Nonsingular Research Articles

Integral-algebraic equations with variable limits of integration. The case when the lower limit is at

The paper is devoted to the study and numerical solution of linear systems of integral equations with an identically singular matrix in the principal part (called integral-algebraic equations). The systems studied in the paper are fundamentally different from those previously studied in that the upper and lower limits of integration are variable. These systems of equations will be referred to as integral-algebraic equations with variable integration limits. The paper develops and justifies an algorithm for a numerical solution of the first order of accuracy and establishes conditions for the existence of a unique continuous solution to the given problem. There are numerical computations provided for model examples.

Adaptive directional estimator of the density in [formula omitted] for independent and mixing sequences

A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, α-mixing and τ-mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. More precisely, the estimator is able to reach the convergence rate induced by the best Sobolev regularity of the density of AX, where A belongs to a class of invertible matrices describing all the possible directions. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas.

FNIRS based study of brain network characteristics in children with cerebral palsy during bilateral lower limb movement.

Motor dysfunctions in children with cerebral palsy (CP) are caused by nonprogressive brain damage. Understanding the functional characteristics of the brain is important for rehabilitation. This paper aimed to study the brain networks of children with CP during bilateral lower limb movement using functional near-infrared spectroscopy (fNIRS) and to explore effective fNIRS indices for reflecting functional brain activity. Using fNIRS, cerebral oxygenation signals in the bilateral prefrontal cortex (LPFC/RPFC) and motor cortex (LMC/RMC) were recorded from fifteen children with spastic CP and seventeen children with typical development (CTDs) in the resting state and during bilateral lower limb movement. Functional connectivity matrices based on phase-locking values (PLVs) were calculated using Hilbert transformation, and binary networks were constructed at different sparsity levels. Network metrics such as the clustering coefficient, global efficiency, local efficiency, and transitivity were calculated. Furthermore, the time-varying curves of network metrics during movement were obtained by dividing the time window and using sparse inverse covariance matrices. Finally, conditional Granger causality (GC) was used to explore the causal relationships between different brain regions. Compared to CTDs, the connectivity between RMC-RPFC (p=0.017) and RMC-LMC (p=0.002) in the brain network was decreased in children with CP, and the clustering coefficient (p=0.003), global efficiency (p=0.034), local efficiency (p=0.015), and transitivity (p=0.009) were significantly lower. The standard deviation of the changes in global efficiency of children with CP during motion was also greater than that of CTDs. Using GC, it was found that there was a significant increase in causal strength from the RMC to the RPFC (p=0.04) and from the RMC to the LMC (p=0.042) in children with CP during motion. Additionally, there were significant negative correlations between the PLV of LMC-RMC (p=0.002) and the Gross Motor Function Classification System (GMFCS) and between the GMFCS and the clustering coefficient (p=0.01). During rehabilitation training of the lower limbs, there were significant differences in brain network indices between children with CP and CTDs. The indicators proposed in this paper are effective at evaluating motor function and the real-time impact of rehabilitation training on the brain network and have great potential for application in guiding clinical motor function assessment and planning rehabilitation strategies.

An automata theoretic approach to observer design for switched linear systems

We present an approach for designing asymptotic observers for discrete-time switched linear systems. We first give an automata theoretic characterization of switching signals containing an infinite number of reconstructible sequences, i.e. sequences allowing to estimate the state of the system. We show that such switching signals can be generated by a deterministic Büchi automaton whose construction is given in the paper. Then, we present a methodology to design switched observers. These observers have an internal discrete state variable whose dynamics is given by the transition map of the Büchi automaton. We then present two approaches to design observer gains such that the observer is convergent for all switching signals whose occurrence rate of reconstructible sequences is higher than a tunable threshold. The first approach gives an explicit construction of the observer gains while the second one is based on linear matrix inequalities. For switched systems with invertible state matrices, we show that the proposed observer structure is universal in the sense that it is always possible to design an observer of the proposed form. We use a simple example to illustrate our methodology and then consider a case study in which we design an observer for a multicellular converter.

Inverse matrix estimations by iterative methods with weight functions and their stability analysis

In this paper, we construct a parametric family of iterative methods to compute the inverse of a nonsingular matrix. This class is free of inverse operators. We prove the third-order of convergence under some conditions involving the parameter of the family. Moreover, a dynamical analysis is made for the first time to a matrix iterative method, finding intervals of stability, that include but are wider than those found in the convergence analysis. Numerical tests on large random matrices confirm the results found.

