Square Root Of Matrix Research Articles

Artificial neural network based on strong track and square root UKF for INS/GNSS intelligence integrated system during GPS outage

When INS/GNSS (inertial navigation system/global navigation satellite system) integrated system is applied, it will be affected by the insufficient number of visible satellites, and even the satellite signal will be lost completely. At this time, the positioning error of INS accumulates with time, and the navigation accuracy decreases rapidly. Therefore, in order to improve the performance of INS/GNSS integration during the satellite signals interruption, a novel learning algorithm for neural network has been presented and used for intelligence integrated system in this article. First of all, determine the input and output of neural network for intelligent integrated system and a nonlinear model for weighs updating during neural network learning has been established. Then, the neural network learning based on strong tracking and square root UKF (unscented Kalman filter) is proposed for iterations of the nonlinear model. In this algorithm, the square root of the state covariance matrix is used to replace the covariance matrix in the classical UKF to avoid the filter divergence caused by the negative definite state covariance matrix. Meanwhile, the strong tracking coefficient is introduced to adjust the filter gain in real-time and improve the tracking capability to mutation state. Finally, an improved calculation method of strong tracking coefficient is presented to reduce the computational complexity in this algorithm. The results of the simulation test and the field-positioning data show that the proposed learning algorithm could improve the calculation stability and robustness of neural network. Therefore, the error accumulation of INS/GNSS integration is effectively compensated, and then the positioning accuracy of INS/GNSS intelligence integrated system has been improved.

A novel resampling-free update framework-based cubature Kalman filter for robust estimation

The resampling-free update (RFU) framework avoids discarding the higher-order moment information of the state probability distribution in Gaussian approximation filters. Still, it suffers from the problem of numerical instability and estimation optimality being corrupted caused by non-closed mapping without Gaussian reconstruction. This study proposes a novel robust RFU framework-based cubature Kalman filter. The maximum correntropy criterion is adopted as the optimization cost to exploit the non-Gaussian moments caused by non-closed mapping in RFU. An RFU update is reconstructed based on the square root of a posterior error matrix to improve the numerical stability. In addition, the periodic resampling operation is implemented to mitigate the non-Gaussianity while keeping higher-order moments. The illustrative example demonstrates that the proposed method can improve the estimation robustness and consistency of the RFU framework compared to other state-of-the-art RFU-based filters.

Online Adaptive Model Identification and State of Charge Estimation for Vehicle-Level Battery Packs

Accurate state of charge (SOC) estimation of traction batteries plays a crucial role in energy and safety management for electric vehicles. Existing studies focus primarily on cell battery SOC estimation. However, numerical instability and divergence problems might occur for a large-size lithium-ion battery pack consisting of many cells. This paper proposes a high-performance online model identification and SOC estimation method based on an adaptive square root unscented Kalman filter (ASRUKF) and an improved forgetting factor recursive least squares (IFFRLS) for vehicle-level traction battery packs. The model parameters are identified online through the IFFRLS, where the conventional method might encounter numerical stability problems. By updating the square root of the covariance matrix, the divergence problem in the traditional unscented Kalman filter is solved in the ASRUKF algorithm, where the positive semi-definiteness of the covariance matrix is guaranteed. Combined with the adaptive noise covariance matched filtering algorithm and real-time compensation of system error, the proposed method solves the problem of ever-degrading estimation accuracy in the presence of time-varying noise with unknown statistical characteristics. Using a 66.2-kWh vehicle battery pack, we experimentally verified that the proposed algorithm could achieve high estimation accuracy with guaranteed numerical stability. The maximum error of SOC estimation can be bounded by 1%, and the root-mean-square error is as low as 0.47% under real-world vehicle operating conditions.

Positivity preserving density matrix minimization at finite temperatures via square root.

