Symmetric And Positive Definite Research Articles

Multivariate phase space reconstruction and Riemannian manifold for sleep stage classification

Background and ObjectiveSleep was highly imperative in human daily life. However, an increasing number of people were undergoing sleep deprivation and sleep disorders. Sleep stage classification became a highly essential process in sleep scoring. Nevertheless, the visual scoring of arousals during sleep routinely performed by sleep specialists was a challenging exercise. MethodsThis paper introduced a novel approach for sleep stage classification using covariance feature matrix architecture with multivariate phase space reconstruction (MPSR). The goal was to capture the geometric properties and the hidden dynamic characteristics of multiple physiological signals. The covariance matrices constructed through the MPSR approach were considered as symmetric and positive definite (SPD) matrices, forming a Riemannian manifold space. These SPD matrices in the Riemannian manifold were mapped to the matrices in the tangent space, allowing them to be vectorized and treated as feature vectors in Euclidean space. Finally, an ensemble learning classifier was applied to perform various sleep stage tasks. ResultsOur proposed method was evaluated on three benchmark datasets to assess its effectiveness and robustness. For the tasks of both Five class and Six class sleep stages, the proposed approach in ten-fold cross validation achieved high accuracy of 93.57% and 92.56% for the Sleep EDF dataset, 86.36% and 84.18% for the DREAMS Subjects dataset, as well as 88.93% and 88.42% for the Sleep EDF Expanded dataset (95 subjects), respectively. In leave-one-subject-out cross validation, our proposed approach for the tasks of both Five class and Six class sleep stages yielded an accuracy of 84.46% and 80.73% for the Sleep EDF dataset, 82.50% and 79.51% for the DREAMS Subjects dataset, as well as 93.25% and 92.06% for the Sleep EDF Expanded-20 dataset (20 subjects), respectively. ConclusionCompared to the traditional sample covariance matrix (SCM), the covariance feature matrix using the MPSR method successfully captured the distinction of spatial information among various sleep stages. Moreover, our proposed method obtained good performance without requiring computationally large artifact suppression or a long signal decomposition process.

Marquardt Extended Technique for Solving Unconstrained Optimization Problems

The objective of this article is to solve the unconstrained optimization problem by using Marquardt method together with MQ-N (modified quasi-Newton) method. The Hessian matrix will be computed numerically by using modified Broyden-Flechert-Goldfarb-Shanno (BFGS) update ( H-version) after convert it to the B-version by using Sherman-Morisson-Woodburge (SMW) formula which guarantee the two important properties SPD (symmetric and positive definite), and hence the exact second derivative of OF (objective function) does not be needed to compute. The line search technique is very important to accelerate the method to terminate at the minimum value of OF, so in this article the line search technique is instead by used the search technique of Marquardt method together with MQ-N method (especially modified BFGS method where the step size is equal one) to solve the unconstrained optimization problem. Test problems are solved by Matlab software to prove the effective of this new technique so called the Marquardt extended technique (MET).

Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

Automatic detection of epileptic seizures using Riemannian geometry from scalp EEG recordings.

This paper proposes a new framework for epileptic seizure detection using non-invasive scalp electroencephalogram (sEEG) signals. The major innovation of the current study is using the Riemannian geometry for transforming the covariance matrices estimated from the EEG channels into a feature vector. The spatial covariance matrices are considered as features in order to extract the spatial information of the sEEG signals without applying any spatial filtering. Since these matrices are symmetric and positive definite (SPD), they belong to a special manifold called the Riemannian manifold. Furthermore, a kernel based on Riemannian geometry is proposed. This kernel maps the SPD matrices onto the Riemannian tangent space. The SPD matrices, obtained from all channels of the segmented sEEG signals, have high dimensions and extra information. For these reasons, the sequential forward feature selection method is applied to select the best features and reduce the computational burden in the classification step. The selected features are fed into a support vector machine (SVM) with an RBF kernel to classify the feature vectors into seizure and non-seizure classes. The performance of the proposed method is evaluated using two long-term scalp EEG (CHB-MIT benchmark and private) databases. Experimental results on all 23 subjects of the CHB-MIT database reveal an accuracy of 99.87%, a sensitivity of 99.91%, and a specificity of 99.82%. In addition, the introduced algorithm is tested on the private sEEG signals recorded from 20 patients, having 1380 seizures. The proposed approach achieves an accuracy, a sensitivity, and a specificity of 98.14%, 98.16%, and 98.12%, respectively. The experimental results on both sEEG databases demonstrate the effectiveness of the proposed method for automated epileptic seizure detection, especially for the private database which has noisier signals in comparison to the CHB-MIT database. Graphical Abstract Block diagram of the proposed epileptic seizure detection algorithm.

An adaptive BDDC preconditioner for advection-diffusion problems with a stabilized finite element discretization

A BDDC preconditioner with adaptive coarse space for advection-diffusion problems discretized by stabilized finite element method is proposed. Since the bilinear form of the corresponding variational form is nonsymmetric and positive definite (NSPD), the adaptive BDDC preconditioner, which is always used for solving the symmetric and positive definite (SPD) problems, is extended to solve the nonsymmetric problems. By decomposing the original bilinear form to the symmetric part and the skew-symmetric part, a series of local generalized eigenvalue problems with respect to the symmetric part of the original bilinear form for the common faces/edges are designed and analyzed to form the adaptive coarse components. Numerical results are presented for model problems with various viscosities to show the performance of the proposed preconditioner.

