Generalized Inverse Of Matrix Research Articles

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

Conditions for the existence and representations of 2-, 1-, and 1,2-inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.

Improved ADRC for a Maglev Planar Motor with a Concentric Winding Structure

In the semiconductor industry, positioning accuracy and acceleration are critical parameters. To improve the acceleration speed of a motor, this paper proposes the moving-coil maglev planar motor with a concentric winding structure. The coordinate system has been built for the multiple degrees of freedom movement system. The Lorenz force method has been applied to solve its electromagnetic model. The real-time solving of the generalized inverse matrix of factors can realize the decoupling of the winding current. When the maglev height changes, the electromagnetic force and torque decreases exponentially with the increase of the air gap. To decrease the influence on control system performance by the internal model change and the external disturbance, this paper proposes an improved active disturbance rejection control (ADRC) to design the controller. This new controller overcomes the jitter phenomenon due to the turning point for the traditional ADRC, thus it is more suitable for the maglev control system. The comparison between ADRC and the improved ADRC has been conducted, the result of which shows the improved ADRC has greater robustness.

Successive optimization Tomlinson-Harashima precoding strategies for physical-layer security in wireless networks

In this paper, we propose novel non-linear precoders for the downlink of a multi-user MIMO system in the existence of multiple eavesdroppers. The proposed non-linear precoders are designed to improve the physical-layer secrecy rate. Specifically, we combine the non-linear successive optimization Tomlinson-Harashima precoding (SO-THP) with the generalized matrix inversion (GMI) technique to maximize the physical-layer secrecy rate. For the purpose of comparison, we examine different traditional precoders with the proposed algorithm in terms of secrecy rate as well as bit error rate (BER) performance. We also investigate simplified generalized matrix inversion (S-GMI) and lattice-reduction (LR) techniques in order to efficiently compute the parameters of the precoders. We further conduct computational complexity and secrecy-rate analysis of the proposed and existing algorithms. In addition, in the scenario without knowledge of the channel state information (CSI) to the eavesdroppers, a strategy of injecting artificial noise (AN) prior to the transmission is employed to enhance the physical-layer secrecy rate. Simulation results show that the proposed non-linear precoders outperform existing precoders in terms of BER and secrecy-rate performance.

Perturbation bounds of generalized inverses

Let complex matrices A and B have the same sizes. We characterize the generalized inverse matrix B(1, i), called an {1, i}-inverse of B for each i=3 and 4, such that the distance between a given {1, i}-inverse of a matrix A and the set of all {1, i}-inverses of the matrix B reaches minimum under 2-norm (spectral norm) and Frobenius norm. Similar problems are also studied for {1, 2, i}-inverse. In practice, the matrix B is often considered as the perturbed matrix of A, and hence based on the previous results, the additive perturbation bounds for the {1, i}- and {1, 2, i}-inverses and multiplicative perturbation bounds for the {1}-, {1, i}- and {1, 2, i}-inverses are proposed. Numerical examples show that these multiplicative perturbation bounds can be achieved respective under 2-norm and Frobenius norm.

The computation of key properties of Markov chains via perturbations

Computational procedures for the stationary probability distribution, the group inverse of the Markovian kernel and the mean first passage times of a finite irreducible Markov chain, are developed using perturbations. The derivation of these expressions involves the solution of systems of linear equations and, structurally, inevitably the inverses of matrices. By using a perturbation technique, starting from a simple base where no such derivations are formally required, we update a sequence of matrices, formed by linking the solution procedures via generalised matrix inverses and utilising matrix and vector multiplications. Four different algorithms are given, some modifications are discussed, and numerical comparisons are made using a test example. The derivations are based upon the ideas outlined by Hunter [14].

Weighted Moore–Penrose generalized matrix inverse: MySQL vs. Cassandra database storage system

The research in this paper refers to two areas: programming and data storage (data base) for computing the weighted Moore–Penrose inverse. The main aim of this paper analysis of the execution speed of computing using PHP programming language and data store. The research shows that the speed of execution gives considerable difference between the Procedural programming and Object Oriented PHP language, on the middle layer in the three tier of the web architecture. Also, the research concerning the comparison of relation database system, MySQL and NoSQL, key value store system, ApacheCassandra, on the database layer. The CPU times are compared and discussed.

