Relative Eigenvalue Research Articles

Utilizing Principal Component Analysis and Hierarchical Clustering to Develop Driving Cycles: A Case Study in Zhenjiang

Accurate driving cycles are key for effectively evaluating electric vehicle performance. The K-means algorithm is widely used to construct driving cycles; however, this algorithm is sensitive to outliers, and determining the K value is difficult. In this paper, a novel driving cycle construction method based on principal component analysis and hierarchical clustering is proposed. Real road vehicle data were collected, denoised, and divided into vehicle microtrip data. The eigenvalues of the microtrips were extracted, and their dimensions were reduced through principal component analysis. Hierarchical clustering was then performed to classify the microtrips, and a representative set of microtrips was randomly selected to construct the driving cycle. The constructed driving cycle was verified and compared with a driving cycle constructed using K-means clustering and the New European Driving Cycle. The average relative eigenvalue error, maximum speed acceleration probability distribution difference rate, average cycle error, and simulated relative power consumption error per 100 km between the hierarchical driving cycle and the real road data were superior to those of the K-means driving cycle, which indicated the effectiveness of the proposed method. Though the methodology proposed in this paper has not been verified in other regions, it provided a certain reference value for other research of the developing driving cycle.

An effect size for comparing the strength of morphological integration across studies.

Understanding how and why phenotypic traits covary is a major interest in evolutionary biology. Biologists have long sought to characterize the extent of morphological integration in organisms, but comparing levels of integration for a set of traits across taxa has been hampered by the lack of a reliable summary measure and testing procedure. Here, we propose a standardized effect size for this purpose, calculated from the relative eigenvalue variance, . First, we evaluate several eigenvalue dispersion indices under various conditions, and show that only remains stable across samples size and the number of variables. We then demonstrate that accurately characterizes input patterns of covariation, so long as redundant dimensions are excluded from the calculations. However, we also show that the variance of the sampling distribution of depends on input levels of trait covariation, making unsuitable for direct comparisons. As a solution, we propose transforming to a standardized effect size (Z-score) for representing the magnitude of integration for a set of traits. We also propose a two-sample test for comparing the strength of integration between taxa, and show that this test displays appropriate statistical properties. We provide software for implementing the procedure, and an empirical example illustrates itsuse.

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fast real arithmetic algorithm and compare it with alternative published approaches.

Statistics of eigenvalue dispersion indices: Quantifying the magnitude of phenotypic integration.

Analysis of trait covariation plays a pivotal role in the study of phenotypic evolution. The magnitude of covariation is often quantified with statistics based on dispersion of eigenvalues of a covariance or correlation matrix-eigenvalue dispersion indices. This study clarifies the statistical justifications of these statistics and elaborates on their sampling properties. The relative eigenvalue variance of a covariance matrix is known in the statistical literature a test statistic for sphericity, and thus is an appropriate measure of eccentricity of variation. The same of a correlation matrix is equal to the average squared correlation, which has a straightforward interpretation as a measure of integration. Here, expressions for the mean and variance of these statistics are analytically derived under multivariate normality, clarifying the effects of sample size N, number of variables p, and parameters on sampling bias and error. Simulations confirm that approximations involved are reasonably accurate with a moderate sample size (N ≥ 16-64). Importantly, sampling properties of these indices are not adversely affected by a high p:N ratio, promising their utility in high-dimensional phenotypic analyses. They can furthermore be applied to shape variables and phylogenetically structured data with appropriate modifications.

Research on Ultrasonic Testing with Wavelet Packet Analysis for Shotcrete

Shotcrete structures are widely used in tunnel engineering. Quality inspection is difficult, and the traditional ultrasonic testing (UT) method based on first arrival velocity has limitations. In this paper, shotcrete-rock specimens were made in a laboratory and evaluated using UT. Wavelet packet decomposition is introduced for better frequency analysis of the condition evaluation. Two methods, including calculation of the energy eigenvalues and machine learning, are used to describe the contact quality at the interface between the shotcrete and rock. The relative energy eigenvalue increases with the gradual reduction of contact quality, which can become a quantitative index of the contact quality. Machine learning performed well in the rapid recognition of discontinuities in the multiple-classification models. Both methods based on wavelet packet decomposition achieved good results in identifying discontinuities and have the potential to be used in practical engineering applications.

