Structure Of Covariance Matrix Research Articles

Persymmetric adaptive detection with improved robustness to steering vector mismatches

We exploit persymmetry to study the adaptive detection problem with multiple observations in partially homogeneous environments where noise shares the same covariance matrix up to different power levels between the test and training data. A persymmetric subspace model is designed for taking into account steering vector mismatches. Based on the persymmetric subspace model, we propose adaptive detectors in partially homogeneous environments, according to the criteria of two-step generalized likelihood ratio test (GLRT), Wald test, and Rao test. It is found that the proposed GLRT and Wald test coincide, while the Rao test does not exist. The proposed detector is proved to exhibit a constant false alarm rate property against both the covariance matrix structure and the scaling factor. Numerical examples show that the proposed detector, compared to its counterparts, is more robust to steering vector mismatches.

Asymptotic Distributions of Empirical Interaction Information

Interaction Information is one of the most promising interaction strength measures with many desirable properties. However, its use for interaction detection was hindered by the fact that apart from the simple case of overall independence, asymptotic distribution of its estimate has not been known. In the paper we provide asymptotic distributions of its empirical versions which are needed for formal testing of interactions. We prove that for three-dimensional nominal vector normalized empirical interaction information converges to the normal law unless the distribution coincides with its Kirkwood approximation. In the opposite case the convergence is to the distribution of weighted centred chi square random variables. This case is of special importance as it roughly corresponds to interaction information being zero and the asymptotic distribution can be used for construction of formal tests for interaction detection. The result generalizes result in Han (Inf Control 46(1):26–45 1980) for the case when all coordinate random variables are independent. The derivation relies on studying structure of covariance matrix of asymptotic distribution and its eigenvalues. For the case of 3 × 3 × 2 contingency table corresponding to study of two interacting Single Nucleotide Polymorphisms (SNPs) for prediction of binary outcome, we provide complete description of the asymptotic law and construct approximate critical regions for testing of interactions when two SNPs are possibly dependent. We show in numerical experiments that the test based on the derived asymptotic distribution is easy to implement and yields actual significance levels consistently closer to the nominal ones than the test based on chi square reference distribution.

Rank-Ordered Analysis of Consumer Preferences for the Attributes of a Value-Added Biofuel Co-Product

Biochar is a co-product of the production of advanced biofuels that sequesters carbon when used as a soil amendment. Gardening consumers are a potential market for biochar and their purchase of biochar-amended products could provide biofuel producers with an additional revenue stream. To better understand this opportunity, preferences for the attributes of potting soils amended with biochar were elicited using a best-worst scaling experiment administered in a survey of 880 Tennessee households. The attributes analyzed were whether the biochar was produced in Tennessee, certified as biobased, a coproduct of biofuel production, and produced from food waste, wood waste, agricultural by-product, or a non-food energy crop feedstock. The effects of consumer demographics and attitudes on preferences for the biochar attributes were also estimated. We tested the independence of irrelevant alternative assumption using a structured covariance matrix designed specifically to the survey’s structure. The results suggest that the attributes most likely to influence favorably consumers are production from agricultural by-product or wood waste feedstock. On the other hand, the attributes least likely to entice consumers are biochar produced in Tennessee or produced as a co-product of renewable fuel.

Robust Low-Rank Change Detection for Multivariate SAR Image Time Series

This article derives a new change detector for multivariate synthetic aperture radar (SAR) image time series (ITS). Classical statistical change detection methodologies based on covariance matrix analysis are usually built upon the Gaussian assumption, as well as an unstructured signal model. Both of these hypotheses may be inaccurate for high-dimension/resolution images, where the noise can be heterogeneous (non-Gaussian) and where the relevant signals usually lie in a low-dimensional subspace (low-rank structure). These two issues are tackled by proposing a new generalized likelihood ratio test based on a robust (compound Gaussian) low-rank (structured covariance matrix) model. The interest of the proposed detector is assessed on two SAR-ITS set from UAVSAR.

