Complete Covariance Matrix Research Articles

Compressive stochastic subspace identification: A modal parameter identification method suitable for compressive sampling

Stochastic subspace identification (SSI) and its variants are widely used in conventional structural health monitoring (SHM) owing to their high computation robustness and estimation accuracy. Conventional SHM is based on a Nyquist (high-rate and uniform) sampling dictated by Shannon–Nyquist sampling theorem. As data quantity of SHM is increasing with duration and equipment at an ever-growing rate and new applications of SHM with strict constricts on sampling are emerging one by one, sub-Nyquist sampling, also known as compressive sampling has increasingly been used in SHM in recent years. Therefore, despite the success of SSI method and its algorithmic variants, a significant drawback lies in their handling of sub-Nyquist data. To overcome this drawback, we set out to extend the applicable scenarios of SSI from Nyquist (high-rate/uniform) sampling to sub-Nyquist (low-rate/nonuniform) sampling. In this paper, we propose a compressive stochastic subspace identification (CSSI) which can identify modal parameters from sub-Nyquist/compressed data. Specifically, CSSI involves two aspects: covariance matrix reconstruction and system parameter identification. We find the complete covariance matrix in classic SSI can also be estimated/reconstructed by sub-Nyquist samples. More importantly, this reconstruction can be physically guaranteed through special sampling patterns without sparsity prior. Based on the above findings, we develop a compressed covariance sensing-based covariance matrix reconstruction technique and derive the optimal sub-Nyquist sampling pattern such that the reconstruction is hassle-free and efficient. Then, we perform modal parameter identification on the basis of the reconstructed covariance matrix. Finally, the effectiveness of CSSI is validated by simulations and experiments. In short, CSSI, as a new extension of SSI, can extract modal parameters at lower sampling rates than what is prescribed by the classic SSI. The implication of this is the possibility that CSSI for operational modal analysis will require fewer measurements thus it would reduce the storage requirement and the bandwidth for transmitting data.

Conditional hazard for simplified multi‐site seismic hazard and risk analyses

AbstractThe joint probability distribution of different intensity measures at different sites in one earthquake is needed to define the stochastic process regulating the exceedance (in time) of the at the ensemble of the sites, which in turn is a funding element of the so‐called multi‐site probabilistic seismic hazard analysis (MSPSHA). The simplest model for the joint distribution of the in one seismic event requires the mean vector and the covariance matrix of the at all sites, conditional to the magnitude and location of the earthquake, which may need a large amount of data to be calibrated. The conditional hazard (CH) approach, originally developed for single‐site surrogate vector‐valued probabilistic seismic hazard analysis, may be a simplified option for MSPSHA, as it explicitly models only part of the covariance matrix of the at the sites, while the rest forcefully follows the working assumptions. The presented paper compares the CH approach for MSPSHA against the benchmark in which the complete covariance matrix is modelled, using a testbed in which one‐hundred sites are considered. When the comparison metric is the probability of a given number of exceedances observed at the sites in some time intervals, it is found that CH is a viable alternative to MSPSHA, although the degree of approximation is sensitive to which and how many are considered (e.g., which spectral acceleration). When the analysis is taken all the way to the risk, considering the fragilities of a portfolio of hypothetical buildings, and taking as the metric the probability of observing a given number of structural failures in a time interval, it is found that the approximation introduced by CH with respect to the benchmark is further reduced.

2D-DOA Estimation in Switching UCA Using Deep Learning-Based Covariance Matrix Completion.

In this paper, we study the two-dimensional direction of arrival (2D-DOA) estimation problem in a switching uniform circular array (SUCA), which means performing 2D-DOA estimation with a reduction in the number of radio frequency (RF) chains. We propose a covariance matrix completion algorithm for 2D-DOA estimation in a SUCA. The proposed algorithm estimates the complete covariance matrix of a fully sampled UCA (FUCA) from the sample covariance matrix of the SUCA through a neural network. Afterwards, the MUSIC algorithm is performed for 2D-DOA estimation with the completed covariance matrix. We conduct Monte Carlo simulations to evaluate the performance of the proposed algorithm in various scenarios; the performance of 2D-DOA estimation in the SUCA gradually approaches that in the FUCA as the SNR or the number of snapshots increases, which means that the advantages of a FUCA can be preserved with fewer RF chains. In addition, the proposed algorithm is able to implement underdetermined 2D-DOA estimation.

