-dimensional Markowitz portfolio optimization problem: empirical comparison of covariance matrix estimators

ABSTRACTWe compare the performance of recently developed regularized covariance matrix estimators for Markowitz's portfolio optimization and of the minimum variance portfolio (MVP) problem in particular. We focus on seven estimators that are applied to the MVP problem in the literature; three regularize the eigenvalues of the sample covariance matrix, and the other four assume the sparsity of the true covariance matrix or its inverse. Comparisons are made with two sets of long-term S&P 500 stock return data that represent two extreme scenarios of active and passive management. The results show that the MVPs with sparse covariance estimators have high Sharpe ratios but that the naive diversification (also known as the ‘uniform (on market share) portfolio’) still performs well in terms of wealth growth.

Discriminant Analysis for the Eigenvalues of Variance Covariance Matrix of FFT Scaling of DNA Sequences: An Empirical Study of Some Organisms

Many studies discussed different numerical representations of DNA sequences. One naive approach for exploring the nature of a DNA sequence is to assign numerical values (or scales) to the nucleotides and then proceed with standard time series methods. The analysis will depend actually on the particular assignment of numerical values.Discriminant analysis aims to examine the dependence of one qualitative (classification) variable from several quantitative variables according to number of variations of qualitative variable we can distinction. Actually, there is a discriminant analysis for two or more groups. The essential work of discriminant analysis is to get the optimal assigning rules that will minimize the likelihood of incorrect classification of elements. In this paper, we discussed the discriminant analysis of the first, second, third and fourth eigenvalues of variance covariance matrix of Fast Fourier Transform (FFT) for numerical values representation of DNA sequences of five organisms, Human, E. coli, Rat, Wheat and Grasshopper. The analysis is based on three methods (All Variables, Forward Selection and Backward Selection) of discrimination. Functions have been reached whereby discrimination is made among organisms under consideration. Empirical studies are conducted to show the value of our point of view and the applications based on. Therefore, we recommended that, other empirical studies should be done for other organisms and statistical methods by using the point of view adopted here. Also, aspects stated here must be used in an applied manner for DNA sequences discrimination.

Low-Dose CT Image Denoising Model Based on Sparse Representation by Stationarily Classified Sub-Dictionaries

Low-dose computed tomography (LDCT) technique is an important imaging modality, but LDCT images are always severely degraded by mottle noise and streak artifacts. The recently proposed nonlocally centralized sparse representation (NCSR) algorithm has good performance in natural image denoising, but it suffers from residual streak artifacts and can't preserve edges structure information well when implemented in LDCT image denoising. In addition, it has high computational complexity. To address this problem, in this paper, we propose an improved model, i.e. SNCSR model, based on the stationary PCA sub-dictionaries, nonlocally centralized sparse representation and relative total variation. In the SNCSR model, in order to learn more accurate sub-dictionaries, the LDCT image is preprocessed by the improved total variation (ITV) model in which the weighted coefficient of the regularization term is constructed depending on a clipped and normalized local activity. In addition, the maximum eigenvalue of the gradient covariance matrix of the image patch is used to distinguish edge structure information from background region so that the restored image can be represented more sparsely. Moreover, unlike the NCSR model that needs to learn sub-dictionaries in each outer loop, the proposed model learns stationary sub-dictionaries only once before iteration starts, which shorten the computation time significantly. At last, the relative total variation (RTV) algorithm is applied to further reduce the residual artifacts in the recovered image more thoroughly. The experiments are performed on the simulated pelvis phantom, the actual thoracic phantom and the clinical abdominal data. Compared with several other competitive denoising algorithms, both subjective visual effect and objective evaluation criteria show that the proposed SNCSR model has lower computational complexity and can improve LDCT images quality more effectively.