Bayesian Multi-Task Relationship Learning with Link Structure

In this paper, we study the multi-task learning problem with a new perspective of considering the link structure of data and task relationship modeling simultaneously. In particular, we first introduce the Matrix Gaussian (MG) distribution and Matrix Generalized Inverse Gaussian (MGIG) distribution, then define a Matrix Gaussian Matrix Generalized Inverse Gaussian (MG-MGIG) prior. Based on this prior, we propose a novel multi-task learning algorithm, the Bayesian Multi-task Relationship Learning (BMTRL) algorithm. To incorporate the link structure into the framework of BMTRL, we propose link constraints between samples. Through combining the BMTRL algorithm with the link constraints, we propose the Bayesian Multi-task Relationship Learning with Link Constraints (BMTRL-LC) algorithm. Further, we apply the manifold theory to provide an extension of BMTRL-LC to data with no link structure. Specifically, BMTRL-LC is effective for multi-task learning with only limited training samples, which is not addressed in the existing literature. To make the computation tractable, we simultaneously use a convex optimization method and sampling techniques. In particular, we adopt two stochastic EM algorithms for BMTRL and BMTRL-LC, respectively. The experimental results on three real datasets demonstrate the promise of the proposed algorithms.

Synergetics based damage detection of frame structures using piezoceramic patches

This paper investigates the Synergetics based Damage Detection Method (SDDM) for frame structures by using surface-bonded PZT (Lead Zirconate Titanate) patches. After analyzing the mechanism of pattern recognition from Synergetics, the operating framework with cooperation-competition-update process of SDDM was proposed. First, the dynamic identification equation of structural conditions was established and the adjoint vector (AV) set of original vector (OV) set was obtained by Generalized Inverse Matrix (GIM).Then, the order parameter equation and its evolution process were deduced through the strict mathematics ratiocination. Moreover, in order to complete online structural condition update feature, the iterative update algorithm was presented. Subsequently, the pathway in which SDDM was realized through the modified Synergetic Neural Network (SNN) was introduced and its assessment indices were confirmed. Finally, the experimental platform with a two-story frame structure was set up. The performances of the proposed methodology were tested for damage identifications by loosening various screw nuts group scenarios. The experiments were conducted in different damage degrees, the disturbance environment and the noisy environment, respectively. The results show the feasibility of SDDM using piezoceramic sensors and actuators, and demonstrate a strong ability of anti-disturbance and anti-noise in frame structure applications. This proposed approach can be extended to the similar structures for damage identification.

Parallel Algorithm for Finding Inverse of a Matrix and its Application in Message Sharing (Coding Theory)

A parallel algorithm for finding the inverse of the matrix using Gauss Jordan method in OpenMP. The Gauss Jordan method has been chosen for this project because it provides a direct method for obtaining inverse matrix and requires approx. 50% fewer operations unlike other methods. Hence forth it is suitable for massive parallelization. Then, authors have analyzed the parallel algorithm for computing the inverse of the matrix and compared it with its perspective sequential algorithm in terms of run time, speed-up and efficiency. Further, the obtained result is used to propose a new method of Message Sharing (called Coding Theory). The proposed method is simple and has a great potential to be applied to other situation where the exchange of messages is done confidentially such as in military operation, banking transactions etc. General Terms Matrix Inversion, Security, Digital Image Processing

物体力対の逆解析に関する研究（せん断型物体力対の導入による残留応力表現の高度化）

The present work proposes the shear type body force dipole (BFD) in addition to the normal type to simulate residual stress fields more properly. Expressions for the displacements and stresses induced by the BFD are derived and formulated in the boundary integral equations which govern the elastic field. In the first step of the numerical approach, the sensitivity matrix is constructed to correlate the BFD distributions with the boundary stresses, and is transformed into the generalized inverse matrix by the singular value decomposition technique. Then the generalized inverse matrix is operated on the boundary stresses so that the unknown BFD distributions are evaluated. Based on the study for the effect of the shear type BFD on stress and displacement, discussions are focused to the accuracy of the inverse analysis and the influencing factors such as the number of stress data. The use of the boundary displacement besides the stress data is not important to improve the accuracy in the present problem. It is also suggested that the accuracy of the evaluated BFD distribution can be improved by the following iteration of forward stress analysis so as to minimize the stress error norm at the boundary.

Identifying Classes of Explanations for Crime Drop: Period and Cohort Effects for New York State