Bootstraps regularize singular correlation matrices

I show analytically that the average of k bootstrapped correlation matrices rapidly becomes positive-definite as k increases, which provides a simple approach to regularize singular Pearson correlation matrices. If n is the order of the matrix and t the number of features, the averaged correlation matrix is almost surely positive-definite if k>ee−1nt≃1.58nt in the limit of large t and n. The probability of obtaining a positive-definite correlation matrix with k bootstraps is also derived for finite n and t. Finally, I demonstrate that the number of required bootstraps is always smaller than n. This method is particularly relevant in fields where n is orders of magnitude larger than the size of data points t, e.g., in finance, genetics, social science, or image processing.

Uncertainty Computation at Finite Distance in Nonlinear Mixed Effects Models-a New Method Based on Metropolis-Hastings Algorithm.

The standard errors (SE) of the maximum likelihood estimates (MLE) of the population parameter vector in nonlinear mixed effect models (NLMEM) are usually estimated using the inverse of the Fisher information matrix (FIM). However, at a finite distance, i.e. far from the asymptotic, the FIM can underestimate the SE of NLMEM parameters. Alternatively, the standard deviation of the posterior distribution, obtained in Stan via the Hamiltonian Monte Carlo algorithm, has been shown to be a proxy for the SE, since, under some regularity conditions on the prior, the limiting distributions of the MLE and of the maximum a posterior estimator in a Bayesian framework are equivalent. In this work, we develop a similar method using the Metropolis-Hastings (MH) algorithm in parallel to the stochastic approximation expectation maximisation (SAEM) algorithm, implemented in the saemix R package. We assess this method on different simulation scenarios and data from a real case study, comparing it to other SE computation methods. The simulation study shows that our method improves the results obtained with frequentist methods at finite distance. However, it performed poorly in a scenario with the high variability and correlations observed in the real case study, stressing the need for calibration.

Non-square internal model control for mode transition of hybrid electric vehicles with multiple time delays

Mode transitions of multi-mode hybrid electric vehicles need careful coordination among the engine torque, motor torque, and clutch transmitted torque. However, the three torques have different actuation delays which make it difficult to ensure mode transition performance. A non-square internal model controller (IMC) is proposed in this paper for mode transition during which the input number is greater than the output number and each input has a different actuation delay. At first, the Moore-Penrose generalized inverse matrix is introduced to solve the inverse of the non-square matrix and make the IMC applicable. Secondly, the all-pole method is adopted to approximate the three delay models. Based on these specialized techniques, the tracking controller and anti-disturbance controller of the IMC are derived. Experiment results verify the effectiveness of the proposed controller.

Symbol-level fiber-longitudinal power profile estimation.

Symbol-level fiber-longitudinal power profile estimation (PPE) greatly reduces the implementation complexity compared with the waveform-level PPE using oversampled data. However, symbol-rate data cannot account for the inter-sample interaction, which leads to inaccuracy of the absolute power estimation. To realize an accurate symbol-level PPE, we provide an in-depth analysis of the differences between symbol-level and waveform-level perturbation matrices and propose a power calibration method based on the trace of the inverse matrix. Evaluated in the probabilistic constellation shaping (PCS) 64QAM 130 Gbaud 5 × 50 km optical links, the root mean squared error (RMSE) of the symbol-level PPE decreases by 0.98 and 0.62 dB at erbium-doped fiber amplifier (EDFA) positions and all estimated positions with the aid of matrix calibration.

A new matrix representation for the Heisenberg group

For any parameter vector [Formula: see text] in [Formula: see text], the representation matrix [Formula: see text] of the Heisenberg group [Formula: see text] is extended to a new form [Formula: see text] with nonsingular matrices [Formula: see text] which are related to the directional translation [Formula: see text], directional modulation [Formula: see text] and dilation [Formula: see text] if [Formula: see text], the identity matrix of order [Formula: see text]. In the case [Formula: see text], an intrinsic group structure under the matrix [Formula: see text] is investigated and a unitary operator representation is offered.

Signed graphs and inverses of their incidence matrices

The Laplacian matrix L of a signed graph G may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of L. This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an incidence matrix of G. This work generalizes related known results on incidence and Laplacian matrices of an unsigned graph. Several examples are provided to show the usefulness of these combinatorial formulas.

