Eigenvectors Of Matrix Research Articles

Principal component analysis with drop rank covariance matrix

<p style='text-indent:20px;'>This paper considers the principal component analysis when the covariance matrix of the input vectors drops rank. This case sometimes happens when the total number of the input vectors is very limited. First, it is found that the eigen decomposition of the covariance matrix is not uniquely defined. This implies that different transform matrices could be obtained for performing the principal component analysis. Hence, the generalized form of the eigen decomposition of the covariance matrix is given. Also, it is found that the matrix with its columns being the eigenvectors of the covariance matrix is not necessary to be unitary. This implies that the transform for performing the principal component analysis may not be energy preserved. To address this issue, the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived. Moreover, since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem, the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived. In fact, the unitary matrix with its columns being the eigenvectors of the covariance matrix is only a particular case of the condition. Furthermore, the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is derived. It is found that the unitary matrix with its columns being the eigenvectors of the covariance matrix does not satisfy this condition. Computer numerical simulation results are given to valid the results.

Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation

We consider the performance analysis of the multiple signal classification (MUSIC) algorithm for multiple incident signals when the uniform linear array (ULA) is adopted for estimation of the azimuth of each incident signal. We derive closed-form expression of the estimation error for each incident signal. After some approximations, we derive closed-form expression of the mean square error (MSE) for each incident signal. In the MUSIC algorithm, the eigenvectors of covariance matrix are used for calculation of the MUSIC spectrum. Our derivation is based on how the eigenvectors of the sample covariance matrix are related to those of the true covariance matrix. The main contribution of this paper is the reduction in computational complexity for the performance analysis of the MUSIC algorithm in comparison with the traditional Monte–Carlo simulation-based performance analysis. The validity of the derived expressions is shown using the numerical results. Future work includes an extension to performance analysis of the MUSIC algorithm for simultaneous estimation of the azimuth and the elevation.

Distinguishing cell phenotype using cell epigenotype.

The relationship between microscopic observations and macroscopic behavior is a fundamental open question in biophysical systems. Here, we develop a unified approach that-in contrast with existing methods-predicts cell type from macromolecular data even when accounting for the scale of human tissue diversity and limitations in the available data. We achieve these benefits by applying a k-nearest-neighbors algorithm after projecting our data onto the eigenvectors of the correlation matrix inferred from many observations of gene expression or chromatin conformation. Our approach identifies variations in epigenotype that affect cell type, thereby supporting the cell-type attractor hypothesis and representing the first step toward model-independent control strategies in biological systems.

Iterated partial summations applied to finite-support discrete distributions

Abstract The problem of iterated partial summations is solved for some discrete distributions defined on finite supports. The power method, usually used as a computational approach to the problem of finding matrix eigenvalues and eigenvectors, is in some cases an effective tool to prove the existence of the limit distribution, which is then expressed as a solution of a system of linear equations. Some examples are presented.

Recovery of eigenvectors from eigenvalues in systems of coupled harmonic oscillators

The eigenvector-eigenvalue identity relates the eigenvectors of a Hermitian matrix to its eigenvalues and the eigenvalues of its principal submatrices in which the jth row and column have been removed. We show that one-dimensional arrays of coupled resonators, described by square matrices with real eigenvalues, provide simple physical systems where this formula can be applied in practice. The subsystems consist of arrays with the jth resonator removed, and thus can be realized physically. From their spectra alone, the oscillation modes of the full system can be obtained. This principle of successive single resonator deletions is demonstrated in two experiments of coupled radiofrequency resonator arrays with greater-than-nearest neighbor couplings, in which the spectra are measured with a network analyzer. Both the Hermitian as well as a non-Hermitian case are covered in the experiments. In both cases the experimental eigenvector estimates agree well with numerical simulations if certain consistency conditions imposed by system symmetries are taken into account. In the Hermitian case, these estimates are obtained from resonance spectra alone without knowledge of the system parameters. It remains an interesting problem of physical relevance to find conditions under which the full non-Hermitian eigenvector set can be obtained from the spectra alone.

Numerical instability of the C method when applied to coated gratings and methods to avoid it.

