Matrix Operations Research Articles

Using land surface phenology and information theory to assess and map complex landscape dynamics

Abstract Context Characterizing landscape ecological complexity and change requires integrated description of spatial and temporal landscape organization and dynamics, as suggested by the shifting mosaic concept. Remotely sensed land surface phenology allows the detection of even small differences among landscape patches and through time, allowing for the analysis of landscapes as shifting mosaics. Objectives We sought to quantify aspects of the complex landscape behaviors that are implied by spatiotemporal variation in land surface phenology. We adapted an information-theoretic (IT) framework from ecosystem ecology to capture landscape-level spatiotemporal complexity and organization and map these properties across large areas. Methods Phenology data were derived from remotely sensed, pixel-level time series of a vegetation greenness index, across a large portion of North America. We summarized multi-year, multi-pixel dynamics in transition matrices, calculated IT metrics from the matrices, and used matrix projection to quantify disequilibrium dynamics and long-term trajectories of the metrics. Results Mapping the IT metrics and their disequilibria revealed gradients in the spatiotemporal complexity and organization of multi-year land surface phenology dynamics at continental to local scales. These gradients suggest influences of biophysical and biogeographic setting, ecological development and disturbances, land use, and other drivers of landscape ecological dynamics. The spatiotemporal IT metrics were influenced by both year-to-year dynamics and spatial landscape heterogeneity, but correlations with spatial and temporal complexity measures varied among the IT metrics. Landscapes showing the strongest disequilibrium dynamics were mostly in the western part of the continent and appeared to be associated with large-scale disturbances including severe fire, forest pathogens, climate variability, and land use change—important subjects for further study. Conclusions This approach reveals novel features of the shifting landscape mosaic, with implications for understanding landscape resilience and sustainability. Resulting spatial data products describing long-term landscape dynamics have potential applications in broad-scale ecological modeling, monitoring, assessment, and prediction.

Operational matrix method approach for fractional partial differential-equations

Abstract Fractional partial differential equations (FPDEs) have become very popular to model and analyze numerous different physical phenomena in recent years. However, it is generally complicated to find the exact solutions of those FPDEs. The objective of this study is to find the approximate numerical solution of FPDEs by introducing a wavelet-based operational matrix technique. In this study we employ Hermite wavelets (HWs) and the operational matrices of the fractional integration for Hermite wavelets. The sparsity of the obtained operational matrices provides fast and efficient computation of the proposed method. The original FPDE equations are converted to Sylvester equations, which then are solved to obtain the final solution. We provide a few numerical examples to demonstrate the versatility and efficiency of the proposed method.

Joint Sparse Local Linear Discriminant Analysis for Feature Dimensionality Reduction of Hyperspectral Images

Although linear discriminant analysis (LDA)-based subspace learning has been widely applied to hyperspectral image (HSI) classification, the existing LDA-based subspace learning methods exhibit several limitations: (1) They are often sensitive to noise and demonstrate weak robustness; (2) these methods ignore the local information inherent in data; and (3) the number of extracted features is restricted by the number of classes. To address these drawbacks, this paper proposes a novel joint sparse local linear discriminant analysis (JSLLDA) method by integrating embedding regression and locality-preserving regularization into the LDA model for feature dimensionality reduction of HSIs. In JSLLDA, a row-sparse projection matrix can be learned, to uncover the joint sparse structure information of data by imposing a L2,1-norm constraint. The L2,1-norm is also employed to measure the embedding regression reconstruction error, thereby mitigating the effects of noise and occlusions. A locality preservation term is incorporated to fully leverage the local geometric structural information of the data, enhancing the discriminability of the learned projection. Furthermore, an orthogonal matrix is introduced to alleviate the limitation on the number of acquired features. Finally, extensive experiments conducted on three hyperspectral image (HSI) datasets demonstrated that the performance of JSLLDA surpassed that of some related state-of-the-art dimensionality reduction methods.