Strain field reconstruction from helical-winding fiber distributed acoustic sensing and its application in anisotropic elastic reverse time migration

Optical fiber-based distributed acoustic sensing (DAS) technology has been a popular seismic acquisition tool due to its easy deployment, wide bandwidth, and dense sampling. However, the sensitivity of straight optical fiber to only single-axis strain presents challenges in fully characterizing multicomponent seismic wavefields, making it difficult to use these data in elastic reverse time migration (ERTM). The helical-winding fiber receives projecting signals projected onto the fiber from all seismic strain field components and has the potential to reconstruct those strain components for ERTM imaging. Here, we give detailed mathematical principles of helical fiber-based DAS with crucial parameters such as pitch angle, gauge length, and rotating angle. At least six points of DAS responses are required in one or several winding periods to rebuild the strain fields within the seismic wavelength. The projecting matrix of conventional regular helical-winding fiber is singular and ill conditioned, which results in computation challenges for the inverse of the Hessian matrix for strain component reconstruction. To tackle this problem, we develop a nonregular variant pitch-angle winding configuration for helical fiber. Our winding design is validated using the rank and condition number of the projecting matrix, which is proven to be an important tool in the reconstruction of the original seismic strains. The recovered strain components from the DAS response are then used to backward propagate the receiver wavefields in ERTM with an efficient P/S decoupled approach. To summarize, we develop a novel winding design of helical fiber to recover the strain fields and then develop an efficient 3D anisotropic P/S wave-mode decomposition method for generating vector P and S wavefields during their propagation. Both methods are applied to build an anisotropic DAS-ERTM workflow for producing PP and PS images. Two synthetic examples demonstrate the effectiveness of our approach.

Pemodelan Regresi Semiparametrik B Spline (Studi Kasus: Pengaruh Harga Emas dan Minyak Mentah Dunia Terhadap Indeks Harga Saham Gabungan)

<p dir="ltr"><span>The increases of the world gold and crude oil prices have a big role as a main factor that effect composite stock price index, the effect can make investors to buy stock from Bursa Efek Indonesia. Regression semiparametric used in this research for a purpose to get combined parametric and nonparametric with B Spline approach. B Spline is a development of spline to overcome weaknesses in making singular matrix at a high order spline with many knot points and close together. Variable parametric component is composite stock price index with crude oil price, and variable nonparametric component is composite stock price index with gold price that got obtained from January 2015 until December 2022. The result from this research is best regression semiparametric B-Spline modelling can be obtained using some combination of order and knot points. The optimal point is obtained on 2nd order using 4 knot point (1.135;1.319,15;1.320,75;1.323,25) with a minimum GCV value is 100.227,8. The best measure of goodness with a coefficient of determination value (R-Square) obtained a value 78,8%, because the value is more than 67% make it as a strong model. MAPE value is 3,37% that has a value less than 10 %, make this model have a perfect forecasting ability.</span></p><p><span id="docs-internal-guid-09fba3ae-7fff-a25a-8197-7dd90ea8b1e1"><span>Keywords: </span><span>Gold; Crude Oil; Composite Stock Price Index; Semiparametric B Spline; GCV</span></span></p>

Semi-blind hyperspectral and multispectral image fusion via generalized inverse matrix optimization

Combining a low spatial resolution hyperspectral image (HSI) with a high spatial resolution multispectral image (MSI) is an economical way to obtain a high spatial resolution hyperspectral image (HR-HSI). However, existing fusion methods often rely on a critical variable, the point spread function (PSF), which is difficult to obtain in practice. Additionally, these methods often require a long fusion time and have many parameters that need re-debugging on different datasets to achieve satisfactory results. Many existing methods lack robustness in handling complex scenarios, such as complex PSF, changing spatial downsampling factors, and large-scale datasets. These issues can result in degraded performance or render the methods inapplicable to practical scenarios. Therefore, we propose an efficient and parameterless semi-blind fusion method based on generalized inverse matrix optimization (GIMO), which is robust for multiple complex situations. Our method uses the definition of generalized inverse matrix and the relationship among HSI, MSI, and HR-HSI to approximate the function module of PSF. Then, we calculate the generalized inverse matrix of the spectral response function (SRF). Finally, we optimize the generalized inverse matrix of SRF using the Sylvester equation and multiply it with MSI to obtain HR-HSI. This is the first proposed method to achieve hyperspectral image super-resolution by optimizing the inverse matrix of the spectral response function. Experiments on multiple simulated datasets and one real dataset have validated that our GIMO method outperforms existing state-of-the-art methods, especially regarding fusion time and robustness for various complex scenarios.

