Symmetric Covariance Matrix Research Articles

Weight Variability, Weight Gain Goals, and Biopsychosocial Factors Among Pregnant Women.

This study investigated the pattern of weight variability over 8 weeks and its associations with achieving weight gain goals and five biopsychosocial factors among pregnant women. We conducted a secondary analysis of 117 weeks of data from 16 pregnant women with a body mass index (BMI) ≥25. Weight variability was calculated from the difference of ending and beginning and maximum and minimum weights in a week and percent of each difference from baseline weight. Loess smoother, repeated measures model, and compound symmetric covariance matrix were used for analysis. The variability measure of maximum-minimum weight (overall mean: 2.1 ± 0.4 lbs.) was greater than the ending-beginning weight measure (overall mean: 0.7 ± 0.6 lbs.). Weight variability was negatively associated with achieving weight gain goals but not with biopsychosocial factors. Assessing weight variability is important during pregnancy so that preventive measures or lifestyle counseling can be instituted immediately to prevent excessive weight gain.

Two-Step Self-Calibration of LiDAR-GPS/IMU Based on Hand-Eye Method

Multi-line LiDAR and GPS/IMU are widely used in autonomous driving and robotics, such as simultaneous localization and mapping (SLAM). Calibrating the extrinsic parameters of each sensor is a necessary condition for multi-sensor fusion. The calibration of each sensor directly affects the accurate positioning control and perception performance of the vehicle. Through the algorithm, accurate extrinsic parameters and a symmetric covariance matrix of extrinsic parameters can be obtained as a measure of the confidence of the extrinsic parameters. As for the calibration of LiDAR-GPS/IMU, many calibration methods require specific vehicle motion or manual calibration marking scenes to ensure good constraint of the problem, resulting in high costs and a low degree of automation. To solve this problem, we propose a new two-step self-calibration method, which includes extrinsic parameter initialization and refinement. The initialization part decouples the extrinsic parameters from the rotation and translation part, first calculating the reliable initial rotation through the rotation constraints, then calculating the initial translation after obtaining a reliable initial rotation, and eliminating the accumulated drift of LiDAR odometry by loop closure to complete the map construction. In the refinement part, the LiDAR odometry is obtained through scan-to-map registration and is tightly coupled with the IMU. The constraints of the absolute pose in the map refined the extrinsic parameters. Our method is validated in the simulation and real environments, and the results show that the proposed method has high accuracy and robustness.

Eigen analysis of flipped Toeplitz covariance matrix for very low SNR sinusoidal signals detection and estimation

Detection and estimation of feeble signals in noise is of great importance in practice. In this paper, we use the properties of eigen-space and eigen-spectrum of symmetric and Toeplitz covariance matrix, to address the problem of detecting/estimating very low signal-to-noise ratio (SNR) sinusoidal signals. We show that as the input signal vector length increases, the sign of the dominant eigenvalue of flipped covariance matrix changes with the same frequency as the input sinusoidal signal. Based on this fact, we introduce two algorithms to detect weak sinusoidal signal buried in additive white Gaussian noise. These algorithms have similarities with the bordered-autocorrelation-method Karhunen-Loeve transform (BAM-KLT) method, which is previously proposed in the literature for weak signal detection. However, the BAM-KLT has some limitations and bottlenecks in computational burden, performance and implementation. Moreover, its exact analysis in the discrete domain is missed in the literature. In this paper, we provide the detailed analysis of discrete BAM-KLT. We also show analytically that if the discrete BAM-KLT is applied to flipped covariance matrix instead of covariance matrix itself, its limitations and bottlenecks can be overcome. This idea is the basis of the proposed algorithms. The algorithms are studied analytically and evaluated through numerical simulation. The results show that the proposed methods effectively increase the capability of existing methods, in terms of detection/estimation performance, root-mean-square frequency error, contrast, as well as the number of detectable signal sources.

Structure Parameter Optimized Kernel Based Online Prediction With a Generalized Optimization Strategy for Nonstationary Time Series

In this paper, sparsification techniques aided online prediction algorithms in a reproducing kernel Hilbert space are studied for nonstationary time series. The online prediction algorithms as usual consist of the selection of kernel structure parameters and the kernel weight vector updating. For structure parameters, the kernel dictionary is selected by some sparsification techniques with online selective modeling criteria, and moreover the kernel covariance matrix is intermittently optimized in the light of the covariance matrix adaptation evolution strategy (CMA-ES). Optimizing the real symmetric covariance matrix can not only improve the kernel structure's flexibility by the cross relatedness of the input variables, but also partly alleviate the prediction uncertainty caused by the kernel dictionary selection for nonstationary time series. In order to sufficiently capture the underlying dynamic characteristics in prediction-error time series, a generalized optimization strategy is designed to construct the kernel dictionary sequentially in multiple kernel connection modes. The generalized optimization strategy provides a more self-contained way to construct the entire kernel connections, which enhances the ability to adaptively track the changing dynamic characteristics. Numerical simulations have demonstrated that the proposed approach has superior prediction performance for nonstationary time series.

