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Covariance matrix tapers have a long history in signal processing and related fields. Examples of applications include autoregressive models (promoting a banded structure) or beamforming (widening the spectral null width associated with an interferer). In this paper, the focus is on high-dimensional setting where the dimension $p$ is high, while the data aspect ratio $n/p$ is low. We propose an estimator called Tabasco (TApered or BAnded Shrinkage COvariance matrix) that shrinks the tapered sample covariance matrix towards a scaled identity matrix. We derive optimal and estimated (data adaptive) regularization parameters that are designed to minimize the mean squared error (MSE) between the proposed shrinkage estimator and the true covariance matrix. These parameters are derived under the general assumption that the data is sampled from an unspecified elliptically symmetric distribution with finite 4th order moments (both real- and complex-valued cases are addressed). Simulation studies show that the proposed Tabasco outperforms all competing tapering covariance matrix estimators in diverse setups. An application to space-time adaptive processing (STAP) also illustrates the benefit of the proposed estimator in a practical signal processing setup. [ABSTRACT FROM AUTHOR]

A simple closed-form expression is derived for the Cramer-Rao bound (CRB) of the demixing matrix directly from its definition. The independent component analysis (ICA) estimators are compared with the CRB.

A stochastic maximum-likelihood (ML) method is derived for estimating spatio-temporal parameters for multiple-input multiple-output (MIMO) channels. It is found that the estimator is robust when applied to data generated by channel models other than the one used in its derivation.

We consider the problem of estimating high-dimensional covariance matrices of $K$ -populations or classes in the setting where the sample sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an asymptotically unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data. [ABSTRACT FROM AUTHOR]

The spatial sign covariance matrix (SSCM), also known as the normalized sample covariance matrix (NSCM), has been widely used in signal processing as a robust alternative to the sample covariance matrix (SCM). It is well-known that the SSCM does not provide consistent estimates of the eigenvalues of the shape matrix (normalized scatter matrix). To alleviate this problem, we propose BASIC (Bias Adjusted SIgn Covariance), which performs an approximate bias correction to the eigenvalues of the SSCM under the assumption that the samples are generated from zero mean unspecified complex elliptically symmetric distributions (the real-valued case is also addressed). We then use the bias correction in order to develop a robust regularized SSCM based estimator, BASIC Shrinkage estimator (BASICS), which is suitable for high dimensional problems, where the dimension can be larger than the sample size. We assess the proposed estimator with several numerical examples as well as in a linear discriminant analysis (LDA) classification problem with real data sets. The simulations show that the proposed estimator compares well to competing robust covariance matrix estimators but has the advantage of being significantly faster to compute. [ABSTRACT FROM AUTHOR]

We derive the form of the variance-covariance matrix for any affine equivariant matrix-valued statistics when sampling from complex elliptical distributions. We then use this result to derive the variance-covariance matrix of the sample covariance matrix (SCM) as well as its theoretical mean squared error (MSE) when finite fourth-order moments exist. Finally, illustrative examples of the formulas are presented. [ABSTRACT FROM AUTHOR]

A highly popular regularized (shrinkage) covariance matrix estimator is the shrinkage sample covariance matrix (SCM) which shares the same set of eigenvectors as the SCM but shrinks its eigenvalues toward the grand mean of the eigenvalues of the SCM. In this paper, a more general approach is considered in which the SCM is replaced by an M-estimator of scatter matrix and a fully automatic data adaptive method to compute the optimal shrinkage parameter with minimum mean squared error is proposed. Our approach permits the use of any weight function such as Gaussian, Huber's, Tyler's, or t weight functions, all of which are commonly used in M-estimation framework. Our simulation examples illustrate that shrinkage M-estimators based on the proposed optimal tuning combined with robust weight function do not loose in performance to shrinkage SCM estimator when the data is Gaussian, but provide significantly improved performance when the data is sampled from an unspecified heavy-tailed elliptically symmetric distribution. Also, real-world and synthetic stock market data validate the performance of the proposed method in practical applications. [ABSTRACT FROM AUTHOR]

Ultrasound imaging (UI) is characterized by the presence of multiplicative speckle noise and various acquisition artefacts. Designing ultrasound (US) similarity measures thus requires a particular attention. In the specific context of motion estimation, incorporating US characteristics does not only benefit traditional methods but also learning-based approaches, which are highly sensitive to the quality of training data. Deriving similarity measures from a maximum likelihood (ML) perspective allows us to take these specificities into account. As opposed to the classical Rayleigh modelling, the proposed similarity measures incorporate more realistic scattering conditions, such as, varying speckle densities and shadowing. Specifically, the deviations from the Rayleigh statistics are modelled using the $t$ -distribution for the complex radio-frequency (RF) signals and the Nakagami-Gamma (NG) compound model for the echo amplitudes. Furthermore, the model parameters are learnt patch-wise, which leads to data-adaptive similarity measures. The proposed criteria are investigated in the context of motion estimation using synthetic, phantom, as well as 2D and 3D in vivo images. The experimental results show an improvement in performance and robustness in comparison to the classical Rayleigh-based approach and state-of-the-art similarity measures. [ABSTRACT FROM AUTHOR]