We recently found that the coordinate transformation method (the C method) equipped with well-established recursive algorithms for solving the system of linear equations is numerically instable when it is applied to thinly coated gratings. The origin of this new kind of numerical instability is not the exponential dependence of the field in the coated layers but the ill condition of the eigenvector matrix of the C method when the truncation number is high. Two simple and effective methods to circumvent the new instability are recommended. We also found that the popular recursive matrix algorithms have different (poor) immunities to the new instability, and they all perform inferiorly to the full matrix (nonrecursive) algorithm.

Determinate perfect foresight forecasting in overlapping generations models

In this paper, we show the existence of the perfect foresight forecast functions which generate unique and stable equilibrium price dynamics for comparative static analysis and calibration in multi-period overlapping generations models with cohort heterogeneity, multiple goods, and fiat money. We also show how to recover such forecast functions up to the first order using the eigenvalues and eigenvectors of the Jacobian matrix of the equilibrium conditions. In a special one good case, our construction of the forecast functions requires information about only the eigenvalues. In addition, we demonstrate that the stable subspace where the price sequences generated by the linearized forecast functions exist is also an invariant cyclic subspace.

Distribution of rare saddles in the p -spin energy landscape

We compute the statistical distribution of index-1 saddles surrounding a given local minimum of the p -spin energy landscape, as a function of their distance to the minimum in configuration space and of the energy of the latter. We identify the saddles also in the region of configuration space in which they are subdominant in number (i.e. rare) with respect to local minima, by computing large deviation probabilities of the extremal eigenvalues of their Hessian. As an independent result, we determine the joint large deviation probability of the smallest eigenvalue and eigenvector of a GOE matrix perturbed with both an additive and multiplicative finite-rank perturbation.

Decoupling of hyperbolic conservation laws using an improved eigenvector matrix

The resolution of Harten’s non-oscillatory explicit second-order accurate total variation diminishing scheme was relatively high for computations of nonlinear wave equations. However, this scheme shows false dissipation for sharp discontinuities, such as shocks and contacts, when applied to 1D Euler equation. The selection of the entropy fixing parameter, flux limiter, and appropriate eigenvector matrix is crucial to control the amount of such dissipation. This study focuses on the formulation of eigenvector matrix, which is used for the decoupling of a highly coupled nonlinear 1D Euler equation into a nonlinear wave equation. The selection of the eigenvector matrix is crucial to limit the unwanted false dissipation, which is inherent to the numerical schemes derived for nonlinear wave equations. A generalized form of the scaling factor for the eigenvector matrix is then derived, which is satisfied by the scaling factors used by Harten, Hofmann, and Yee. LAX, SOD, and inverse shock test cases of the shock tube problem are solved to examine and analyze the corresponding physical aspects. The findings show that multiplying the scaling factor by itself or by a constant does not affect the results. However, the variables inside the scaling factor, as well as powers of these variables play a vital role in controlling false dissipation. These scaling factors can effectively capture physics in one case, but might not exhibit satisfactory performance for other problems. In addition, Yee’s scaling factor can effectively capture shock and contact discontinuities, especially for the inverse shock test case.

Multilevel Image Thresholding Using the Complement Feature

A novel Eigen formulation is proposed for image segmentation. Each pixel is represented by a unit vector having the x-component as the normalized gray value of the pixel. The axes of inertia are simply the Eigen vectors of the auto-correlation matrix. The largest Eigen vector is used as the point of split. Each sub-image is further split in the same way. The ratio of the smallest Eigen value to the sum of Eigen values represents the percentage of the minority and was used to control further splitting. The process continues until no sub-image is larger in size than this ratio. The results are very encouraging on a wide range of images.

Deformation mode analysis by eigenvectors of atomic elastic stiffness in static uniaxial tension of various fcc, bcc, and hcp metals

In order to clarify the physical meaning of the eigenvector of the atomic elastic stiffness matrix, Bija=Δσia/Δεj, static calculations of uniaxial tension are performed on various fcc, bcc, and hcp metals with four different embedded atom method (EAM) potentials. Many fcc metals show instability for the constant volume mode, or the eigenvector of (Δεxx, Δεyy, Δεzz) = (±1, ∓1, 0), under the [001] tension. Bcc also loses resistance against other constant volume mode, (Δεxx, Δεyy, Δεzz) = (±1, ±1, ∓2), in the [001] tension. Hcp shows shear modes Δγyz and Δγzx under the [0001] tension, which correspond to atom migration by dislocation on the slip plane. Similar shear modes appear in the [111] tension of fcc and [110] tension of bcc. Hcp also changes the mode to constant volume and shear in the [1¯010] tension, which imply the deformation in the pyramidal and prismatic planes.