Correction to: Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions

Correction to: Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions

Hybrid projection methods for large-scale inverse problems with mixed Gaussian priors

When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub–Kahan process, which is an extension of the generalized Golub–Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods.

Gridless DOA estimation with finite rate of innovation reconstruction based on symmetric Toeplitz covariance matrix

Due to the rapid development and wide application of compressed sensing and sparse reconstruction theory, there exists a series of sparsity-based methods for the antenna sensor array direction of arrival (DOA) estimation with excellent performance. However, it is known that this kind of algorithms always suffers from the problem of grid mismatch. To overcome this shortcoming, a gridless DOA estimation algorithm with finite rate of innovation (FRI) based on a symmetric Toeplitz covariance matrix is proposed for uniform linear array (ULA) in this paper. In particular, a multiple measurement vector (MMV) FRI reconstruction model is built by exploiting the covariance data denoised according to covariance fitting criteria rather than the direct data or the original covariance data, which is commonly used in other representative gridless DOA estimation methods. Next, DOA can be retrieved from the recovered covariance matrix by utilizing an annihilating filter because each covariance data is a linear combination of complex exponentials. It guarantees to produce an exact spatial sparse estimate without discretization required by existing sparsity-based DOA estimation methods. Finally, the effectiveness and superiority of the proposed algorithm are demonstrated by numerical simulations.

On the Matrix Condition of Phylogenetic Tree.

Phylogenetic comparative analyses use trees of evolutionary relationships between species to understand their evolution and ecology. A phylogenetic tree of n taxa can be algebraically transformed into an n by n squared symmetric phylogenetic covariance matrix C where each element in C represents the affinity between extant species i and extant species j. This matrix C is used internally in several comparative methods: for example, it is often inverted to compute the likelihood of the data under a model. However, if the matrix is ill-conditioned (ie, if , defined by the ratio of the maximum eigenvalue of C to the minimum eigenvalue of C, is too high), this inversion may not be stable, and thus neither will be the calculation of the likelihood or parameter estimates that are based on optimizing the likelihood. We investigate this potential issue and propose several methods to attempt to remedy this issue.

Multispectral, multiazimuth, and multioffset coherence attribute applications

The coherence attribute computation is typically carried out as a poststack application on 3D prestack migrated seismic data volumes. However, since its inception, interpreters have applied coherence to band-pass-filtered data, azimuthally limited stacks, and offset-limited stacks to enhance discontinuities seen at specific frequencies, azimuths, and offsets. The limitation of this approach is the multiplicity of coherence volumes. Of the various coherence algorithms that have evolved over the past 25 years, the energy ratio coherence computation stands apart from the others, being more sensitive to the seismic waveform changes rather than changes in their amplitude. The energy ratio algorithm is based on the crosscorrelation of five or more adjacent traces to form a symmetric covariance matrix that can then be decomposed into eigenvalues and eigenvectors. The first eigenvector represents a vertically variable, laterally consistent pattern that best represents the data in the analysis window. The first eigenvalue represents the energy of the data represented by this pattern. Coherence is then defined as the ratio of the energy represented by the first eigenvalue to the sum of the energy of the original data. An early generalization of this algorithm was to compute the sum of two covariance matrices, one from the original data and the other from the 90° phase rotated data, thereby eliminating artifacts about low-amplitude zero crossings. More recently, this concept has been further generalized by computing a sum of covariance matrices of traces represented by multiple spectral components, by their azimuthally limited stacks, and by their offset-limited stacks. These more recently developed algorithms capture many of the benefits of discontinuities seen at specific frequencies, azimuths, and offsets, but they present the interpreter with a single volume. We compare the results of multispectral, multiazimuth, and multioffset coherence volumes with the traditional coherence computation, and we find that these newer coherence computation procedures produce superior results.

Invariance Theory for Adaptive Detection in Interference With Group Symmetric Covariance Matrix

This paper describes the adaptive detection of point-like targets in a Gaussian environment assuming a group symmetric structure for the interference covariance matrix. This special configuration enforces block-sparsity and permits splitting of the original observation space into lower-dimensional subspaces, each characterized by its own nuisance parameters. Hence, the Principle of Invariance is invoked to select decision rules enjoying some symmetry defined by a suitable group of transformations which leave the decision problem unaltered. All the invariant tests can be expressed as functions of a maximal invariant statistic whose derivation is among the key results of this paper. Finally, some common design criteria are applied to come up with adaptive architectures that share invariance, and their behavior is assessed to highlight the interplay between sample size and detection performance in comparison with conventional decision schemes.

Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions

In this paper, the authors study the problem of testing the hypothesis of a block compound symmetry covariance matrix with two-level multivariate observations, taken for m variables over u sites or time points. Through the use of a suitable block-diagonalization of the hypothesis matrix, it is possible to obtain a decomposition of the main hypothesis into two sub-hypotheses. Using this decomposition, it is then possible to obtain the likelihood ratio test statistic as well as its exact moments in a much simpler way. The exact distribution of the likelihood ratio test statistic is then analyzed. Because this distribution is quite elaborate, yielding a non-manageable distribution function, a manageable but very precise near-exact distribution is developed. Numerical studies conducted to evaluate the closeness between this near-exact distribution and the exact distribution show the very good performance of this approximation even for very small sample sizes and the approach followed allows us to extend its validity to situations where the population distributions are elliptically contoured. A real-data example is presented and a simulation study is also conducted.

Testing the equality of mean vectors for paired doubly multivariate observations in blocked compound symmetric covariance matrix setup

In this article we develop a test statistic for testing the equality of mean vectors for paired doubly multivariate observations with q response variables and u sites in blocked compound symmetric covariance matrix setting. We obtain a natural extension of the Hotelling’s T2 statistic, the Block T2 (BT2) statistic, a convolution of two T2’s, which uses unbiased estimates of the component matrices of the orthogonally transformed blocked compound symmetric covariance matrix that is present in a data set. The new test statistic is implemented with two real data sets. We compare our findings with the conventional Hotelling’s T2 statistic.

Sampling Gaussian Distributions in Krylov Spaces with Conjugate Gradients

This paper introduces a conjugate gradient sampler that is a simple extension of the method of conjugate gradients (CG) for solving linear systems. The CG sampler iteratively generates samples from a Gaussian probability density, using either a symmetric positive definite covariance or precision matrix, whichever is more convenient to model. Similar to how the Lanczos method solves an eigenvalue problem, the CG sampler approximates the covariance or precision matrix in a small dimensional Krylov space. As with any iterative method, the CG sampler is efficient for high dimensional problems where forming the covariance or precision matrix is impractical, but operating by the matrix is feasible. In exact arithmetic, the sampler generates Gaussian samples with a realized covariance that converges to the covariance of interest. In finite precision, the sampler produces a Gaussian sample with a realized covariance that is the best approximation to the desired covariance in the smaller dimensional Krylov space. In this paper, an analysis of the sampler is given, and we give examples showing the usefulness and limitations of the Krylov approximations.

Improved likelihood ratio test based voice activity detector applied to speech recognition

Nowadays, the accuracy of speech processing systems is strongly affected by acoustic noise. This is a serious obstacle regarding the demands of modern applications. Therefore, these systems often need a noise reduction algorithm working in combination with a precise voice activity detector (VAD). The computation needed to achieve denoising and speech detection must not exceed the limitations imposed by real time speech processing systems. This paper presents a novel VAD for improving speech detection robustness in noisy environments and the performance of speech recognition systems in real time applications. The algorithm is based on a Multivariate Complex Gaussian (MCG) observation model and defines an optimal likelihood ratio test (LRT) involving multiple and correlated observations (MCO) based on a jointly Gaussian probability distribution (jGpdf) and a symmetric covariance matrix. The complete derivation of the jGpdf-LRT for the general case of a symmetric covariance matrix is shown in terms of the Cholesky decomposition which allows to efficiently compute the VAD decision rule. An extensive analysis of the proposed methodology for a low dimensional observation model demonstrates: (i) the improved robustness of the proposed approach by means of a clear reduction of the classification error as the number of observations is increased, and (ii) the trade-off between the number of observations and the detection performance. The proposed strategy is also compared to different VAD methods including the G.729, AMR and AFE standards, as well as other recently reported algorithms showing a sustained advantage in speech/non-speech detection accuracy and speech recognition performance using the AURORA databases.

Generalized indirect covariance NMR formalism for establishment of multidimensional spin correlations.

Multidimensional nuclear magnetic resonance (NMR) experiments measure spin-spin correlations, which provide important information about bond connectivities and molecular structure. However, direct observation of certain kinds of correlations can be very time-consuming due to limitations in sensitivity and resolution. Covariance NMR derives correlations between spins via the calculation of a (symmetric) covariance matrix, from which a matrix-square root produces a spectrum with enhanced resolution. Recently, the covariance concept has been adopted to the reconstruction of nonsymmetric spectra from pairs of 2D spectra that have a frequency dimension in common. Since the unsymmetric covariance NMR procedure lacks the matrix-square root step, it does not suppress relay effects and thereby may generate false positive signals due to chemical shift degeneracy. A generalized covariance formalism is presented here that embeds unsymmetric covariance processing within the context of the regular covariance transform. It permits the construction of unsymmetric covariance NMR spectra subjected to arbitrary matrix functions, such as the square root, with improved spectral properties. This formalism extends the domain of covariance NMR to include the reconstruction of nonsymmetric NMR spectra at resolutions or sensitivities that are superior to the ones achievable by direct measurements.

