Structure Of Covariance Matrix Research Articles

Bayesian estimation in maternally ancestral animal models for weaning weight of beef cattle1

The Bayesian approach was implemented for fitting several maternally ancestral models for weaning weight data of Angus calves. The goal was to evaluate to what extent genetic evaluation models with additive grand maternal effects (G), or with an ancestrally structured covariance matrix for maternal environmental effects (E), or with a sire × year interaction (ISY), or combinations thereof (GE, GSY, ESY, GESY), redistribute the additive variability and reduce the negative magnitude of the additive correlation between direct and maternal effects (r(AoAm)), when compared with the regular maternal animal model (I). All animals with records had known dams and maternal granddams. The sampling scheme induced low autocorrelations among all variables and tended to converge quickly. The signs of the estimates of r(AoAm) were consistently negative for all models fitted. The magnitudes of the estimates of r(AoAm) from models E, G, GE, ESY, and GESY were almost one-third of those from models I and ISY. Inclusion of the sire × year interaction had some effect in reducing the negative magnitude of r(AoAm), but also reduced the size of the estimates of direct (h(0)(2)) and maternal (h(m)(2)) heritabilities. In comparison, models E or G reduced the negative magnitude of r(AoAm) by 0.50 units and produced more favorable estimates of H(0)(2) and h(m)(2) than models I and ISY. The estimate of h(0)(2) from G was similar to the one from I; however, the estimated h(m)(2) was 0.04 units greater, whereas the estimate of r(AoAm) was much less negative (-0.21 vs. -0.71) than the respective estimates from I. The environmental correlation between the weaning weights of dams and their daughters (λ) was estimated to be -0.28 ± 0.03 in E and ESY, and -0.21 ± 0.03 in GE and GESY. Inclusion of the sire × year interaction effect by itself did not have much of an impact in the reduction of the estimated magnitude of r(AoAm). Rank correlations among EBV for direct effects were larger than 0.94 and did not show any appreciable difference among models, whereas the rank correlation among maternal breeding values displayed differences in the ranking between I and the other models. Models E and ESY recovered the largest amount of total additive variability with maternal effects.

Estimating and Testing a Structured Covariance Matrix for Three-Level Multivariate Data

This article considers an approach to estimating and testing a new Kronecker product covariance structure for three-level (multiple time points (p), multiple sites (u), and multiple response variables (q)) multivariate data. Testing of such covariance structure is potentially important for high dimensional multi-level multivariate data. The hypothesis testing procedure developed in this article can not only test the hypothesis for three-level multivariate data, but also can test many different hypotheses, such as blocked compound symmetry, for two-level multivariate data as special cases. The tests are implemented with two real data sets.

High resolution radar target adaptive detector and performance assessment

The high resolution radar target detection is addressed in the non-Gaussian clutter. An adaptive detector is derived for range-spread target based on a novel covariance matrix estimator. It is proved that the new detector is constant false alarm rate (CFAR) to both of the clutter covariance matrix structure and power level theoretically for match cases. The simulation results show that the new detector is almost CFAR for mismatch cases, and it outperforms the existing adaptive detector based on the sample covariance matrix. It also shows that the detection performance improves, as the number of pulses, the number of secondary data or the clutter spike increases. In addition, the derived detector is robust to different subsets, estimated clutter group sizes and correlations of clutter. Importantly, the number of iterations for practical application is just one.

On parameter identifiability of MIMO radar with waveform diversity

Parameter identifiability is one of the most interesting issues in multiple-input multiple-output (MIMO) radar systems. Previous works have presented conditions for parameter identifiability of the systems with uncorrelated waveforms. In this paper, under general waveform assumptions, we investigate conditions for parameter identifiability of MIMO radar systems with more flexible waveform diversity, where the transmitted waveforms are not restricted to be uncorrelated. The conditions specify the maximum number of targets that can be uniquely localized, which are obtained based on the rule of time-bandwidth product. Moreover, it is found that the conditions are in terms of the rank and structure of the waveform covariance matrix (WCM) and the geometry of MIMO radar systems. Numerical examples are provided to verify the effectiveness of the conditions.

Noisy Constrained Maximum-Likelihood Linear Regression for Noise-Robust Speech Recognition

Adaptive training is a widely used technique for building speech recognition systems on nonhomogeneous training data. Recently, there has been interest in applying these approaches for situations where there is significant levels of background noise in the training data. Various schemes for adaptive training are based on noise-, or speaker-, specific transforms of features to yield estimates of the clean speech. However, when there are high levels of background noise, these clean speech estimates may be poor resulting in degradations in performance. In this paper, a new approach for adaptive training on noise-corrupted training data is presented. It extends a popular form of linear transform for model-based adaptation and adaptive training, constrained MLLR (CMLLR), to reflect additional uncertainty from noise-corrupted observations. This new form of adaptation transform is called noisy CMLLR (NCMLLR). NCMLLR uses a modified version of generative model between clean speech and noisy observation, similar to factor analysis (FA). However, in contrast to FA here the generative model describes an adaptation transform, rather than a covariance matrix structure. The use of NCMLLR for adaptive training using an expectation-maximization approach is described. Discriminative adaptive training with NCMLLR is also described based on the minimum phone error criterion. Experimental results comparing NCMLLR with standard adaptive training schemes are given on a noise-corrupted version of Resource Management, the ARPA 1994 CSRNAB Spoke 10 task, and in-car recorded data.