We present a Wave Operator Minimization (WOM) method for calculating the Fermi-Dirac density matrix for electronic structure problems at finite temperature while preserving physicality by construction using the wave operator, i.e., the square root of the density matrix. WOM models cooling a state initially at infinite temperature down to the desired finite temperature. We consider both the grand canonical (constant chemical potential) and canonical (constant number of electrons) ensembles. Additionally, we show that the number of steps required for convergence is independent of the number of atoms in the system. We hope that the discussion and results presented in this article reinvigorate interest in density matrix minimization methods.

An Efficient Differential Grouping Algorithm for Large-Scale Global Optimization

Cooperative co-evolution (CC) is a practical and efficient evolutionary framework for solving large-scale global optimization problems (LSGOPs). The performance of CC depends on how variables are being grouped and can be improved through guided variable decomposition for various optimization problems. However, achieving a proper variable decomposition is computationally expensive. This paper proposes an effective yet efficient differential grouping (EDG) method to reduce the associated computational cost. Our method exploits the historical interrelationship information of previous variable groups to examine interactions between the remnant variable groups. This allows us to spend less computing resources without compromising the accuracy of the final grouping result. Our proposal utilizes the covariance matrix adaptation evolution strategy (CMA-ES) algorithm, in conjunction with EDG, to solve LSGOPs. Further, to reduce time complexity and improve the stability of CMA-ES, we substitute the complex matrix decomposition step with simpler matrix operations to compute the square root of the covariance matrix. Results from our experiments and analysis indicate that EDG is a competitive method to solve LSGOPs and improve the performance of CC. The proposed schemes significantly enhance the searchability of CMA-ES compared to the other large-scale variants of CMA-ES and state-of-the-art large-scale optimizers. Moreover, our EDG could be integrated with evolutionary optimizers of different flavors like Differential Evolution (DE).

Optimal Tuning-Free Convex Relaxation for Noisy Matrix Completion

This paper is concerned with noisy matrix completion—the problem of recovering a low-rank matrix from partial and noisy entries. Under uniform sampling and incoherence assumptions, we prove that a tuning-free square-root matrix completion estimator (square-root MC) achieves optimal statistical performance for solving the noisy matrix completion problem. Similar to the square-root Lasso estimator in high-dimensional linear regression, square-root MC does not rely on the knowledge of the size of the noise. While solving square-root MC is a convex program, our statistical analysis of square-root MC hinges on its intimate connections to a nonconvex rank-constrained estimator.

A Novel Adaptive Square Root UKF with Forgetting Factor for the Time-Variant Parameter Identification

The unscented Kalman filter (UKF) serves as an efficient estimator widely utilized for the recursive identification of parameters. However, the UKF is not well suited for tracking time-variant parameters. Moreover, the unscented transformation (UT) used in the UKF typically relies on Cholesky decomposition to perform the square root operation of the covariance matrix. This method necessitates the matrix to maintain symmetry and positive definiteness. Due to the adverse influence of rounding error and noise, it becomes challenging to guarantee the positive definiteness of the matrix in each recursive step for practical engineering. The square root UKF (SRUKF) eliminates the need for the square root operation in the UT by directly updating the square root of the covariance matrix during each recursion. However, the SRUKF still relies on the rank 1 update to the Cholesky factorization to perform the recursive process, which also necessitates the matrix to be positive definite. Furthermore, the SRUKF is ineffective in the identification of time-variant parameters. Therefore, this paper proposes a modification to the SRUKF that ensures unconditional numerical stability by utilizing QR decomposition. Subsequently, the modified square root UKF (MSRUKF) method is enhanced by incorporating an adaptive forgetting factor that can be adjusted based on the residual information from each recursive step. This adaptation leads to the development of the adaptive SRUKF with forgetting factor (ASRUKF-FF) method, which significantly improves the tracking capability for time-variant parameters. To validate the effectiveness of the proposed method, this paper demonstrates its application in identifying the time-variant stiffness and damping parameters of a three-story frame structure. In addition, the method is employed to estimate the time-variant stiffness of the bridge excited by vehicles. The simulation results show that the proposed method has the superiority of high accuracy, strong robustness, and widespread applicability, even with incomplete measurements and inappropriate parameter settings.

