Eigenvector Method Research Articles

Significance of the order of pair‐wise comparisons in Analytic Hierarchy Process: an experimental study

AbstractThe article describes an approach, ensuring higher credibility of expert estimation results, based on specific order of pair‐wise comparisons. The order of pair‐wise comparisons is, in its turn, based on the distance between estimated objects in the ranking. According to the suggested approach (and some human psychophysiological features), the most ordinally distant objects should be compared before ordinally closer ones. In order to empirically confirm this assumption, a special experiment involving real experts has been conducted. The results of the experiment indicate that if objects are presented to the expert for comparison in the suggested order, then in the majority of cases relative weights of objects, obtained using eigenvector method, most adequately reflect this expert's priorities. Moreover, pair‐wise comparison matrices constructed using the suggested comparison order tend to be slightly more consistent. The suggested approach to re‐ordering of pair‐wise comparisons can be applied as part of the AHP algorithm in weakly structured subject domains, influenced by multiple intangible criteria. It also provides conceptual basis for reduction of the number of pair‐wise comparisons, required to obtain credible results, in AHP without loss or distortion of expert data. It can also be used for modification of combinatorial pair‐wise comparison aggregation method, based on spanning tree enumeration. And, finally, it will improve the overall multi‐criteria decision‐making process in diverse subject domains, characterized by high uncertainty levels.

Augmenting input-output system under the curse of dimensionality: AHP-RAS integration and algorithm simulation

ABSTRACT In the estimation of input-output tables, one frequently encountered challenge is the aggregation of sectors in initial tables, which complicates the estimation of more detailed tables due to missing data—a phenomenon referred to as the “curse of dimensionality” when augmenting from lower to higher dimensions. We propose an AHP-RAS method to address the substantial data deficiencies encountered in the expansion of input-output tables. The method applies the Analytical Hierarchy Process (AHP) to transform the subjective relative importance of sectors into input coefficients using Pairwise Comparison Matrices (PCM) and eigenvector methods. Subsequently, it incorporates the RAS approach to supplement objective data, thereby integrating subjective judgement and objective information to tackle the curse of dimensionality. Unlike traditional AHP methods, this study employs binary integer programming to decompose large-scale matrices into subsets, making the AHP technique applicable to input-output tables with a broader range of sectors. We employ Monte Carlo simulation techniques to evaluate the robustness of the proposed methodology, specifically assessing the impacts of subjective judgement, objective data utilization, and the variation in sector numbers. Overall, the method overcomes the curse of dimensionality, yielding coefficients within an acceptable error margin, thereby demonstrating a significant advancement in the field of input-output analysis.

Controlled generation of OAM-switchable vortex beams at 639 nm from a Pr3+: YLF laser

In this paper, we reported and demonstrated a blue LD-pumped Pr3+: YLF laser for the controlled generation of orbital angular momentum (OAM) switchable vortex beams at 639 nm. To begin with, the quantitative tilt imbalance of the resonator was introduced to select the higher-order Hermite-Gaussian (HG) modes by utilizing the eigenvector method and introducing the concept of mode discriminant factor. Moreover, the mode converter, composed of two cylindrical mirrors, was applied to convert the Hermite- Gaussian (HG) modes into the Laguerre-Gaussian (LG) modes with the OAM. Then, the experiment setup was built to validate the correctness and effectiveness of the quantitative tilt imbalance. The results showed that the vortex beams with OAM of 0 to ±14 at 639 nm can be obtained by adjusting the tilted angle of the output mirror, which agreed well with the simulation results, and the slope efficiencies of the vortex beams with OAM = 1, 2, and 3 are 11.64%, 8.01%, and 6.05%, respectively. The method provided a viable solution for the generation of visible vortex beams with switchable OAM from the Pr3+: YLF laser.

Robust Tensor-Based DOA and Polarization Estimation in Conformal Polarization Sensitive Array with Bad Data.

