Decomposition Of Covariance Matrix Research Articles

Efficient Random Field Generation With Rotational Anisotropy for Probabilistic SPH Analysis of Slope Failure

ABSTRACTDue to geological processes such as sedimentation, tectonic movement, and backfilling, natural soil often exhibits characteristics of rotated anisotropy. Recent studies have shown the significant impact of rotated anisotropy on slope stability. However, little research has explored how this rotated anisotropy affects the large deformations occurring after slope failure. Therefore, this study integrates rotated random field theory with smoothed particle hydrodynamics (SPH) to investigate its influence on post‐failure slope behavior. Focusing on a typical slope scenario, this research utilizes graphics processing unit (GPU)–accelerated covariance matrix decomposition (CMD) method to create rotated anisotropy random fields and applies the SPH framework for analysis. It examines the influence of rotated anisotropy angles and the cross‐correlation between cohesion and internal friction angle on landslides. The results indicate that the rotational anisotropy of the slope significantly influences post‐failure behavior. When the rotation angle is close to the slope surface, it tends to amplify both the magnitude and variability of slope failure. Furthermore, the study evaluates the efficiency of generating these random fields and emphasizes the substantial computational speed improvements achieved with GPU acceleration. These findings offer a robust approach for probabilistic analysis of slope large deformations considering rotated anisotropy. They provide a theoretical foundation for accurately assessing the risk of slope collapse, holding significant practical implications for geotechnical engineering.

Influence of spatial variability of soil shear-wave velocity considering intralayer correlation on seismic response of engineering site

Shear-wave velocity uncertainty significantly impacts seismic response analysis at engineering sites. Previous studies focused on the correlations of shear-wave velocities within soil layers. However, the influence of the vertical spatial variability of shear-wave velocity within individual soil layers was not considered. This study selected a typical Site Class II engineering site to investigate this influence based on the equivalent linear seismic response analysis method of layered soil deposits. Existing field test data on shear-wave velocity were used to discretize the shear-wave velocity within soil layers into a one-dimensional (1D) random field through covariance matrix decomposition and local average process theory. Monte Carlo simulations generated random shear-wave velocity profiles with different vertical correlation distances. Three artificial records with different earthquake return periods were used as input motions at the engineering bedrock for the 1D free-field model. Considering soil shear-wave velocity uncertainty, the peak shear strain increased by 19–27% with the input ground motion amplitude. Moreover, the site peak shear strain variability increased by 25–34% with the vertical correlation distance. The proposed 1D random field model effectively simulated the interlayer correlation and spatial variability of shear-wave velocity within soil layers, demonstrating its significance in seismic response analysis.

Seepage safety evaluation of high earth-rockfill dams considering spatial variability of hydraulic parameters via subset simulation

Seepage failure of high earth-rockfill dams have devastating consequences, and its safety analysis is of great importance in the design phase. However, the highest safety standards in high dams pose a challenge to the calculated efficiency of safety assessment. To address this problem, a random seepage safety assessment method which can consider the spatial variability of hydraulic parameters is proposed. Firstly, the spatial variability of the hydraulic parameters in the impervious materials is simulated using the covariance matrix decomposition under the framework of the non-intrusive finite element method. Then, according to the safety index of critical hydraulic gradient, the seepage stability formula is established to evaluate the safety state. Finally, considering that the highest safety standards in high dams make the probability of seepage hazard small (low failure probability event), it may require more than 10,000 deterministic calculations to complete the assessment process. Therefore, an efficient method based on subset simulation is proposed to optimize the computational efficiency. After the seepage safety evaluation of a 315 m high dam, the results show that the hydraulic conductivity of the core wall has the greatest influence on the seepage safety of the dam, and even under the influence of strong parameter spatial variability, its seepage failure probability is lower than 10-3. Compared with Monte Carlo simulation, the proposed method requires only 1/11 of the computation time at the 10-3 failure level. In addition, the results of grey relational analysis show that the coefficient of variation and correlation distance have obvious effects on seepage safety, and the autocorrelation distance has a greater effect.

FRMDN: Flow-based Recurrent Mixture Density Network

The class of recurrent mixture density networks is an important class of probabilistic models used extensively in sequence modeling and sequence-to-sequence mapping applications. In this class of models, the density of a target sequence in each time-step is modeled by a Gaussian mixture model with the parameters given by a recurrent neural network. In this paper, we generalize recurrent mixture density networks by using a normalizing flow to non-linearly transform the target space. Furthermore to improve the modeling power, we adopting a suitable covariance matrix decomposition involving a summation of a low-rank and a diagonal matrix. Using these two techniques, we still have a tractable log-likelihood. We also applied the proposed model on some speech and image data, and observed that the model has significant modeling power outperforming other state-of-the-art methods in terms of the log-likelihood on some data. The log-likelihood improvement over other methods is 3523 units for TIMIT speech dataset, and is 1118 and 176 units for MNIST and CIFAR10 image datasets. We were only underperformed on one of the speech datasets by 1209 units.

