Moore-Penrose Research Articles

Statistical Linearization of Nonlinear Structural Systems with Singular Matrices

AbstractA generalized statistical linearization technique is developed for determining approximately the stochastic response of nonlinear dynamic systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Specifically, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, a family of optimal and response-dependent equivalent linear matrices is derived. This set of equations in conjunction with a generalized excitation-response relationship for linear systems leads to an iterative determination of the system response mean vector and covariance matrix. Further, it is proved ...

Modeling Drinking Behavior Progression in Youth with Cross-sectional Data: Solving an Under-identified Probabilistic Discrete Event System.

The probabilistic discrete event systems (PDES) method provides a promising approach to study dynamics of underage drinking using cross-sectional data. However, the utility of this approach is often limited because the constructed PDES model is often non-identifiable. The purpose of the current study is to attempt a new method to solve the model. A PDES-based model of alcohol use behavior was developed with four progression stages (never-drinkers [ND], light/moderate-drinker [LMD], heavy-drinker [HD], and ex-drinker [XD]) linked with 13 possible transition paths. We tested the proposed model with data for participants aged 12-21 from the 2012 National Survey on Drug Use and Health (NSDUH). The Moore-Penrose (M-P) generalized inverse matrix method was applied to solve the proposed model. Annual transitional probabilities by age groups for the 13 drinking progression pathways were successfully estimated with the M-P generalized inverse matrix approach. Result from our analysis indicates an inverse "J" shape curve characterizing pattern of experimental use of alcohol from adolescence to young adulthood. We also observed a dramatic increase for the initiation of LMD and HD after age 18 and a sharp decline in quitting light and heavy drinking. Our findings are consistent with the developmental perspective regarding the dynamics of underage drinking, demonstrating the utility of the M-P method in obtaining a unique solution for the partially-observed PDES drinking behavior model. The M-P approach we tested in this study will facilitate the use of the PDES approach to examine many health behaviors with the widely available cross-sectional data.

Linear Random Vibration of Structural Systems with Singular Matrices

AbstractA framework is developed for determining the stochastic response of linear multi-degree-of-freedom (MDOF) structural systems with singular matrices. This system modeling can arise when using more than the minimum number of coordinates, and can be advantageous, for instance, in cases of complex multibody systems whose dynamics satisfy a number of constraints. In such cases the explicit formulation of the equations of motion can be a nontrivial task, whereas the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, standard concepts, relationships, and equations of the linear random vibration theory are extended and generalized herein to account for systems with singular matrices. Adopting a state-variable formulation, equations governing the system response mean vector and covariance matrix are formed and solved...

On matrices whose Moore-Penrose inverses are ray unique

Let denote the set of matrices with the same ray pattern as a complex matrix . If the Moore-Penrose (M-P) inverses of and have the same ray pattern for every , then is said to have ray M-P inverse. In this paper, we give complete characterizations of matrices with ray M-P inverse, which extends some results in [Shao JY, Shan HY, Matrices with signed generalized inverses. Linear Algebra Appl. 2001;322:105–127].

An improved cuckoo search based extreme learning machine for medical data classification

Machine learning techniques are being increasingly used for detection and diagnosis of diseases for its accuracy and efficiency in pattern classification. In this paper, improved cuckoo search based extreme learning machine (ICSELM) is proposed to classify binary medical datasets. Extreme learning machine (ELM) is widely used as a learning algorithm for training single layer feed forward neural networks (SLFN) in the field of classification. However, to make the model more stable, an evolutionary algorithm improved cuckoo search (ICS) is used to pre-train ELM by selecting the input weights and hidden biases. Like ELM, Moore–Penrose (MP) generalized inverse is used in ICSELM to analytically determines the output weights. To evaluate the effectiveness of the proposed model, four benchmark datasets, i.e. Breast Cancer, Diabetes, Bupa and Hepatitis from the UCI Repository of Machine Learning are used. A number of useful performance evaluation measures including accuracy, sensitivity, specificity, confusion matrix, Gmean, F-score and norm of the output weights as well as the area under the receiver operating characteristic (ROC) curve are computed. The results are analyzed and compared with both ELM based models like ELM, on-line sequential extreme learning algorithm (OSELM), CSELM and other artificial neural networks i.e. multi-layered perceptron (MLP), MLPCS, MLPICS and radial basis function neural network (RBFNN), RBFNNCS, RBFNNICS. The experimental results demonstrate that the ICSELM model outperforms other models.