A study for analysing predator–prey ecological system using Legendre polynomial

This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on Caputo operators, we present a newly developed system of differential equations for the predator–prey system using wavelet method. It is well known that a system of nonlinear singular models cannot operate smoothly since they are singular and nonlinear. Therefore, with the help of this numerical approach, we have converted the system into a nonlinear system of algebraic equations by extending it through operational matrix of Legendre wavelets. Using the wavelet collocation scheme, we have calculated these unknown coefficients. It has been demonstrated in tables and graphs that the developed approach is consistent and proficient. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. In addition, a nonlinear functional analysis have used to establish uniformly boundedness for the proposed model. Also we have discussed residual error analysis and Lyapunov exponent. The applicability and efficacy of this methodology have been demonstrated through this nonlinear system. Additionally, a comparison with existing results highlights the advantages of our numerical approach. All calculations have been done using MATLAB.

A Rapid Numerical Method for Nonlinear Generalized Time-Fractional Kawahara Equations via Domination Polynomials of Complete Graph.

Abstract The key objective of this study is to examine the nonlinear Time-fractional generalized Kawahara equation (N-TF-GKE), a significant nonlinear equation that appears in a variety of physical domains, including ion-acoustic waves in plasmas, shallow water waves, and nonlinear acoustics. This Equation exhibits complex dynamic properties, most notably the appearance of shock waves, solitary waves, and chaotic solutions. We use a graph theoretic polynomial approach to obtain numerical solutions of the N-TF-GKE. We present the dominance polynomials of complete graphs (DPC), a new class of graph polynomials, to develop an efficient operational integration matrix to approximate the N-TF-GKE. The N-TF-GKE transformed into a collection of nonlinear algebraic equations with the standard collocation points. We acquire the numerical solutions for the Domination polynomial using Newton's approach. We validate the efficiency and robustness of our domination polynomial collocation method (DCM) with a set of numerical experiments and comparative analyses.

Linking Electricity and Air Quality Models by Downscaling: Weather-Informed Hourly Dispatch of Generation Accounting for Renewable and Load Temporal Variability Scenarios.

National models of the electric sector typically consider a handful of generator operating periods per year, while pollutant fate and transport models have an hourly resolution. We bridge that scale gap by introducing a novel fundamental-based temporal downscaling method (TDM) for translating national or regional energy scenarios to hourly emissions. Optimization-based generator dispatch is used to account for variations in emissions stemming from weather-sensitive power demands and wind and solar generation. The TDM is demonstrated by downscaling emissions from the electricity market module in the National Energy Model System. As a case study, we implement the TDM in the Virginia-Carolinas region and compare its results with traditional statistical downscaling used in the Sparse Matrix Operator Kernel Emissions (SMOKE) processing model. We find that the TDM emission profiles respond to weather and that nitrogen oxide emissions are positively correlated with conditions conducive to ozone formation. In contrast, SMOKE emission time series, which are rooted in historical operating patterns, exhibit insensitivity to weather conditions and potential biases, particularly with high renewable penetration and climate change. Relying on SMOKE profiles can also obscure variations in emission patterns across different policy scenarios, potentially downplaying their impacts on power system operations and emissions.

Accelerated strategies for impact identification of composite laminate using sparse sensor networks and model-based virtual time reversal

Barely visible impact damages are notorious threats to the health of composite structures. Detection of the impact is critical to promise the performance of composite structures. Recently, the authors proposed an impact identification method based on sparse sensor networks and virtual time reversal, which is verified on composite laminates. However, the computational demands of virtual time reversal simulations pose a notable limitation to this approach. To improve the efficiency of the method, several acceleration measures are developed in this article. First, the re-emitted wavefield analysis is accelerated using a frequency-domain method rather than the time-domain simulation, which largely reduces the computational costs. Subsequently, the GPU parallel acceleration algorithm is configured to further improve the efficiency. The proposed acceleration strategy is experimentally validated by various impact events on a composite laminate. The results render that the proposed acceleration strategy can promise the accuracy of the results of impact identification while significantly promoting computational efficiency, in which the time consumption plummets to only 1.94 s. Moreover, factors affecting the efficiency of the proposed method are discussed. The time consumption is positively correlated with the number of sensors involved in the identification. In addition, as the number of candidate points increases, the GPU encounters difficulty in handling all complex matrix operations simultaneously, leading to a substantial rise in computation time. Furthermore, the effect of signal lengths on the efficiency is minimal and largely negligible.

