Matrix Estimation Research Articles

Minimum-variance portfolio optimization: an asset pricing model approach

ABSTRACT This paper employs novel asset pricing model methods to construct the global minimum-variance portfolio G. Out-of-sample analyses of U.S. stock returns show that optimized G portfolios have relatively higher expected returns, lower variance, and higher Sharpe ratios than those based on traditional variance-covariance matrix estimation methods. Robustness checks confirm these findings and show that our G portfolio has lower variance than currently available minimum-variance exchange traded funds (ETFs). We conclude that asset pricing models can be used to build high performing G portfolios. Implications to portfolio management are discussed.

Satellite Image Restoration via an Adaptive QWNNM Model

Due to channel noise and random atmospheric turbulence, retrieved satellite images are always distorted and degraded and so require further restoration before use in various applications. The latest quaternion-based weighted nuclear norm minimization (QWNNM) model, which utilizes the idea of low-rank matrix approximation and the quaternion representation of multi-channel satellite images, can achieve image restoration and enhancement. However, the QWNNM model ignores the impact of noise on similarity measurement, lacks the utilization of residual image information, and fixes the number of iterations. In order to address these drawbacks, we propose three adaptive strategies: adaptive noise-resilient block matching, adaptive feedback of residual image, and adaptive iteration stopping criterion in a new adaptive QWNNM model. Both simulation experiments with known noise/blurring and real environment experiments with unknown noise/blurring demonstrated that the effectiveness of adaptive QWNNM models outperformed the original QWNNM model and other state-of-the-art satellite image restoration models in very different technique approaches.

Buckling analysis of single‐walled carbon nanotubes using the initial value method and approximate transfer matrix based on nonlocal elasticity theory

AbstractIn this study, a novel method is presented to analyze the buckling behavior of single‐walled carbon nanotubes (SWCNTs) using the initial value method (IVM) in conjunction with the approximate transfer matrix, within the framework of nonlocal elasticity theory. The study aims to accurately approximate critical buckling load parameters under various boundary conditions, without encountering high computational requirements. IVM enables the computation of displacements and stress resultants along the entire beam from given initial conditions. The approximate transfer matrix is employed to analyze system states at different points through successive integration of solutions, generating the principal matrix needed for IVM and ensuring systematic and precise results that optimize the accuracy of the analysis. A convergence study confirms the effectiveness and precision of the proposed method, revealing a decrease in the critical buckling load parameters as the nonlocal parameter increases, applicable across all boundary conditions studied (simply supported, clamped–clamped, clamped–simply supported, and clamped‐free). These results underscore the need to incorporate nonlocal effects for more accurate nanostructure mechanics predictions. The integration of IVM and the approximate transfer matrix provides a computationally efficient alternative to traditional numerical and semi‐analytical methods, aiding researchers and engineers working with SWCNTs and other nanomaterials.

Exploring Bounded Component Analysis Using an ℓ∞ Norm Criterion

In this paper we propose a new criterion for the Blind Source Separation (BSS) of antisparse bounded sources, based on the sum of the ℓ∞-norm of the sources. Based on the observation that the mixing process of bounded sources with any mixing matrix with unitary Frobenius norm will increase the ℓ∞-norm of the sources, unless it is the identity matrix, the minimization of the sum of the ℓ∞-norm of the sources can be used for the estimation of a separation matrix. To that, a Principle Component Analysis technique followed by a Givens Rotations based optimization method can be used for the separation of independent bounded sources. Also, the Givens Rotations based optimization method can be used for the separation of correlated bounded sources mixed by a rotation matrix. We theoretically analyze the proposed criterion and assess its performance through numerical simulations involving three distinct types of bounded signals. Our theoretical and experimental findings underscore the efficacy of the ℓ∞ norm as a suitable contrast function for antisparse bounded sources, showcasing its superior performance relative to a state-of-the-art algorithm.

