Estimation Of Covariance Matrices Research Articles

How Close to the Mark Might Published Heritability Estimates Be?

The behavioural scientist who requires an estimate of narrow heritability, h2, will conduct a twin study, and input the resulting estimated covariance matrices into a particular mode of estimation, the latter derived under supposition of the standard biometric model (SBM). It is known that the standard biometric model can be expected to misrepresent the phenotypic (genetic) architecture of human traits. The impact of this misrepresentation on the accuracy of h2 estimation is unknown. We aimed to shed some light on this general issue, by undertaking three simulation studies. In each, we investigated the parameter recovery performance of five modes- Falconer’s coefficient and the SEM models, ACDE, ADE, ACE, and AE- when they encountered a constructed, non-SBM, architecture, under a particular informational input. In study 1, the architecture was single-locus with dominance effects and genetic-environment covariance, and the input was a set of population covariance matrices yielded under the four twin designs, monozygotic-reared together, monozygotic-reared apart, dizygotic-reared together, and dizygotic-reared apart; in study 2, the architecture was identical to that of study 1, but the informational input was monozygotic-reared together and dizygotic-reared together; and in study 3, the architecture was multi-locus with dominance effects, genetic-environment covariance, and epistatic interactions. The informational input was the same as in study 1. The results suggest that conclusions regarding the coverage of h2 must be drawn conditional on a) the general class of generating architecture in play; b) specifics of the architecture’s parametric instantiations; c) the informational input into a mode of estimation; and d) the particular mode of estimationemployed. The results showed that the more complicated the generating architecture, the poorer a mode’s h2 recovery performance. Random forest analyses furthermore revealed that, depending on the genetic architecture, h2, the dominance and locus additive parameter, and proportions of alleles were involved in complex interaction effects impacting on h2 parameter recovery performance of a mode of estimation. Data and materials: https://osf.io/aq9sx/

Element Aggregation for Estimation of High-Dimensional Covariance Matrices

This study addresses the challenge of estimating high-dimensional covariance matrices in financial markets, where traditional sparsity assumptions often fail due to the interdependence of stock returns across sectors. We present an innovative element-aggregation method that aggregates matrix entries to estimate covariance matrices. This method is designed to be applicable to both sparse and non-sparse matrices, transcending the limitations of sparsity-based approaches. The computational simplicity of the method’s implementation ensures low complexity, making it a practical tool for real-world applications. Theoretical analysis then confirms the method’s consistency and effectiveness with its convergence rate in specific scenarios. Additionally, numerical experiments validate the method’s superior algorithmic performance compared to conventional methods, as well as the reduction in relative estimation errors. Furthermore, empirical studies in financial portfolio optimization demonstrate the method’s significant risk management benefits, particularly its ability to effectively mitigate portfolio risk even with limited sample sizes.

Multilevel superposition for deciphering the conformational variability of protein ensembles.

The dynamics and variability of protein conformations are directly linked to their functions. Many comparative studies of X-ray protein structures have been conducted to elucidate the relevant conformational changes, dynamics and heterogeneity. The rapid increase in the number of experimentally determined structures has made comparison an effective tool for investigating protein structures. For example, it is now possible to compare structural ensembles formed by enzyme species, variants or the type of ligands bound to them. In this study, the author developed a multilevel model for estimating two covariance matrices that represent inter- and intra-ensemble variability in the Cartesian coordinate space. Principal component analysis using the two estimated covariance matrices identified the inter-/intra-enzyme variabilities, which seemed to be important for the enzyme functions, with the illustrative examples of cytochrome P450 family 2 enzymes and class A $\beta$-lactamases. In P450, in which each enzyme has its own active site of a distinct size, an active-site motion shared universally between the enzymes was captured as the first principal mode of the intra-enzyme covariance matrix. In this case, the method was useful for understanding the conformational variability after adjusting for the differences between enzyme sizes. The developed method is advantageous in small ensemble-size problems and hence promising for use in comparative studies on experimentally determined structures where ensemble sizes are smaller than those generated, for example, by molecular dynamics simulations.