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.

The Role of Pseudo Measurements in Equality-Constrained State Estimation

The pseudo measurement method is a main approach to equality-constrained state estimation due to its simplicity. It is, however, not popular due to possible numerical problems and increased computational complexity. The work presented here further develops the pseudo measurement method. To avoid numerical problems resulting from singular measurement noise when a matrix inverse is used, the Moore-Penrose (MP) inverse is used instead. Also, to reduce the computational load without performance loss and to simplify the analysis of this type of estimation problem, two sequential forms are obtained. They differ only in the processing order of the physical measurement and the pseudo measurement (i.e., the equality constraint). Although form 1 is the same as some existing results, form 2 is new. This motivates the discussion of processing order for this type of estimation problem, especially in the extension to the nonlinear case. It is found that under certain conditions, the use of the pseudo measurement for filtering is redundant. This differs in effect from update by the physically error-free measurement. However, if there exists model mismatch, update by the pseudo measurement is necessary and helpful. Supporting numerical examples are provided.

An improved evolutionary extreme learning machine based on particle swarm optimization

Recently Extreme Learning Machine (ELM) for single-hidden-layer feedforward neural networks (SLFN) has been attracting attentions for its faster learning speed and better generalization performance than those of traditional gradient-based learning algorithms. However, ELM may need high number of hidden neurons and lead to ill-condition problem due to the random determination of the input weights and hidden biases. In this paper, a hybrid learning algorithm is proposed to overcome the drawbacks of ELM, which uses an improved particle swarm optimization (PSO) algorithm to select the input weights and hidden biases and Moore–Penrose (MP) generalized inverse to analytically determine the output weights. In order to obtain optimal SLFN, the improved PSO optimizes the input weights and hidden biases according to not only the root mean squared error (RMSE) on validation set but also the norm of the output weights. The proposed algorithm has better generalization performance than traditional ELM and other evolutionary ELMs, and the conditioning of the SLFN trained by the proposed algorithm is also improved. Experiment results have verified the efficiency and effectiveness of the proposed method.

Robust extreme learning machine

The output weights computing of extreme learning machine (ELM) encounters two problems, the computational and outlier robustness problems. The computational problem occurs when the hidden layer output matrix is a not full column rank matrix or an ill-conditioned matrix because of randomly generated input weights and biases. An existing solution to this problem is Singular Value Decomposition (SVD) method. However, the training speed is still affected by the large complexity of SVD when computing the Moore–Penrose (MP) pseudo inverse. The outlier robustness problem may occur when the training data set contaminated with outliers then the accuracy rate of ELM is extremely affected. This paper proposes the Extended Complete Orthogonal Decomposition (ECOD) method to solve the computational problem in ELM weights computing via ECODLS algorithm. And the paper also proposes the other three algorithms, i.e. the iteratively reweighted least squares (IRWLS-ELM), ELM based on the multivariate least-trimmed squares (MLTS-ELM), and ELM based on the one-step reweighted MLTS (RMLTS-ELM) to solve the outlier robustness problem. However, they also encounter the computational problem. Therefore, the ECOD via ECODLS algorithm is also used successfully in the three proposed algorithms. The experiments of regression problems were conducted on both toy and real-world data sets. The outlier types are one-sided and two-sided outliers. Each experiment was randomly contaminated with outliers, of one type only, with 10%, 20%, 30%, 40%, and 50% of the total training data size. Meta-metrics evaluation was used to measure the outlier robustness of the proposed algorithms compared to the existing algorithms, i.e. the minimax probability machine regression (MPMR) and the ordinary ELM. The experimental results showed that ECOD can effectively replace SVD. The ECOD is robust to the not full column rank or the ill-conditional problem. The speed of the ELM training using ECOD is also faster than the ordinary training algorithm. Moreover, the meta-metrics measure showed that the proposed algorithms are less affected by the increasing number of outliers than the existing algorithms.