Incremental Classification for High-Dimensional EEG Manifold Representation Using Bidirectional Dimensionality Reduction and Prototype Learning.

In brain-computer interface (BCI) systems, symmetric positive definite (SPD) manifold within Riemannian space has been frequently utilized to extract spatial features from electroencephalogram (EEG) signals. However, the intrinsic high dimensionality of SPD matrices introduces too much computational burden to hinder the real-time applications of such BCI, especially in handling dynamic tasks, like incremental learning. Directly reducing the dimensionality of SPD matrices with conventional dimensionality reduction (DR) methods will alter the fundamental properties of SPD matrices. Moreover, current DR methods for incremental learning always necessitate retaining old data to update their representations under new mapping. To this end, a bidirectional two-dimensional principal component analysis for SPD manifold (B2DPCA-SPD) is proposed to reduce the dimensionality of SPD matrices, in such way that the reduced matrices remain on SPD manifold. Afterwards, the B2DPCA-SPD is extended to adapt to incremental learning tasks without saving old data. The incremental B2DPCA-SPD can be seamlessly integrated with the matrix-formed growing neural gas network (MF-GNG) to achieve an incremental EEG classification, where the new low-dimensional representations of the prototypes in old classifiers can be easily recalculated with the updated projection matrix. Extensive experiments are conducted on two public datasets to perform the EEG classification. The results demonstrate that our method significantly reduces computation time by 38.53% and 35.96%, and outperforms conventional methods in classification accuracy by 4.21% to 19.59%.

A nonlinear manifold model order reduction approach for thermo‐mechanically coupled plasticity

AbstractThis study employs a nonlinear manifold mode order reduction (MOR) approach to address the high computational cost inherent in multi‐physical nonlinear models, which are discretized by the finite element method (FEM), with a particular focus on thermo‐mechanically coupled plasticity under finite strain conditions. Utilizing nodal snapshots obtained from a full‐order simulation, a projection matrix is constructed through singular value decomposition (SVD), facilitating the construction of a reduced system within a lower‐dimensional space. This reduction process entails the multiplication of the discretized residual vector and stiffness matrix by a linear projection matrix, complemented by the incorporation of a nonlinear projection component to ensure accurate reconstruction within the nonlinear part. Subsequently, a comprehensive three‐dimensional benchmark test is conducted to assess the accuracy and efficacy of the reduced‐order model.

Incremental feature selection: Parallel approach with local neighborhood rough sets and composite entropy

Rough set theory is a powerful mathematical framework for managing uncertainty and is widely utilized in feature selection. However, traditional rough set-based feature selection algorithms encounter significant challenges, especially when processing large-scale incremental data and adapting to the dynamic nature of real-world scenarios, where both data volume and feature sets are continuously changing. To overcome these limitations, this study proposes an innovative algorithm that integrates local neighborhood rough sets with composite entropy to measure uncertainty in information systems more accurately. By incorporating decision distribution, composite entropy enhances the precision of uncertainty quantification, thereby improving the effectiveness of the algorithm in feature selection. To further improve performance in handling large-scale incremental data, matrix operations are employed in place of traditional set-based methods, allowing the algorithm to fully utilize modern hardware capabilities for accelerated processing. Additionally, parallel computing technology is integrated to further enhance computational speed. An incremental version of the algorithm is also introduced to better adapt to dynamic data environments, increasing its flexibility and practicality. Comprehensive experimental evaluations demonstrate that the proposed algorithm significantly surpasses existing methods in both effectiveness and efficiency.