Iterative Coordinate Reduction Algorithm of Flexible Multibody Dynamics Using a Posteriori Eigenvalue Error Estimation

Coordinate reduction has been widely used for efficient simulation of flexible multibody dynamics. To achieve the reduction of flexible bodies with reasonable accuracy, the appropriate number of dominant modes used for the reduction process must be selected. To handle this issue, an iterative coordinate reduction strategy is introduced. In the iteration step, more dominant modes of flexible bodies are selected than the ones in the previous step. Among the various methods, the conventional frequency cut-off rule is here considered. As a stop criterion, a novel a posteriori error estimator that can evaluate the relative eigenvalue error between full and reduced flexible bodies is proposed. Through the estimated relative eigenvalue error obtained, the number of dominant modes is automatically selected to satisfy the error tolerance up to the desired mode range. The applicability to the automation process is verified through numerical examples. It is also evaluated that efficient and accurate flexible multibody dynamics simulation is available with the reduced flexible body, generated by the proposed algorithm.

Measuring the magnitude of morphological integration: The effect of differences in morphometric representations and the inclusion of size.

The magnitude of morphological integration is a major aspect of multivariate evolution, providing a simple measure of the intensity of association between morphological traits. Studies concerned with morphological integration usually translate phenotypes into morphometric representations to quantify how different morphological elements covary. Geometric and classic morphometric representations translate biological form in different ways, raising the question if magnitudes of morphological integration estimates obtained from different morphometric representations are compatible. Here we sought to answer this question using the relative eigenvalue variance of the covariance matrix obtained for both geometric and classical representations of empirical and simulated datasets. We quantified the magnitude of morphological integration for both shape and form and compared results between representations. Furthermore, we compared integration values between shape and form to evaluate the effect of the inclusion or not of size on the quantification of the magnitude of morphological integration. Results show that the choice of morphological representation has significant impact on the integration magnitude estimate, either for shape or form. Despite this, ordination of the integration values within representations is relatively the same, allowing for similar conclusions to be reached using different methods. However, the inclusion of size in the dataset significantly changes the estimates of magnitude of morphological integration, hindering the comparison of this statistic obtained from different spaces. Morphometricians should be aware of these differences and must consider how biological hypothesis translate into predictions about integration in each particular choice ofrepresentation.

Study on a Simplified Structure of a Two-Stage Grid-Connected Photovoltaic System for Parameter Design Optimization

Conventional parameter designs of two-stage grid-connected photovoltaic (PV) system relied on its mathematical model of the cascade structure (CS), but the procedure is excessively cumbersome to implement. Besides, for a two-stage converter system, the coupling interaction between the power converters can directly lead to a poor parameter design. To overcome this drawback, this paper uses a simplified structure (SS) of single-phase two-stage grid-connected PV system to better design the parameters of the front-stage dc-dc converter. After establishing the small-signal model for SS and CS in the PV system, the relative eigenvalue sensitivity is used as the criterion for judging the influence of some parameters on the stability of the two structures. The stable boundary of MPPT control parameters is compared and discussed in SS and CS, respectively. In addition, the relationship between the front-stage dc-dc converter and the rear-stage dc-ac inverter is analyzed by the modal participation factor calculated in CS. An experiment is also performed at the end of this paper to further verify the feasibility of using SS to design the parameters of the dc-dc converter in the PV system.

Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

In this paper, when A and B are {1;1}-quasiseparable matrices, we consider the structured generalized relative eigenvalue condition numbers of the pair $$(A, \, B)$$ with respect to relative perturbations of the parameters defining A and B in the quasiseparable and the Givens-vector representations of these matrices. A general expression is derived for the condition number of the generalized eigenvalue problems of the pair $$(A,\, B)$$ , where A and B are any differentiable function of a vector of parameters with respect to perturbations of such parameters. Moreover, the explicit expressions of the corresponding structured condition numbers with respect to the quasiseparable and Givens-vector representation via tangents for $$\{1; 1\}$$ -quasiseparable matrices are derived. Our proposed condition numbers can be computed efficiently by utilizing the recursive structure of quasiseparable matrices. We investigate relationships between various condition numbers of structured generalized eigenvalue problem when A and B are {1;1}-quasiseparable matrices. Numerical results show that there are situations in which the unstructured condition number can be much larger than the structured ones.

Iterative Component Mode Synthesis Using a Priori and a Posteriori Criteria

An iterative algorithm of component mode synthesis for a rational reduced-order modeling is proposed. The present algorithm consists of three steps: initial mode selection with a priori criterion, model reduction using the selected modes, and a posteriori error evaluation. The algorithm is based on the Craig–Bampton method, which is one of the most widely used techniques. Specifically, the present algorithm uses, in a judicious manner, both an a priori mode selection criterion employing the moment-matching strategy and an a posteriori error estimation technique in evaluating the relative eigenvalue errors. If desired, an optional step of boundary interface reduction offers an additional possibility of obtaining a more compact model. The proposed iterative algorithm is recommended to researchers as well as commercialized software users in developing linear reduced-order structural dynamics models. Numerical examples are presented for its practical usage and performance.