Ranking of nuclear data contributions to uncertainties in core physics parameters

Nuclear data uncertainties lead to substantial uncertainties in the predictions of LWR core calculations. Such problems are typically solved using lattice calculations and nodal codes, such as the WIMS-PANTHER system. It is desirable to derive uncertainties of core simulation output parameters (e.g. enthalpy deposition in an REA) to nuclear data uncertainties for specific nuclide-reaction pairs. This allows the key uncertainty contributions to be identified and ranked and facilitates use of data assimilation techniques. Key challenges in ranking the contributions to uncertainty are (1) the large number of input parameters (2) burnup-dependence of nuclide populations and microscopic cross sections, which challenges the use of adjoint methods (3) propagation of uncertainties through a multi-step calculation sequence. In this paper, an integrated methodology for deriving such uncertainties is presented. This comprises the following steps: (1) sensitivity analysis to identify the key reactions in the problem and exclude insignificant reactions (2) orthogonal decomposition of the nuclear data covariance matrix to calculate and rank eigenvectors (3) direct and consistent perturbation of the microscopic cross sections for the eigenvectors with significant eigenvalues (4) separate lattice (WIMS) and core (PANTHER) calculations with nuclear data perturbed for each eigenvector. This method (which assumes linearity of perturbations) is compared to a direct Sobol sample of the nuclear data and for the cases examined gives consistent results for the importance ranking of reactions, while greatly reducing the number of calculations employed. Hence, it can complement sampling-based approaches in developing an overall understanding of the magnitude, distribution, and contributors to uncertainty in core calculations. Both the linear method and the Sobol sampling method are integrated within the Visual Workshop graphical user environment to automate the process with the WIMS and PANTHER codes.

Sampling Gaussian stationary random fields: A stochastic realization approach

Generating large-scale samples of stationary random fields is of great importance in the fields such as geomaterial modeling and uncertainty quantification. Traditional methodologies based on covariance matrix decomposition have the difficulty of being computationally expensive, which is even more serious when the dimension of the random field is large. This paper proposes an efficient stochastic realization approach for sampling Gaussian stationary random fields from a systems and control point of view. Specifically, we take the exponential and squared exponential covariance functions as examples and make a decoupling assumption when there are multiple dimensions. Then a rational spectral density is constructed in each dimension using techniques from covariance extension, and the corresponding autoregressive moving-average (ARMA) model is obtained via spectral factorization. As a result, samples of the random field with a specific covariance function can be generated very efficiently in the space domain by implementing the ARMA recursion using a Gaussian white noise input. Such a procedure is computationally cheap due to the fact that the constructed ARMA model has a low order. Furthermore, the same method is integrated to multiscale simulations where interpolations of the generated samples are achieved when one zooms into finer scales. Both theoretical analysis and simulation results show that our approach performs favorably compared with covariance matrix decomposition methods.

Core shrinkage covariance estimation for matrix-variate data

Abstract A separable covariance model can describe the among-row and among-column correlations of a random matrix and permits likelihood-based inference with a very small sample size. However, if the assumption of separability is not met, data analysis with a separable model may misrepresent important dependence patterns in the data. As a compromise between separable and unstructured covariance estimation, we decompose a covariance matrix into a separable component and a complementary ‘core’ covariance matrix. This decomposition defines a new covariance matrix decomposition that makes use of the parsimony and interpretability of a separable covariance model, yet fully describes covariance matrices that are non-separable. This decomposition motivates a new type of shrinkage estimator, obtained by appropriately shrinking the core of the sample covariance matrix, that adapts to the degree of separability of the population covariance matrix.