Contribution of lean production to the resilience of the Brazilian automotive supply chain

We propose a method to assess companies’ ability to cope with different types of disruptive events by developing resilience capabilities through Lean Production (LP) implementation. We use direct and reverse mathematical operations with two expert opinion-based matrices mapping relationships between LP practices and resilience capabilities, and resilience capabilities and types of disruptive events, obtained collecting opinions from 12 experts. We apply the method in two instances. First, using results from a survey with 135 practitioners from the Brazilian automotive industry regarding their adoption levels of LP practices. Direct matrix operations using this input indicated the development levels of eight resilience capabilities and the ability degree to cope with four types of disruptive events. Second, presenting four scenarios in which a company is exposed to high- and low-impact disruptive events of temporary and permanent nature, we perform reverse matrix operations to identify resilience capabilities and LP practices best fit to cope with each type of event. From the direct analysis of practitioners’ data two LP practices (supplier feedback and involved customers) and two resilience capabilities (provide a quick response to disruption and adapt to disruption easily) stand out. These practices and capabilities mainly prepare companies to cope with high impact disruptions of permanent nature.

Systematic Analysis of Low-Precision Training in Deep Neural Networks: Factors Influencing Matrix Computations

As Deep Neural Networks (DNNs) continue to increase in complexity, the computational demands of their training have become a significant bottleneck. Low-precision training has emerged as a crucial strategy, wherein full-precision values are quantized to lower precisions, reducing computational overhead while aiming to maintain model accuracy. While prior research has primarily focused on minimizing quantization noise and optimizing performance for specific models and tasks, a comprehensive understanding of the general principles governing low-precision computations across diverse DNN architectures has been lacking. In this paper, we address this gap by systematically analyzing the factors that influence low-precision matrix computations, which are fundamental to DNN training. We investigate three critical factors—accumulation in matrix calculations, the frequency of element usage, and the depth of matrices within the model—and their impact on low-precision training. Through controlled experiments on standard models, as well as customized experiments designed to isolate individual factors, we derive several key insights: layers with higher accumulation and matrices with lower usage frequencies demonstrate greater tolerance to low-precision noise, without significantly compromising the stability of model training. Additionally, while the depth of matrices influences the stability of matrix operations to some extent, it does not have a noticeable effect on the overall training outcomes. Our findings contribute to the development of generalizable principles for low-precision training, offering a systematic framework applicable across various DNN architectures. We provide empirical evidence supporting the strategic allocation of training bit-widths based on the analyzed factors, thereby enhancing the efficiency and effectiveness of DNN training in resource-constrained environments.

Unsupervised Feature Selection via Controllable Adaptive Graph Learning and Discriminative Feature Learning.

Unsupervised feature selection is challenging in machine learning, pattern recognition, and data mining. The crucial difficulty is to learn a moderate subspace that preserves the intrinsic structure and to find uncorrelated or independent features simultaneously. The most common solution is first to project the original data into a lower dimensional space and then force them to preserve the similar intrinsic structure under linear uncorrelation constraint. However, there are three shortcomings. First, the final graph generated by the iterative learning process differs significantly from the initial graph in which the original intrinsic structure is embedded. Second, it requires prior knowledge about a moderate dimension of subspace. Third, it is inefficient when dealing with high-dimensional datasets. The first shortcoming, which is longstanding and undiscovered, makes the previous methods fail to achieve their expected results. The last two ones increase the difficulty of applying in different fields. Therefore, two unsupervised feature selection methods are proposed based on controllable adaptive graph learning and uncorrelated/independent feature learning (CAG-U and CAG-I) to address the abovementioned issues. In the proposed methods, the final graph that preserves intrinsic structure can be adaptively learned while the difference between the two graphs can be well controlled. Besides, relatively uncorrelated/independent features can be selected using a discrete projection matrix. The experimental results on 12 datasets in different fields show the superiority of CAG-U and CAG-I.