This paper considers the problem of estimating a high-dimensional covariance matrix in a low sample support situation where the sample size is smaller, or not much larger, than the dimensionality of the data, which could potentially be very large. We develop a regularized sample covariance matrix (RSCM) estimator which can be applied in commonly occurring high-dimensional data problems. The proposed RSCM estimator is based on estimators of the unknown optimal (oracle) shrinkage parameters that yield the minimum mean squared error between the RSCM and the true covariance matrix when the data is sampled from an unspecified elliptically symmetric distribution. We propose two variants of the RSCM estimator which differ in the approach in which they estimate the underlying sphericity parameter involved in the theoretical optimal shrinkage parameter. The performance of the proposed RSCM estimators are evaluated with numerical simulation studies. In particular, when the sample sizes are low, the proposed RSCM estimators often show a significant improvement over the conventional RSCM estimator by Ledoit and Wolf (2004). We further evaluate the performance of the proposed estimators in a portfolio optimization problem with real data wherein the proposed methods are able to outperform the benchmark methods. [ABSTRACT FROM AUTHOR]

This paper proposes efficient algorithms for accurate recovery of direction-of-arrivals (DoAs) of sources from single-snapshot measurements using compressed beamforming (CBF). In CBF, the conventional sensor array signal model is cast as an underdetermined complex-valued linear regression model and sparse signal recovery methods are used for solving the DoA finding problem. A complex-valued pathwise weighted elastic net (c-PW-WEN) algorithm is developed that finds solutions at the knots of penalty parameter values over a path (or grid) of elastic net (EN) tuning parameter values. c-PW-WEN also computes least absolute shrinkage and selection operator (LASSO) or weighted LASSO in its path. A sequential adaptive EN (SAEN) method is then proposed that is based on c-PW-WEN algorithm with adaptive weights that depend on previous solution. Extensive simulation studies illustrate that SAEN improves the probability of exact recovery of true support compared to conventional sparse signal recovery approaches such as LASSO, EN, or orthogonal matching pursuit in several challenging multiple target scenarios. The effectiveness of SAEN is more pronounced in the presence of high mutual coherence. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of performing statistical inference for large scale data sets i.e., Big Data. The volume and dimensionality of the data may be so high that it cannot be processed or stored in a single computing node. We propose a scalable, statistically robust and computationally efficient bootstrap method, compatible with distributed processing and storage systems. Bootstrap resamples are constructed with smaller number of distinct data points on multiple disjoint subsets of data, similarly to the bag of little bootstrap method (BLB) [A. Kleiner, A. Talwalkar, P. Sarkar, and M. I. Jordan, “A scalable bootstrap for massive data,” J. Roy. Statist. Soc.: Ser. B (Statist. Methodol.), vol. 76, no. 4, pp. 795–816, 2014]. The disjoint subsets are significantly smaller than the original full data set and they may be processed in different storage and computing units in parallel. Then significant savings in computation is achieved by avoiding the recomputation of the estimator for each bootstrap sample. Instead, a computationally efficient fixed-point estimation equation is analytically solved via a smart approximation following the Fast and Robust Bootstrap method (FRB) [M. Salibián-Barrera, S. Van Aelst, and G. Willems, “Fast and robust bootstrap,” Statist. Methods Appl., vol. 17, no. 1, pp. 41–71, 2008]. Our proposed bootstrap method facilitates the use of highly robust statistical methods in analyzing large scale data sets. The favorable statistical properties of the method are established analytically. Numerical examples demonstrate scalability, low complexity and robust statistical performance of the method in analyzing large data sets. [ABSTRACT FROM AUTHOR]

Complex elliptically symmetric (CES) distributions have been widely used in various engineering applications for which non-Gaussian models are needed. In this overview, circular CES distributions are surveyed, some new results are derived and their applications e.g., in radar and array signal processing are discussed and illustrated with theoretical examples, simulations and analysis of real radar data. The maximum likelihood (ML) estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness, and for convergence of the iterative fixed point algorithm are established. Specific ML-estimators for several CES distributions that are widely used in the signal processing literature are discussed in depth, including the complex t-distribution, K-distribution, the generalized Gaussian distribution and the closely related angular central Gaussian distribution. A generalization of ML-estimators, the M-estimators of the scatter matrix, are also discussed and asymptotic analysis is provided. Applications of CES distributions and the adaptive signal processors based on ML- and M-estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for Direction-of-Arrival (DOA) estimation and beamforming. Furthermore, experimental validation of the usefulness of CES distributions for modelling real radar data is given. [ABSTRACT FROM PUBLISHER]

In this correspondence, we provide a multiple hypothesis test to detect the number of latent noncircular signals in a complex Gaussian random vector. Our method sequentially tests the results of individual generalized likelihood ratio test (GLRT) statistics with known asymptotic distributions to form the multiple hypothesis detector. Specifically, we are able to set a threshold yielding a precise probability of error. This test can be used to statistically determine if a given complex observation is circular Gaussian, and if not, how many latent signals in the observation are noncircular. Simulations are used to quantify the performance of the detector as compared to a detector based on the minimum description length (MDL) criterion. The utility of the detector is shown by applying it to a beamforming application using independent component analysis (ICA). [ABSTRACT FROM AUTHOR]