Ship Detection in PolSAR Images Based on a Modified Polarimetric Notch Filter

Ship detection based on synthetic aperture radar (SAR) imagery is one of the key applications for maritime security. Compared with single-channel SAR images, polarimetric SAR (PolSAR) data contains the fully-polarized information, which better facilitates better discriminating between targets, sea clutter, and interference. Therefore, many ship detection methods based on the polarimetric scattering mechanism have been studied. To deal with the false alarms caused by the existence of ghost targets, resulting from azimuth ambiguities and interference from side lobes, a modified polarimetric notch filter (PNF) is proposed for PolSAR ship detection. In the proposed method, the third eigenvalue obtained by the eigenvalue–eigenvector decomposition of the polarimetric covariance matrix is utilized to construct a new feature vector. Then, the target power can be computed to construct the modified PNF detector. On the one hand, the detection rate of ship targets can be enhanced by target-to-clutter contrast. On the other hand, false alarms resulting from azimuth ambiguities and side lobes can be reduced to an extent. Experimental results based on three C-band AIRSAR PolSAR datasets demonstrated the capability of the proposed PNF detector to improve detection performance while reducing false alarms. To be specific, the figure of merit (FoM) of the proposed method is the highest among comparative approaches with results of 80%, 100%, and 100% for the tested datasets, respectively.

Non-Parametric Tomographic SAR Reconstruction via Improved Regularized MUSIC

Height estimation of scatterers in complex environments via the Tomographic Synthetic Aperture Radar (TomoSAR) technique is still a valuable research field. The parametric spectral estimation approach constitutes a powerful tool to identify the superimposed scatterers with different complex reflectivities, located at different heights in the same range–azimuth resolution cell. Unfortunately, this approach requires prior knowledge about the number of scatterers for each pixel, which is not possible in practical situations. In this paper, we propose a method that analyzes the scree plot, generated from the spectral decomposition of the multidimensional covariance matrix, in order to estimate automatically the number of scatterers for each resolution cell. In this context, a properly improved regularization step is included during the reconstruction process, transforming the parametric MUSIC estimator into a non-parametric method. The experimental results on two data sets covering high elevation towers, with different facade coating characteristics, acquired by the TerraSAR-X satellite highlighted the effectiveness of the proposed regularized MUSIC for the reconstruction of high man-made structures compared with classical approaches.

High-Precision Time Delay Estimation Based on Closed-Form Offset Compensation

To improve the estimation accuracy, a novel time delay estimation (TDE) method based on the closed-form offset compensation is proposed. Firstly, we use the generalized cross-correlation with phase transform (GCC-PHAT) method to obtain the initial TDE. Secondly, a signal model using normalized cross spectrum is established, and the noise subspace is extracted by eigenvalue decomposition (EVD) of covariance matrix. Using the orthogonal relation between the steering vector and the noise subspace, the first-order Taylor expansion is carried out on the steering vector reconstructed by the initial TDE. Finally, the offsets are compensated via simple least squares (LS). Compared to other state-of-the-art methods, the proposed method significantly reduces the computational complexity and achieves better estimation performance. Experiments on both simulation and real-world data verify the efficiency of the proposed approach.

Composite forecasting of vast-dimensional realized covariance matrices using factor state-space models

We propose a dynamic factor state-space model for the prediction of high-dimensional realized covariance matrices of asset returns. Using a block LDL decomposition of the joint covariance matrix of assets and factors, we express the realized covariance matrix of the individual assets similar to an approximate factor model. We model the individual parts, i.e., the factor and residual covariances as well as the factor loadings, independently via a tractable state-space approach. This results in closed-form Matrix-F predictive densities for the distinct covariance elements and Student’s t predictive densities for the factor loadings. In an out-of-sample forecasting and portfolio selection exercise we compare the performance of the proposed factor model under different specifications of the residual dynamics. These includes block diagonal residuals based on the GICS sector classifications and strict diagonality assumptions as well as combinations of both using linear shrinkage. We find that the proposed model performs very well in an empirical application to realized covariance matrices for 225 NYSE-traded stocks using the well-known Fama–French factors and sector-specific factors represented by exchange traded funds.

An Adaptive PolSAR Trilateral Filter Based on the Mechanism of Scattering Consistency

Polarimetric Synthetic Aperture Radar (PolSAR) can record the backscattering information of ground objects by capturing the phase difference of echo signal in different polarization combinations. Due to the characteristics of coherent imaging, there is a lot of coherent speckle noise in PolSAR images. However, most of the current PolSAR image despeckling algorithms are difficult to balance the ability in aspects of speckle suppression, polarization information retention and edge structure preservation. This paper proposes an adaptive PolSAR trilateral filter (PTF) based on the mechanism of scattering consistency, which consists of following three steps. (1) The marker map of ground object scattering category is firstly obtained based on the dominant scattering mechanism determined by Freeman decomposition of the polarization covariance matrix. (2) Next, the neighborhood pixels with the same scattering mechanism as the center pixel are screened out within an adaptive filtering window. (3) A PolSAR trilateral filter is established through integration of the spatial proximity, polarization similarity and local structural characteristics of the center pixel and the screened pixels to effectively suppress the speckle noises. Experimental results upon the data of Radarsat-2 and AIRSAR demonstrate that PTF can not only suppress coherent speckle noise effectively without sacrificing the edge structure information, but also retain the unique polarization information of PolSAR image. In particular, compared with the traditional six widely used PolSAR despeckling algorithms, PTF have been significantly improved in terms of Equivalent Number of Looks (ENL) and Degree of difference (Dod) indexes by 2.49 times and 6 orders of magnitude, respectively.