A general adaptive unsupervised feature selection with auto-weighting

Feature selection (FS) is essential in machine learning and data mining as it makes handling high-dimensional data more efficient and reliable. More attention has been paid to unsupervised feature selection (UFS) due to the extra resources required to obtain labels for data in the real world. Most of the existing embedded UFS utilize a sparse projection matrix for FS. However, this may introduce additional regularization terms, and it is difficult to control the sparsity of the projection matrix well. Moreover, such methods may seriously destroy the original feature structure in the embedding space. Instead, avoiding projecting the original data into the low-dimensional embedding space and identifying features directly from the raw features that perform well in the process of making the data show a distinct cluster structure is a feasible solution. Inspired by this, this paper proposes a model called A General Adaptive Unsupervised Feature Selection with Auto-weighting (GAWFS), which utilizes two techniques, non-negative matrix factorization, and adaptive graph learning, to simulate the process of dividing data into clusters, and identifies the features that are most discriminative in the clustering process by a feature weighting matrix Θ. Since the weighting matrix is sparse, it also plays the role of FS or a filter. Finally, experiments comparing GAWFS with several state-of-the-art UFS methods on synthetic datasets and real-world datasets are conducted, and the results demonstrate the superiority of the GAWFS.

The Analysis and Compensation Experiment of Linear Feed Axis Positioning Error Based on Matrix Operation

The positioning accuracy of the linear feed axis is a key factor affecting the machining accuracy of CNC machine tools. The generation process of the positioning error of the linear feed axis is expounded. The Z axis of the vertical linear feed axis of the machining center is the research object, the positioning error analytical matrix is established, and the positioning error calculation of the Z axis of the vertical linear feed axis is completed based on the principle of solving the inhomogeneous linear equation. The positioning error detection of the vertical linear feed axis Z axis of the CNC machining center is carried out. Based on the characteristics of the error detection data, an error compensation model is constructed. , the variance of the positioning error detection point after compensation is reduced to 1.562, which meets the needs of precision machining.

Cortical Alterations Associated with Executive Function Deficits in Youth with a Congenital Heart Defect

Abstract Adolescents and young adults born with a complex congenital heart defect (CHD) are at risk for executive function (ExF) impairments, which contribute to the psychological and everyday burden of CHD. Cortical dysmaturation has been well described in fetuses and neonates with CHD and early evidence suggests that cortical alterations in thickness, surface area, and gyrification index are non-transient and can be observed in adolescents with CHD. However, cortical alterations have yet to be correlated with ExF deficits in youth with CHD. This study aims to use a data-driven approach to identify the most important cortical features associated with ExF deficits in adolescents and young adults with CHD. To do so, we combined two comparable datasets acquired at the Research Institute of the McGill University Health Centre and the University Children’s Hospital Zurich each including both youth with CHD and healthy controls. For each participant, a high-resolution T1-weighted magnetic resonance image, a self-reported ExF assessment (the Behaviour Rating Inventory of Executive Function— Adult Scale), and their clinical and demographic characteristics were available. Corticometric Iterative Vertex-Based Estimation of Thickness (CIVET) was used to extract cortical thickness, cortical surface area, and local gyrification index measures. Using orthogonal projective non-negative matrix factorization (OPNMF), we identified non-overlapping spatial components that integrate cortical thickness, cortical surface area, and local gyrification index and capture structural covariance across these features. Behavioural partial least squares correlation (bPLS) analysis was then used to compute correlations between the individual variability in the OPNMF covariance patterns and ExF outcomes for each subject. A total of 56 youth with CHD who underwent cardiopulmonary bypass surgery before three years of age and 56 age- and sex-matched healthy controls were included in our analyses. Cortical grey matter volume, cortical thickness, and cortical surface area were found to be significantly reduced in CHD patients compared to controls. OPNMF identified 12 stable cortex-wide components summarizing the inter-subject variability in cortical thickness, cortical surface area, and local gyrification index. bPLS revealed two significant latent variables (LV) accounting for a total of 82.8% of the variance in the sample, each describing distinct patterns between the brain and cognitive data. LV1 summarized a pattern of belonging to the CHD group, worse scores on most BRIEF-A scales, younger age at MRI, and female sex. This pattern was associated with increased cortical thickness, local gyrification index, and decreased cortical surface area in several OPNMF components. Finally, we identified a positive relationship between the LV1 brain-behaviour pattern and total aortic cross-clamp time in the CHD group, indicating that longer aortic cross-clamp time was associated with worse neurodevelopmental outcomes. In this study, we uncover novel multivariate relationships between ExF and alterations in cortical thickness, surface area, and local gyrification index in adolescents and young adults with CHD using a data-driven approach. Although our findings highlight the important role played by the cortex in higher order cognitive processes, future studies are needed to elucidate the individual contribution of individual and clinical attributes into the deficits observed in this population.

