Eigenvalue Decomposition Research Articles

A dynamical systems analysis of change in PTSD symptoms, depression symptoms, and suicidal ideation among military personnel during treatment for PTSD

ObjectiveThe connections among posttraumatic stress disorder (PTSD), depression, and suicidal ideation are elusive because of an overreliance on cross-sectional studies. In this secondary analysis of pooled data from three clinical trials of 742 military personnel, we examined the dynamic relationships among PTSD, depression, and suicidal ideation severity assessed repeatedly during and after outpatient treatment for PTSD. MethodsWe conducted dynamical systems analyses to explore the potential for coordinated change over time in psychotherapy for PTSD. ResultsOver the course of psychotherapy, PTSD, depression, and suicidal ideation severity changed in coordinated ways, consistent with an interdependent network. Results of eigenvalue decomposition analysis indicated the dominant change dynamic involved high stability and resistance to change but indicators of cycling were also observed, indicating participants “switched” between states that resisted change and states that promoted change. Depression (B = 0.48, SE = 0.11) and suicidal desire (B = 0.15, SE = 0.01) at a given assessment were associated with greater change in PTSD symptom severity at the next assessment. Suicidal desire (B = 0.001, SE < 0.001) at a given assessment was associated with greater change in depression symptom severity at the next assessment. Neither PTSD (B = −0.004, SE = 0.007) nor depression symptom severity (B = 0.000, SE = 0.001) was associated with subsequent change in suicidal ideation severity. ConclusionsIn a sample of treatment-seeking military personnel with PTSD, change in suicidal ideation and depression may precede change in PTSD symptoms but change in suicidal ideation was not preceded by change in PTSD or depression symptoms.

An extended Merola–Ragnisco–Tu lattice integrable hierarchy and infinite conservation laws

By means of the zero-curvature equation and Lenard recursive operators, a novel differential-difference integrable hierarchy is derived, which is related to a discrete 3 × 3 matrix spectral problem with five potentials. The first nontrivial member in this hierarchy under specific reduction is the Merola–Ragnisco–Tu lattice equation whose continuum limit is the nonlinear Schrödinger equation. The infinite conservation laws of the first two nontrivial members in the hierarchy, namely, the extended Merola–Ragnisco–Tu lattice equation and its high-order generalization are constructed with the help of two Ricatti-type equations and the spectral parameter expansions.

A fast algorithm to factorize high-dimensional tensor product matrices used in genetic models.

Many genetic models (including models for epistatic effects as well as genetic-by-environment) involve covariance structures that are Hadamard products of lower rank matrices. Implementing these models requires factorizing large Hadamard product matrices. The available algorithms for factorization do not scale well for big data, making the use of some of these models not feasible with large sample sizes. Here, based on properties of Hadamard products and (related) Kronecker products, we propose an algorithm that produces an approximate decomposition that is orders of magnitude faster than the standard eigenvalue decomposition. In this article, we describe the algorithm, show how it can be used to factorize large Hadamard product matrices, present benchmarks, and illustrate the use of the method by presenting an analysis of data from the northern testing locations of the G × E project from the Genomes to Fields Initiative (n ∼ 60,000). We implemented the proposed algorithm in the open-source "tensorEVD" R package.

A Geometric Algorithm for Contrastive Principal Component Analysis in High Dimension

Principal component analysis (PCA) has been widely used in exploratory data analysis. Contrastive PCA (Abid et al.), a generalized method of PCA, is a new tool used to capture features of a target dataset relative to a background dataset while preserving the maximum amount of information contained in the data. With high dimensional data, contrastive PCA becomes impractical due to its high computational requirement of forming the contrastive covariance matrix and associated eigenvalue decomposition for extracting leading components. In this article, we propose a geometric curvilinear-search method to solve this problem and provide a convergence analysis. Our approach offers significant computational efficiencies. Specifically, it reduces the time complexity from O ( ( n ∨ m ) p 2 ) to a more manageable O ( ( n ∨ m ) p r ) , where n, m are the sample sizes of the target data and background data, respectively, p is the data dimension and r is the number of leading components. Additionally, we streamline the space complexity from O ( p 2 ) , necessary for storing the contrastive covariance matrix, to a more economical O ( ( n ∨ m ) p ) , sufficient for storing the data alone. Numerical examples are presented to show the merits of the proposed algorithm. Supplementary materials for this article are available online.