BootCMatch

This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch , which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the goal of adaptivity is to achieve a method with a prescribed convergence rate. The presented method exploits a general coarsening process based on aggregation of unknowns, obtained by a maximum weight matching in the adjacency graph of the system matrix. More specifically, a maximum product matching is employed to define an effective smoother subspace (complementary to the coarse space), a process referred to as compatible relaxation, at every level of the recursive two-level hierarchical AMG process. Results on a large variety of test cases and comparisons with related work demonstrate the reliability and efficiency of the method and of the software.

On the geometrical structure of symmetric matrices

The identity ray, αI for α>0, can be seen as the center ray of the cone of symmetric and positive definite (SPD) matrices. In that sense, the angle that any SPD matrix forms with the identity plays a very important role to understand the geometrical structure of the cone. In this work, we extend this relationship, and analyze the geometrical structure of symmetric matrices including the location of all orthogonal matrices, not only the identity matrix. This geometrical understanding leads to new results in the subspace of symmetric matrices. We also extend some of the geometrical results for the case of general (not necessarily symmetric) nonsingular matrices.

Multiclass Brain–Computer Interface Classification by Riemannian Geometry

This paper presents a new classification framework for brain-computer interface (BCI) based on motor imagery. This framework involves the concept of Riemannian geometry in the manifold of covariance matrices. The main idea is to use spatial covariance matrices as EEG signal descriptors and to rely on Riemannian geometry to directly classify these matrices using the topology of the manifold of symmetric and positive definite (SPD) matrices. This framework allows to extract the spatial information contained in EEG signals without using spatial filtering. Two methods are proposed and compared with a reference method [multiclass Common Spatial Pattern (CSP) and Linear Discriminant Analysis (LDA)] on the multiclass dataset IIa from the BCI Competition IV. The first method, named minimum distance to Riemannian mean (MDRM), is an implementation of the minimum distance to mean (MDM) classification algorithm using Riemannian distance and Riemannian mean. This simple method shows comparable results with the reference method. The second method, named tangent space LDA (TSLDA), maps the covariance matrices onto the Riemannian tangent space where matrices can be vectorized and treated as Euclidean objects. Then, a variable selection procedure is applied in order to decrease dimensionality and a classification by LDA is performed. This latter method outperforms the reference method increasing the mean classification accuracy from 65.1% to 70.2%.

New parallel symmetric SOR preconditioners by multi-type partitioning

A new parallel symmetric successive over-relaxation (PSSOR) preconditioner is proposed in this paper by the multi-type partition techniques introduced in SIAM J. Scientific Computing 20, 2006, pp. 1513–1533. In a general matrix expression, it is proved to be symmetric and positive-definite (SPD) if the coefficient matrix of a linear system is SPD. It is also proved to be equivalent to the SSOR preconditioner using the multi-type ordering. Thus, it works for the preconditioned conjugate gradient method (PCG) and can be analysed by the classic SOR theory. Numerical tests on an anisotropic model problem show that the PSSOR preconditioner can make the PCG have a faster rate of convergence and better parallel performance than the red–black SSOR preconditioner. They also confirm that the PSSOR preconditioner can have a rate of convergence that is nearly the same as the classic sequential SSOR preconditioner when the problem has large anisotropy.

Solution of shallow-water equations using least-squares finite-element method

A least-squares finite-element method (LSFEM) for the non-conservative shallow-water equations is pre- sented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercritical flows, and flows with smooth and sharp gradient changes. Advantages of the model include: (1) sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; (2) upwind scheme is no needed; (3) a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) condition; and (4) the resulting system of equations is symmetric and positive-definite (SPD) which can be solved efficiently with the preconditioned conjugate gradient method. The model is verified with flow over a bump, tide induced flow, and dam-break. Computed results are compared with analytic solutions or other numerical results, and show the model is conservative and accurate. The model is then used to simulate flow past a circular cylinder. Important flow characteristics, such as variation of water surface around the cylinder and vortex shedding behind the cylinder are investigated. Computed results compare well with experiment data and other numerical results.

A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis

In this paper, a stabilized radial point collocation method (RPCM) that uses locally supported nodes is proposed using least-squares stabilization technique. The focus of this work is to stabilize the solution of the RPCM in order to perform adaptive analysis. Good stiffness matrix properties such as symmetric and positive definite (SPD) are gained via a least-squares procedure, and yet the formulation procedure of the stabilized LS-RPCM is still kept simple and straightforward. Adaptive analysis study has then been successfully carried out using the stabilized LS-RPCM. Error indicator based on interpolation error is adopted in the adaptive scheme for nodal refinement. Thanks for the meshfree features of the RPCM, refinement process can be easily done by inserting additional nodes based on the Voronoi diagram, without worrying about nodal connectivity in the formulation of system equations. Good numerical performance has been shown in the numerical examples presented in this paper.