Surface subsidence monitoring with an improved distributed scatterer interferometric SAR time series method in a filling mining area

Statistically homogeneous pixel (SHP) selection and DS phase optimization are two critical steps in the distributed scatterer interferometric SAR (DS-InSAR) time series method. In this paper, a new algorithm named dynamic hypothesis test of confidence interval (D-HTCI) is proposed, which reduces the wrong selection rate of SHP and increases the number of SHP selections. Using adaptive spatial nonlocal filtering method for DS phase optimization, the phase standard deviation (PSD) and the sum of phase differences (SPD) show that compared with the traditional covariance matrix decomposition method, the optimization quality is improved by 2.1 and 1.8 times, respectively. Combining 24 scenes Sentinel-1A data from September 17, 2017 to July 14, 2018, the method is applied to monitor surface subsidence of the Daizhuang filling mining area (Jining, Shandong, China). The results show that the proposed method has mm-level accuracy for monitoring of surface subsidence in a filling mining area.

A DOA Estimation Method for UCA Based on Orthogonal Subspace Compensation

In this letter, the problem of 2-D direction of arrival (DOA) estimation using uniform circular array (UCA) is addressed. The performance of the beamspace transform (BT) based algorithm is degraded when shrinking the quantity of antennas because the BT raises the residual error. To overcome this problem, we propose a 2-D compensation method. Firstly, we estimate the biased 2-D DOAs via UCA-ESPRIT algorithm. Secondly, the eigenvalue decomposition (EVD) of covariance matrix without the BT is conducted to extract the noise subspace in element space. Thereafter, inspired by the relationship that the steering vector is orthogonal to the noise subspace, the first-order Taylor expansion is performed for the direction matrix which is reconstructed by the biased DOA estimates. Finally, the offsets are compensated via least squares (LS). Compared to virtual array transformation methods, the proposed method possesses better estimation performance and needs less computation cost. Multiple simulation experiments are presented to verify the efficiency of the approach.

Flexible Covariance Matrix Decomposition Method for Data Augmentation and Its Application to Brainwave Signals

The acquisition of a large-volume brainwave database is challenging because of the stressful experiments that are required; however, data synthesis techniques can be used to address this limitation. Covariance matrix decomposition (CMD), a widely used data synthesis approach, generates artificial data using the correlation between features and random noise. However, previous CMD methods constrain the stochastic characteristics of artificial datasets because the random noise used follows a standard distribution. Therefore, this study has improved the performance of CMD by releasing such constraints. Specifically, a generalized normal distribution (GND) was used as it can alter the kurtosis and skewness of the random noise, affecting the distribution of the artificial data. For the validation of GND performance, a motor imagery brainwave classification was conducted on the artificial dataset generated by GND. The GND-based data synthesis increased the classification accuracy obtained with the original data by approximately 8%.

Autonomous Quadrotor Landing on Inclined Surfaces Using Perception-Guided Active Asymmetric Skids

We present an autonomous quadrotor capable of safely landing on sloped surfaces up to 40 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> , intended for emergency scenarios where the only terrain available for landing may be sloped. This system uses a downward-facing depth perception system to determine the direction, angle, and smoothness of the slope. Two robotic landing skids of different lengths actively conform to the slope in order to maintain level body attitude upon landing. We developed an analytical model to design leg lengths, conform to the slope surface angle, and ensure both zero tilt angle of the quadrotor body and clearance of the propellers from the surface. The selection of skid angles is framed as an optimization to match the slope angle while maximizing the buffer between the propellers and the surface. An eigenvalue decomposition of the point cloud covariance matrix provides a surface normal vector, which is used to determine the proper skid angle and yaw angle of the quadrotor. The ratio of the eigenvalues is used to determine whether the surface is sufficiently smooth for safe landing. The proposed system and method were validated in a motion capture environment by conducting five autonomous takeoff and land missions over different sloped surfaces ranging from 0 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> to 40 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> . The detected slope angle and direction of all trials were within 1 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> and 3.3 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> , respectively, of ground truth. No failures or crashes occurred during testing, which demonstrates the viability and robustness of this system for use in real-world scenarios.