Simultaneous image denoising and completion through convolutional sparse representation and nonlocal self-similarity

Low rank matrix approximation (LRMA) has been widely studied due to its capability of approximating original image from the degraded image. According to the characteristics of degraded images, image denoising and image completion have become research objects. Existing methods are usually designed for a single task. In this paper, focusing on the task of simultaneous image denoising and completion, we propose a weighted low rank sparse representation model and the corresponding efficient algorithm based on LRMA. The proposed method integrates convolutional analysis sparse representation (ASR) and nonlocal statistical modeling to maintain local smoothness and nonlocal self-similarity (NLSM) of natural images. More importantly, we explore the alternating direction method of multipliers (ADMM) to solve the above inverse problem efficiently due to the complexity of simultaneous image denoising and completion. We conduct experiments on image completion for partial random samples and mask removal with different noise levels. Extensive experiments on four datasets, i.e., Set12, Kodak, McMaster, and CBSD68, show that the proposed method prevents the transmission of noise while completing images and has achieved better quantitative results and human visual quality compared to 17 methods. The proposed method achieves (1.9%, 1.8%, 4.2%, and 3.7%) gains in average PSNR and (4.2%, 2.9%, 6.7%, and 6.6%) gains in average SSIM over the sub-optimal method across the four datasets, respectively. We also demonstrate that our method can handle the challenging scenarios well. Source code is available at https://github.com/weimin581/demo_CSRNS.

Integrating Factor Models Using Bayesian Model Averaging: Evidence from Korea

This study applies the Bayesian model averaging approach to the Korean stock market data to yield an integrated factor model that can address model uncertainty. During the period from 2004 to 2022, we choose 14 factors and 10 macro predictors and generate a total of 10,485,760 candidate factor models by considering all possible combinations of each variable and whether to allow for model mispricing. The integrated model estimates the expected return and covariance matrix by combining the predictions of all the candidate models using Bayesian posterior probabilities as weights. In the Korean stock market, there is no clear winner model with dominant posterior probabilities, implying that model uncertainty is substantial. The optimal portfolio of the integrated model has a higher out-of-sample Sharpe ratio and lower downside risk than benchmark factor models. Model uncertainty has a significant impact on the estimation of the covariance matrix of the integrated model, and model disagreement about expected returns is particularly acute during periods of sharp market declines. In the presence of model uncertainty, Bayesian investors perceive equities to be riskier than historical volatility.

Adaptive Learning for Dynamic Features and Noisy Labels.

Applying current machine learning algorithms in complex and open environments remains challenging, especially when different changing elements are coupled and the training data is scarce. For example, in the activity recognition task, the motion sensors may change position or fall off due to the intensity of the activity, leading to changes in feature space and finally resulting in label noise. Learning from such a problem where the dynamic features are coupled with noisy labels is crucial but rarely studied, particularly when the noisy samples in new feature space are limited. In this paper, we tackle the above problem by proposing a novel two-stage algorithm, called Adaptive Learning for Dynamic features and Noisy labels (ALDN). Specifically, optimal transport is firstly modified to map the previously learned heterogeneous model to the prior model of the current stage. Then, to fully reuse the mapped prior model, we add a simple yet efficient regularizer as the consistency constraint to assist both the estimation of the noise transition matrix and the model training in the current stage. Finally, two implementations with direct (ALDN-D) and indirect (ALDN-ID) constraints are illustrated for better investigation. More importantly, we provide theoretical guarantees for risk minimization of ALDN-D and ALDN-ID. Extensive experiments validate the effectiveness of the proposed algorithms.

Overview of Tensor-Based Cooperative MIMO Communication Systems—Part 2: Semi-Blind Receivers

Cooperative MIMO communication systems play an important role in the development of future sixth-generation (6G) wireless systems incorporating new technologies such as massive MIMO relay systems, dual-polarized antenna arrays, millimeter-wave communications, and, more recently, communications assisted using intelligent reflecting surfaces (IRSs), and unmanned aerial vehicles (UAVs). In a companion paper, we provided an overview of cooperative communication systems from a tensor modeling perspective. The objective of the present paper is to provide a comprehensive tutorial on semi-blind receivers for MIMO one-way two-hop relay systems, allowing the joint estimation of transmitted symbols and individual communication channels with only a few pilot symbols. After a reminder of some tensor prerequisites, we present an overview of tensor models, with a detailed, unified, and original description of two classes of tensor decomposition frequently used in the design of relay systems, namely nested CPD/PARAFAC and nested Tucker decomposition (TD). Some new variants of nested models are introduced. Uniqueness and identifiability conditions, depending on the algorithm used to estimate the parameters of these models, are established. Two families of algorithms are presented: iterative algorithms based on alternating least squares (ALS) and closed-form solutions using Khatri–Rao and Kronecker factorization methods, which consist of SVD-based rank-one matrix or tensor approximations. In a second part of the paper, the overview of cooperative communication systems is completed before presenting several two-hop relay systems using different codings and configurations in terms of relaying protocol (AF/DF) and channel modeling. The aim of this presentation is firstly to show how these choices lead to different nested tensor models for the signals received at destination. Then, by capitalizing on these models and their correspondence with the generic models studied in the first part, we derive semi-blind receivers to jointly estimate the transmitted symbols and the individual communication channels for each relay system considered. In a third part, extensive Monte Carlo simulation results are presented to compare the performance of relay systems and associated semi-blind receivers in terms of the symbol error rate (SER) and channel estimate normalized mean-square error (NMSE). Their computation time is also compared. Finally, some perspectives are drawn for future research work.