A Review of Closed-Form Cramér-Rao Bounds for DOA Estimation in the Presence of Gaussian Noise Under a Unified Framework

The Cramer-Rao Bound (CRB) for direction of arrival (DOA) estimation has been extensively studied over the past four decades, with a plethora of CRB expressions reported for various parametric models. In the literature, there are different methods to derive a closed-form CRB expression, but many derivations tend to involve intricate matrix manipulations which appear difficult to understand. Starting from the Slepian-Bangs formula and following the simplest derivation approach, this paper reviews a number of closed-form Gaussian CRB expressions for the DOA parameter under a unified framework, based on which all the specific CRB presentations can be derived concisely. The results cover three scenarios: narrowband complex circular signals, narrowband complex noncircular signals, and wideband signals. Three signal models are considered: the deterministic model, the stochastic Gaussian model, and the stochastic Gaussian model with the a priori knowledge that the sources are spatially uncorrelated. Moreover, three Gaussian noise models distinguished by the structure of the noise covariance matrix are concerned: spatially uncorrelated noise with unknown either identical or distinct variances at different sensors, and arbitrary unknown noise. In each scenario, a unified framework for the DOA-related block of the deterministic/stochastic CRB is developed, which encompasses one class of closed-form deterministic CRB expressions and two classes of stochastic ones under the three noise models. Comparisons among different CRBs across classes and scenarios are presented, yielding a series of equalities and inequalities which reflect the benchmark for the estimation efficiency under various situations. Furthermore, validity of all CRB expressions are examined, with some specific results for linear arrays provided, leading to several upper bounds on the number of resolvable Gaussian sources in the underdetermined case.

Adaptive Persymmetric Subspace Detectors in the Partially Homogeneous Environment

This paper addresses the adaptive detection of subspace signals in the noise whose covariance matrix is unknown. The partially homogeneous scenario, where the primary data have the same noise covariance matrix with the training data up to an unknown scaling factor is considered. We exploit the persymmetric structure of the noise covariance matrix to enhance the matched detection performance in the case of limited number of training data. Three persymmetric subspace detectors are proposed by applying the generalized likelihood ratio (GLR), Rao and Wald design criteria, respectively. It is proved that the three persymmetric subspace detectors can ensure the constant false alarm rate (CFAR) property. Experimental results show that the new persymmetric subspace detectors significantly outperform the conventional subspace detector in terms of the matched detection performance. Compared with the persymmetric rank-one signal detectors, the proposed persymmetric subspace detectors are more robust in the mismatched signal case.

Toeplitz Structured Covariance Matrix Estimation for Radar Applications

Following a geometric paradigm, the estimation of a Toeplitz structured covariance matrix is considered. The estimator minimizes the distance from the Sample Covariance Matrix (SCM) while complying with some specific constraints modeling the covariance structure. The resulting constrained optimization problem is solved globally resorting to the Dykstra’ projection framework. Each step of the procedure involves the solution of two convex sub-problems, whose minimizers are available in closed form. Simulation results related to typical radar environments highlight the effectiveness of the devised method.

Parsimonious Mean-Covariance Modeling for Longitudinal Data with ARMA Errors

Based on the generalized estimating equation approach, the authors propose a parsimonious mean-covariance model for longitudinal data with autoregressive and moving average error process, which not only unites the existing autoregressive Cholesky factor model and moving average Cholesky factor model but also provides a wide variety of structures of covariance matrix. The resulting estimators for the regression coefficients in both the mean and the covariance are shown to be consistent and asymptotically normally distributed under mild conditions. The authors demonstrate the effectiveness, parsimoniousness and desirable performance of the proposed approach by analyzing the CD4+ cell counts data set and conducting extensive simulations.

Adaptive Radar Detection in Gaussian Disturbance With Structured Covariance Matrix via Invariance Theory

This paper deals with adaptive radar detection of targets in the presence of Gaussian disturbance sharing a block-diagonal covariance structure. The problem is formulated according to a very general signal model, which contains the point-like, range-spread, and subspace target (or targets) as special instances. Hence, a unified study on the resulting adaptive detection problem is handled with the use of the invariance theory. The obtained results, including an appropriate transformation group, a maximal invariant and an induced maximal invariant, are proven consistent with those existing in the literature for some simple scenarios. Meanwhile, since the widely-used generalized likelihood ratio detector does not admit a closed form expression, new invariant detectors and their CFAR versions are proposed in this general scenario. Finally, their detection performance is assessed and validated via numerical examples.

HYPOTHESIS TESTING ON LINEAR STRUCTURES OF HIGH DIMENSIONAL COVARIANCE MATRIX.