State-of-Charge Estimation of Lithium-Ion Batteries Based on Dual-Coefficient Tracking Improved Square-Root Unscented Kalman Filter

Accurate state of charge (SOC) estimation is helpful for battery management systems to extend batteries’ lifespan and ensure the safety of batteries. However, due to the pseudo-positive definiteness of the covariance matrix and noise statistics error accumulation, the SOC estimation of lithium-ion batteries is usually inaccurate or even divergent using Kalman filters, such as the unscented Kalman filter (UKF) and the square-root unscented Kalman filter (SRUKF). To resolve this problem, an SOC estimation method based on the dual-coefficient tracking improved square-root unscented Kalman filter for lithium-ion batteries is developed. The method is composed of an improved square-root unscented Kalman filter (ISRUKF) and a dual-coefficient tracker. To avoid the divergence of SOC estimation due to the covariance matrix with pseudo-positive definiteness, an ISRUKF based on the QR decomposition covariance square-root matrix is presented. Moreover, the dual-coefficient tracker is designed to track and correct the state noise error of the battery, which can reduce the SOC estimation error caused by the accumulation of the battery model error using the ISRUKF. The accuracy and robustness of the SOC estimation method using the developed method are validated by the comparison with the UKF and SRUKF. The developed algorithm shows the highest SOC estimation accuracy with the SOC error within 1.5%.

Online Dominant Generalized Eigenvectors Extraction via a Randomized Algorithm

This paper is concerned with online algorithms for the generalized Hermitian eigenvalue problem (GHEP). We first present an algorithm based on randomization, termed alternate-projections randomized eigenvalue decomposition (APR-EVD), to solve the standard eigenvalue problem. The APR-EVD algorithm is computationally efficient and can be computed by making only one pass through the input matrix. We then develop two online algorithms based on APR-EVD for the dominant generalized eigenvectors extraction. Our proposed algorithms use the fact that GHEP is transformed into a standard eigenvalue problem, however to avert computations of a matrix inverse and inverse of the square root of a matrix, which are prohibitive, they exploit the rank-1 strategy for the transformation. Our algorithms are devised for extracting generalized eigenvectors for scenarios in which observed stochastic signals have unknown covariance matrices. The effectiveness and practical applicability of our proposed algorithms are validated through numerical experiments with synthetic and real-world data.

Adaptive Mask Sampling and Manifold to Euclidean Subspace Learning With Distance Covariance Representation for Hyperspectral Image Classification

For the abundant spectral and spatial information recorded in hyperspectral images (HSIs), fully exploring spectral-spatial relationships has attracted widespread attention in hyperspectral image classification (HSIC) community. However, there are still some intractable obstructs. For one thing, in the patch based processing pattern, some spatial neighbor pixels are often inconsistent with the central pixel in land-cover class. For another thing, linear and nonlinear correlations between different spectral bands are vital yet tough for representing and excavating. To overcome these mentioned issues, an adaptive mask sampling and manifold to Euclidean subspace learning (AMS-M2ESL) framework is proposed for HSIC. Specifically, an adaptive mask based intra-patch sampling (AMIPS) module is firstly formulated for intra-patch sampling in an adaptive mask manner based on central spectral vector oriented spatial relationships. Then, based on distance covariance descriptor, a dual channel distance covariance representation (DC-DCR) module is proposed for modeling unified spectral-spatial feature representations and exploring spectral-spatial relationships, especially linear and nonlinear interdependence in spectral domain. Furthermore, considering that distance covariance matrix lies on the symmetric positive definite (SPD) manifold, we implement a manifold to Euclidean subspace learning (M2ESL) module respecting Riemannian geometry of SPD manifold for high-level spectral-spatial feature learning. Additionally, we introduce an approximate matrix square-root (ASQRT) layer for efficient Euclidean subspace projection. Extensive experimental results on three popular HSI data sets with limited training samples demonstrate the superior performance of the proposed method compared with other state-of-the-art methods. The source code is available at https://github.com/lms-07/AMS-M2ESL.