Covariance Matrix Estimation for High-Throughput Biomedical Data with Interconnected Communities

Estimating a covariance matrix is central to high-dimensional data analysis. Empirical analyses of high-dimensional biomedical data, including genomics, proteomics, microbiome, and neuroimaging, among others, consistently reveal strong modularity in the dependence patterns. In these analyses, intercorrelated high-dimensional biomedical features often form communities or modules that can be interconnected with others. While the interconnected community structure has been extensively studied in biomedical research (e.g., gene co-expression networks), its potential to assist in the estimation of covariance matrices remains largely unexplored. To address this gap, we propose a procedure that leverages the commonly observed interconnected community structure in high-dimensional biomedical data to estimate large covariance and precision matrices. We derive the uniformly minimum variance unbiased estimators for covariance and precision matrices in closed forms and provide theoretical results on their asymptotic properties. Our proposed method enhances the accuracy of covariance- and precision-matrix estimation and demonstrates superior performance compared to the competing methods in both simulations and real data analyses.

Bayesian estimation of cluster covariance matrices of unknown form

We develop a flexible Bayesian model for cluster covariance matrices in large dimensions where the number of clusters and the assignment of cross-sectional units to a cluster are a-priori unknown and estimated from the data. In a cluster covariance matrix, the variances and covariances are equal within each diagonal block, while the covariances are equal in each off-diagonal block. This reduces the number of parameters by pooling those parameters together that are in the same cluster. In order to treat the number of clusters and the cluster assignments as unknowns, we build a random partition model which assigns a prior distribution over the space of partitions of the data into clusters. Sampling from the posterior over the space of partitions creates a flexible estimator because it averages across a wide set of cluster covariance matrices. We illustrate our methods on linear factor models and large vector autoregressions.

Sequential management of energy and low-carbon portfolios

This study explores the ability of clean energy and European Union Allowance (EUA) assets to diminish portfolio risk when mixed with unclean energy assets. We use a family of Asymmetric Dynamic Conditional Correlation-Generalized AutoRegressive Conditional Heteroskedastic (ADCC-GARCH) models and provide a flexible and adaptive estimation and model selection framework based on a sequential strategy with differently sized estimation and validation windows, as well as different model update frequencies. Through this procedure, we obtain accurate estimations of the conditional covariance matrices of day-to-day asset returns and build adequate optimal minimum variance portfolios. The analyzed period (Jan. 2010–May. 2022) includes the latest crisis episodes (Sovereign debt crisis, Brexit, COVID-19, and the Russian–Ukrainian war). Our findings show that since the 2015 Paris Agreement (the only exception being the pandemic period), investing in clean energy companies and EUAs is an attractive investment in terms of return-risk. These results should provide investors with more incentives to decarbonize their portfolios.

Joint Learning of Full-Structure Noise in Hierarchical Bayesian Regression Models.

We consider the reconstruction of brain activity from electroencephalography (EEG). This inverse problem can be formulated as a linear regression with independent Gaussian scale mixture priors for both the source and noise components. Crucial factors influencing the accuracy of the source estimation are not only the noise level but also its correlation structure, but existing approaches have not addressed the estimation of noise covariance matrices with full structure. To address this shortcoming, we develop hierarchical Bayesian (type-II maximum likelihood) models for observations with latent variables for source and noise, which are estimated jointly from data. As an extension to classical sparse Bayesian learning (SBL), where across-sensor observations are assumed to be independent and identically distributed, we consider Gaussian noise with full covariance structure. Using the majorization-maximization framework and Riemannian geometry, we derive an efficient algorithm for updating the noise covariance along the manifold of positive definite matrices. We demonstrate that our algorithm has guaranteed and fast convergence and validate it in simulations and with real MEG data. Our results demonstrate that the novel framework significantly improves upon state-of-the-art techniques in the real-world scenario where the noise is indeed non-diagonal and full-structured. Our method has applications in many domains beyond biomagnetic inverse problems.

Efficient nonparametric estimation of Toeplitz covariance matrices

Abstract A new efficient nonparametric estimator for Toeplitz covariance matrices is proposed. This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression. The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data driven and computationally very fast. Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices. These results are readily extended to estimation of inverses of Toeplitz covariance matrices. Also, an alternative version of the Whittle likelihood for the spectral density based on the discrete cosine transform is proposed.