This paper advances current understanding of the contemporary crime drop by focusing on the changes in the age distribution of arrests from 1990 to 2010. Using the New York State Computerized Criminal History (CCH) file, which tracks every arrest in the state, we apply standard demographic methods to examine age-specific arrest rates over time. We test whether the 25 % drop in the felony arrest rate can be best explained by period or cohort effects with special attention to how the phenomenon varies across crime types and regions within the state. Following the analytic approach of O’Brien and Stockard (J Quant Criminol 25(1):79–101, 2009), we fit the age–period–cohort (APC) model using the generalized inverse matrix, which creates an estimable model. We partition the model variation into each factor by subtracting the variation of the two-factor model from the variation of the three-factor model to provide a direct comparison of the two different classes of explanations for crime drop: period and cohort. Our analysis supports a cohort explanation over a period explanation. Controlling for the (substantial) variation due to age, the cohort effect accounts for twice as much of the remaining variation as the period effect. Specifically, the drop in arrest rates is concentrated in more recent birth cohorts across all ages. Although we found statistically significant age–period interaction effects for the younger age group (ages 16–20) in 1990 and 1995, the cohort effect was still a much stronger predictor of felony arrest rates than the period explanation, even with the age–period interaction. The current study reports that the overall drop in felony arrest rates from 1990 to 2010 is mostly due to decreased arrests among those who were born after 1970 rather than a universal drop across different age groups. We discuss but do not test two potential explanations—the legalization of abortion and the ban on leaded gasoline—for the underlying factors associated with a different criminal propensity among birth cohorts.

Systemize the Probabilistic Discrete Event Systems with Moorepenrose Generalized-inverse Matrix Theory for Cross-sectional Behavioral Data

Moore-Penrose (M-P) generalized inverse matrix theory provides a powerful approach to solve an admissible linear-equation system when the inverse of the coefficient matrix does not exist. M-P matrix theory has been used in different areas to solve challenging research questions, including operations research, signal process, and system controls. In this study, we report our work to systemize a probability discrete event systems (PDES) modeling in characterizing the progression of health risk behaviors. A novel PDES model was devised by Lin and Chen to extract and investigate longitudinal properties of smoking multi-stage behavioral progression with cross-sectional survey data. Despite its success, this PDES model requires extra exogenous equations for the model to be solvable and practically implementable. However, exogenous equations are often difficult if not impossible to obtain. Even if the additional exogenous equations are derived, the data used to generate the equations are often error-prone. By applying the M-P theory, our research demonstrates that Lin and Chen’s PDES model can be solved without using exogenous equations. For practical application, we demonstrate the M-P approach using the open-source R software with real data from 2000 National Survey of Drug Use and Health. The removal of extra data facilitate researchers to use the novel PDES method in examining human behaviors, particularly, health related behaviors for disease prevention and health promotion. Successful application of the M-P matrix theory in solving the PDES model suggests potentials of this method in system modeling to solve challenge problems for other medical and health related research.

An eigenvalue filtering based subspace approach for speech enhancement

In this paper, a subspace approach based on eigenvalue filtering is proposed for enhancement of corrupted speech. The new method firstly simultaneously diagonalizes the covariance matrix of clean speech and noise signal based on GEVD (generalized eigenvalues decomposition), and then filters the smaller components whose eigenvalues are less than zero. Because the remainder eigenvector matrix after filtering is irreversible, we introduce the generalized inverse matrix transform to solve this problem for recovery of speech signal. Experimental results show the proposed method performs better than many conventional methods under strong noise conditions, in terms of yielding less residual noise and lower speech distortion.

Mathematical probit and logistic mortality models of the Khapra beetle fumigated with plant essential oils.

In the current study, probit and logistic models were employed to fit experimental mortality data of the Khapra beetle, Trogoderma granarium (Everts) (Coleoptera: Dermestidae), when fumigated with three plant oils of the gens Achillea. A generalized inverse matrix technique was used to estimate the mortality model parameters instead of the usual statistical iterative maximum likelihood estimation. As this technique needs to perturb the observed mortality proportions if the proportions include 0 or 1, the optimal perturbation in terms of minimum least squares (L2) error was also determined. According to our results, it was better to log-transform concentration and time as explanatory variables in modeling mortality of the test insect. Estimated data using the probit model were more accurate in terms of L2 errors, than the logistic one. Results of the predicted mortality revealed also that extending the fumigation period could be an effective control strategy, even, at lower concentrations. Results could help in using a relatively safe and effective strategy for the control of this serious pest using alternative control strategy to reduce the health and environmental drawbacks resulted from the excessive reliance on the broadly toxic chemical pesticides and in order to contribute safeguard world-wide grain supplies.

Load Identification of Acoustic and Vibration Sources Following Linear Regression and Least-squares of Generalized Matrix Inverse Method

In order to overcome the uncertainty and unclear physical meanings of multi-source load identication, a new load identication algorithm based on linear regression identication model of liner system transfer function and least-squares of generalized matrix inverse is proposed. Using least-squares of generalized matrix inverse method, this new algorithm transforms the multi-objective optimization problem into single-objective optimization problem. Load identication result of this algorithm is the optimal solution for minimum square sum of error objective function of multi-spot response signals. Load identication experiment results on vibration and acoustic multi-source data set in cylindrical shell validate the feasibility and accuracy of this algorithm.