A new polarimetric phase optimization method based on eigenvalue decomposition and adaptive filtering

A new polarimetric phase optimization method based on eigenvalue decomposition and adaptive filtering

Monitoring land subsidence through improved CAESAR algorithm in time-series InSAR processing

Time-series SAR interferometry (InSAR) combining permanent scatterer and distributed scatterer (DS), has been strongly developed in subsidence monitoring. It is known that the Component extrAction and sElection SAR (CAESAR) is a recently presented approach of selecting and filtering scattering mechanisms for DS. This article proposes an improved CAESAR algorithm in InSAR processing. Phase optimization is performed by eigenvalue decomposition and principal component analysis of the coherence matrix that constructed based on the identified homogeneous pixels. In addition, only the interferometric phases with low noise are used to calculate the goodness-of-fit value. The improved method has been tested for subsidence monitoring over a nonurban area located in Xiongxian, China using 25 Sentinel-1A images. The results show that the improved method can provide the high spatial density of deformation measurements with accuracy ensured. Noticeable land subsidence is revealed widely within the north of Xiongxian county, particularly in Daying, Mijiawun and Beishakou towns.

Mordo2: A Personalization Framework for Silent Command Recognition.

Wearable human-computer interactions in daily life are increasingly encouraged by the prevalence of intelligent wearables. It poses a demanding requirement of micro-interaction and minimizing social awkwardness. Our previous work demonstrated the feasibility of recognizing silent commands through around-ear biosensors with the limitation of user adaptation. In this work, we ease the limitation by a personalization framework that integrates spectral factorization of signals, temporal confidence rejection and commonly used transfer learning algorithms. Specifically, we first empirically formulate the user adaptation issue by presenting the accuracies of applying transfer learning algorithms to our previous method. Second, we improve the signal-to-noise ratio by proposing the supervised spectral factorization method that learns the amplitude and phase mappings between around-ear signals and the signals of articulated facial muscles. Third, we leverage the time continuity of commands and introduce the time decay into confidence rejection. Finally, extensive experiments have been conducted to evaluate the feasibility and improvements. The results indicate an average accuracy of 92.38% which is significantly larger than solely using transfer learning algorithms. And a comparable accuracy can be achieved with significantly reduced data of new users. The overall performance shows the framework can significantly improve the accuracy of user adaptations. The work would aid a further step toward commercial products for silent command recognition and inspire the solution to the user adaptation challenge of wearable human-computer interactions.

Further Properties of an Operator Commuting with an Injective Quasi-Nilpotent Operator

In (Aiena et al., Math. Proc. R. Irish Acad. 122A(2):101–116, 2022), it has been shown that a bounded linear operator T∈L(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\in L(X)$$\\end{document}, defined on an infinite-dimensional complex Banach space X, for which there exists an injective quasi-nilpotent operator that commutes with it, has a very special structure of the spectrum. In this paper, we show that we have much more: if a such quasi-nilpotent operator does exist, then some of the spectra of T originating from B-Fredholm theory coalesce. Further, the spectral mapping theorem holds for all the B-Weyl spectra. Finally, the generalized version of Weyl type theorems hold for T assuming that T is of polaroid type. Our results apply to the operators that belong to the commutant of Volterra operators.

Graph Convolutional Neural Networks With Diverse Negative Samples via Decomposed Determinant Point Processes.

