Components Of Eigenvectors Research Articles

Node Centrality Based on Edge Dynamics in a Chaotic Network

This paper introduces a novel node centrality index based on edge dynamics in a complex network: adjacent edge index. Through the analysis of network dynamics and topology, we provide a neat index to qualify node importance, which is calculated by the eigenvector components corresponding to the Fiedler value of the network. Notably, it identifies nodes whose adjacent edge sets play a significant role in network synchronization. The index indicates that not only a node’s degree but also its location should be considered. To demonstrate the effectiveness of the proposed index, consider a double-star network and a Jazz Musicians network in which chaotic Chua’s circuits as nodes are coupled. By factoring in the dynamic information exchange between nodes and their neighbors, this index offers valuable insights for optimizing strategies in engineering areas such as efficient information dissemination.

Investigation on Traffic Carbon Emission Factor Based on Sensitivity and Uncertainty Analysis

The premise for formulating effective emission control strategies is to accurately and reasonably evaluate the actual emission level of vehicles. Firstly, the active subspace method is applied to set up a low-dimensional model of the relationship between CO2 emission and multivariate vehicle driving data, in which the vehicle specific power (VSP) is identified as the most significant factor on the CO2 emission factor, followed by speed. Additionally, acceleration and exhaust temperature had the least impact. It is inferred that the changes in data sampling transform the establishment of subspace matrices, affecting the calculation of eigenvector components and the fitting of the final quadratic response surface, so that the emission sensitivity and final fitting accuracy are impressionable by the data distribution form. For the VSP, the best fitting result can be obtained when the VSP conforms to a uniform distribution. Moreover, the Bayesian linear regression method accounts for fitting parameters between the VSP and CO2 emission factor with uncertainties derived from heteroscedastic measurement errors, and the values and distributions of the intercept and slope α and β are obtained. In general, the high-resolution inventory of the carbon emission factor of the tested vehicle is set up via systematically analyzing it, which brings a bright view of data processing in further counting the carbon footprint.

Knowledge-Data-Based Synchronization States Analysis for Process Monitoring and Its Application to Hydrometallurgical Zinc Purification Process

Modern industrial processes generate many inter-associated variables, which are more likely to implicit associations knowledge for describing irregular changes at different times to accurately describe behaviour changes. Motivated by this issue, a novel knowledge-data-based synchronization states analysis method is proposed for process monitoring. Its advantage mainly refers to integrating physical-chemical mechanism knowledge to handle the representation of associated relationships between numerous monitor variables. Furthermore, this method utilizes the trend distributions of variable changes to observe the differences between operation states and their parents online, which can maintain the simple, practical, and efficient advantage of data-driven process monitoring. Specifically, global process monitoring can be achieved by the synchronization status exceeding its corresponding threshold ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\chi ^{2}$</tex-math></inline-formula> distribution). At the same time, the local cause of backtracking can also be identified by whether the weighting of eigenvector components of each variable exceeds their corresponding thresholds ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\chi ^{2}$</tex-math></inline-formula> distribution). This novel proposed process monitoring method is applied to one practical hydrometallurgical zinc purification process consisting of copper and cobalt removal processes. The application's comparable performance shows the applicability and effectiveness of this proposed method.

Looking into the Market Behaviors through the Lens of Correlations and Eigenvalues: An Investigation on the Chinese and US Markets Using RMT.

This research systematically analyzes the behaviors of correlations among stock prices and the eigenvalues for correlation matrices by utilizing random matrix theory (RMT) for Chinese and US stock markets. Results suggest that most eigenvalues of both markets fall within the predicted distribution intervals by RMT, whereas some larger eigenvalues fall beyond the noises and carry market information. The largest eigenvalue represents the market and is a good indicator for averaged correlations. Further, the average largest eigenvalue shows similar movement with the index for both markets. The analysis demonstrates the fraction of eigenvalues falling beyond the predicted interval, pinpointing major market switching points. It has identified that the average of eigenvector components corresponds to the largest eigenvalue switch with the market itself. The investigation on the second largest eigenvalue and its eigenvector suggests that the Chinese market is dominated by four industries whereas the US market contains three leading industries. The study later investigates how it changes before and after a market crash, revealing that the two markets behave differently, and a major market structure change is observed in the Chinese market but not in the US market. The results shed new light on mining hidden information from stock market data.