This paper is concerned with test of significance on high dimensional covariance structures, and aims to develop a unified framework for testing commonly-used linear covariance structures. We first construct a consistent estimator for parameters involved in the linear covariance structure, and then develop two tests for the linear covariance structures based on entropy loss and quadratic loss used for covariance matrix estimation. To study the asymptotic properties of the proposed tests, we study related high dimensional random matrix theory, and establish several highly useful asymptotic results. With the aid of these asymptotic results, we derive the limiting distributions of these two tests under the null and alternative hypotheses. We further show that the quadratic loss based test is asymptotically unbiased. We conduct Monte Carlo simulation study to examine the finite sample performance of the two tests. Our simulation results show that the limiting null distributions approximate their null distributions quite well, and the corresponding asymptotic critical values keep Type I error rate very well. Our numerical comparison implies that the proposed tests outperform existing ones in terms of controlling Type I error rate and power. Our simulation indicates that the test based on quadratic loss seems to have better power than the test based on entropy loss.

DOA Estimation for UCA in the Presence of Mutual Coupling via Error Model Equivalence

In this letter, a direction-of-arrival (DOA) estimation method for a uniform-circular-array (UCA) with mutual coupling is presented. Due to the mutual coupling model of UCA, the actual steering vector in the mode domain is considered to correspond to a virtual uniform-linear-array (ULA) with gain-phase errors. This consideration indicates the calibration for mutual coupling in UCA can be transformed to that for gain-phase errors in the ULA: the gain errors are estimated with the main diagonal of the actual covariance matrix of the virtual ULA, and the phase errors are obtained via the Toeplitz structure of the ideal covariance matrix. In addition, due to the property of the ULA, there is an ambiguity in the estimation of the phase errors, which is analyzed and eliminated via least-square-method. Ultimately, the DOA and the coefficient of mutual coupling are estimated using estimating- signal-parameter-via-rotational-invariance-techniques and rank- reduction theory, respectively. The proposed method works on line without multidimensional parameter search. Furthermore, it avoids the ambiguity of the DOA estimation and simulations verify its effectiveness.

Moving target detection with polarimetric distributed MIMO radar in heterogeneous clutter

In this study, the authors consider the problem of moving target detection in the presence of heterogeneous clutter. The knowledge-aided method has been extended to derive the polarimetric multiple-input multiple-output (MIMO) radar detector. And the polarimetric clutter covariance matrix was modelled as a complex inverse Wishart distribution. More precisely, the clutter covariance matrices between different transmitter–receiver (Tx–Rx) pairs are assumed to follow the distribution which shares a prior clutter covariance matrix structure while with different values of the power levels. Under this assumption, the generalised likelihood ratio test (GLRT) approach has been adopted to develop the polarimetric knowledge-aided detector without secondary data. Numerical results are shown to illustrate the effectiveness of the polarimetric knowledge-aided GLRT detector in heterogeneous clutter.

Direction Finding for Bistatic MIMO Radar with Unknown Spatially Colored Noise

In this paper, we investigate the problem of joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation for bistatic multiple-input multiple-output radar with unknown spatially colored noise. By exploiting the sparse structure of the noise covariance matrix, a new de-noising scheme is designed. The signal covariance matrix is recast as a low-rank matrix with missing entries, which can be approximately tackled via solving an optimization problem. Thereafter, DODs and DOAs are obtained with the traditional subspace techniques. The proposed method does not bring any virtual aperture loss; thus, it achieves more accurate estimation performance than several state-of-the-art de-noising methods. Finally, the stochastic Cramer–Rao bound for joint direction finding is derived. Numerical computer simulations verify the effectiveness of the proposed algorithm.

Training Data Classification Algorithms for Radar Applications

In this letter, the problem of environment classification in the radar context is addressed. Specifically, adaptive architectures are conceived to classify training data, used for covariance estimation, as either homogeneous or heterogeneous. Such architectures are based upon the generalized likelihood ratio test criterion and exploit three covariance matrix structures (i.e., Hermitian, persymmetric, and symmetric structures). Numerical examples based on both synthetic and real data confirm the effectiveness of the proposed algorithms. It is important to highlight that the proposed architectures might represent a preliminary stage whose decisions can be used to select a suitable covariance estimate for target detection purposes.