LATIS: Constraints on the Galaxy–Halo Connection at z ∼ 2.5 from Galaxy–Galaxy and Galaxy–Lyα Clustering

The connection between galaxies and dark matter halos is often quantified using the stellar mass–halo mass (SMHM) relation. Optical and near-infrared imaging surveys have led to a broadly consistent picture of the evolving SMHM relation based on measurements of galaxy abundances and angular correlation functions. Spectroscopic surveys at z ≳ 2 can also constrain the SMHM relation via the galaxy autocorrelation function and through the cross-correlation between galaxies and Lyα absorption measured in transverse sight lines; however, such studies are very few and have produced some unexpected or inconclusive results. We use ∼3000 spectra of z ∼ 2.5 galaxies from the Lyα Tomography IMACS Survey (LATIS) to measure the galaxy–galaxy and galaxy–Lyα correlation functions in four bins of stellar mass spanning 109.2 ≲ M */M ⊙ ≲ 1010.5. Parallel analyses of the MultiDark N-body and ASTRID hydrodynamic cosmological simulations allow us to model the correlation functions, estimate covariance matrices, and infer halo masses. We find that results of the two methods are mutually consistent and broadly accord with standard SMHM relations. This consistency demonstrates that we are able to measure and model Lyα transmission fluctuations δ F in LATIS accurately. We also show that the galaxy–Lyα cross-correlation, a free by-product of optical spectroscopic galaxy surveys at these redshifts, can constrain halo masses with similar precision to galaxy–galaxy clustering.

Two New Algorithms for Maximum Likelihood Estimation of Sparse Covariance Matrices With Applications to Graphical Modeling

Two New Algorithms for Maximum Likelihood Estimation of Sparse Covariance Matrices With Applications to Graphical Modeling

Optimal Estimation of High-Dimensional Covariance Matrices with Missing and Noisy Data

Optimal Estimation of High-Dimensional Covariance Matrices with Missing and Noisy Data

Robust Estimation of Covariance Matrices: Adversarial Contamination and Beyond

Robust Estimation of Covariance Matrices: Adversarial Contamination and Beyond

Estimation of large covariance matrices with mixed factor structures

Summary We extend the principal orthogonal complement thresholding (POET) framework by J. Fan, Y. Liao, and M. Mincheva (2013) to estimate large covariance matrices with a ‘mixed’ structure of observable and unobservable strong/weak factors, and we call this method the extended POET (ePOET). Especially, the weak factor structure allows the existence of slowly divergent eigenvalues of the covariance matrix that are frequently observed in real data. Under some mild conditions, we derive the uniform consistency of the proposed estimator for the cases with or without observable factors. Furthermore, several simulation studies show that the ePOET achieves good finite-sample performance regardless of data with strong, weak, or mixed factors structure. Finally, we conduct empirical studies to present the practical usefulness of the ePOET.

Bayesian estimation of non-linear centroid moment tensors using multiple seismic data sets

SUMMARY Centroid moment tensor (CMT) parameters of earthquakes are routinely estimated to gain information on structures and regional tectonics. However, for small earthquakes (M < 4), it is still challenging to determine CMTs due to the lack of high-quality waveform data. In this study, we propose to improve solutions for small earthquakes by incorporating multiple seismic data types in Bayesian joint inversion: polarities picked on broad-band signals, amplitude spectra for intermediate frequency bands (0.2–2.0 Hz), and waveforms at low frequencies (0.05–0.2 Hz). Both measurement and theory errors are accounted for by iterative estimation of non-Toeplitz covariance matrices, providing objective weightings for the different data types in the joint parameter estimation. Validity and applicability of the method are demonstrated using simulated and field data. Results demonstrate that combinations of data, such as a single high-quality waveform, a few amplitude spectra and many waveform polarities, are able to resolve CMT parameters to comparable quality as if many high-quality waveforms were available. Results of 10 induced seismic events that occurred in northeastern British Columbia, Canada, between January 2020 and February 2022 indicate predominantly strike-slip focal mechanisms with low non-double-couple components. These events appear to be located at shallow depths with short time duration, as expected for induced seismicity. These results are consistent with previous studies, indicating that this method reduces the dependence of source inversion on high-quality waveforms, and can provide resolution of CMT parameters for earthquakes as small as ML 1.6.