Extension of Judd-Ofelt theory: Application on Eu3+, Nd3+ and Er3+

We present a modified version of the Judd-Ofelt theory, which describes the intensities of f-f transitions for trivalent lanthanide ions (Ln3+) in solids. In our model, the properties of the dopant are calculated with well-established atomic-structure techniques, while the influence of the crystal-field potential is described as a perturbation, by three adjustable parameters. Compared to our previous work (Hovhannesyan et al., 2022 [28]), the spin-orbit interaction within the first excited configuration 4fw−15d is described in a perturbative way, whereas it is exactly taken into account in the ground configuration 4fw, using all the eigenvector components of the free-ion levels. Moreover, the wavelength-dependence of the refractive index of the host material is also accounted for. We test the validity of our model on three ions: Eu3+, Nd3+ and Er3+. The results of the extension are satisfactory, we are able to give a physical insight into all the transitions within the ground electronic configuration, and also to reproduce quantitatively experimental absorption oscillator strengths. We also performed calculations of standard JO parameters, and the results are in good agreement with the values reported in the literature. The code used to make the calculations is available on GitLab.

Spiked multiplicative random matrices and principal components

In this paper, we study the eigenvalues and eigenvectors of the spiked invariant multiplicative models when the randomness is from Haar matrices. We establish the limits of the outlier eigenvalues λ̂i and the generalized components (〈v,ûi〉 for any deterministic vector v) of the outlier eigenvectors ûi with optimal convergence rates. Moreover, we prove that the non-outlier eigenvalues stick with those of the unspiked matrices and the non-outlier eigenvectors are delocalized. The results also hold near the so-called BBP transition and for degenerate spikes. On one hand, our results can be regarded as a refinement of the counterparts of Belinschi et al. (2017) under additional regularity conditions. On the other hand, they can be viewed as an analog of Ding and Yang (2021) by replacing the random matrix with i.i.d. entries with Haar random matrix.

Lattice-dynamics study of Raman-active modes in LaTiO3

The lattice dynamics plays an important role in the determining the orbital and magnetic ordering phenomena in rare-earth (R) RTiO3 titanates. In the present study, the assignment and vibrational character of Ag and B2g-symmetry phonons observed in the Raman spectra of LaTiO3 crystal is verified by shell-model calculations of eigenfrequencies, relative intensities, and eigenvector components of the normal modes. The eigenvector pattern of the Ag mode located around the most pronounced central mode at 296cm−1 is turned out to be in full agreement with the La–O covalency in LaTiO3, where the concomitant GdFeO3-type distortion is decisive. The GdFeO3-type mode is located in the close proximity to the Jahn-Teller (JT) tetragonal E-type Qε mode. An interplay between the R–O covalency and JT effects can determine the subtle conditions for the electronic ground state of the Ti3+ ion and associated orbital and magnetic ordering phenomena in RTiO3 titanates.

Multiple-bipartite consensus for networked Lagrangian systems without using neighbours' velocity information in the directed graph

This paper investigates the multiple-bipartite consensus problem without considering neighbours' velocity information in networked Lagrangian systems (NLSs). A distributed adaptive control algorithm without using neighbours' velocity information is proposed, which facilitates the practical configuration deployments in the coopetition networks. By borrowing a subtle vector composed of the eigenvector components associated with zero eigenvalue of Laplacian matrix, a novel reference estimated vector is introduced to conduct the stability analysis step-by-step in the coopetition networks. Finally, simulations are provided to show the effectiveness of the proposed algorithm.

High spectral covariation between frequency channels contributes to clear speech intelligibility

Speech signals are acoustically redundant, which could explain why sentence intelligibility is fairly robust even when sentences are acoustically degraded. We investigated the contributions to sentence intelligibility of clear speech redundancy encoded as patterns of spectrotemporal covariation between frequency channels. Participants (N = 16) transcribed 120 clear-speech English sentences acoustically degraded to 5, 8, or 15 frequency bands derived from an ERB-scaled filter bank. Before the acoustic degradation, each sentence was expressed as a linear combination of principal component eigenvectors representing different patterns of covariation between channels. Half of the sentences preserved the channels providing larger score magnitudes for the eigenvector accounting for more spectral covariance (high-covariance condition). These channels represented the spectral covariation patterns that were more dominant in each sentence. The other half of the sentences preserved the bands conveying larger score magnitudes for the eigenvector accounting forless spectral covariance (low-covariance condition). These bands represented the spectral covariation patterns that were less dominant. Participants yielded significantly better transcription accuracy in the high-covariance condition (mixed-effects, ps &amp;lt; 0.0021). Critically, accuracy in this condition was higher than 56% on average for as few as 5 bands. These findings indicate that clear speech intelligibility is supported by patterns of spectral covariation between frequency bands.

Central limit theorem for the principal eigenvalue and eigenvector of Chung–Lu random graphs

A Chung–Lu random graph is an inhomogeneous Erdős–Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung–Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph.