Graph convolutional neural networks (GCNs) have achieved great success in graph representation learning by extracting high-level features from nodes and their topology. Since GCNs generally follow a message-passing mechanism, each node aggregates information from its first-order neighbor to update its representation. As a result, the representations of nodes with edges between them should be positively correlated and thus can be considered positive samples. However, there are more non-neighbor nodes in the whole graph, which provide diverse and useful information for the representation update. Two non-adjacent nodes usually have different representations, which can be seen as negative samples. Besides the node representations, the structural information of the graph is also crucial for learning. In this article, we used quality-diversity decomposition in determinant point processes (DPPs) to obtain diverse negative samples. When defining a distribution on diverse subsets of all non-neighboring nodes, we incorporate both graph structure information and node representations. Since the DPP sampling process requires matrix eigenvalue decomposition, we propose a new shortest-path-base method to improve computational efficiency. Finally, we incorporate the obtained negative samples into the graph convolution operation. The ideas are evaluated empirically in experiments on node classification tasks. These experiments show that the newly proposed methods not only improve the overall performance of standard representation learning but also significantly alleviate over-smoothing problems.

A Novel Normalized-Cut Solver With Nearest Neighbor Hierarchical Initialization.

Normalized-Cut (N-Cut) is a famous model of spectral clustering. The traditional N-Cut solvers are two-stage: 1) calculating the continuous spectral embedding of normalized Laplacian matrix; 2) discretization via K-means or spectral rotation. However, this paradigm brings two vital problems: 1) two-stage methods solve a relaxed version of the original problem, so they cannot obtain good solutions for the original N-Cut problem; 2) solving the relaxed problem requires eigenvalue decomposition, which has O(n3) time complexity ( n is the number of nodes). To address the problems, we propose a novel N-Cut solver designed based on the famous coordinate descent method. Since the vanilla coordinate descent method also has O(n3) time complexity, we design various accelerating strategies to reduce the time complexity to O(|E|) ( |E| is the number of edges). To avoid reliance on random initialization which brings uncertainties to clustering, we propose an efficient initialization method that gives deterministic outputs. Extensive experiments on several benchmark datasets demonstrate that the proposed solver can obtain larger objective values of N-Cut, meanwhile achieving better clustering performance compared to traditional solvers.

Bayesian Spectral Decomposition for Efficient Modal Identification Using Ambient Vibration

Modal parameter identification via ambient vibration is popular but faces challenges from uncertainties due to unknown inputs and low signal‐to‐noise ratio. Bayesian methods are gaining increasing attention for operational modal identification due to their ability to quantify uncertainties. However, improvements in computational efficiency are needed, particularly when addressing numerous modes and degrees of freedom. To address this challenge, this study proposes an innovative approach, termed the “Bayesian spectral decomposition” method (BSD), employing the decompose‐and‐conquer strategy. This novel method, operating within the frequency domain, identifies each mode individually by exploiting their inherent separated modal characteristics. For each mode, the response spectrum matrix undergoes an eigenvalue decomposition, yielding crucial eigenvalues (incorporating frequency and damping information) and eigenvectors (containing mode shape information). Subsequently, statistical properties of the eigenvalues and eigenvectors are utilized to establish likelihood functions for Bayesian parameter identification. By combining prior information, the posterior probability distribution functions of modal parameters are derived. The optimal solution is then obtained by resolving the maximum posterior probability distribution function problem. To further quantify the uncertainty of modal parameters, Gaussian distributions are employed to approximate the posterior probability distribution functions. The adoption of the decomposition approach circumvents the joint identification of all modal parameters, substantially reducing the parameter dimensions for optimization. Consequently, this strategy leads to decreased computational complexity and significantly improved computational stability. The effectiveness of the BSD is confirmed through simulated data generated from an 8‐story shear building as well as measured data collected from both an experimental shear frame and the Canton Tower. The results demonstrate that the proposed method achieves high accuracy in identifying modal parameters, greatly improves computational efficiency, and effectively quantifies the uncertainties in modal parameters.

Efficient Approximations for Matrix-Based Rényi's Entropy on Sequential Data.