An Approximate Formula linking Active Share And Tracking Error

The concept of Active Share in portfolio management has gained widespread acceptance since it was proposed in 2009 by Cremers and Petajisto (2009). An explicit relationship between Active Share and the other popular measure of relative risk, Tracking Error, was proposed by Stowe in 2014 but necessitates the full knowledge of all the weights of the portfolio and of the complete covariance matrix of the assets. I derive an approximation in a simple setup where the exact weights are unknown, the pairwise correlation between the returns of the assets is fixed, and the volatility of the assets is described by its mean and variance. This approximation is really a “toy model” and works for the purpose of understanding changes in Tracking error stemming from changes in a portfolio, and to show what the drivers of the link between Active Share and Tracking error are. Using the approximation and the Fundamental Law of Active Management, I compute the total expected return and variance of a fund with a given active share in our setup. A full derivation of the main results is provided in the appendix.

Augmented Covariance Matrix Reconstruction for DOA Estimation Using Difference Coarray

As is well known, nonuniform linear arrays have significant advantages in array aperture and degrees of freedom over uniform linear arrays. Using their difference coarrays, subspace-based approaches can be utilized to perform underdetermined and high-resolution direction-of-arrival (DOA) estimation. However, the subspace-based approaches depend on the covariance matrix reconstruction in the coarray domain, which are not statistically efficient when the number of sources is more than one and less than the number of sensors. In this paper, to overcome this drawback, we devise an augmented covariance matrix reconstruction algorithm for DOA estimation in the coarray domain. The proposed algorithm recovers the complete augmented covariance matrix by solving a rank-minimization problem. But unlike the conventional schemes, it exploits the estimation error distribution of the incomplete augmented covariance matrix to derive the constraint condition of the rank-minimization problem. Based on the reconstructed augmented covariance matrix, we can enhance the DOA estimation performance for multiple source scenario at high signal-to-noise ratio. Although our algorithm is developed based on the non-consecutive coarray, it is also suitable for the consecutive coarray. Numerical results demonstrate the superiority of the proposed algorithm over several existing approaches.

From nuclear physics to displacement damage calculation and uncertainty propagation in CONRAD

Due to the importance of radiation damage of materials in nuclear facilities, it is of interest to accurately calculate the damage and the corresponding uncertainty. The CONRAD code is developed at CEA Cadarache for evaluating nuclear data and corresponding uncertainties using various methods. Different Displacement per Atom (DPA) models have been recently implemented in CONRAD to perform the calculation, the sensitivity analysis, and the uncertainty propagation of radiation damage cross sections. The current version of CONRAD is able to compute damage cross section and the associated covariance matrix from nuclear model parameters and from DPA model parameters. In addition to the sensitivities of damage cross section of 56Fe to different nuclear model parameters and DPA model parameters, the present work illustrates the importance of a complete covariance matrix between different nuclear data, while these correlations are normally not given in the current evaluated nuclear data libraries.

Weighted Estimation of AMMI and GGE Models

The AMMI/GGE model can be used to describe a two-way table of genotype–environment means. When the genotype–environment means are independent and homoscedastic, ordinary least squares (OLS) gives optimal estimates of the model. In plant breeding, the assumption of independence and homoscedasticity of the genotype–environment means is frequently violated, however, such that generalized least squares (GLS) estimation is more appropriate. This paper introduces three different GLS algorithms that use a weighting matrix to take the correlation between the genotype–environment means as well as heteroscedasticity into account. To investigate the effectiveness of the GLS estimation, the proposed algorithms were implemented using three different weighting matrices, including (i) an identity matrix (OLS estimation), (ii) an approximation of the complete inverse covariance matrix of the genotype–environment means, and (iii) the complete inverse covariance matrix of the genotype–environment means. Using simulated data modeled on real experiments, the different weighting methods were compared in terms of the mean-squared error of the genotype–environment means, interaction effects, and singular vectors. The results show that weighted estimation generally outperformed unweighted estimation in terms of the mean-squared error. Furthermore, the effectiveness of the weighted estimation increased when the heterogeneity of the variances of the genotype–environment means increased.

Target parameter estimation for spatial and temporal formulations in MIMO radars using compressive sensing

Conventional algorithms used for parameter estimation in colocated multiple-input-multiple-output (MIMO) radars require the inversion of the covariance matrix of the received spatial samples. In these algorithms, the number of received snapshots should be at least equal to the size of the covariance matrix. For large size MIMO antenna arrays, the inversion of the covariance matrix becomes computationally very expensive. Compressive sensing (CS) algorithms which do not require the inversion of the complete covariance matrix can be used for parameter estimation with fewer number of received snapshots. In this work, it is shown that the spatial formulation is best suitable for large MIMO arrays when CS algorithms are used. A temporal formulation is proposed which fits the CS algorithms framework, especially for small size MIMO arrays. A recently proposed low-complexity CS algorithm named support agnostic Bayesian matching pursuit (SABMP) is used to estimate target parameters for both spatial and temporal formulations for the unknown number of targets. The simulation results show the advantage of SABMP algorithm utilizing low number of snapshots and better parameter estimation for both small and large number of antenna elements. Moreover, it is shown by simulations that SABMP is more effective than other existing algorithms at high signal-to-noise ratio.