Structured Parallel Architecture for Displacement MIMO Kalman Equalizer in CDMA Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A reduced complexity MIMO Kalman equalizer architecture is proposed in this brief by jointly considering the displacement structure and the block-Toeplitz structure. Numerical matrix–matrix multiplications with <formula formulatype="inline"> <tex>${\cal O}(F^{3})$</tex></formula> complexity are eliminated by simple data loading process, where <formula formulatype="inline"><tex>$F$</tex></formula> is the spreading factor. Finally, an iterative Conjugate-Gradient based algorithm is proposed to avoid the inverse of the Hermitian symmetric innovation covariance matrix in Kalman gain processor. The proposed architecture not only reduces the numerical complexity from <formula formulatype="inline"><tex>${\cal O}(F^{2})$</tex> </formula> to <formula formulatype="inline"><tex>${\cal O}(F\log_{2}F)$</tex> </formula> per chip, but also facilitates the parallel and pipelined VLSI implementation in real-time processing. </para>

Nonnegative quadratic estimation and quadratic sufficiency in general linear models

Notions of linear sufficiency and quadratic sufficiency are of interest to some authors. In this paper, the problem of nonnegative quadratic estimation for β ′ H β + h σ 2 is discussed in a general linear model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are investigated. As a direct consequence, the result on (ordinary) quadratic sufficiency is obtained. In addition, we pose a practical problem and extend a special situation to the multivariate case. Moreover, a simulated example is conducted, and applications to a model with compound symmetric covariance matrix are given. Finally, we derive a remark which indicates that our main results could be extended further to the quasi-normal case.

A computational procedure to estimate the stochastic dynamic response of large non-linear FE-models

An algorithm for the computation of the stochastic non-stationary non-linear response of large FE-models is presented. In stochastic analysis, the response is described by the mean and covariance function and possibly by the probability distribution of the non-linear response. To estimate the covariance matrix, the method of equivalent statistical linearization is applied for linearizing all non-linear elements. The large fully populated and symmetric covariance matrix of dimension ⩾2 n is described by the so called Karhunen–Loéve expansion representation, which allows one to employ a feasible description. In this context, an efficient procedure is suggested to determine the minimal number of Karhunen–Loéve vectors necessary to assure a sufficiently accurate representation. This method in fact allows one to employ any available deterministic integration scheme to compute the Karhunen–Loéve vectors. To increase the computational efficiency, modal coordinates are used to represent the linear sub-system. Special attention is given to the effects of mode truncation which are generally not negligible for large FE-systems. Moreover, the suggested approach has no limitation w.r.t. the size of the model in terms of degrees of freedom (DOFs) or the number of non-linear elements. The feasibility of the proposed procedure is demonstrated in the numerical examples where the methodology is applied to an office building with approximately 25, 50 and 100 thousand DOFs containing hysteretic damping elements.

Perspectives on errors-in-variables estimation for dynamic systems

The paper gives an overview of various methods for identifying dynamic errors-in-variables systems. Several approaches are classified by how the original information in time-series data of the noisy input and output measurements is condensed before further processing. For some methods, such as instrumental variable estimators, the information is condensed into a nonsymmetric covariance matrix as a first step before further processing. In a second class of methods, where a symmetric covariance matrix is used instead, the Frisch scheme and other bias-compensation approaches appear. When dealing with the estimation problem in the frequency domain, a milder data reduction typically takes place by first computing spectral estimators of the noisy input–output data. Finally, it is also possible to apply maximum likelihood and prediction error approaches using the original time-domain data in a direct fashion. This alternative will often require quite high computational complexity but yield good statistical efficiency. The paper is also presenting various properties of parameter estimators for the errors-in-variables problem, and a few conjectures are included, as well as some perspectives and experiences by the authors.

Adaptive antenna array using real symmetric arraycovariance matrix

A novel scheme using a real symmetric array covariance matrix for an adaptive antenna array in DS/CDMA is proposed. A real symmetric array covariance matrix is estimated from a complex array covariance matrix using a unitary transformation and Toeplitz matrix approximation methods. Especially, the Hermitian Toeplitzation method not only estimates a persymmetric matrix form for a real symmetric array covariance matrix but also enhances the performance of the received signal by removing the undesired effect obtained from a complex array covariance matrix. Simulation results demonstrate the effectiveness of the proposed scheme.