The matrix-based Rényi's entropy (MBRE) has recently been introduced as a substitute for the original Rényi's entropy that could be directly obtained from data samples, avoiding the expensive intermediate step of density estimation. Despite its remarkable success in a broad of information-related tasks, the computational cost of MBRE, however, becomes a bottleneck for large-scale applications. The challenge, when facing sequential data, is further amplified due to the requirement of large-scale eigenvalue decomposition on multiple dense kernel matrices constructed by sliding windows in the region of interest, resulting in O(mn3) overall time complexity, where m and n denote the number and the size of windows, respectively. To overcome this issue, we adopt the static MBRE estimator together with a variance reduction criterion to develop randomized approximations for the target entropy, leading to high accuracy with substantially lower query complexity by utilizing the historical estimation results. Specifically, assuming that the changes of adjacent sliding windows are bounded by β << 1 , which is a trivial case in domains, e.g., time-series analysis, we lower the complexity by a factor of √{β} . Polynomial approximation techniques are further adopted to support arbitrary α orders. In general, our algorithms achieve O(mn2√{β}st) total computational complexity, where s, t << n denote the number of vector queries and the polynomial degrees, respectively. Theoretical upper and lower bounds are established in terms of the convergence rate for both s and t , and large-scale experiments on both simulation and real-world data are conducted to validate the effectiveness of our algorithms. The results show that our methods achieve promising speedup with only a trivial loss in performance.

CONDITIONS FOR EQUICONVERGENCE OF EXPANSION IN EIGENFUNCTIONS OF ONE SCHRÖDINGER OPERATOR WITH GENERALIZED POTENTIAL

Abstract. The paper considers the Schrödinger operator with a generalized potential. Conditions have been found under which the operator under consideration has neither a discrete spectrum nor spectral singularities. A theorem on the equiconvergence of the spectral expansion in solutions of the scattering theory problem for the operator under consideration is proved.

Multi server queuing model with dynamic power shifting and performance efficiency factor

The need for high performance models in the queuing systems; availability in the fields of computing and communication systems and logistics management poses extensive challenges in design and development of the appropriate modeling. In this work, we propose a robust probabilistic model to mitigate these problems by appropriately choosing a multi-server queuing system with dynamic service facility. The arrival substances, such as packets or jobs or customers or consumers, follow a Poisson arrival process and these arrivals enter a queuing system according to FIFO discipline. The system designed in the system is armed with a fixed number of service stations, in which some servers are capable of allocating additional service capacity by adjusting dynamically. When a customer arrives at the system, if a free server is available, it is immediately served by one of the free processors; if no server is free, that is all are busy, the arrived job is accommodated in the queue and waits for service in the system. In this paper, we proposed a stochastic model to handle peak loads efficiently, boosting service capacity during burst arrivals. Numerical results presented in this paper are generated by the spectral expansion method to demonstrate the model’s performance, offering insights into its efficiency and accuracy. Furthermore, we derived some important special cases of speedup factor, which provide the mathematical estimation of system performance in terms of computation and communication times.

Spectral Expansion for Conformable Fractional Sturm- Liouville Problem on the Whole Line

In this article, we discuss a conformable fractional Sturm-Liouville boundary-value problem on the whole line. We prove the existence of a spectral function for the singular conformable fractional Sturm-Lioville problem. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function for conformable fractional Sturm-Liouville problem on the whole line.

Periodicity and chaos of thermal convective flows in annular cylindrical domains using the method of isolation by spectral expansions

The dynamical and chaotic behaviors of natural convection flow in the semi-annular and full-annular cylindrical domains are studied using the method of isolation by spectral expansion. For each of the flow cases, spectral models with different orders are obtained and dynamical characteristics of the flow are investigated using numerical simulations. For the semi-annular case of study, results reveal that in all the generated spectral models, chaotic solution appears, but the onset of chaos between each system requires different set of control parameters. Bifurcation diagrams are provided for all the systems which show the ranges of periodic and chaotic behavior of the flow. Strange attractors are captured for all the systems with different orders. For the lowest-order model, a Lorenz-like attractor is produced, which has two distinguishable scrolls similar to Lorenz's famous attractor and for the other high-order models, the shape of strange attractor gets different. For the full-annular case of study, the physical characteristics of the flow are obtained. The results show that the produced thermal plume on the top of annulus gets higher fluctuations as the flow Rayleigh number gets higher. At a specific Rayleigh number, the flow behavior gets completely unstable, which shows a transitional regime change. At this state of flow, the domain is dominated by several vortices.