Partially impaired sensor arrays pose a significant challenge in accurately estimating signal parameters. The occurrence of bad data is highly probable, resulting in random loss of source information and substantial performance degradation in parameter estimation. In this paper, a tensor variational sparse Bayesian learning (TVSBL) method is proposed for the estimate of direction of arrival (DOA) and polarization parameters jointly based on a conformal polarization sensitive array (CPSA), taking into account scenarios with the partially impaired sensor array. First, a sparse tensor-based received data model is developed for CPSAs that incorporates bad data. Then, a column vector detection method is proposed to diagnose the positions of the impaired sensors. In scenarios involving partially impaired sensor arrays, a low-rank matrix completion method is employed to recover the random loss of signal information. Finally, variational sparse Bayesian learning (VSBL) and minimum eigenvector methods are utilized sequentially to obtain the DOA and polarization parameters estimation, successively. Furthermore, the Cramér-Rao bound is given for the proposed method. Simulation results validated the effectiveness of the proposed method.

Estimating accurate and precise strain in low-magnitude penetratively deformed rocks: Integrating forward strain modeling and natural data

Estimating grain-scale strain using commonly used strain methods from a low-magnitude penetratively deformed rock is typically fraught with uncertainties as the measured strain is influenced by the pre-deformed grain shapes and grain-scale strain partitioning. Therefore, developing kinematic and mechanical models of orogenic wedge evolution based on such strain data can be misleading without knowing how accurate and precise those data are. To determine accuracy we created five 2-D synthetic rock images and deformed them with known strain under pure shear, simple shear, and general shear. We generated 4160 strain ratio (Rs) and long-axis orientation (φ) data. To determine precision we calculated 2σ (95%) uncertainties in Rs and φ from 102 quartz-rich rocks in Sikkim and generated 977 uncertainty data. Results show that the shape matrix eigenvector and stereographic projection methods record the lowest root mean square error, mean absolute error, and 2σ uncertainty values suggesting they are the most accurate and precise, regardless of strain ratio and deformation types. However, the accuracy and precision of φ deteriorate at Rs < 1.5. For most accurate and precise results, using the above mentioned methods, we recommend a minimum of 100 and 75 markers for Rs < 1.5 and Rs > 1.5, respectively.

Static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix

A higher-order shear deformable model is extended in this paper for static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix. Principle of virtual work is used to derive governing equations of static bending through the computation of strain energy and external work. A higher-order shear deformable model is used to develop the kinematic relations. The sandwich cylindrical panel is subjected to mechanical and thermal loads. The governing equations are solved using Eigenvalue–Eigenvector method for clamped–clamped boundary conditions. The eigenvalues of the characteristic equation are obtained, and the solution is obtained for real and complex roots. The boundary conditions are applied to obtain integration constants. The axial and radial variation of the displacement, strain, and stress components are presented in terms of significant input parameters such as volume fraction, and folding angle of the graphene origami, various distribution patterns, porosity coefficient, thermal and mechanical loads, and geometric parameters. The results indicate that the stress, strain, and deformation components are increased with an increase in porosity coefficient and folding angle of the graphene. Furthermore, an enhancement in the volume fraction of the graphene leads to a significant decrease in various stress, strain, and deformation components. The results of this work can be used in aerospace science and technology specially in airplanes.

F-EVM: improving routing in Internet of Things using fuzzy MAIRCA approach and fuzzy eigenvector method

F-EVM: improving routing in Internet of Things using fuzzy MAIRCA approach and fuzzy eigenvector method

Elastic analysis of functionally graded porous cylinders under uniform internal pressure by using shear deformation theory

This research aims to analyze cylindrical shells made of functionally graded porous materials, focusing on three types of monotonous, symmetric, and nonsymmetric porosity distributions. Material properties are defined for different porosity distributions, individually. Due to the slight variation of Poisson’s ratio, it is assumed to be constant throughout the thickness of the cylinder. The governing equations are derived using the first-order shear deformation theory and the virtual work principle. The governing equations are solved using the eigenvalue–eigenvector method for clamped–clamped boundary conditions. The solution’s accuracy and convergence are validated by conducting a comparative study. Finally, the numerical results include the components of displacement, axial displacement, radial displacement, circumferential stress, radial stress, and shear stress are reported according to porosity coefficient and different porosity distributions.