Multivariate GARCH with dynamic beta

We investigate a solution for the problems related to the application of multivariate GARCH models to markets with a large number of stocks by restricting the form of the conditional covariance matrix and by introducing a system of recursion formals. The model is based on a decomposition of the conditional covariance matrix into two components and requires only six parameters to be estimated. The first component can be interpreted as the market factor, all remaining components are assumed to be equal. This allow the analytical calculation of the inverse covariance matrix. The factors are dynamic and therefore enable to describe dynamic beta coefficients. We compare the estimated covariances for the S&P500 market with those of other GARCH models and find that they are competitive, despite the low number of parameters. As applications we use the daily values of beta coefficients to confirm a transition of the market in 2006. Furthermore we discuss the relationship of our model with the leverage effect.

Centralized spectrum sensing based on covariance matrix decomposition and particle swarm clustering

As the essential technology of cognitive radio networks, numerous spectrum sensing approaches have been established to date. However, some spectrum sensing methods seem to fail in providing an effective signal feature and an accurate decision threshold, which affects the detection accuracy of the spectrum holes. In this article, to further ameliorate the recognition rate of the spectrum holes, a novel centralized spectrum sensing method based on covariance matrix decomposition and particle swarm clustering is presented. A novel signal feature vector is extracted by the IQ decomposition method and the Cholesky decomposition technique. Moreover, particle swarm clustering algorithm is trained to obtain a decision classifier, which can be used to identify whether the licensed spectrum is available or not. Additionally, two different application scenarios are considered here. Indeed, simulations examples are provided to demonstrate that the proposed algorithm can considerably ameliorate the sensing performance for centralized spectrum sensing.

Scalable ESPRIT Processor for Direction-of-Arrival Estimation of Frequency Modulated Continuous Wave Radar

The estimation of signal parameters via rotational invariance techniques (ESPRIT) is an algorithm that uses the shift-invariant properties of the array antenna to estimate the direction-of-arrival (DOA) of signals received in the array antenna. Since the ESPRIT algorithm requires high-complexity operations such as covariance matrix and eigenvalue decomposition, a hardware processor must be implemented such that the DOA is estimated in real time. Additionally, the ESPRIT processor should support a scalable number of antenna configuration for DOA estimation in various applications because the performance of ESPRIT depends on the number of antennas. Therefore, we propose an ESPRIT processor that supports two to eight scalable antenna configuration. In addition, since the proposed ESPRIT processor is based on multiple invariances (MI) algorithm, it can achieve a much better performance than the existing ESPRIT processor. The execution time is reduced by simplifying the Jacobi method, which has the most significant computational complexity for calculating eigenvalue decomposition (EVD) in ESPRIT. Moreover, the ESPRIT processor was designed using hardware description language (HDL), and an FPGA-based verification was performed. The proposed ESPRIT processor was implemented with 10,088 slice registers, 18,207 LUTs, and 80 DSPs, and the slice register, LUT, and DSP were reduced by up to 71.45%, 54.5%, and 68.38%, respectively, compared to the existing structure.

First-Order Error-Adapted Eigen Perturbation for Real-Time Modal Identification of Vibrating Structures

Abstract A new computationally efficient error adaptive first-order eigen-perturbation technique for real-time modal identification of linear vibrating systems is proposed. The existence of error terms in the approximation of the eigenvalue problem of response covariance matrix in a perturbative framework often hinders the convergence of response-only modal identification. In the proposed method, the error in first-order eigen-perturbation is incorporated using a feedback, formulated by exploiting the generalized eigenvalue decomposition of the real-time covariance matrix of streaming response data. Since the incorporation of the higher-order perturbation terms in the total perturbation is mathematically challenging, the proposed feedback approach provides a computationally efficient framework yet in a more elegant manner. A new criterion for the quality of updated eigenspace is proposed in the present work utilizing the concept of diagonal dominance. Numerical case studies and validation using a standard ASCE benchmark problem have shown applicability of the proposed approach in faster estimation of real-time modal properties and anomaly identification with minimal number of initially required batch data. The applicability of the proposed approach toward real-time under-determined modal identification problems is demonstrated using a real-time decentralized framework. The advantage of rapidly converging online mode-shapes is demonstrated using a passive vibration control problem, where a multi-tuned-mass-damper (MTMD) for a multi-degrees-of-freedom system is tuned online. An extension for online retuning of the detuned MTMD system further demonstrates the fidelity of the proposed algorithm in online passive control.