Solutions for the MIMO Gaussian Wiretap Channel With a Cooperative Jammer

We study the Gaussian MIMO wiretap channel with a transmitter, a legitimate receiver, an eavesdropper and an external helper, each equipped with multiple antennas. The transmitter sends confidential messages to its intended receiver, while the helper transmits jamming signals independent of the source message to confuse the eavesdropper. The jamming signal is assumed to be treated as noise at both the intended receiver and the eavesdropper. We obtain a closed-form expression for the structure of the artificial noise covariance matrix that guarantees a secrecy rate larger or at least equal to the secrecy capacity of the wiretap channel with no jamming signal. We also describe how to find specific realizations of this covariance matrix expression that provide good secrecy rate performance, even when there is no nontrivial null space between the helper and the intended receiver. Unlike prior work, our approach considers the general MIMO case, and is not restricted to SISO or MISO scenarios.

Neopolyploidy and diversification in Heuchera grossulariifolia.

Newly formed polyploid lineages must contend with several obstacles to avoid extinction, including minority cytotype exclusion, competition, and inbreeding depression. If polyploidization results in immediate divergence of phenotypic characters these hurdles may be reduced and establishment made more likely. In addition, if polyploidization alters the phenotypic and genotypic associations between traits, that is, the P and G matrices, polyploids may be able to explore novel evolutionary paths, facilitating their divergence and successful establishment. Here, we report results from a study of the perennial plant Heuchera grossulariifolia in which the phenotypic divergence and changes in phenotypic and genotypic covariance matrices caused by neopolyploidization have been estimated. Our results reveal that polyploidization causes immediate divergence for traits relevant to establishment and results in significant changes in the structure of the phenotypic covariance matrix. In contrast, our results do not provide evidence that polyploidization results in immediate and substantial shifts in the genetic covariance matrix.

More on the Kronecker Structured Covariance Matrix

In this article, the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly focused is the estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The suggested estimation generalizes the procedure proposed by Srivastava et al. (2008) for a separable covariance matrix. The restrictions imposed by separability and double separability are also discussed.

On Space–Frequency Correlation of UWB MIMO Channels

Conventional multiple-input-multiple-output (MIMO) channel correlation models focus on describing the complex amplitude correlation of resolvable multipath components (MPCs). It has recently been shown that the time-of-arrival (ToA) correlation of resolvable MPCs, which has largely been ignored in the literature so far, is important for characterizing the space-frequency (SF) correlation of MIMO channels in the ultrawideband (UWB) regime. The proposal of ToA correlation brings forth the following open questions: What is the fundamental difference between amplitude and ToA correlations? Are they mutually replaceable? What impact do they have on system performance? Is this impact related to channel bandwidth? This paper systematically answers the above questions. We demonstrate that, compared with a conventional model that only considers amplitude correlation, a UWB MIMO channel model taking into account both amplitude and ToA correlations leads to different theoretical maximum diversity order, numerical structure of the SF channel covariance matrix, spatial correlation function, and diversity performance. We conclude that ToA correlation is a necessary complement to amplitude correlation when the channel bandwidth exceeds 500 MHz, i.e., UWB channels. This new insight serves as the motivation for more realistic modeling of UWB MIMO channels and the inspiration for designing advanced UWB MIMO systems to fully exploit the SF channel correlation properties.

Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices

Many statistical applications require an estimate of a covariance matrix and/or its inverse. When the matrix dimension is large compared to the sample size, which happens frequently, the sample covariance matrix is known to perform poorly and may suffer from ill-conditioning. There already exists an extensive literature concerning improved estimators in such situations. In the absence of further knowledge about the structure of the true covariance matrix, the most successful approach so far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to a multiple of the identity, by taking a weighted average of the two, turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean, while retaining the sample eigenvectors. Our paper extends this approach by considering nonlinear transformations of the sample eigenvalues. We show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations, the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage.