Interference Covariance Matrix Structure Classification in Heterogeneous Environment

In this letter, an adaptive approach to classify the structure of the Interference Covariance Matrix (ICM) is proposed. It extends the framework of [1] to the heterogeneous environment where the secondary radar data used to estimate the ICM share the same covariance structure but different power levels. In particular, the considered classification problem is formulated in terms of a multiple hypothesis test and the Principle of Invariance is exploited to replace original data with a suitable statistic whose distribution is independent of the power scaling factors. Then, classification schemes are devised resorting to model order selection rules. At the analysis stage, the effectiveness of the newly devised classifiers is illustrated over simulated data as well as radar measured data.

Multilevel simultaneous equation model: A novel specification and estimation approach

Conventional simultaneous equation models assume that the error terms are serially independent. In some situations, data may present hierarchical or grouped structure and this assumption may be invalid. A new multivariate model referred as to Multilevel Simultaneous Equation Model (MSEM) is developed under this motivation. The maximum likelihood estimation of the parameters of an MSEM is considered. A matrix-valued distribution, namely, the matrix normal distribution, is introduced to incorporate an among-row and an among-column covariance matrix structure in the specification of the model. In the absence of an analytical solution of the system of likelihood equations, a general-purpose optimization solver is employed to obtain the maximum likelihood estimators. In a first approach to the solution of the problem, the adequacy of the matrix normal distribution is evaluated empirically in the case in which the double covariance structure is known. Using simulated data under the model assumptions, the performance of the maximum likelihood estimator (MLE) is assessed with regard to other conventional alternatives such as two-stage least squares estimator (2SLS).

Bayesian Deep Net GLM and GLMM

Deep feedforward neural networks (DFNNs) are a powerful tool for functional approximation. We describe flexible versions of generalized linear and generalized linear mixed models incorporating basis functions formed by a DFNN. The consideration of neural networks with random effects is not widely used in the literature, perhaps because of the computational challenges of incorporating subject specific parameters into already complex models. Efficient computational methods for high-dimensional Bayesian inference are developed using Gaussian variational approximation, with a parsimonious but flexible factor parameterization of the covariance matrix. We implement natural gradient methods for the optimization, exploiting the factor structure of the variational covariance matrix in computation of the natural gradient. Our flexible DFNN models and Bayesian inference approach lead to a regression and classification method that has a high prediction accuracy, and is able to quantify the prediction uncertainty in a principled and convenient way. We also describe how to perform variable selection in our deep learning method. The proposed methods are illustrated in a wide range of simulated and real-data examples, and compare favorably to a state of the art flexible regression and classification method in the statistical literature, the Bayesian additive regression trees (BART) method. User-friendly software packages in Matlab, R, and Python implementing the proposed methods are available at https://github.com/VBayesLab.

Persymmetric generalised adaptive matched filter for range‐spread targets in homogeneous environment

This study deals with the problem of adaptively detecting range-spread targets in Gaussian environment, by assuming a persymmetric structure of covariance matrix. A persymmetric detector based on the two-step generalised likelihood ratio test design scheme is devised to reduce secondary data requirement, which ensures the constant false alarm rate property. Moreover, the general closed-form expressions of the probabilities of the false alarm and detection for the proposed detector are derived, in both cases of deterministic and fluctuating scatter models. Furthermore, the compact versions of forenamed expressions are presented in a special case when the joint space-time dimension of the received signal is odd. Finally, the numerical results of these derived statistical expressions are confirmed with those of Monte Carlo trials.

Simultaneous hypotheses testing in models with blocked compound symmetric covariance structure

In this article, authors consider simultaneous testing hypotheses about structure of mean vector and covariance matrix in models with some special covariance structure, so-called blocked compound symmetric (BCS) covariance structure, for variate observations over u levels of some factor on each of n individual under the assumption of multivariate normality. Authors, using framework of ratio of positive and negative parts of best unbiased estimators, obtain test statistic for simultaneous test and prove that under null hypothesis the statistic has exact F distribution. Simulation study is conducted to show strong and weak sides of considered test which is compared with two other F tests for testing single hypotheses about mean and covariance matrix.

Persymmetric Rao test for MIMO radar in Gaussian disturbance

This paper considers the target detection problem in multiple-input multiple-output radar where the disturbance covariance matrix is unknown. Based on the Rao criterion, a persymmetric Rao test is designed by exploiting the persymmetric structure of the disturbance covariance matrix. Particularly, the proposed detector does not require training data, and exhibits strong selectivity to mismatched signals. Another contribution in this paper is to propose a two-stage detector whose second stage coincides with the proposed persymmetric Rao test. The selectivity or robustness of the two-stage detector can be tuned by selecting different threshold pairs.