The Covariance Adjustment Approaches for Combining Incomparable Cox Regressions Caused by Unbalanced Covariates Adjustment: A Multivariate Meta-Analysis Study.

Background. Univariate meta-analysis (UM) procedure, as a technique that provides a single overall result, has become increasingly popular. Neglecting the existence of other concomitant covariates in the models leads to loss of treatment efficiency. Our aim was proposing four new approximation approaches for the covariance matrix of the coefficients, which is not readily available for the multivariate generalized least square (MGLS) method as a multivariate meta-analysis approach. Methods. We evaluated the efficiency of four new approaches including zero correlation (ZC), common correlation (CC), estimated correlation (EC), and multivariate multilevel correlation (MMC) on the estimation bias, mean square error (MSE), and 95% probability coverage of the confidence interval (CI) in the synthesis of Cox proportional hazard models coefficients in a simulation study. Result. Comparing the results of the simulation study on the MSE, bias, and CI of the estimated coefficients indicated that MMC approach was the most accurate procedure compared to EC, CC, and ZC procedures. The precision ranking of the four approaches according to all above settings was MMC ≥ EC ≥ CC ≥ ZC. Conclusion. This study highlights advantages of MGLS meta-analysis on UM approach. The results suggested the use of MMC procedure to overcome the lack of information for having a complete covariance matrix of the coefficients.

Zero-correlation transformation and threshold for efficient GNSS carrier phase ambiguity resolution

The key point of accurate and precise applications of Global Navigation Satellite Systems lies in knowing how to efficiently obtain correct integer ambiguity. One of the methods in solving the ambiguity resolution problem is applying the ambiguity searching technique coupled with an ambiguity decorrelation technique. Traditionally, an integer-valued limitation of the transformation matrix ensures that the integer characteristic of candidates exists after the inverse transformation, but this also makes the decorrelation imperfect. In this research, the float transformation matrix will be considered. To ensure both the integer characteristic and perfect decorrelation can be reached, the float transformation is used indirectly. To solve the ambiguity resolution problem, the problem is transformed by integer and float transformation matrices. The objective of integer transformation is reducing the number of candidates. The target of float transformation is validating these reduced candidates. A zero correlation domain or a near complete diagonalization covariance matrix can be obtained via the float transformation. A space in this domain will be used as the threshold; hence the zero correlation domain is called the threshold domain. The number of ambiguity candidates based on integer transformation can be reduced once again through the proposed method. The experiments in this paper prove that the method can make the ambiguity resolution become more efficient without any drop in the accuracy.

Error characterization of theGaiaastrometric solution

Context. Accurate characterization of the astrometric errors in the forthcoming Gaia Catalogue will be essential for making optimal use of the data. This includes the correlations among the estimated astrometric parameters of the stars as well as their standard uncertainties, i.e., the complete (variance-)covariance matrix of the relevant astrometric parameters. Aims. Because a direct computation of the covariance matrix is infeasible due to the large number of parameters, approximate methods must be used. The aim of this paper is to provide a mathematical basis for estimating the variance-covariance of any pair of astrometric parameters, and more generally the covariance matrix for multidimensional functions of the astrometric parameters. The validation of this model by means of numerical simulations will be considered in a forthcoming paper. Methods. Based on simplifying assumptions (in particular that calibration errors can be neglected), we derive and analyse a series expansion of the covariance matrix of the least-squares solution. A recursive relation for successive terms is derived and interpreted in terms of the propagation of errors from the stars to the attitude and back. We argue that the expansion should converge rapidly to useful precision. The recursion is vastly simplified by using a kinematographic (step-wise) approximation of the attitude model. Results. Low-order approximations of arbitrary elements from the covariance matrix can be computed efficiently in terms of a limited amount of pre-computed data representing compressed observations and the structural relationships among them. It is proposed that the user interface to the Gaia Catalogue should provide the tools necessary for such computations. (Less)