An iterated tangential filtering decomposition

AbstractLarge linear systems arising from the discretization of partial differential equations with finite differences or finite elements on structured grids in dimension d(d ⩾ 3) require efficient preconditioners. For a symmetric and positive definite (SPD) matrix, we propose a SPD block LDLT preconditioner whose factorized form requires a smaller amount of memory than the original matrix. Moreover, the computing time for the preconditioner solves is linear with respect to the number of unknowns. The preconditioner is built in d stages: in a first stage, we use the tangential filtering decomposition of Wittum et al. and obtain a preconditioner which remains rather difficult to factorize. Then, in a second stage, we apply tangential filtering decomposition recursively to the diagonal blocks of this first preconditioner. The final stage consists of factorizing exactly the blocks corresponding to one dimensional problems. Such preconditioners can also be computed adaptively and combined in a multiplicative way. A generic programming implementation is discussed and numerical tests are presented, in particular for problems with highly heterogeneous media. Copyright © 2003 John Wiley & Sons, Ltd.

An iterative method for the positive real linear system

In this paper a new iterative method is given for the linear system of equations Au = b, where A is large, sparse and nonsymmetrical and AT + A is symmetric and positive definite (SPD) or equivalently A is positive real. The new iterative method is based on a mixed-type splitting of the matrix A. The iterative method contains an auxiliary matrix D 1. It is shown that by proper chxoice of D 1 the new iterative method is convergent. It is also shown that by special choice of D 1, the new iterative method becomes the well-known (point) successive overrelaxiation (SOR) [1] method. Hence, it is shown that the (point) SOR method applied to the positive real system is convergent if the overrelaxiation parameter ω is in (0,ω U ). The upper bound ω U is also given in terms of the norm and smallest eigenvalue of related matrices (see Eq. (23)).

Qualitative Analysis of One-Step Algebraic Models with Tridiagonal Toeplitz Matrices

One-step algebraic models with symmetric and positive definite (SPD) tridiagonal Toeplitz matrices are introduced and their qualitative properties are studied. Theoretical results are applied to the numerical solution of parabolic differential equations, with illustrations by numerical examples. Possible extensions as well as arising open problems are discussed in concluding remarks.

S-step iterative methods for symmetric linear systems

In this paper we introduce s-step Conjugate Gradient Method for Symmetric and Positive Definite (SPD) linear systems of equations and discuss its convergence. In the s-step Conjugate Gradient Method iteration s new directions are formed simultaneously from ≎ri, Ari,…,As−1ri≎ and the preceding s directions. All s directions are chosen to be A-orthogonal to the preceding s directions. The approximation to the solution is then advanced by minimizing an error functional simultaneously in all s directions. This intuitively means that the progress towards the solution in one iteration of the s-step method equals the progress made over s consecutive steps of the one-step method. This is proven to be true.

D1/2-Norms of the sor and related methods for a class of nonsymmetric matrices

The SOR, Modified SOR (MSOR), Sheldon, Modified Sheldon, Cyclic Chebyshev Semi-Iterative (CCSI) methods have been extensively studied for linear systems Ax = b with A symmetric and positive definite (SPD) and having the form, Many interesting results have been produced. However, if the matrix A is not SPD, then how do these methods behave? Are the methods which are efficient and effective for the SPD case still appropriate for the nonsymmetric case? Here we consider these methods under the assumptions that A has the form (1.2) and the associated Jacobi matrix B has either purely imaginary or zero eigenvalues. For the SOR method, it is shown that the Jordan canonical form is not diagonal but the maximum order of the sub-Jordan block is 2, that ‖ L ωb ‖ D 1/2 < 1 if and only if ρ(B)<1, while for the SPD case ρ(B)<1√2, and that ‖L ω| D 1/2 where ω + = 1/(1 + ρ 2(B)). The explicit expression for ‖L k ωb ‖ D 1/2 is also given. For the MSOR method with fixed parameter ω and ω′ a. sufficient and necessary condit...

On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systems

The conjugate-gradient (CC) method, developed by Hestenes and Stiefel in 1952, can be effectively used to solve the linear system Au = b when A is symmetrixable in the sense that ZA and Z are symmetric and positive definite (SPD) for some Z. A number of generalizations of the CG method have been proposed by the authors and by others for handling the nonsymmetrizable case. For many problems the amount of computer memory and computational effort required may be so large as to make the procedures not feasible. Truncated schemes are often used, but in some cases the truncated methods may not converge even though the nontruncated schemes converge. However, it is well known that if A is symmetric, the generalized CG schemes can be greatly simplified, even though A is not SPD, so that the truncated schemes are equivalent to the nontruncated schemes. In the present paper it is shown that such a simplification can occur if a nonsingular matrix H is available such that HA = A T H. (Of course, if A = A T , then H can be taken to be the identity matrix.) It is also shown that such an H always exists; however, it may not be practical to compute H. These results are used to derive three variations of the Lanczos method for solving nonsymmetrizable systems. Two of the forms are well known, but the third appears to be new. An argument is given for choosing the third form over the other two.