Construction of the similarity matrix for the spectral clustering method: Numerical experiments

Spectral clustering is a powerful method for finding structure in a dataset through the eigenvectors of a similarity matrix. It often outperforms traditional clustering algorithms such as k-means when the structure of the individual clusters is highly non-convex. Its accuracy depends on how the similarity between pairs of data points is defined. Two important items contribute to the construction of the similarity matrix: the sparsity of the underlying weighted graph, which depends mainly on the distances among data points, and the similarity function. When a Gaussian similarity function is used, the choice of the scale parameter σ can be critical. In this paper we examine both items, the sparsity and the selection of suitable σ’s, based either directly on the graph associated to the dataset or on the minimal spanning tree (MST) of the graph. An extensive numerical experimentation on artificial and real-world datasets has been carried out to compare the performances of the methods.

Hill-top branching: Its asymptotically expanded and visually solved bifurcation equations

Hill-top branching (HTB) is an exceptional stability problem in which the bifurcation point (BP) coincides exactly with a limit point (LP). In such cases, snap-through and branching instability occur simultaneously. Then, HTB is often obscured in snap-through behavior and is often overlooked in stability design. The compound instability is in fact not well realized in the community of computational mechanics, when the rank deficiency and number of negative eigenvalues of the stiffness matrix are carelessly examined. No report of the relevant literature has established diagnosis of path branching or proposed a branch-switching procedure. Furthermore, few HTB models are available in the literature.This report proposes a set of asymptotic bifurcation equations to address this academically important and exciting issue. There might be a single or two critical eigenvectors of the singular stiffness matrix. Higher-derivatives of the stiffness matrix with respect to nodal displacements can be computed exactly using hyper-dual numbers (HDNs). The resulting simultaneous equations can be solved visually using graphical tools available in popular mathematical software packages such as MATLAB®. From the locations of existing real roots displayed on a graphic monitor, the crossover situation of path branching is visually clear above and below the hill-top. For the present study, three HTB examples are computed to verify the proposed algorithm strategy using HDNs and graphical tools. The obtained numerical results show excellent agreement with analytical and finite element predictions. The proposed asymptotically expanded and visually solved bifurcation equations reliably diagnose asymmetric and unstable symmetric HTB in structural stability problems and function well in computational engineering problems.

Truss optimization using eigenvectors of the covariance matrix

In this paper, by combining eigenvectors of the covariance matrix and random normal distribution, a new method has been presented to solve optimization problems. This method has been inspired by CMA-ES method and has been named eigenvectors of covariance matrix (ECM). ECM generates some solutions in each stage; then it assigns a value to all solutions by utilizing a dynamic penalty function. The best solution in each stage is selected as the answer of optimization problem and ECM tries to push towards a better point. According to the penalty function, the solutions with no violation along with the solutions with little violation are considered as good solutions and named desirable data. By utilizing desirable data, a square matrix is formed and its covariance matrix can be calculated. The eigenvectors of the covariance matrix are orthogonal and they show directions of distribution of desirable points that have been chosen before. We can hopefully get a better answer by moving from the best current answer towards the points which are distributed normally and randomly around the weighted combination of eigenvectors. To ensure the validity and accuracy of ECM method, weights of six truss structures have been optimized through this method and the results have been compared with previous studies.

Study of Tangle on the Three Qubit Werner States by Using of Twirl Operation

In this paper we study the Tangle of three qubit Werner state using Twirl operation and association scheme. To do this, we introduce the invariants of Twirl operation using total spin representation. Then by using commutator property of this invariant with total spin, the general form of three qubit density matrix of Werner state is obtained. The restricted density matrix to the subspaces of the irreducible representations are diagonal4×4 and Block2×2. Then it provided that this given system has not Tangle. Finally, it is shown that common eigenvectors of this density matrix are equal to the idempotents of the group of association scheme S3.