Complex elliptically symmetric (CES) distributions constitute a flexible and broad class of distributions for many engineering applications and include the widely used complex Gaussian, complex t-, complex generalized Gaussian and symmetric \alpha-stable distributions for example. Their careful statistical characterization is needed. Moreover, the mechanisms for generating random variables from these distributions are not well defined in literature. In this paper we provide such treatment in order to provide a better insight on their statistical properties and simplifying proofs and derivations. For example, insightful expressions for complex kurtosis coefficients of CES distributions are derived providing an interpretation on what complex kurtosis really measures. Also derived are asymptotic distributions of circularity measures, the sample circularity quotient and coefficient, assuming i.i.d. samples from an unspecified CES distribution. Also new Wald's type detectors of circularity are proposed that are valid within CES distributions with finite fourth-order moments. These results are accompanied with examples that are of interest in developing signal processing algorithms for complex-valued signals. [ABSTRACT FROM PUBLISHER]

This paper provides a rigorous statistical analysis of the deflation-based FastICA estimator, where the independent components (ICs) are extracted sequentially. The focus is on two aspects of the estimator: robustness against outliers as measured by the influence function (IF) and on its asymptotic relative efficiency (ARE) as measured by the ratio of the asymptotic variance of the FastICA w.r.t. the optimal maximum likelihood estimator (MLE). The derived compact closed-form expression of the IF reveals the vulnerability of the FastICA estimator to outliers regardless of the used nonlinearity. A cautionary finding is that even a moderate observation towards certain directions can render the estimator deficient in the sense that its separation performance degrades worse than a plain guess. The IF allows the derivation of a compact closed-form expression for the asymptotic covariance matrix of the FastICA estimator and subsequently its asymptotic relative efficiencies (AREs). The ARE figures calculated for some selected source distributions illustrate the fact that the order which the ICs are found is crucial as the accuracy of the previously extracted components can dominantly affect the accuracy of the successive deflation stages. [ABSTRACT FROM AUTHOR]

In this paper, we consider array processors that are scale-invariant functions of the array covariance matrix. The emphasis is on Capon's MVDR beamformer. We call such an array processor as scatter matrix based (SMB) array processor since the covariance matrix is required only up to a constant scalar and thus a scatter matrix (proportional to covariance under finite co- variance assumption) provides sufficient information. In order to establish interesting statistical robustness and large sample properties, we derive a general expression for the influence function and the asymptotic covariance structure of SMB-MVDR beam- former weights. Our results apply under the class of complex elliptically symmetric distributions, which includes the commonly used complex normal distribution as a special case. We illustrate the theory by deriving the IF and asymptotic relative efficiencies of the conventional SMB-MVDR beamformer that employs the sample covariance matrix and beamformers that employ robust M-estimators of scatter. Theoretical findings are confirmed by simulations. Our findings favor beamformers based upon M-estimators of scatter, since they combine a high efficiency with appealing robustness properties. [ABSTRACT FROM AUTHOR]

An important characteristic of a complex random variable z is the so-called circularity property or lack of it. We study the properties of the degree of circularity based on second-order moments, called circularity quotient, that is shown to possess an intuitive geometrical interpretation: the modulus and phase of its principal square-root are equal to the eccentricity and angle of orientation of the ellipse defined by the covariance matrix of the real and imaginary part of z. Hence, when the eccentricity approaches the minimum zero (ellipse is a circle), the circularity quotient vanishes; when the eccentricity approaches the maximum one, the circularity quotient lies on the unit complex circle. Connection with the correlation coefficient p is established and bounds on p given the circularity quotient land vice versa) are derived. A generalized likelihood ratio test (GLRT) of circularity assuming complex normal sample is shown to be a function of the modulus of the circularity quotient with asymptotic X²2distribution. [ABSTRACT FROM AUTHOR]

• We formulate graph error models for the adjacency matrix, which help to quantify the deviation from the true matrix using a few parameters. • We study the structural effects of the proposed error models on the adjacency matrix. • The effects of different type of errors in adjacency matrix specification are illustrated in filtering of graph signal and ICA of graph signals. The first step for any graph signal processing (GSP) procedure is to learn the graph signal representation, i.e., to capture the dependence structure of the data into an adjacency matrix. Indeed, the adjacency matrix is typically not known a priori and has to be learned. However, it is learned with errors. A little attention has been paid to modelling such errors in the adjacency matrix, and studying their effects on GSP methods. However, modelling errors in the adjacency matrix will enable both to study the graph error effects in GSP and to develop robust GSP algorithms. In this paper, we therefore introduce practically justifiable graph error models. We also study, both analytically when possible and numerically, the graph error effect on the performance of GSP methods in different types of problems such as filtering of graph signals and independent component analysis of graph signals (graph decorrelation). [ABSTRACT FROM AUTHOR]