Network community detection from the perspective of time series

We present a quasi-isometric mapping to transform complex networks into time series, which enables the network distance to be strictly preserved and allows to solve the network clustering problem from the perspective of its time series. In order to reconstruct the network distance characteristics exactly, we weight the network links in several ways and then convert the weighted networks into time series via classical multidimensional scaling (CMDS). Given such a transformation framework, we utilize the criterion of relative eigenvalue gap (REG) to estimate the number of communities of a network. Further, we enunciate that the distributions of two time series from two isomorphic networks are identical. We then apply the distance-based k-means algorithm to the generated time series to detect the community structures of complex networks with success. The results of diverse simulated and real networks demonstrate the superiority of quasi-isometry-based time series in network community detection.

Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure

We describe a method for exactly diagonalizing clean $D$-dimensional lattice systems of independent fermions subject to arbitrary boundary conditions in one direction, as well as systems composed of two bulks meeting at a planar interface. Our method builds on the generalized Bloch theorem [A. Alase et al., Phys. Rev. B 96, 195133 (2017)] and the fact that the bulk-boundary separation of the Schrodinger equation is compatible with a partial Fourier transform operation. Bulk equations may display unusual features because they are relative eigenvalue problems for non-Hermitian, bulk-projected Hamiltonians. Nonetheless, they admit a rich symmetry analysis that can simplify considerably the structure of energy eigenstates, often allowing a solution in fully analytical form. We illustrate our extension of the generalized Bloch theorem to multicomponent systems by determining the exact Andreev bound states for a simple SNS junction. We then analyze the Creutz ladder model, by way of a conceptual bridge from one to higher dimensions. Upon introducing a new Gaussian duality transformation that maps the Creutz ladder to a system of two Majorana chains, we show how the model provides a first example of a short-range chiral topological insulator hosting topological zero modes with a power-law profile. Additional applications include the complete analytical diagonalization of graphene ribbons with both zigzag-bearded and armchair boundary conditions, and the analytical determination of the edge modes in a chiral $p+ip$ two-dimensional topological superconductor. Lastly, we revisit the phenomenon of Majorana flat bands and anomalous bulk-boundary correspondence in a two-band gapless $s$-wave topological superconductor. We analyze the equilibrium Josephson response of the system, showing how the presence of Majorana flat bands implies a substantial enhancement in the $4\pi$-periodic supercurrent.

An interpolation-based parametric reduced order model combined with component mode synthesis

This paper presents an efficient interpolation-based parametric reduced order model with component mode synthesis. The off-line sampling of a large-scale structural dynamic system containing many parameters requires many computations to investigate the parameter-dependency of a dynamical system. Therefore, we introduce a dynamic substructuring scheme to execute off-line sampling in the subdomain level. In addition, we suggest discriminating the interpolation of the subsystem depending on the characteristics of each substructural mode. We synthesize the substructures, and we then construct a two-level, semi-parametrized ROM by reducing the degrees of freedom of the interface in the on-line stage. We then demonstrate the efficiency and accuracy of the present method by computing the relative eigenvalue errors and the transient responses of the system with random values for the parameters, and we conduct the design optimization of large-scale systems under dynamic loading conditions. A comparison of the present method with the full order model and a conventional reduced order model indicates that the present method is useful in carrying out an efficient analysis and design optimization for various large-scale structural dynamic systems.

Morphological integration and changes in pelvic dimensions with age in adult female humans

This study examines morphological integration (MI) in the human pelvis with respect to sex and age. Anatomists and biological anthropologists have often cited the flair of the iliac blades, sub‐pubic angle, and breadth between the ischia among the morphologies that distinguish females and males, though the magnitude of difference between the sexes varies among groups. Despite these differences, it is assumed that patterns of MI within the human pelvis are the same between the sexes; the few studies of evolvability and integration in the pelvis to date remove effects of sexual dimorphism on the variance‐covariance matrix. In addition, the age of adult individuals is never considered a factor in quantitative genetic studies of the evolution of the pelvis, as the pelvis is thought to reach its definitive morphology with the cessation of primary growth. However, studies by Tague (1994) and Auerbach et al. (in review) show that older adult females have mediolaterally wider pelvic outlets (MLPO) and anteroposteriorly deeper pelvic inlets (APPI), while no age‐related differences in pelvic dimensions are observed among males. Though covariation within the pelvis should be similar between the sexes and among age groups within the sexes, no assessment of overall integration within the pelvis has been compared in light of these factors.In this study, 18 linear dimensions were measured from the rearticulated pelves of 327 adult human skeletons (188 females, 139 males) recovered from archaeological sites in North America dating from the last millennium. Fusion of the iliac crest was the criterion for inclusion. We aged the skeletons based on diagnostic changes in pelvic articular morphology, and placed the skeletons into two groups: “Young” (<~25 years) and “Not Young” (>~25 years). MI was measured by comparing relative eigenvalue variances between sex and age‐and‐sex groups, following Pavličev et al. (2009. Evol Biol 36:157–170), as well as by comparing mean‐scaled covariance matrices of the traits within each group. Relative eigenvalue variances have a scale of zero to one and higher values indicate more integration.Results indicate that MI differs notably between age groups within sex, but not between the sexes. Both females and males have similar relative eigenvalues (females = 0.162; males = 0.172). The slightly lower value for females may be driven by the influence of the Young female group, which has a relative eigenvalue variance of 0.139. In contrast, the Not Young female group has relative eigenvalue variance of 0.202. This is a stark difference, and indicates that younger females have less integration than older females. Examination of the covariance matrices indicates lower covariances among most traits, but especially between MLPO and APPI. This indicates that covariation of traits within the pelvis continue to change in adults, and older females have more integrated pelves. Further developmental and evolutionary implications are considered.Support or Funding InformationNational Science Foundation, BCS Grant #0962752