Clustering, Spatial Correlations, and Randomization Inference

It is a standard practice in regression analyses to allow for clustering in the error covariance matrix if the explanatory variable of interest varies at a more aggregate level (e.g., the state level) than the units of observation (e.g., individuals). Often, however, the structure of the error covariance matrix is more complex, with correlations not vanishing for units in different clusters. Here, we explore the implications of such correlations for the actual and estimated precision of least squares estimators. Our main theoretical result is that with equal-sized clusters, if the covariate of interest is randomly assigned at the cluster level, only accounting for nonzero covariances at the cluster level, and ignoring correlations between clusters as well as differences in within-cluster correlations, leads to valid confidence intervals. However, in the absence of random assignment of the covariates, ignoring general correlation structures may lead to biases in standard errors. We illustrate our findings using the 5% public-use census data. Based on these results, we recommend that researchers, as a matter of routine, explore the extent of spatial correlations in explanatory variables beyond state-level clustering.

Explicit estimators of parameters in the Growth Curve model with linearly structured covariance matrices

Estimation of parameters in the classical Growth Curve model, when the covariance matrix has some specific linear structure, is considered. In our examples maximum likelihood estimators cannot be obtained explicitly and must rely on optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linear structured covariance matrix.

Improved statistical analysis of moclobemide dose effects on panic disorder treatment

Clinical trials with several measurement occasions are frequently analyzed using only the last available observation as the dependent variable [last observation carried forward (LOCF)]. This ignores intermediate observations. We reanalyze, with complete data methods, a clinical trial previously reported using LOCF, comparing placebo and five dosage levels of moclobemide in the treatment of outpatients with panic disorder to illustrate the superiority of methods using repeated observations. We initially analyzed unprovoked and situational, major and minor attacks as the four dependent variables, by repeated measures maximum likelihood methods. The model included parameters for linear and curvilinear time trends and regression of measures during treatment on baseline measures. Significance tests using this method take into account the structure of the error covariance matrix. This makes the sphericity assumption irrelevant. Missingness is assumed to be unrelated to eventual outcome and the residuals are assumed to have a multivariate normal distribution. No differential treatment effects for limited attacks were found. Since similar results were obtained for both types of major attack, data for the two types of major attack were combined. Overall downward linear and negatively accelerated downward curvilinear time trends were found. There were highly significant treatment differences in the regression slopes of scores during treatment on baseline observations. For major attacks, all treatment groups improved over time. The flatter regression slopes, obtained with higher doses, indicated that higher doses result in uniformly lower attack rates regardless of initial severity. Lower doses do not lower the attack rate of severely ill patients to those achieved in the less severely ill. The clinical implication is that more severe patients require higher doses to attain best benefit. Further, the significance levels obtained by LOCF analyses were only in the 0.05-0.01 range, while significance levels of <0.00001 were obtained by these repeated measures analyses indicating increased power. The greater sensitivity to treatment effect of this complete data method is illustrated. To increase power, it is often recommended to increase sample size. However, this is often impractical since a major proportion of the cost per subject is due to the initial evaluation. Increasing the number of repeated observations increases power economically and also allows detailed longitudinal trajectory analyses.

Generalized Consistent Estimation on Low-Rank Krylov Subspaces of Arbitrarily High Dimension

The problem of Krylov subspace estimation based on the sample covariance matrix is addressed. The focus is on signal processing applications where the Krylov subspace is defined by the unknown second-order statistics of the observed samples and the signature vector associated with the desired parameter. In particular, the consistency of traditionally optimal sample estimators is revised and analytically characterized under a practically more relevant asymptotic regime, according to which not only the number of samples but also the observation dimension grow without bound at the same rate. Furthermore, an improved construction of a class of Krylov subspace estimators is proposed based on the generalized consistent estimation of a set of vector-valued power functions of the observation covariance matrix. To that effect, an extension of some known results from random matrix theory on the estimation of certain spectral functions of the covariance matrix to the convergence of not only the covariance eigenspectrum but also the associated eigensubspaces is provided. A new family of estimators is derived that generalizes conventional implementations by proving to be consistent for observations of arbitrarily high dimension. The proposed estimators are shown to outperform traditional constructions via the numerical evaluation of the solution to two fundamental problems in sensor array signal processing, namely the problem of estimating the power of an intended source and the estimation of the principal eigenspace and dominant eigenmodes of a structured covariance matrix.

Non‐Gaussian state space models in decomposition of ice core time series in long and short time‐scales

AbstractStatistical modelling of six time series of geological ice core chemical data from Greenland is discussed. We decompose the total variation into long time‐scale (trend) and short time‐scale variations (fluctuations around the trend), and a pure noise component. Too heavy tails of the short‐term variation makes a standard time‐invariant linear Gaussian model inadequate. We try non‐Gaussian state space models, which can be efficiently approximated by time‐dependent Gaussian models. In essence, these time‐dependent Gaussian models result in a local smoothing, in contrast to the global smoothing provided by the time‐invariant model. To describe the mechanism of this local smoothing, we utilise the concept of a local variance function derived from a heavy‐tailed density. The time‐dependent error variance expresses the uncertainty about the dynamical development of the model state, and it controls the influence of observations on the estimates of the model state components. The great advantage of the derived time‐dependent Gaussian model is that the Kalman filter and the Kalman smoother can be used as efficient computational tools for performing the variation decomposition. One of the main objectives of the study is to investigate how the distributional assumption on the model error component of the short time‐scale variation affects the decomposition. Copyright © 2009 John Wiley &amp; Sons, Ltd.