FINDING THE BEST-FIT POLYNOMIAL APPROXIMATION IN EVALUATING DRILL DATA: THE APPLICATION OF A GENERALIZED INVERSE MATRIX / POSZUKIWANIE NAJLEPSZEJ ZGODNOŚCI W PRZYBLIŻENIU WIELOMIANOWYM WYKORZYSTANEJ DO OCENY DANYCH Z ODWIERTÓW – ZASTOSOWANIE UOGÓLNIONEJ MACIERZY ODWROTNEJ

Abstract In mining, various estimation models are used to accurately assess the size and the grade distribution of an ore body. The estimation of the positional properties of unknown regions using random samples with known positional properties was first performed using polynomial approximations. Although the emergence of computer technologies and statistical evaluation of random variables after the 1950s rendered the polynomial approximations less important, theoretically the best surface passing through the random variables can be expressed as a polynomial approximation. In geoscience studies, in which the number of random variables is high, reliable solutions can be obtained only with high-order polynomials. Finding the coefficients of these types of high-order polynomials can be computationally intensive. In this study, the solution coefficients of high-order polynomials were calculated using a generalized inverse matrix method. A computer algorithm was developed to calculate the polynomial degree giving the best regression between the values obtained for solutions of different polynomial degrees and random observational data with known values, and this solution was tested with data derived from a practical application. In this application, the calorie values for data from 83 drilling points in a coal site located in southwestern Turkey were used, and the results are discussed in the context of this study.

An Improved Pruning Algorithm for Fuzzy Neural Network

The number of fuzzy rules directly determines the complexity and efficiency of Fuzzy Neural Network (FNN). The Iterative Pruning (IP) algorithm belongs to the Pruning Method, and it spends much time computing adjusting factors of the remaining weights. So the Improved Iterative Pruning (IIP) algorithm is put forward, which adopts dividing blocks strategy and uses the Generalized Inverse Matrix (GIM) algorithm to replace the Conjugate Gradient Precondition Normal Equation (CGPCNE) algorithm for updating the remaining weights. The IIP algorithm is applied in the rule-reasoning layer of FNN to simplify its rules and structure in a great extent and preserve a good level of accuracy and generalization ability without retraining after pruning. The simulation results demonstrate the effectiveness and the feasibility of the algorithm.

Construction of Reduced Order Observer for Linear Time Invariant System using Generalized Matrix Inverse

In this paper a method has been developed to construct a reduced-order observer for a linear time-invariant control system defined by a state space description. The method developed uses the concept of generalized matrix inverse to construct the observer without presupposing the observer structure as well as on the form and rank of the output matrix. Of course, the system must be observable. Illustrative examples covering different cases of the output matrix are included.

Approximate inversion formula for structural dynamics and control

In a recent paper the authors presented an iterative method and an approximate formula for predicting the response of a system modified by a finite set of rank-one modifications. This method was developed based on the successive application of the Sherman–Morrison matrix inversion formula to calculate the inverse of a matrix changed by a rank-k modification and provides an easier method to calculate the change in the transfer matrix of a dynamic system when it undergoes a number of k simply connected modifications. This paper presents a new and more interesting justification of the method which allows an extension of the approximate formula from a formula for frequency response approximation to a more general approximate matrix inversion formula applicable to particularly shaped matrices generally encountered in structural dynamics and control. This new approach extends the area of the application of the previously presented method. An example of a simple application of this method to the problem of feedback control of structures known only through their estimated receptances is presented.

Common nature of learning between back-propagation and Hopfield-type neural networks for generalized matrix inversion with simplified models.

In this paper, two simple-structure neural networks based on the error back-propagation (BP) algorithm (i.e., BP-type neural networks, BPNNs) are proposed, developed, and investigated for online generalized matrix inversion. Specifically, the BPNN-L and BPNN-R models are proposed and investigated for the left and right generalized matrix inversion, respectively. In addition, for the same problem-solving task, two discrete-time Hopfield-type neural networks (HNNs) are developed and investigated in this paper. Similar to the classification of the presented BPNN-L and BPNN-R models, the presented HNN-L and HNN-R models correspond to the left and right generalized matrix inversion, respectively. Comparing the BPNN weight-updating formula with the HNN state-transition equation for the specific (i.e., left or right) generalized matrix inversion, we show that such two derived learning-expressions turn out to be the same (in mathematics), although the BP and Hopfield-type neural networks are evidently different from each other a great deal, in terms of network architecture, physical meaning, and training patterns. Numerical results with different illustrative examples further demonstrate the efficacy of the presented BPNNs and HNNs for online generalized matrix inversion and, more importantly, their common natures of learning.