Estimation of Weak Factor Models

This paper investigates estimation of sparsity-induced weak factor (sWF) models, with large cross-sectional and time-series dimensions (N and T, respectively). It assumes that the kth largest eigenvalue of a data covariance matrix grows proportionally to N^alpha_k with unknown exponents 0 < alpha_k <= 1 for k=1,...,r. Employing the same rotation of the principal components (PC) estimator, the growth rate alpha_k is linked to the degree of sparsity of kth factor loadings. This is much weaker than the typical assumption on the recent factor models, in which all the r largest eigenvalues diverge proportionally to N. We apply the method of sparse orthogonal factor regression (SOFAR) by Uematsu et al (2018) to estimate the sWF models and derive the estimation error bound. Importantly, our method also yields consistent estimation of alpha_k. A finite sample experiment shows that the performance of the new estimator uniformly dominates that of the PC estimator. We apply our method to forecasting bond yields and the results demonstrate that our method outperforms that based on the PC. We also analyze S&P500 firm security returns and find that the first factor is consistently near strong while the others are weak.

Continuous Estimation Using Context-Dependent Discrete Measurements

This paper considers the problem of continuous state estimation from discrete context-based measurements. Context measurements provide binary information as obtained from the system's environment, e.g., a medical alarm indicating that a vital sign is above a certain threshold. Since they provide state information, these measurements can be used for estimation purposes, similar to standard continuous measurements, especially when standard sensors are biased or attacked. Context measurements are assumed to have a known probability of occurring given the state; in particular, we focus on the probit function to model threshold-based measurements, such as the medical-alarm scenario. We develop a recursive context-aware filter by approximating the posterior distribution with a Gaussian distribution with the same first two moments as the true posterior. We show that the filter's expected uncertainty is bounded when the probability of receiving context measurements is lower bounded by some positive number for all system states. Furthermore, we provide an observability-like result—all eigenvalues of the filter's covariance matrix converge to 0 after repeated updates if and only if a persistence of excitation condition holds for the context measurements. Finally, in addition to simulation evaluations, we applied the filter to the problem of estimating a patient's blood oxygen content during surgery using real-patient data.

Analysis of Semi-Blind Channel Estimation in Multiuser Massive MIMO Systems With Perturbations

In the massive multiple-input multiple-output (MIMO) systems, pilot contamination and signal perturbation are two important issues in the semi-blind channel estimation methods. To evaluate the effects of pilot contamination and signal perturbation, the covariance matrix of the received signal should be obtained. In specific, the eigenvectors associated with the K largest eigenvalues of the ideal covariance matrix, contain the subspace information of the desired signals, in which the interference and partial noise are eliminated because of asymptotic orthogonality of the fading matrix. Since it is quite difficult to acquire the ideal covariance matrix, the sample covariance matrix calculated by the finite data samples is adopted to approximate the ideal covariance matrix in practice. However, the gap between the ideal covariance matrix and the sample covariance matrix leads to signal perturbation. In this paper, the perturbation analysis of a semi-blind channel estimation in the massive MIMO systems is presented. Simulation results verify the validity of our perturbation analysis. Furthermore, it is demonstrated by simulations that signal perturbation affects system performance by reshaping the eigenvectors and eigenvalue distributions of the target cell and interfering cells.

Dynamic Conditional Eigenvalue GARCH

In this paper we consider a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models where the eigenvalues of the conditional covariance matrix are time-varying. The proposed dynamics of the eigenvalues is based on applying the general theory of dynamic conditional score models as proposed by Creal, Koopman and Lucas (2013) and Harvey (2013). We denote the obtained GARCH model with dynamic conditional eigenvalues (and constant conditional eigenvectors) as the λ-GARCH model. We provide new results on asymptotic theory for the Gaussian QMLE, and for testing of reduced rank of the (G)ARCH loading matrices of the time-varying eigenvalues. The theory is applied to US data, where we find that the eigenvalue structure can be reduced similar to testing for the number in factors in volatility models.

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.