Shrinkage estimation of gene interaction networks in single-cell RNA sequencing data

BackgroundGene interaction networks are graphs in which nodes represent genes and edges represent functional interactions between them. These interactions can be at multiple levels, for instance, gene regulation, protein-protein interaction, or metabolic pathways. To analyse gene interaction networks at a large scale, gene co-expression network analysis is often applied on high-throughput gene expression data such as RNA sequencing data. With the advance in sequencing technology, expression of genes can be measured in individual cells. Single-cell RNA sequencing (scRNAseq) provides insights of cellular development, differentiation and characteristics at the transcriptomic level. High sparsity and high-dimensional data structures pose challenges in scRNAseq data analysis.ResultsIn this study, a sparse inverse covariance matrix estimation framework for scRNAseq data is developed to capture direct functional interactions between genes. Comparative analyses highlight high performance and fast computation of Stein-type shrinkage in high-dimensional data using simulated scRNAseq data. Data transformation approaches also show improvement in performance of shrinkage methods in non-Gaussian distributed data. Zero-inflated modelling of scRNAseq data based on a negative binomial distribution enhances shrinkage performance in zero-inflated data without interference on non zero-inflated count data.ConclusionThe proposed framework broadens application of graphical model in scRNAseq analysis with flexibility in sparsity of count data resulting from dropout events, high performance, and fast computational time. Implementation of the framework is in a reproducible Snakemake workflow https://github.com/calathea24/ZINBGraphicalModel and R package ZINBStein https://github.com/calathea24/ZINBStein.

Concentration of a Sparse Bayesian Model With Horseshoe Prior in Estimating High‐Dimensional Precision Matrix

ABSTRACTPrecision matrices are crucial in many fields such as social networks, neuroscience and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an off‐diagonal position of the precision matrix indicates conditional independence between nodes. In high‐dimensional settings where the dimension of the precision matrix exceeds the sample size and the matrix is sparse, methods like graphical Lasso, graphical SCAD and CLIME are popular for estimating GGMs. While frequentist methods are well‐studied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate, applying the global‐local horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in high‐dimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior. A concise set of simulations is performed to validate our theoretical findings.

Optimisation models for estimating public transport OD matrices using different data types

Efficient public transport systems rely on origin–destination matrices (ODMs) estimation to accurately capture passenger travel patterns, enabling adjustments to frequencies and lines as needed. In this study, we address the ODM estimation problem by employing multiple bi-level programs that consider an outdated ODM and observed passenger flows on specific transit line arcs. Additionally, we consider various optional data types, including boarding and alighting data, as well as the structure of the outdated ODM and passenger flows, either all, separately, in combination, or none. In our study, we reformulate these bi-level programs into single-level models, and we use a commercial solver to address the problem in benchmark instances. In our analysis, we focus on the impact of incorporating different types of information into the estimation process, leading to valuable insights. We find that considering all the data types leads to a higher accuracy than only a subset of these data types. In particular, focussing only on boarding and alighting data leads to improvements in the estimation process, whereas considering only the structure of the outdated ODM and passenger flows leads to reduced accuracy compared to not incorporating either. The latter highlights the significance of data selection in ODM estimations.