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

Решение оптимизационной задачи оценивания моделей полносвязной линейной регрессии

<p>This article is devoted to the problem of estimating fully connected linear regression models using the maximum likelihood method. Previously, a special numerical method was developed for this purpose, based on solving a nonlinear system using the method of simple iterations. At the same time, the issues of choosing initial approximations and fulfilling sufficient conditions for convergence were not studied. This article proposes a new method for solving the optimization problem of estimating fully connected regressions, similar to the method of estimating orthogonal regressions. It has been proven that, with equal error variances of interconnected variables, estimates of b-parameters of fully connected regression are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse covariance matrix. And if the ratios of the error variances of the variables are equal to the ratios of the variances of the variables, then the b-parameter estimates are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse correlation matrix, multiplied by the specific ratios of the standard deviations of the variables. A numerical experiment was carried out to confirm the correctness of the developed mathematical apparatus. The proposed method for solving the optimization problem of estimating fully connected regressions can be effectively used when solving problems of constructing multiple fully connected linear regressions.</p>

Improved input‐to‐state stability criteria for time‐varying switched singular systems

AbstractThis paper is dedicated to proposing improved input‐to‐state stability (ISS) criteria for a class of continuous‐time time‐varying switched singular systems (CTSSS) under slow/fast/slow‐fast switching signals. Specifically, for a CTSSS with time‐varying singular matrices, the authors start with an improved sufficient condition for verifying its ISS under slow switching signals based on the average dwell time method and a relaxed multiple Lyapunov function. Then, considering several or all subsystems may destroy the overall ISS, relaxed sufficient conditions are presented to ensure the ISS of the CTSSS under fast switching signals and slow‐fast switching signals, respectively. Note that the improvement of the proposed ISS criteria is embodied in the derivative of the required multiple Lyapunov functions along with the state trajectories of CTSSS can be positive or negative. Finally, three numerical examples are provided to illustrate the feasibility of the obtained results.

A New Observer Design for the Joint Estimation of States and Unknown Inputs for a Class of Nonlinear Fractional-Order Systems

This article designs an observer for the joint estimation of the state and the unknown input for a class of nonlinear fractional-order systems (FOSs) such that one portion satisfies the Lipschitz condition and the other does not necessarily satisfy such a condition. Firstly, by reconstructing system dynamics, the observer design is transformed equivalently into the tracking problem between the original nonlinear FOSs and the designed observer. Secondly, the parameterized matrices of the desired observer are derived by use of the property of the generalized inverse matrices and the linear matrix inequality (LMI) technique combined with the Schur complement lemma. Moreover, an algorithm is presented to determine the desired observer for the nonlinear FOSs effectively. Finally, a numerical example is reported to verify the correctness and efficiency of the proposed algorithm.

Lightweight super-resolution multimode fiber imaging with regularized linear regression

Super-resolution multimode fiber imaging provides the means to image samples quickly with compact and flexible setups finding many applications from biology and medicine to material science and nanolithography. Typically, fiber-based imaging systems suffer from low spatial resolution and long measurement times. State-of-the-art computational approaches can achieve fast super-resolution imaging through a multimode fiber probe but currently rely on either per-sample optimised priors or large data sets with subsequent long training and image reconstruction times. This unfortunately hinders any real-time imaging applications. Here we present an ultimately fast non-iterative algorithm for compressive image reconstruction through a multimode fiber. The proposed approach helps to avoid many constraints by determining the prior of the target distribution from a simulated set and solving the under-determined inverse matrix problem with a mathematical closed-form solution. We have demonstrated theoretical and experimental evidence for enhanced image quality and sub-diffraction spatial resolution of the multimode fiber optical system.

Diagonal entries of inverses of diagonally dominant matrices

We consider invertible, row diagonally dominant real matrices and give inequalities on their minors and diagonal entries of their inverses. A very special case is that all diagonal entries of an inverse, of a row stochastic, row diagonally dominant and invertible matrix, are at least 1, with strict inequality at least when the dominance is strict. This was conjectured in international trade theory in economics and motivated the present work (though much more is proven). Some of the results generalize previously known facts for M-matrices.