Modified Dominant Mode Rejection Beamformer for Localizing Brain Activities When Data Covariance Matrix Is Rank Deficient

Minimum variance beamformer (MVB) and its extensions fail in localizing short time brain activities particularly evoked potentials because of rank deficiency or inaccurate estimation of a data covariance matrix. In this paper, the conventional dominant mode rejection (DMR) adaptive beamformer is modified to localize brain short time activities. In the modified DMR, it is attempted to obtain a well-conditioned covariance matrix by dividing the eigenvalues of the data covariance matrix into dominant, medium, and small eigenvalues and then modifying medium and small parts. The performance of the proposed approach is compared with diagonal loading MVB (DL_MVB) and fast fully adaptive (FFA) beamformer by using simulated event-related potentials and real event-related field data. Eigenspace versions of DL_MVB and modified DMR are also implemented. In all simulations, the modified DMR obtains the least localization error (0-5mm) and spread radius (0-8mm) when the signal-to-noise ratio (SNR) varies from 0 to 10dB with step 1dB. In real data, the new approach in comparison to two other ones attains the most concentrated power spectrum. Eigenspace projection of DL_MVB presents better results than DL_MVB but worse results than the modified DMR. Applying eigenspace projection on the proposed method improves its performance at high SNR levels. Empirical results illustrate the superiority of the proposed DMR method to the DL_MVB and FFA in localizing brain short time activities. The proposed method can be utilized in source localization of epilepsy for presurgical clinical evaluation purpose and also in applications dealing with the localization of evoked potentials and fields.

Consistent Estimation of Dimensionality for Data-Driven Methods in fMRI Analysis.

Data-driven methods, such as principal component analysis and independentcomponent analysis, have been successfully applied to functionalmagnetic resonance imaging (fMRI) data in particular and neuro-imaging data in general. A central issue of thesemethods is the importance of correctly selecting the number of components to be used in the factor model. This issue is often addressed using a model selection criterion, where the goodness-of-fit term is obtained from the log-likelihood function. In this paper, an alternative criterion is proposed for selecting the number of components. Unlike existingmodel selection criteria that use the log-likelihood function, the proposed goodness-of-fit termuses the sum of squares of the smallest eigenvalues of the sample covariance matrix. The proposed criterion is obtained from the asymptotic distribution of the goodness-of-fit term, for which consistency is established. This criterion has a straight-forward implementation and is shown to outperform conventional model selection criteria used in fMRI data analysis. Experiments are conducted using simulated and real fMRI data, in which improved performance is obtained by the proposed criterion, both in terms of accuracy and consistency under data variabilities.

Parallel Direct Solution of the Covariance-Localized Ensemble Square Root Kalman Filter Equations with Matrix Functions

Abstract Recently, the serial approach to solving the square root ensemble Kalman filter (ESRF) equations in the presence of covariance localization was found to depend on the order of observations. As shown previously, correctly updating the localized posterior covariance in serial requires additional effort and computational expense. A recent work by Steward et al. details an all-at-once direct method to solve the ESRF equations in parallel. This method uses the eigenvectors and eigenvalues of the forward observation covariance matrix to solve the difficult portion of the ESRF equations. The remaining assimilation is easily parallelized, and the analysis does not depend on the order of observations. While this allows for long localization lengths that would render local analysis methods inefficient, in theory, an eigenpair-based method scales as the cube number of observations, making it infeasible for large numbers of observations. In this work, we extend this method to use the theory of matrix functions to avoid eigenpair computations. The Arnoldi process is used to evaluate the covariance-localized ESRF equations on the reduced-order Krylov subspace basis. This method is shown to converge quickly and apparently regains a linear scaling with the number of observations. The method scales similarly to the widely used serial approach of Anderson and Collins in wall time but not in memory usage. To improve the memory usage issue, this method potentially can be used without an explicit matrix. In addition, hybrid ensemble and climatological covariances can be incorporated.

Asymptotics for empirical eigenvalue processes in high-dimensional linear factor models

When vector-valued observations are of high dimension N relative to the sample size T, it is common to employ a linear factor model in order to estimate the underlying covariance structure or to further understand the relationship between coordinates. Asymptotic analyses of such models often consider the case in which both N andT tend jointly to infinity. Within this framework, we derive weak convergence results for processes of partial sample estimates of the largest eigenvalues of the sample covariance matrix. It is shown that if the effect of the factors is sufficiently strong, then the processes associated with the largest eigenvalues have Gaussian limits under general conditions on the divergence rates of N andT, and the underlying observations. If the common factors are “weak”, then N must grow much more slowly in relation to T in order for the largest eigenvalue processes to have a Gaussian limit. We apply these results to develop general tests for structural stability of linear factor models that are based on measuring the fluctuations in the largest eigenvalues throughout the sample, which we investigate further by means of a Monte Carlo simulation study and an application to US treasury yield curve data.