On the computational efficiency of the error estimator for Guyan reduction

An improved error estimator for Guyan reduction is presented for efficient calculation of the relative eigenvalue error. In this work, the original error estimator is simply redefined in the level of the component matrix by neglecting its component matrices that do not affect the performance of estimation. Consequently, a new simple formulation of the error estimator is developed. Compared with the original formulation, it leads to better computational efficiency without significant loss of the estimating accuracy. The performance of the present error estimator is demonstrated theoretically and numerically.

Error estimation for the automated multi‐level substructuring method

SummaryIn this study, we propose an effective method to estimate the reliability of finite element models reduced by the automated multi‐level substructuring (AMLS) method. The proposed error estimation method can accurately predict relative eigenvalue errors in reduced finite element models. A new, enhanced transformation matrix for the AMLS method is derived from the original transformation matrix by properly considering the contribution of residual substructural modes. The enhanced transformation matrix is an important prerequisite to develop the error estimation method. Adopting the basic concept of the error estimation method recently developed for the Craig–Bampton method, an error estimation method is developed for the AMLS method. Through various numerical examples, we demonstrate the accuracy of the proposed error estimation method and explore its computational efficiency. Copyright © 2015 John Wiley & Sons, Ltd.

A simplified error estimator for the CB method and its application to error control

In this paper, we simplify the error estimation technique developed for the Craig–Bampton (CB) method (Kim et al., 2014). The original formulation is simplified by neglecting insignificant terms, a new error estimator is obtained, and thus computational cost is significantly reduced with negligible accuracy loss. In addition, the contribution of a specific substructure to a relative eigenvalue error can be estimated using the new formulation, in which the estimated relative eigenvalue error is represented by a simple summation of the substructural errors estimated. Therefore, the new formulation can be employed for error control by using the detailed errors estimated for a certain substructure. Through various numerical examples, we verify the accuracy and computational efficiency of the new formulation, and demonstrate an error control strategy.

Structured eigenvalue condition numbers for parameterized quasiseparable matrices

The development of fast algorithms for performing computations with $$n\times n$$n×n low-rank structured matrices has been a very active area of research during the last two decades, as a consequence of the numerous applications where these matrices arise. The key ideas behind these fast algorithms are that low-rank structured matrices can be described in terms of O(n) parameters and that these algorithms operate on the parameters instead on the matrix entries. Therefore, the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer, since this determines the maximum accuracy of a given fast computation. In other words, it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low-rank structured matrices, but, as far as we know, this has not yet been accomplished in any case. In this paper, we derive structured relative eigenvalue condition numbers for the important class of low-rank structured matrices known as $$\{1;1\}$${1ź1}-quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the Givens-vector representations of these matrices, and we provide fast algorithms for computing them. Comparisons among the new structured condition numbers and the unstructured one are also presented, as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge. In addition, the approach presented in this paper is general and may be extended to other problems and classes of low-rank structured matrices.

Estimating relative eigenvalue errors in the Craig-Bampton method

In this study, we propose an accurate error estimator for the Craig-Bampton (CB) method, which is a widely used component mode synthesis (CMS) method. The proposed error estimator can precisely predict relative eigenvalue errors in finite element models reduced by the CB method. To develop the error estimator, we propose an enhanced transformation matrix for the CB method and, using the transformation matrix, the error estimator is derived from the global (original) eigenvalue problem. In this paper, we demonstrate the robustness of the proposed error estimator through various numerical examples.

An accurate error estimator for Guyan reduction

The objective of this study is to develop an error estimator that accurately predicts relative eigenvalue errors in finite element models reduced by Guyan reduction. We present a derivation procedure for the error estimator, in which Kidder’s transformation matrix for Guyan reduction is employed. We demonstrate the excellent performance of the proposed error estimator through various numerical examples: rectangular plate, cylindrical panel, hyperboloid panel, and shaft–shaft interaction problems.