Evolutionary RBF classifier for polarimetric SAR images

In this paper, a robust radial basis function (RBF) network based classifier is proposed for polarimetric synthetic aperture radar (SAR) images. The proposed feature extraction process utilizes the covariance matrix elements, the H/α/A decomposition based features combined with the backscattering power (span), and the gray level co-occurrence matrix (GLCM) based texture features, which are projected onto a lower dimensional feature space using principal components analysis. For the classifier training, both conventional backpropagation (BP) and multidimensional particle swarm optimization (MD-PSO) based dynamic clustering are explored. By combining complete polarimetric covariance matrix and eigenvalue decomposition based pixel values with textural information (contrast, correlation, energy, and homogeneity) in the feature set, and employing automated evolutionary RBF classifier for the pattern recognition unit, the overall classification performance is shown to be significantly improved. An experimental study is performed using the fully polarimetric San Francisco Bay and Flevoland data sets acquired by the NASA/Jet Propulsion Laboratory Airborne SAR (AIRSAR) at L-band to evaluate the performance of the proposed classifier. Classification results (in terms of confusion matrix, overall accuracy and classification map) compared with the major state of the art algorithms demonstrate the effectiveness of the proposed RBF network classifier.

Inversion of levelling data: how important is error treatment?

Even if proper treatment of error statistics is potentially essential for the reliability of experimental data inversion, a critical evaluation of its effects on levelling data inversion is still lacking. In this paper, we consider the complete covariance matrix for levelling measurements, obtained by combining the covariance matrix due to measurement errors and the covariance matrix due to non-measurement errors, under the simple hypothesis of uncorrelated non-measurement errors on bench mark vertical displacements. The complete covariance matrix is reduced to diagonal form by means of a rotation matrix; the same rotation transforms the data to independent form. The eigenvalues of the complete covariance matrix give the uncertainties of the transformed independent data. This procedure can be used also with non-normal distributions of errors, in which case misfit functions other than χ2 (e.g. the mean absolute deviation) are minimized. Here we focus on two test cases (the 1989 Loma Prieta earthquake and the 1908 Messina earthquake) inverting both real data and synthetics. The inversion of synthetic data sets does not evidence any systematic dependence of retrieved parameter values on the covariance matrix. Most retrieved fault parameter values are close to what used in the forward model, whatever covariance matrix is used. As a consequence, large discrepancies among results obtained using covariance matrices including different combinations of measurement and non-measurement errors when inverting measured and synthetic data sets would possibly indicate the need for further investigations. While measurement errors can be a priori evaluated, it is difficult to estimate non-measurement errors. Our synthetic tests using a uniform-slip rectangular fault in a homogeneous elastic half-space show that, if measurement errors have been correctly evaluated, average non-measurement errors can be estimated by choosing their weight inside the covariance matrix so that the ratio between the total square residual for the χ2 best-fitting model and the number of degrees of freedom is about one. This result holds even if non-measurement errors are different from bench mark to bench mark and/or not normally distributed. In case of more complex models (e.g. one distributed-slip fault) it is difficult to estimate the actual number of degrees of freedom. Our synthetic tests show that this difficulty can be surmounted by using the Akaike Information Criterion, which actually decreases when turning from a uniform-slip inverse model to a distributed-slip inverse model only if the correct amount of non-measurement errors is included in the covariance matrix.

Estimates of the complete genetic covariance matrix for traits in multi-trait genetic evaluation of Australian Hereford cattle

Estimates of covariance components among all 22 traits considered in the current multi-trait genetic evaluation of Australian Hereford cattle were obtained. Traits included 5 weight traits, 8 traits measured through live ultrasound scanning, 3 traits related to reproductive performance, and 6 carcass traits. Estimates were obtained by restricted maximum likelihood, carrying out a series of bivariate analyses. Data for each analysis were selected attempting to maximise the number of animals or animal–parent pairs that had both traits recorded. Estimates were pooled using a weighted 'iterative summing of expanded part matrices' procedure, which ensured positive semi-definite covariance matrices. Models of analyses for individual traits closely resembled those used in genetic evaluation. Results generally agreed with literature results, although estimates of genetic parameters for carcass traits that had few records available tended to fluctuate. Except for 'days to calving', heritability estimates were moderate to high for all traits. Genetic parameters for early growth were different to those for other breeds, with maternal effects for weaning weight being considerably more important and the heritability somewhat lower.

Some efficiency properties of best linear unbiased estimators

In this paper, we show that the best linear unbiased estimators of the location and scale parameters of a location-scale parameter distribution based on a general Type-II censored sample are in fact trace-efficient linear unbiased estimators as well as determinant-efficient linear unbiased estimators. More generally, we show that the best linear unbiased estimators possess complete covariance matrix dominance in the class of all linear unbiased estimators of the location and scale parameters.

Optimization of the kernel functions in a probabilistic neural network analyzing the local pattern distribution.