Anisotropic computational modelling of bony structures from CT data: An almost automatic procedure

Background and objectiveThe use of modelling techniques that combine CT data and bone tissue micromechanics is spreading in computational biomechanics. Finite Element models show great potential in surgical planning of intervention and in prediction of stress and strain fields through a non-invasive method. The main challenge pertains to the reliable characterization of bone mechanical behaviour. An almost automatic procedure is here defined, which provides computational models of bony structures considering the actual anisotropy of bone tissue response. The innovative aspect resides on the automatic detection of the directions of anisotropy as the eigenvectors of a three-dimensional distribution matrix of HU values. Methods:The procedure combines CT data and micromechanics modelling techniques. Regarding a specific location, the procedure reports both the orthotropic elastic constants, by the analysis of the local HU value, and the anisotropic material directions, by the analysis of the HU values distribution around the specific location. ResultsThe procedure returns the distribution of bone tissue orthotropic elasticity tensor. The procedure proves to correctly respect the differentiation between cortical and trabecular bone. Principal directions show to be consistent with experimental data from ultrasound measurements. Regarding the material mapping from voxel to FE model, the developed strategies show to be reliable, leading to marginal errors (lower than 10%) for most of CT voxels (more than 90%). The computational analyses of typical structural loading conditions lead to strain values that are comparable with results from strain gauges experimentations. The development and the exploitation of FE models of different bony structures allow assessing the reliability of the procedure for cortical bone. ConclusionsThe results highlight the potentialities of the procedure in providing accurate patient-specific biomechanical models of bony structures starting from CT data. The accuracy and the automatism of the procedure are important factors for the development of real time clinical tools. The main limitations of this work remain the not fully automatism and the reliability assessment, which is based mainly on cortical bone regions only.

Sensor Selection with Nonsmooth Design Criteria Based on Semi-Infinite Programming

A problem of optimal node activation in large-scale sensor networks is considered. The resulting measurements are supposed to be used to estimate unknown parameters of a spatiotemporal process described by a partial differential equation. In this setting, the sensor subset selection problem may quickly become computationally intractable when an excessively complex sensor location algorithm is employed. The is even more pronounced when the design criterion is nondifferentiable. A vital example of this criterion is the sum of an arbitrary number of smallest eigenvalues of the Fisher information matrix, being a generalization of the well-known E-optimality criterion. A simple branch-and-bound algorithm is exposed here to maximize this criterion. Its key component to produce upper bounds to the maximum of the objective function implements a relaxation procedure for solving semi-infinite programming problems. It alternates between solving a linear programming subproblem and evaluation of the eigenvalues and eigenvectors of the current information matrix, which makes it extremely easy to implement. The paper is complemented with a numerical example of computing actual sensor locations.

Method for efficiently orthogonalizing the eigenvectors of the Laplacian matrix to estimate social network structure

Method for efficiently orthogonalizing the eigenvectors of the Laplacian matrix to estimate social network structure

Consensus Based Distributed Spectral Radius Estimation

A consensus based distributed algorithm to compute the spectral radius of a network is proposed. The spectral radius of the graph is the largest eigenvalue of the adjacency matrix, and is a useful characterization of the network graph. Conventionally, centralized methods are used to compute the spectral radius, which involves eigenvalue decomposition of the adjacency matrix of the underlying graph. Our distributed algorithm uses a simple update rule to reach consensus on the spectral radius, using only local communications. We consider time-varying graphs to model packet loss and imperfect transmissions, and provide the convergence characteristics of our algorithm, for both static and time-varying graphs. We prove that the convergence error is a function of principal eigenvector of adjacency matrix of the graph and reduces as $\mathcal {O}(1/t)$ , where $t$ is the number of iterations. The algorithm works for any connected graph structure. Simulation results supporting the theory are also presented.

Rigid 3D Registration: A Simple Method Free of SVD and Eigen-Decomposition

The 3-D registration is extensively required in many industrial applications involving 3-D perception and reconstruction. However, conventional methods consisting of singular value decomposition (SVD) or eigendecomposition are all hard to be implemented and are difficult to be ported using simple digital circuit prototypes. A novel solution is obtained to solve the rigid 3-D registration problem, motivated by previous eigendecomposition approaches. Different from existing solvers, the proposed algorithm does not require sophisticated matrix operations, e.g., SVD or eigenvalue decomposition. Instead, the optimal eigenvector of the point cross-covariance matrix can be computed within several iterations. It is also proven that the optimal rotation matrix can be directly computed for cases without the need for a quaternion. The simple framework provides a very easy approach, even without floating-number processing. Simulations on noise-corrupted point clouds have verified the robustness and computation speed of the proposed method. The final results indicate that the proposed algorithm is accurate, robust, and owns over 60%–80% less computation time than representatives. It has also been applied to real-world applications for faster robotic visual perception.