Nonlocal buckling analysis of double-walled carbon nanotubes using initial value method and matricant approach

In this study, the buckling behavior of double-walled carbon nanotubes (DWCNTs) is investigated using the initial value method (IVM) combined with the approximate transfer matrix (ATM) approach within the framework of nonlocal elasticity theory. The aim is to provide a method that best estimates the critical buckling loads with improved accuracy and computational efficiency. To validate the method, the critical buckling loads obtained without considering nonlocal effects were compared with results from the literature, showing a high degree of agreement. Various boundary conditions, including simply supported (S-S), clamped-simply supported (C-S), clamped-free (C-F), and clamped-clamped (C-C), have been considered for investigation. The effects of nonlocal parameters, aspect ratios, and boundary conditions on the critical buckling loads are analyzed. The findings emphasize the importance of accounting for these parameters in mechanical analyses to ensure accurate predictions. The proposed framework provides an efficient alternative to existing numerical methods and offers valuable benchmarks for future research on the mechanical behavior of nanostructures.

Ridge-type covariance and precision matrix estimators of the multivariate normal distribution

AbstractWe consider ridge-type estimation of the multivariate normal distribution’s covariance matrix and its inverse, the precision matrix. While several ridge-type covariance and precision matrix estimators have been presented in the literature, their respective inverses are often not considered as precision and covariance matrix estimators even though their estimands are one-to-one related through the matrix inverse. We study which estimator is to be preferred in what case. Hereto we compare the ridge-type covariance matrix estimators and their properties to that of the inverse of the ridge-type precision matrix estimators, and vice versa. The comparison, in which we take all ridge-type estimators along, is limited to a specific case that is illustrative of the difference between the covariance and precision matrix estimators. The comparison addresses the estimators’ estimating equation, analytic expression, analytic properties like positive definiteness and penalization limit, mean squared error, consistency, Bayesian formulation, and their loss and potential for marginal and partial correlation screening.

Representing turbulent statistics with partitions of state space. Part 1. Theory and methodology

This is the first of a two-part paper. We formulate a data-driven method for constructing finite-volume discretizations of an arbitrary dynamical system's underlying Liouville/Fokker–Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centres) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We illustrate the method on the Lorenz equations (a three-dimensional ordinary differential equation) saving a fluid dynamical application for Part 2 (Souza, J. Fluid Mech., vol. 997, 2024, A2).

On robust inference in time series regression

Abstract Least squares regression with heteroskedasticity consistent standard errors (“OLS-HC regression”) has proved very useful in cross section environments. However, several major difficulties, which are generally overlooked, must be confronted when transferring the HC technology to time series environments via heteroskedasticity and autocorrelation consistent standard errors (“OLS-HAC regression”). First, in plausible time-series environments, OLS parameter estimates can be inconsistent, so that OLS-HAC inference fails even asymptotically. Second, most economic time series have autocorrelation, which renders OLS parameter estimates inefficient. Third, autocorrelation similarly renders conditional predictions based on OLS parameter estimates inefficient. Finally, the structure of popular HAC covariance matrix estimators is ill-suited for capturing the autoregressive autocorrelation typically present in economic time series, which produces large size distortions and reduced power in HAC-based hypothesis testing, in all but the largest samples. We show that all four problems are largely avoided by the use of a simple and easily-implemented dynamic regression procedure, which we call DURBIN. We demonstrate the advantages of DURBIN with detailed simulations covering a range of practical issues.

GNSS-assisted optimal alignment method for low-cost SINS motion of vehicle

Abstract To improve the alignment accuracy and environmental applicability of the micro-electromechanical system (MEMS)-based strapdown inertial navigation system (SINS), this study proposes a global navigation satellite system (GNSS)-assisted multi-vector determination of the attitude optimal indirect coarse alignment. Using the inertial information output from the inertial measurement unit and the velocity information from the GNSS, a simplified velocity observation vector is constructed and the velocity lever arm stemming from the mounting position inconsistencies of the GNSS and SINS is fed back into the velocity of the carrier system. When constructing the observation vectors, the integration interval is shortened by reconstructing the two-vector integration formula for reducing the cumulative error of the inertial device. The attitude matrix problem is defined as the Wahba problem, which is solved using the singular value decomposition method. Based on the relationship between gyro zero bias and misalignment angle, the corresponding state and measurement equations are designed. Furthermore, owing to the measurement noise uncertainty, the adaptive traceless Kalman filtering algorithm is introduced to realize the effective adaptation processing of the measurement noise. More accurate attitude matrix estimates are obtained by continuously correcting the carrier system transformation matrix. The running car experiment results show that the proposed method exhibits higher alignment accuracy and environmental applicability than the current MEMS strapdown inertial navigation coarse alignment method and the traditional optimization-based alignment method.