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.

RGBW LED mixing temperature compensation method with high output consistency

This paper proposes a temperature compensation method for RGBW LED mixing based on fast non-dominated sorting genetic algorithm(NSGA-II). The proposed method can achieve compensation for temperature-induced changes in LED correlated color temperature (CCT), color fidelity (Rf) and color gamut index (Rg) by predicting the spectral power distribution (SPD) at different temperatures. The SPD-temperature model of the RGBW LED light source is established by measuring the spectral power distribution of LEDs at different temperatures. Based on the established model, NSGA-II is used to compensate the mixing temperature of the RGBW LED light source according to the spectral superposition theorem. The experimental results show that the fit of the established temperature-spectral model is R2 > 0.98, and the deviation of the compensated mixing results from the initial state of the light source is less than 10 K in CCT; the deviation value of Rf is less than 4 % in the range of 2000 K-7000 K, and less than 2.15 % in the range of 3000 K-7000 K; and the deviation value of Rg in the range of 2000 K-7000 K is less than 4.46 %.

Analytic prediction of the effective reaction rate for methane cracking in molten catalysts: Transition from kinetics-dominated to diffusion-limited regimes

Molten catalytic media are increasingly attracting interest as a suitable reaction environment where cracking of hydrocarbons and separation of the solid carbon product can be achieved simultaneously, ideally avoiding catalyst regeneration. In conditions relevant to industrial production, multiscale approaches are needed to predict the reactor performance, where the macroscopic dynamics of two-phase flow is combined with the evolution of the reaction inside individual bubbles. We here develop a thermodynamically-consistent, analytically solvable model for predicting the effective reaction rate resulting from the interplay between diffusive transport within the bubble and the surface reaction at the gas/liquid interface. The analytical solution is based on a spectral (eigenvalue/eigenfunction) expansion of the concentration profiles. We show how in practically affordable conditions, the effective characteristic time of the reaction can be estimated based only on the dominant eigenvalue of the diffusion-reaction problem. The closed-form knowledge of the dominant eigenvalue provides key insight in establishing the range of existence of diffusion-limited, mixed, and kinetics-controlled regimes. Based on the limiting solutions of the model, an approximate model for the conversion, yielding the explicit dependence of conversion on the equilibrium, kinetics and transport parameters is derived. The approach proposed is validated by comparing its predictions with the solution of the fully-coupled mass/momentum local balance equations obtained through a finite-element commercial solver enforced on a moving boundary domain to account for the bubble expansion due to the increasing number of moles in the gas phase.

PRINCIPAL COMPONENTS VERSUS AUTOENCODERS FOR DIMENSIONALITY REDUCTION: A CASE OF SUPER-RESOLVED OUTPUTS FROM PRISMA HYPERSPECTRAL MISSION DATA

Abstract. This study attempts to solve these issues associated with hyperspectral (HS) data, i.e., coarse spatial resolution and high volume, by understanding the effect of deep learning and traditional dimensionality reduction on super-resolved products generated from the recently launched PRecursore IperSpettrale della Missione Applicativa (PRISMA) HS mission. Four single-frame super-resolution (SR) algorithms have been used to super-resolve a 30 m PRISMA scene of Ahmedabad, India and generate 15 m spatial resolution images with both spatial and spectral fidelity. Iterative back projection (IBP) and sparse representation (SIS) are the best and worst-performing SR algorithms following a comparative assessment and validation protocol. Next, denoising autoencoders and PCT computed using singular and eigenvalue decompositions have been executed on the original PRISMA, IBP and SIS-based super-resolved datasets. The resulting low-dimensional representations have been assessed to preserve the original dataset's topology using label-independent Lee and Verleysen's co-ranking matrix and loss of quality measure. Findings suggest that autoencoders are computationally expensive and require a higher neighbourhood size than PCT and its variants to produce a high-quality encoding. These insights remain significant for urban information extraction as there are few direct comparative assessments between machine learning-based linear and non-linear data compression methods in earlier studies.