A novel method for generating public keys involving matrix operations.

With the rise of the Internet of things, the limitations of traditional PKI certificate authentication technology have progressively emerged. Hence, identity-based public key algorithms have been proposed. In this paper, we propose a new approach to generate an identity key, named identity public key (IPK). IPK identity key generation protocol is based on SM2 elliptic curve cryptography and random matrix theory. It builds upon the concept of combined public key (CPK) and resolves the linear collusion issue of CPK as well as the authenticity verification problem of the declared public key of the simplified TF-CPK by enhancing the identity mapping approach. Another main aim of this paper is to prove the security of the IPK identity key generation protocol. Roughly speaking, we verify the security of each part of the private key and the security of the composite private key. We also increase the function and performance comparison with other schemes, such as IBC(SM9) and TF-CPK. Our scheme has the lowest computation cost, which demonstrates its rationality.

A novel hybrid method with convergence analysis for approximation of HTLV-I dynamics model

This paper presents a novel numerical approach for approximating the solution of the model describing the infection of CD4+T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$CD4^+T$$\\end{document}-cells by the human T-cell lymphotropic virus I (HTLV-I).The proposed method utilizes the operational matrix along with spectral method to convert the fractional model into a system of nonlinear algebraic equations. The Levenberg-Marquardt algorithm efficiently solves these equations. The study includes theoretical convergence analysis and error bounds to establish the validity of the proposed method. Through several test problems, we demonstrate the effectiveness and accuracy of the approach. We compare its performance and reliability to other existing methods in the literature. The results indicate that the proposed method is a reliable and efficient approach for solving the model.

ScatterX： A software for fast processing of high-throughput small-angle scattering data

Abstract Scattering experiments become increasingly popular in modern scientific research, including the areas of materials, biology, chemistry, physics, etc. Besides, various types of scattering facilities have been developed recently, such as lab-based X-ray scattering equipment, national synchrotron facilities and large neutron facilities. These above-mentioned trends bring up fast-increasing data amounts of scattering data, as well as different scattering types (X-ray, neutron, laser and even microwaves). This struggles with data analysis software of the last generation, where data are usually analyzed assuming certain first-principle models and given data formats. To help researchers process and analyze scattering data more efficiently, we developed a general and model-free scattering data analysis software based on matrix operation, which has the unique advantage of high throughput scattering data processing, analysis and visualization. To maximize generality and efficiency, data processing is performed based on a three-dimensional matrix, where scattering curves are saved as matrices or vectors, rather than the traditional definition of paired values. It can not only realize image batch processing, background subtraction and correction, but also analyze data according to scattering theory and model, such as radius of gyration, fractal dimension and other physical quantities. In the aspect of visualization, the software allows the modify the color maps of two-dimensional scattering images and the gradual color variation of one-dimensional curves to suit efficient data communications. In all, this new software can work as a standalone platform for researchers to process, analyze and visualize scattering data from different research facilities without considering different file types or formats. All codes in this manuscript are open-sourced and can be easily implemented in matrix-based software, such as MATLAB, Python and Igor.

An Error Analysis Study for the Distributed-Order Time-Fraction Model Using an Efficient Numerical Method

In this paper, we propose a numerical scheme based on the shifted Legendre polynomials for solving the forced Korteweg–de Vries (fKdV) equation including a Caputo fractional operator of a distributed order. To obtain numerical solutions of these types of equations, we derive an operational matrix based on the shifted Legendre polynomials, and using this operational matrix, their equations change to a set of nonlinear algebraic systems. Then, by calculating these systems in the collocation points, we solve systems. Also, convergence and error are investigated in this paper. Finally, several numerical examples to show the applicability of our scheme are displayed.