Bayesian inference for link travel time correlation of a bus route

Estimation of link travel time correlation of a bus route is essential to many bus operation applications. Link travel time on a bus route could exhibit complex correlation structures, such as long-range correlations, negative correlations, and time-varying correlations. This paper develops a Bayesian Gaussian model to estimate the link travel time correlation matrix of a bus route using smart-card-like data. Our method overcomes the small-sample-size problem in correlation matrix estimation by borrowing/integrating those incomplete observations from other bus routes. We first conduct a synthetic experiment and results show that the proposed method produces an accurate estimation for correlations with credible intervals. Next, we perform experiments on a real-world bus route with in-out-stop record data; results show that both local and long-range correlations exist on this bus route. Finally, we demonstrate an application of using the estimated covariance matrix to make probabilistic forecasting of link and trip travel time.

Jacobian vs. disturbance method for UMATs in ABAQUS: An application to isotropic damage mechanics

This paper compares the computational efficiency of different methodologies to estimate the tangent stiffness applied to ABAQUS standard UMAT user-subroutine, using implicit formulation with predictor corrector methodology. It uses two new methodologies to estimate the stiffness matrix for the predictor-corrector iterative process, named Disturbance Method and Implicit to Explicit Method. With the Disturbance Method, it is enough to build UMATs using only a secant matrix, since an approximate tangential material matrix is automatically computed, promoting a faster implementation of new material constitutive relations in the ABAQUS standard. It is also studied the possibility of the use of a new algorithm that transforms the UMAT implicit formulation to an explicit formulation using a new secant predictor corrector algorithm. In this study, a new concrete damage model that considers fracture energy regularization for both tensile and compressive behaviour with Mode I Fracture is utilized. Classical and known benchmarks to test damage models in the scientific community are used to compare both methods for the calculation of the Jacobian matrix in terms of time and number of iterations.

Functions of a Complex Variable with a Large Parameter and Construction of Regions

The study of the asymptotic behavior of solutions of singularly perturbed equations in complex domains comes down to the study of integrals of exponential functions containing a large parameter. Such integrals differ significantly from the integrals to which the saddle point method is applicable. The saddle point method is not applicable to study such integrals. Thus, the problem arises of constructing domains and choosing integration paths for studying such integrals. In this work, specific examples of integrals show the construction of regions in the complex plane and the choice of integration paths. The chosen integration paths ensure that the integrals are bounded over a large parameter as this parameter tends to infinity. When constructing the domain and choosing integration paths, level lines of some harmonic functions that have zeros and singular points were used. The principle of symmetry is also used. In early works, cases were considered when the eigenvalues ​​of the first approximation matrix of a singularly perturbed equation had only zeros or only poles. Cases where the eigenvalues ​​have zeros and poles were not considered.

Dynamic correction of forest fire spread prediction using observation error covariance matrix estimation technique based on FLC-GRU

BackgroundData assimilation (DA) techniques have played a significant role in improving the prediction accuracy of forest fire spread. The dynamic correction technique weights the predicted and observed values to obtain an analytical value that better reflects the position of the fire perimeter. The weighted importance of each contribution is determined by the magnitude of its associated error. However, as a crucial parameter affecting prediction accuracy, the covariance matrix of observation errors is often inaccurate and neglects its own temporal correlation. This is unfriendly to spread prediction results. To address this issue, we proposed a targeted technique for estimating the observation error covariance matrix (R matrix) based on the Fire Line Convolutional Gated Recurrent Unit (FLC-GRU).ResultsWe integrated this method into the DA framework and validated its applicability and accuracy using Observing System Simulation Experiment (OSSE). Through comparisons with traditional methods, the results indicate that using the FLC-GRU estimated R matrix for correction calculations leads to wildfire prediction locations that are closer to the true values.ConclusionsThe proposed approach learns the covariance matrix directly from time-series observed fire line data, without requiring any prior knowledge or assumptions about the error distribution, in contrast to classical posterior tuning methods. The proposed method significantly improves the rationality and accuracy of R matrix estimation, enhances the utility of observational data, and thereby improves the correction accuracy of forest fire spread predictions. Moreover, the study also demonstrates the applicability of the proposed method within the DA framework.