A NARX network optimized with an adaptive weighted square-root cubature Kalman filter for the dynamic state of charge estimation of lithium-ion batteries

Due to the high nonlinearities and unstable working conditions, accurately estimating the state of charge (SOC) by the battery management system (BMS) is a major challenge in ensuring the safety and reliability of lithium-ion batteries in electric vehicles. This paper presents a deep learning network, a nonlinear autoregressive model with exogenous inputs (NARX) network with a closed-loop architecture and transfer learning mechanism, which is optimized using a proposed adaptive weighted square-root cubature Kalman filter (AWSCKF) with a moving sliding window and an adaptive weighing coefficient for SOC estimation of lithium-ion batteries. The proposed AWSCKF method is established through square-root and cubature updates to optimize the statistical value of the state estimate, error covariance, and measurement noise covariance matrices, with the ability to incorporate high nonlinearities to filter out the noise, stabilize, and optimize the final SOC. To evaluate the effectiveness of the optimized NARX network and verify the proposed AWSCKF method, battery tests are carried out using a lithium cobalt oxide battery at various charge-discharge rates and a lithium nickel cobalt manganese oxide battery at temperatures of 0 and 45 °C under five complex working conditions. The SOC accuracy of lithium-ion batteries is enhanced by the hybrid method estimation process, which is based on sensitivity analysis and adaptation to various working conditions. The comprehensive results show that the proposed NARX-AWSCKF model achieves the overall best mean absolute error, root mean square error, and mean absolute percentage error values of 0.07293%, 0.0912%, and 0.40356%, respectively, under various complex conditions. By effectively utilizing battery domain knowledge for real-world BMS applications, the proposed model outperforms other existing methods in terms of high effectiveness, robustness, and potential to boost the NARX performance.

Hybrid SOC and SOH estimation method with improved noise immunity and computational efficiency in hybrid railroad propulsion system

The fuel-cell and lithium-ion battery (LIB) hybrid railroad propulsion system (HRPS) technologies are actively being researched to reduce greenhouse gas (GHG) emissions. It is becoming increasingly important to accurately estimate the state-of-charge (SOC) and state-of-health (SOH) to enhance the safety and efficiency of the system. For a real-time embedded system, computational efficiency and the effect of noise are crucial challenges for SOC and SOH estimation. To overcome these problems, a method is required that improves computational efficiency and minimizes the effect of noise.To reduce computational cost and the effect of noise, the adaptive extended Kalman filter (AEKF) and specific partial capacity with single forgetting factor recursive least square (SPCSFFRLS) methods were combined to estimate SOC and SOH. In AEKF-based SOC estimation, the state and measurement noise covariance matrices were corrected by the mean square error (MSE) of the difference between measured voltage and modeled voltage. Thus, AEKF is effective in conquering measurement noise. The battery capacity was obtained by the SPCSFFRLS using station-to-station operation data. To calculate true specific partial capacity (SPC) without noise, the partial capacity method and SFFRLS were used to update the battery capacity. However, as the current accumulation period becomes longer, an accumulation error due to measurement noise may occur. Therefore, a specific period was analyzed to reduce accumulation error caused by measurement noise in the HRPS load profile. Finally, SOC and SOH co-estimation was demonstrated for a single cell and a parallel-connected battery pack (1S18P). As a result, it was found that the proposed method can reduce the computational time by about 33.0 % average while keeping the SOC and SOH estimation accuracy compared to DEKF. Thus, the proposed method is suitable for real-time embedded system such as HRPS to estimate SOC and SOH.