The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model

We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $\alpha$-stable limit distributions of the sample covariance matrix. For the case of heavy-tailed innovations, we show that the limiting behavior resembles that of completely independent observations. In contrast to this, for a heavy-tailed volatility sequence the possible limiting behavior is more diverse and allows for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

Transmitter beamforming and weighted image fusion–based multiple signal classification algorithm for corrosion monitoring

With the increase in aging aircrafts, corrosion monitoring has attracted much attention in the structural health monitoring area. Multiple signal classification has been gradually applied to structural health monitoring area as a new promising method because of its ability of directional scanning and the potential to monitor multiple signal sources. However, applying multiple signal classification algorithm to monitor real damage still faces some challenges. First, the scattered Lamb waves obtained using a single actuator is relatively weak, making the signal-to-noise ratio of the scattered signals low and resulting in the low precision of multiple signal classification–based monitoring. Second, linear sensor array–based structural health monitoring methods have the problem of blind area at the angles close to 0° and 180°. To meet these challenges and target at providing monitoring ability of both the position and severity of the damage, a novel transmitter beamforming and weighted image fusion–based multiple signal classification algorithm is proposed using a dual array that consists of two linear sensor arrays to enhance the amplitude of scattered Lamb waves from corrosion, improve its signal-to-noise ratio and eliminate the blind area. The corrosion severity can be evaluated by analyzing the largest eigenvalue of signal covariance matrix developed using the multiple signal classification algorithm. The proposed transmitter beamforming and weighted image fusion–based multiple signal classification algorithm is verified on aluminum plates with real corrosion damages at five stages. Experimental results show that the proposed method can realize corrosion monitoring with a good precision even at the blind monitoring area.

The blended dominant mode rejection adaptive beamformer

Adaptive beamformers balance competing demands of enhancing desired signals, suppressing discrete interferers and attenuating background noise. This requires an accurate estimate of the spatial covariance matrix that is challenging to obtain in nonstationary environments. The dominant mode rejection (DMR) beamformer [Abraham & Owsley, Oceans, 1990] addresses this challenge by replacing the sample covariance matrix (SCM) eigenvalues of the noise subspace with the average of these eigenvalues to improve its conditioning. The DMR beamformer requires an estimate of the interferer subspace dimension, and the DMR beamformer's performance degrades when this estimated dimension is inaccurate. We propose the Blended DMR beamformer, whose array weights are formed as a mixture of fixed-dimension DMR beamformer array weights across a range of interferer dimensions. The contribution of each fixed-dimension DMR beamformer is a function of its recent performance in suppressing interferers and attenuating noise. The perf...

The conditioning of least‐squares problems in variational data assimilation

SummaryIn variational data assimilation a least‐squares objective function is minimised to obtain the most likely state of a dynamical system. This objective function combines observation and prior (or background) data weighted by their respective error statistics. In numerical weather prediction, data assimilation is used to estimate the current atmospheric state, which then serves as an initial condition for a forecast. New developments in the treatment of observation uncertainties have recently been shown to cause convergence problems for this least‐squares minimisation. This is important for operational numerical weather prediction centres due to the time constraints of producing regular forecasts. The condition number of the Hessian of the objective function can be used as a proxy to investigate the speed of convergence of the least‐squares minimisation. In this paper we develop novel theoretical bounds on the condition number of the Hessian. These new bounds depend on the minimum eigenvalue of the observation error covariance matrix and the ratio of background error variance to observation error variance. Numerical tests in a linear setting show that the location of observation measurements has an important effect on the condition number of the Hessian. We identify that the conditioning of the problem is related to the complex interactions between observation error covariance and background error covariance matrices. Increased understanding of the role of each constituent matrix in the conditioning of the Hessian will prove useful for informing the choice of correlated observation error covariance matrix and observation location, particularly for practical applications.