A Bayesian Algorithm for Functional Mapping of Dynamic Complex Traits

Functional mapping of dynamic traits measured in a longitudinal study was originally derived within the maximum likelihood (ML) context and implemented with the EM algorithm. Although ML-based functional mapping possesses many favorable statistical properties in parameter estimation, it may be computationally intractable for analyzing longitudinal data with high dimensions and high measurement errors. In this article, we derive a general functional mapping framework for quantitative trait locus mapping of dynamic traits within the Bayesian paradigm. Markov chain Monte Carlo techniques were implemented for functional mapping to estimate biologically and statistically sensible parameters that model the structures of time-dependent genetic effects and covariance matrix. The Bayesian approach is useful to handle difficulties in constructing confidence intervals as well as the identifiability problem, enhancing the statistical inference of functional mapping. We have undertaken simulation studies to investigate the statistical behavior of Bayesian-based functional mapping and used a real example with F2 mice to validate the utilization and usefulness of the model.

Bayesian Portfolio Selection with MSV Models

In the paper we compare the predictive ability of discrete-time Multivariate Stochastic Volatility (MSV) models to optimal portfolio choice. We consider MSV models, which differ in the structure of the conditional covariance matrix (including the specifications with zero, constant and time-varying conditional correlations). Next, we construct the optimal portfolio under the assumption that the asset returns are described by the multivariate stochastic volatility models. We consider hypothetical portfolios, which consist of two currencies that were the most important for the Polish economy: the US dollar and euro. In the optimization process we use the predictive distributions of future returns and the predictive conditional covariance matrix obtained from the MSV models.

Improved testing inference in mixed linear models

Mixed linear models are commonly used in repeated measures studies. They account for the dependence amongst observations obtained from the same experimental unit. Often, the number of observations is small, and it is thus important to use inference strategies that incorporate small sample corrections. In this paper, we develop modified versions of the likelihood ratio test for fixed effects inference in mixed linear models. In particular, we derive a Bartlett correction to such a test, and also to a test obtained from a modified profile likelihood function. Our results generalize those in [Zucker, D.M., Lieberman, O., Manor, O., 2000. Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood. Journal of the Royal Statistical Society B, 62, 827–838] by allowing the parameter of interest to be vector-valued. Additionally, our Bartlett corrections allow for random effects nonlinear covariance matrix structure. We report simulation results which show that the proposed tests display superior finite sample behavior relative to the standard likelihood ratio test. An application is also presented and discussed.

Virtual-manifold ambiguity in HOS-based direction-finding with electromagnetic vector-sensors

Herein derived are the lower and upper bounds for the number of linearly independent ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2Q</i> )th-order virtual steering vectors of an array of electromagnetic vector-sensors, with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> being any positive integer over one. These bounds help determine the number of non-Gaussian signals whose directions-of-arrival (DOAs) can be uniquely identified from ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2Q</i> )th-order statistics data. The derived lower bounds increase with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> , whereas the derived upper bounds often fall below the maximum number of virtual sensors achievable from ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2Q</i> )th-order statistics manipulation. These bounds are independent of the permutation of the ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2Q</i> )th-order statistics entries in the higher order cumulant matrix that has a similar algebraic structure of the classical covariance matrix used in the second-order subspace-based direction-finding algorithms.

Some Studies on the Structure of Covariance Matrix of Discrete-Time fBm

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <?Pub Dtl=""?>This paper presents some results on the structure of autocovariance matrix of discrete-time fractional Brownian motion. Since these processes are nonstationary, the autocovariance matrix is a function of time. The eigenvalues associated with the autocovariance matrix are dependent on Hurst exponent characterizing the discrete-time fractional Brownian motion. However, it is seen that only one eigenvalue of this autocovariance matrix depends on time index <formula formulatype="inline"><tex>$n$</tex></formula> in the asymptotic sense, and it increases as the time index increases. All other eigenvalues are observed to be invariant with time index in the asymptotic sense. It is observed that the eigenvectors associated with these eigenvalues have an interesting structure and these eigenvectors can be used as filters of an <formula formulatype="inline"> <tex>$M$</tex></formula>-band orthogonal wavelet system. The eigenvector associated with the time-varying eigenvalue is a lowpass filter. The subband signal from this lowpass filtered branch imbibes the nonstationary attribute of the input process and is a nonstationary signal whereas all other subband signals are stationary in nature. </para>