Realized regression with asynchronous and noisy high frequency and high dimensional data

We develop regression for high frequency data. This regression is novel in that it can be for both fixed and increasing dimension. Also, the data may have microstructure noise, and observations (trades, or quotes) can be asynchronous, (i.e., the observations do not need to be synchronized across dimensions). As is customary for high-frequency inference methods, we refer to our method as “realized” regression.In our methodology, spot beta becomes a key quantity in the nonparametric framework of high frequency econometrics. The central contribution of this paper is a feasible estimator of spot beta, which is robust to noise and asynchronicity. With the help of the spot-version of the Smoothed TSRV estimator, spot beta can be consistently estimated. There are two direct applications of the spot beta estimates in the current paper. In the first application, the integrated beta can be consistently estimated by aggregating the spot beta estimates. After a bias-correction procedure, a fixed dimension central limit theorem is established for the bias-corrected estimator, with convergence rate which may be arbitrarily close to Op(n−1/4). In the second application we assume time-varying factor structure and conditional sparsity. The spot beta matrix estimator enables the estimation of high dimensional spot covariance and precision matrices. The latter is obtained by thresholding the spot residual covariance estimates, and convergence rates derived. As an empirical application, this paper explores the hourly change in beta around earnings announcements of the S&P 100 constituents.

Estimation of the trajectory and attitude of railway vehicles using inertial sensors with application to track geometry measurement

This paper describes a novel method for the estimation of the trajectory and orientation of a rigid body moving along a railway track. Compared to other recent developments in the literature, the presented approach has the significant advantage of using inertial sensors only, excluding global position and orientation sensors. The excluded sensors are compensated with an odometry system and previous knowledge of the design track geometry. The procedure is based on a kinematic model of the relative motion of the body with respect to the track, together with a Kalman filter algorithm. Two different approaches are used and compared for the estimation of the noise covariance matrices in the Kalman filter. One is based on the use of experimental results with a known output. The other one relies upon constrained maximum likelihood estimation. The calculated trajectory and orientation are applied in this research to the problem of track geometry measurement. A scale track is used for experimental validation, showing that results are sufficiently accurate for this application. The obtained results also reveal that the constrained maximum likelihood estimation performs similarly to the known-output method. This is very convenient because it allows a straightforward application of the algorithm in different scenarios.

Positive-definite thresholding estimators of covariance matrices with zeros

A positive definite estimator of a covariance matrix with zero entries provides a valid covariance matrix that can be used an input in almost any area of multivariate statistical analysis. However, most current approaches do not yet guarantee positive definiteness or deal with the asymptotic efficiency of the covariance estimator. Focusing on the classical setting when the number of Gaussian variables is fixed and the sample size increases, we construct a positive definite and asymptotically efficient estimator by the iterative conditional fitting algorithm (Chaudhuri et al., 2007) when the location of the zero entries is known. If the location of the zero entries is unknown, we further construct a positive definite thresholding estimator by combining the iterative conditional fitting algorithm with thresholding. We prove our thresholding estimator is asymptotically efficient with probability tending to one. In simulation studies, we show our estimator more closely matches the true covariance and more correctly identifies the non-zero entries than competing estimators. We apply our estimator to a neuroimaging study of Huntington disease to detect non-zero correlations among brain regional volumes. Such correlations are timely for ongoing treatment studies to inform how different brain regions are likely to be affected by these treatments.

Separation-based parameterization strategies for estimation of restricted covariance matrices in multivariate model systems

Many multivariate model systems involve the estimation of a covariance matrix that must be positive-definite. A common strategy to ensure positive definiteness of the covariance matrix is through the use of a Cholesky parameterization of the covariance matrix. However, several model systems require imposing restrictions on the elements of the covariance elements. For instance, modelling systems may require fixing some (or all) of the diagonal elements in the covariance matrix to unity due to identification considerations. However, imposing such restrictions using the traditional Cholesky decomposition approach is not feasible and requires the additional parameterization of the Cholesky elements.In this paper, we explore a separation-based strategy with spherical parameterization of the Cholesky matrix to impose restrictions on the covariance matrix. Importantly, using this separation-based parameterization strategy, we also explore the possibility of restricting some covariance (or correlation) terms to zero. The effectiveness of the proposed strategy is assessed through extensive simulation experiments. The results from the simulation experiments highlight better performance of the separation-based strategy in terms of recovery of model parameters – particularly those in the covariance matrix, than the traditional Cholesky parameterization approach. Finally, the proposed strategy is implemented in a joint multivariate binary probit ordered probit model system to analyze the usage (and the extent of use) of non-private modes of transportation in Bengaluru, India. In doing so, the proposed strategy is implemented to restrict several correlations to zero, thus avoiding the estimation of a profligate correlation matrix and substantially easing the estimation process.