Research on Stock Price Prediction Based on Orthogonal Gaussian Basis Function Expansion and Pearson Correlation Coefficient Weighted LSTM Neural Network

For stock price prediction in quantitative finance, deep learning techniques such as LSTM neural network do not need the stationarity assumption of traditional time series models (such as ARIMA and GARCH) and can forecast medium and long-term time series, so they have attracted much attention. This paper proposes an improved LSTM neural network based on orthogonal Gaussian basis function expansion and Pearson correlation coefficient weighting. The proposed method uses the functional features of intra-day prices to fit the residual series predicted by the LSTM neural network. Considering that the underlying model structure between each component of the function eigenvector and the residual series is unknown, we use the Bagging method to capture and trade off the variance and bias of the prediction model. In addition, since the dimension of the predictive variable of the LSTM neural network is a parameter to be estimated, we use the model averaging method based on Pearson correlation coefficient weighting for tuning. The results of actual data analysis show that the proposed method can significantly improve the prediction accuracy of the original LSTM neural network and has certain robustness. Finally, the proposed method can be further applied to consumer price index (CPI) prediction, daily average temperature prediction, and real-time monitoring of environmental trace elements.

The static potential using trial states from Laplacian eigenmodes

We compute the static potential of a quark-antiquark pair in lattice QCD using a method which is not based on Wilson loops, but where the trial states are formed by eigenvector components of the covariant lattice Laplace operator. The computational effort of this method is significantly lower than the standard Wilson loop calculation, when computing the static potential not only for on-axis, but also for many off-axis quark-antiquark separations, i.e., when a fine spatial resolution is required, e.g., for string breaking calculations. We further improve the signal by using multiple eigenvector pairs, weighted with Gaussian profile functions of the eigenvalues, providing a basis for a generalized eigenvalue problem (GEVP), as it was recently introduced to improve distillation in meson spectroscopy. We show results from the new method for the static potential with dynamical fermions and demonstrate its efficiency compared to traditional Wilson loop calculations.

Enhanced features in principal component analysis with spatial and temporal windows for damage identification

Principal component analysis (PCA) methods have been widely applied to damage identification in the long-term structural health monitoring (SHM) of infrastructure. Usually, the first few eigenvector components derived by PCA methods are treated as damage-sensitive features. In this paper, the effective method of double-window PCA (DWPCA) and novel features are proposed for better damage identification performance. In the proposed method, spatial and temporal windows are introduced to the traditional PCA method. The spatial windows are applied to group damage-sensitive sensors and exclude those sensors insensitive to damage, while the temporal window is applied to better discriminate eigenvectors between the damaged and healthy states. In addition, the length and directional angle of the eigenvector variation between the healthy and damaged states are used as the damage-sensitive features, instead of the components of the eigenvector variation used in previous studies. Numerical simulations based on a large-scale bridge reveal that the proposed features are successful in identifying the damage located far from sensors due to the use of both spatial and temporal windows as well as the length of the eigenvector variation. In addition, compared to the previous PCA and moving PCA methods, the novel features have higher sensitivity and resolution in damage identification.

A simple and efficient pseudo-inverse approximation for real symmetric matrices and applications to elasticity problems

This study proposes an efficient approximation method for the Moore–Penrose pseudo-inverse when, for a matrix, the eigenvectors associated to the eigenvalues zero (referred to herein as zeros eigenvectors) are known in advance. The method reduces the computational cost by several orders of magnitude. The approximation is performed by the addition of a small-amplitude diagonal matrix to regularise the matrix and multiplication with a projection matrix after its regular inversion. The projection removes the components of the zeros eigenvectors. The condition for obtaining a good approximation, is that the amplitude of the small-amplitude matrix should be sufficiently smaller than the smallest non-zero eigenvalue of the matrix. When the matrix is a stiffness matrix in a support-free elasticity problem (a problem whereby the elastic body is unsupported), the zeros eigenvectors indicate rigid-body motions. The method was applied to robust support-free topology optimization, revealing its excellent accuracy and efficiency. The observed computational time was found to be proportional to the size of the stiffness matrix. Furthermore, conducting robust topology optimization for fine-mesh problems resulted in structures that exhibited biological features.

Machine learning in eigensubspace for network path identification and flow forecast

This paper emphasizes the joint time-frequency interpretation of eigensubspace representation for network statistics as features for identification and tracking traffic flows based on the link level activity. Eigencoefficients (frequency domain feature) and eigenvector components (time domain feature) are jointly utilized to quantify their combined significance on the representation of each link data (each component of the link traffic vector) in the eigensubspace. The joint time-frequency method is employed to analyze the traffic data obtained from the Internet2 network. It is shown that the analysis with link-level resolution brings advantages for network traffic engineering applications. A machine learning method is investigated to identify network paths using eigenanalysis of link statistics as the feature set. The merit of the method is validated by experimental studies of the network scenarios considered in the paper. Eigenvectors and eigenflows in the subspace are jointly used as factors (features) for linear regression to forecast the network link traffic. It is demonstrated that the eigensubspace based auto-regressive order two, AR (2), predictor is superior to the time-domain based predictor to forecast the link level traffic of a network.