Representations of the Moore–Penrose inverse for a class of 2-by-2 block operator valued partial matrices

We obtain necessary and sufficient conditions for 2-by-2 block operator valued triangular matrices to be Moore–Penrose (MP) invertible and give new representations of such MP inverses in terms of the individual blocks.

REPRESENTATIONS OF THE MOORE-PENROSE INVERSE OF 2×2 BLOCK OPERATOR VALUED MATRICES

We obtain necessary and sucient conditions for 2 ◊2 block operator valued matrices to be Moore-Penrose (MP) invertible and give new representations of such MP inverses in terms of the individual blocks.

Learning Algorithms for Human–Machine Interfaces

The goal of this study is to create and examine machine learning algorithms that adapt in a controlled and cadenced way to foster a harmonious learning environment between the user and the controlled device. To evaluate these algorithms, we have developed a simple experimental framework. Subjects wear an instrumented data glove that records finger motions. The high-dimensional glove signals remotely control the joint angles of a simulated planar two-link arm on a computer screen, which is used to acquire targets. A machine learning algorithm was applied to adaptively change the transformation between finger motion and the simulated robot arm. This algorithm was either LMS gradient descent or the Moore-Penrose (MP) pseudoinverse transformation. Both algorithms modified the glove-to-joint angle map so as to reduce the endpoint errors measured in past performance. The MP group performed worse than the control group (subjects not exposed to any machine learning), while the LMS group outperformed the control subjects. However, the LMS subjects failed to achieve better generalization than the control subjects, and after extensive training converged to the same level of performance as the control subjects. These results highlight the limitations of coadaptive learning using only endpoint error reduction.

Bacterial Foraging Algorithm을 이용한 Extreme Learning Machine의 파라미터 최적화

최근 단일 은닉층을 갖는 전방향 신경회로망 구조로, 기존의 경사 기반 학습알고리즘들보다 학습 속도가 매우 우수한 ELM(Extreme Learning Machine)이 제안되었다. ELM 알고리즘은 입력 가중치들과 은닉 바이어스들의 초기 값을 무작위로 선택하고 출력 가중치들은 Moore-Penrose(MP) 일반화된 역행렬 방법을 통하여 구해진다. 그러나 입력 가중치들과 은닉층 바이어스들의 초기 값 선택이 어렵다는 단점을 갖고 있다. 본 논문에서는 최적화 알고리즘 중 박테리아 생존(Bacterial Foraging) 알고리즘의 수정된 구조를 이용하여 ELM의 초기 입력 가중치들과 은닉층 바이어스들을 선택하는 개선된 방법을 제안하였다. 실험을 통하여 제안된 알고리즘이 많은 입력 데이터를 가지는 문제들에 대하여 성능이 우수함을 보였다. Recently, Extreme learning machine(ELM), a novel learning algorithm which is much faster than conventional gradient-based learning algorithm, was proposed for single-hidden-layer feedforward neural networks. The initial input weights and hidden biases of ELM are usually randomly chosen, and the output weights are analytically determined by using Moore-Penrose(MP) generalized inverse. But it has the difficulties to choose initial input weights and hidden biases. In this paper, an advanced method using the bacterial foraging algorithm to adjust the input weights and hidden biases is proposed. Experiment at results show that this method can achieve better performance for problems having higher dimension than others.