Properties of eigenvalues and eigenvectors of large-dimensional sample correlation matrices

This paper is to study the properties of eigenvalues and eigenvectors of high-dimensional sample correlation matrices. We first improve the result of Jiang (Sankhyā 66 (2004) 35–48), Xiao and Zhou (J. Theoret. Probab. 23 (2010) 1–20) and the Theorem 1 of El Karoui (Ann. Appl. Probab. 19 (2009) 2362–2405), both concerning the limiting spectral distribution and the extreme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matrices. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influenced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrices is not asymptotically Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigenvectors of large sample covariance matrices. This theorem improves the main results in Bai, Miao and Pan (Ann. Probab. 35 (2007) 1532–1572), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (Ann. Appl. Probab. 18 (2008) 1232–1270), which relaxed the Gaussian like fourth moment requirement but assumes the maximum entries of the projection vector converge to 0 uniformly. We illustrate the usefulness of the theoretical results through an application in communications.

Polynomial whitening for high-dimensional data

The inverse square root of a covariance matrix is often desirable for performing data whitening in the process of applying many common multivariate data analysis methods. Direct calculation of the inverse square root is not available when the covariance matrix is either singular or nearly singular, as often occurs in high dimensions. We develop new methods, which we broadly call polynomial whitening, to construct a low-degree polynomial in the empirical covariance matrix which has similar properties to the true inverse square root of the covariance matrix (should it exist). Our method does not suffer in singular or near-singular settings, and is computationally tractable in high dimensions. We demonstrate that our construction of low-degree polynomials provides a good substitute for high-dimensional inverse square root covariance matrices, in both d < N and d ge N cases. We offer examples on data whitening, outlier detection and principal component analysis to demonstrate the performance of the proposed method.

Constructing a Matrix Mid-Point Iterative Method for Matrix Square Roots and Applications

In this paper, an improvement to the mid-point method is contributed for finding the square root of a matrix as well as its inverse. To this aim, an iteration scheme to find this matrix function is constructed, and its error and stability estimates are provided to show the theoretical rate of convergence. Our higher-order method can compete with the existing iterative methods of a similar nature. This is illustrated in numerical simulations of various sizes.

Efficient jamming attack against MIMO transceiver

This work addresses the issue of point-to-point intelligent jamming against MIMO architecture-based systems. The receiver uses an optimal Maximum Likelihood (ML) detector to extract the data stream from the received signal. Analyzing the receiver performance in the presence of a jamming signal, optimal attack strategies based on the pairwise error probability parameter as an objective function are designed, enabling the achievement of maximum degradation of the service quality. To perform the worst-case jamming of a legitimate transceiver, we assume that the jammer can eavesdrop on the Channel State Information (CSI). Depending on the jamming power budget constraint and the potential of jamming systems to estimate the CSI, different scenarios are envisaged, including imperfect CSI estimation cases. By minimizing the trace of Positive Semi-Definite (PSD) matrix products, the optimal jamming approaches that ensure maximum degradation of the legitimate link are designed. The disruption efficiency of the proposed approaches is comparable to the employment of the brute-force attack, but with a higher power level that is proportional to the number of jamming antennas in the case of partial knowledge of CSI, and inversely to the statistical expectation of the square root of the Wishart matrix's eigenvalues in the case of complete knowledge of CSI. To justify the relevance of the suggested policy, extensive simulations of the effect of these interventions are presented.

Estimation of a partially linear seemingly unrelated regressions model: application to a translog cost system

This article studies a partially linear seemingly unrelated regressions (SUR) model to estimate a translog cost system that consists of a partially linear translog cost function and input share equations. The parametric component is estimated via a simple two-step feasible SUR estimation procedure. We show that the resulting estimator achieves root-n convergence and is asymptotically normal. The nonparametric component is estimated with a nonparametric SUR estimator based on the Cholesky decomposition. We show that this estimator is consistent, asymptotically normal, and more efficient relative to the ones that ignore cross-equation correlation. We emphasize the importance and implication of the choice of square root of the covariance matrix by comparing the Cholesky and Spectral decompositions. A model specification test for parametric functional form is proposed. An Italian banking data set is used to estimate the translog cost system. Results show that marginal effects of risks on cost of production are heterogeneous but increase with risk levels.