Coupling of Homotopy Perturbation Method and Kriging surrogate model for an efficient fuzzy linear buckling analysis: Application to additively manufactured lattice structures

This paper presents a new method to efficiently approximate both linear buckling loads and associated mode shapes of finite element structures subject to perturbations. To achieve this, a coupling between a Reduced Order Model (ROM) based on the Homotopy Perturbation Method (HPM) and a Kriging model is presented here. The ROM maintains the link between eigenvalues, related eigenvectors and the dependencies between each eigenvector components, leading to a high precision level. The computational time is greatly reduced by the surrogate model which avoids the computation of modified finite element matrices for each prediction. Next, the capabilities of the method allow to efficiently handle the prediction, sensitivity and optimization steps of an uncertain propagation problem using fuzzy formalism. Additive Manufacturing is a powerful and impressive process but many factors can be responsible for relatively large discrepancies in the mechanical and geometrical characteristics of the manufactured structure. Lastly, a study shows how the proposed fuzzy strategy allows the prediction of the buckling variability of a set of lattice structures.

Spiked separable covariance matrices and principal components

We study a class of separable sample covariance matrices of the form Q˜1:=A˜1/2XB˜X∗A˜1/2. Here, A˜ and B˜ are positive definite matrices whose spectrums consist of bulk spectrums plus several spikes, that is, larger eigenvalues that are separated from the bulks. Conceptually, we call Q˜1 a spiked separable covariance matrix model. On the one hand, this model includes the spiked covariance matrix as a special case with B˜=I. On the other hand, it allows for more general correlations of datasets. In particular, for spatio-temporal dataset, A˜ and B˜ represent the spatial and temporal correlations, respectively. In this paper, we study the outlier eigenvalues and eigenvectors, that is, the principal components, of the spiked separable covariance model Q˜1. We prove the convergence of the outlier eigenvalues λ˜i and the generalized components (i.e., ⟨v,ξ˜i⟩ for any deterministic vector v) of the outlier eigenvectors ξ˜i with optimal convergence rates. Moreover, we also prove the delocalization of the nonoutlier eigenvectors. We state our results in full generality, in the sense that they also hold near the so-called BBP transition and for degenerate outliers. Our results highlight both the similarity and difference between the spiked separable covariance matrix model and the spiked covariance matrix model in (Probab. Theory Related Fields 164 (2016) 459–552). In particular, we show that the spikes of both A˜ and B˜ will cause outliers of the eigenvalue spectrum, and the eigenvectors can help to select the outliers that correspond to the spikes of A˜ (or B˜).

Signatures of Chaos in Nonintegrable Models of Quantum Field Theories.

We study signatures of quantum chaos in (1+1)D quantum field theory (QFT) models. Our analysis is based on the method of Hamiltonian truncation, a numerical approach for the construction of low-energy spectra and eigenstates of QFTs that can be considered as perturbations of exactly solvable models. We focus on the double sine-Gordon, also studying the massive sine-Gordon and ϕ^{4} model, all of which are nonintegrable and can be studied by this method with sufficiently high precision from small to intermediate perturbation strength. We analyze the statistics of level spacings and of eigenvector components, which are expected to follow random matrix theory predictions. While level spacing statistics are close to the Gaussian orthogonal ensemble (GOE) as expected, on the contrary, the eigenvector components follow a distribution markedly different from the expected Gaussian. Unlike in the typical quantum chaos scenario, the transition of level spacing statistics to chaotic behavior takes place already in the perturbative regime. Moreover, the distribution of eigenvector components does not appear to change or approach Gaussian behavior, even for relatively large perturbations. Our results suggest that these features are independent of the choice of model and basis.

Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra

If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda _1(A),\dots ,$ $\lambda _n(A)$ and $i,j = 1,\dots ,n$, then the $j$th component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda _i(A)$ is related to the eigenvalues $\lambda _1(M_j),\dots ,$ $\lambda _{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j$th row and column by the formula \begin{equation*} |v_{i,j}|^2\prod _{k=1;k\neq i}^{n}\left (\lambda _i(A)-\lambda _k(A)\right )=\prod _{k=1}^{n-1}\left (\lambda _i(A)-\lambda _k(M_j)\right ). \end{equation*} We refer to this identity as the

Localization and Universality of Eigenvectors in Directed Random Graphs.

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree distribution. For delocalized eigenvectors, we recover in this limit the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.