This article proposes a procedure for the automatic determination of the elements of the covariance matrix of the gaussian kernel function of probabilistic neural networks. Two matrices, a rotation matrix and a matrix of variances, can be calculated by analyzing the local environment of each training pattern. The combination of them will form the covariance matrix of each training pattern. This automation has two advantages: First, it will free the neural network designer from indicating the complete covariance matrix, and second, it will result in a network with better generalization ability than the original model. A variation of the famous two-spiral problem and real-world examples from the UCI Machine Learning Repository will show a classification rate not only better than the original probabilistic neural network but also that this model can outperform other well-known classification techniques.

Incoherent scatter experiments at Jicamarca using alternating codes

An incoherent scatter experiment using alternating coded pulses has been implemented at Jicamarca. The experiment is intended to facilitate aeronomy research in the topside and protonosphere. By virtue of utilizing the full duty cycle capabilities of the radar transmitter, the experiment has improved sensitivity over the conventional double pulse experiment and permits multiple parameter estimation at protonospheric altitudes. Error analysis, including the estimation of the complete error covariance matrix for the alternating coded pulses, is discussed. Example density, temperature, and composition data illustrate the capabilities of the new experiment.

Estimating the Propagation Rate of a Viral Infection of Potato Plants via Mixtures of Regressions

SUMMARY A problem arising from the study of the spread of a viral infection among potato plants by aphids appears to involve a mixture of two linear regressions on a single predictor variable. The plant scientists studying the problem were particularly interested in obtaining a 95% confidence upper bound for the infection rate. We discuss briefly the procedure for fitting mixtures of regression models by means of maximum likelihood, effected via the EM algorithm. We give general expressions for the implementation of the M-step and then address the issue of conducting statistical inference in this context. A technique due to T. A. Louis may be used to estimate the covariance matrix of the parameter estimates by calculating the observed Fisher information matrix. We develop general expressions for the entries of this information matrix. Having the complete covariance matrix permits the calculation of confidence and prediction bands for the fitted model. We also investigate the testing of hypotheses concerning the number of components in the mixture via parametric and ‘semiparametric’ bootstrapping. Finally, we develop a method of producing diagnostic plots of the residuals from a mixture of linear regressions.

Influence of Imputation and EM Methods on Factor Analysis when Item Nonresponse in Questionnaire Data is Nonignorable

This study deals with the influence of each of twelve imputation methods and two methods using the EM algorithm on the results of maximum likelihood factor analysis as compared with results obtained from the complete data factor analysis (no missing scores). Complete questionnaire rating scale data were simulated and, next, missing item scores were created under both ignorable and nonignorable nonresponse mechanisms. Next, imputation methods were used to fill the gaps and factor analysis was applied to both the original complete data and to the data sets including imputed scores. Each imputation method was implemented once with residual error and once without residual error. Also, one EM method estimated the factor loadings directly and the other estimated the complete data covariance matrix, which subsequently was factor analyzed. A design was analyzed with design factors Latent Trait Structure (technically called Mixing Configuration), Correlation Between Latent Traits, Nonresponse Mechanism, Percentage of Missingness, Sample Size, and Imputation Method. We found that, in general, methods that impute a score based on a respondent's mean score obtained from his/her observed item scores best recovered the factor loadings structure from the complete data. Moreover, for unidimensional data person mean methods with a residual error gave better results than the other imputation methods, either with or without a residual error component. For the EM methods a smaller design was analyzed. The conclusion was that both EM methods better recovered the complete data factor loadings than the imputation methods.

Explicit, approximate expressions for the resolution anda posterioricovariance of massive tomographic systems

We present an approximate method to estimate the resolution, covariance and correlation matrix for linear tomographic systems Ax=b that are too large to be solved by singular value decomposition. An explicit expression for the approximate inverse matrix A− is found using one-step backprojections on the Penrose condition AA−≈I, from which we calculate the statistical properties of the solution. The computation of A− can easily be parallelized, each column being constructed independently. The method is validated on small systems for which the exact covariance can still be computed with singular value decomposition. Though A− is not accurate enough to actually compute the solution x, the qualitative agreement obtained for resolution and covariance is sufficient for many purposes, such as rough assessment of model precision or the reparametrization of the model by the grouping of correlating parameters. We present an example for the computation of the complete covariance matrix of a very large (69 043 × 9610) system with 5.9 × 106 non-zero elements in A. Computation time is proportional to the number of non-zero elements in A. If the correlation matrix is computed for the purpose of reparametrization by combining highly correlating unknowns xi, a further gain in efficiency can be obtained by neglecting the small elements in A, but a more accurate estimation of the correlation requires a full treatment of even the smaller Aij. We finally develop a formalism to compute a damped version of A−.