A computing method for the Moore-Penrose inverse of some special matrices

A number of methods have been proposed for computing the Moore Penrose (M-P) inverse, X + , of an arbitrary mxn matrix of rank r ≤ m ≤ n. In this paper we have given a simple procedure to find the explicit form of M-P inverse of a matrix that has special form and will consider the ANOVA models. Also computational algorithms for constructing and computing M-P inverse of these special matrices are given.

Evolutionary extreme learning machine

Extreme learning machine (ELM) [G.-B. Huang, Q.-Y. Zhu, C.-K. Siew, Extreme learning machine: a new learning scheme of feedforward neural networks, in: Proceedings of the International Joint Conference on Neural Networks (IJCNN2004), Budapest, Hungary, 25–29 July 2004], a novel learning algorithm much faster than the traditional gradient-based learning algorithms, was proposed recently for single-hidden-layer feedforward neural networks (SLFNs). However, ELM may need higher number of hidden neurons due to the random determination of the input weights and hidden biases. In this paper, a hybrid learning algorithm is proposed which uses the differential evolutionary algorithm to select the input weights and Moore–Penrose (MP) generalized inverse to analytically determine the output weights. Experimental results show that this approach is able to achieve good generalization performance with much more compact networks.

A new approach to the solution of the linear mixing model for a single isotope: application to the case of an opportunistic predator

Mixing models are used to determine diets where the number of prey items are greater than one, however, the limitation of the linear mixing method is the lack of a unique solution when the number of potential sources is greater than the number (n) of isotopic signatures +1. Using the IsoSource program all possible combinations of each source contribution (0-100%) in preselected small increments can be examined and a range of values produced for each sample analysed. We propose the use of a Moore Penrose (M-P) pseudoinverse, which involves the inverse of a 2x2 matrix. This is easily generalized to the case of a single isotope with (p) prey sources and produces a specific solution. The Antarctic leopard seal (Hydrurga leptonyx) was used as a model species to test this method. This seal is an opportunistic predator, which preys on a wide range of species including seals, penguins, fish and krill. The M-P method was used to determine the contribution to diet from each of the four prey types based on blood and fur samples collected over three consecutive austral summers. The advantage of the M-P method was the production of a vector of fractions f for each predator isotopic value, allowing us to identify the relative variation in dietary proportions. Comparison of the calculated fractions from this method with 'means' from IsoSource allowed confidence in the new approach for the case of a single isotope, N.

Additive maps preserving M-P inverses of matrices over Fields*

Suppose F is a field of characteristic not 2 or 3. A characterization is given for all additive maps, on the algebra of all n × n matrices over F. which preserve Moore -Penrose(M-P) Inverses of matrices.

An Alternative Proof of the Greville Formula

A simple proof of the Greville formula for the recursive computation of the Moore–Penrose (MP) inverse of a matrix is presented. The proof utilizes no more than the elementary properties of the MP inverse.

Moore-penrose inverse method of topological variation of finite element systems

A topological modification method for structural variations of finite element system is studied in this paper. The Moore-Penrose (M-P) inverse theory and a new factorization of a stiffness matrix are used. A set of explicit formulations of variations are obtained. Two numerical examples are given to illustrate the validity of the present method.

On the matrix convexity of the moore—penrose inverse and some applications

It is well-known that if A and B are two positive definite matrices of the same order and 0 ≤ λ ≤ 1, then . It is easy to construct an example consisting of two positive semi-definite matrices for which the above inequality is not true when one replaces the inverse operation by Moore-Penrose inverse operation. In this paper we give necessary and sufficient conditions for the validity of the inequality for every 0 ≤ λ ≤ 1. As an application, we give a sufficient condition under which the inequality (EA)+ λ E (A +) is valid, where A is a square matrix of random variables which is almost surely positive semi-definite, generalizing the well-known result (EA)− ≤ EA−1 when A is almost surely positive definite.