A Spatial Feature-Enhanced Attention Neural Network with High-Order Pooling Representation for Application in Pest and Disease Recognition

With the development of advanced information and intelligence technologies, precision agriculture has become an effective solution to monitor and prevent crop pests and diseases. However, pest and disease recognition in precision agriculture applications is essentially the fine-grained image classification task, which aims to learn effective discriminative features that can identify the subtle differences among similar visual samples. It is still challenging to solve for existing standard models troubled by oversized parameters and low accuracy performance. Therefore, in this paper, we propose a feature-enhanced attention neural network (Fe-Net) to handle the fine-grained image recognition of crop pests and diseases in innovative agronomy practices. This model is established based on an improved CSP-stage backbone network, which offers massive channel-shuffled features in various dimensions and sizes. Then, a spatial feature-enhanced attention module is added to exploit the spatial interrelationship between different semantic regions. Finally, the proposed Fe-Net employs a higher-order pooling module to mine more highly representative features by computing the square root of the covariance matrix of elements. The whole architecture is efficiently trained in an end-to-end way without additional manipulation. With comparative experiments on the CropDP-181 Dataset, the proposed Fe-Net achieves Top-1 Accuracy up to 85.29% with an average recognition time of only 71 ms, outperforming other existing methods. More experimental evidence demonstrates that our approach obtains a balance between the model’s performance and parameters, which is suitable for its practical deployment in precision agriculture art applications.

Closed-form solution of non-symmetric algebraic Riccati matrix equation

In this work we obtain closed-form expression for the solution of non-symmetric algebraic Riccati equation (NARE), AX+XB+XCX=D,A,B,C,D∈ℂn×n, under some conditions on the matrices A,B,C and D. We show that the solution (real and complex) of this NARE is X=(A2+DC−A)C−1, where M is square root of matrix M. Finally this formula applied on some NAREs.

A Non-Convex Activation and Noise-Suppressing Zeroing Neural Network for Finding Time-Varying Matrix Square Root

In this paper, a zeroing neural network model with properties of non-convex activation and noise suppressing (NANSZNN) is proposed for finding the square root of a time-varying matrix. In comparison with the existing zeroing neural network models, the proposed NANSZNN model relieves the limit of convex constraint and makes a breakthrough in noise-suppressing. Furthermore, theoretical analyses and strict proofs are provided in detail for the global convergence and noise-suppressing performance of the proposed NANSZNN model under the circumstances of various noises by using Lyaponov stability theory. Finally, numerical experiments and comparative analyses are offered to further illustrate the availability, effectiveness, and robustness against different noises of the proposed NANSZNN model.

An efficient space-time model for the stochastic nuclear reactors

Modelling of physical systems with stochastic variations is mandatory in many applications. In this work, a new stochastic space–time kinetic model for the nuclear reactor is developed. The model is an efficient alternative to existing techniques available in the literature. The main advantage is to avoid square root of the covariance matrix and hence reduces the computational cost. The model is constructed and derived in detail. The mean, statistical properties, and quantification of uncertainties due to noise are obtained. The computational complexity is compared to the existing models to validate the efficiency. The model is tested against several problems and has shown accuracy and efficiency compared with existing models in the literature.

A study on square root control-systems

Starting with the simplest case of an analogue square root control-system, we extend the idea to the discrete-time case and ultimately the multivariable discrete-time case. Several new results then emerge from the multivariable work. A nonlinear system is designed to take the square root of an arbitrary square matrix. The matrix can have real or complex values and an analysis of stability is reached by using matrix calculus and nonlinear theory. The root-locus of the multivariable system exhibits straight-line characteristics which has not been observed before and to achieve stability we introduce the concept of a complex step-size (or gain) in a recursive square root algorithm.