Obviating some of the theoretical barriers of analytical hierarchy process by a revised eigenvector method: a case study in sustainable supplier selection

Obviating some of the theoretical barriers of analytical hierarchy process by a revised eigenvector method: a case study in sustainable supplier selection

Analytic formula for the group velocity and its derivatives with respect to elastic moduli for a general anisotropic medium

Studying the anisotropy of seismic wave velocity can provide a theoretical basis for understanding the characteristics of seismic wave propagation in anisotropic media, such as forward modeling of seismic wavefield and simultaneous inversion of anisotropic parameters. It is an important work to calculate the group velocities of three kinds of seismic waves (qP, qS1, and qS2) and the analytical expressions of the partial derivatives of the group velocities with respect to 21 elastic parameters in general anisotropic media. Therefore, a new relationship between the phase and group velocity in the general anisotropic medium is introduced first, and then the analytical expressions of the group velocity of three kinds of waves are deduced. Then, the analytical formulas of partial derivatives of the group (or phase) velocity with respect to 21 elastic parameters are derived. Finally, the distribution of partial derivatives of group slowness with respect to 21 elastic parameters with varied ray angles is analyzed and discussed. The final analytical expressions of group velocity and its partial derivatives are relatively concise and can deal with the singular points and multivalued problems of qS waves. By drawing comparison to the numerical solution of the eigenvector method, the accuracy of the analytical formula of group velocity and its partial derivatives with respect to elastic parameters are verified. The numerical simulation results indicate that the algorithm can accurately obtain the partial derivatives of group slowness with respect to 21 elastic parameters and would provide a theoretical basis for seismic ray tracing and traveltime inversion in general anisotropic media.

The World Formula and the Theorem of Perron-Frobenius: How to Solve (Almost All) Problems of the World

Transfer Functions relate the output or response of a system such as a filter circuit to the input or stimulus . The response can be measured; the is the unknown. Solving the World Formula not only explains how analog images and voice can be digitally stored, but also many methods and techniques such as Big Data and Artificial Intelligence. For linear functions between vector spaces, the Eigenvector Method makes calculating a solution easy, if it exists. Numerical algorithms are available for solving. Thus, the method is suitable for teaching students who are interested in the foundations of science. However, the system must meet certain conditions to make the eigenvector method applicable. The Theorem of Perron-Frobenius defines these conditions. Keywords: transfer functions, big data, world formula, Eigenvector method, theorem of Perron-Frobenius

Efficient Multi-Identity Full Homomorphic Encryption Scheme on Lattice

Aiming at the problem that the fully homomorphic encryption scheme based on single identity cannot satisfy the homomorphic operation of ciphertext under different identities, as well as the inefficiency of trapdoor function and the complexity of sampling algorithm, an improved lattice MIBFHE scheme was proposed. Firstly, we combined MP12 trapdoor function with dual LWE algorithm to construct a new IBE scheme under the standard model, and prove that the scheme is IND-sID-CPA security under the selective identity. Secondly, we used the eigenvector method to eliminate the evaluation key, and transform the above efficient IBE scheme into a single identity IBFHE scheme to satisfy the homomorphic operation. Finally, we improved the ciphertext extension method of CM15 and constructed a new Link-mask system that supports the transformation of IBFHE scheme under the standard model, and then, converted the above IBFHE scheme into MIBFHE scheme based on this system. The comparative analysis results showed that the efficiency of this scheme is improved compared with similar schemes in the trapdoor generation and preimage sampling, and the dimension of lattice and ciphertext size are significantly shortened.

Some explorations of linear algebra

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture in language of linear algebra. In this paper, the authors explain the important applications, convenience, efficiency of using linear algebra in math physics, and computer science. The authors give some examples of linear algebra used in various areas, including datum coordinate system finding a location, encryption and decryption algorithm, storage of images, classical mechanics, and quantum physics. The authors list the definition of the matrix, real matrix, complex matrix, diagonal matrix, identity matrix, scalar matrix, trace, rank, and determinants of matrices. The authors explored Laplace expansion, transpose, inverse, conjugate of a matrix and addition, multiplication between matrices and between a scalar and a matrix, Kronecker product and their properties of matrices, the definitions and solving method of eigenvalue and eigenvector, and the diagonalisation and its conditions of matrices. The authors introduce the applications of matrix transformations and operations in programs. The authors explain two encryption and decryption algorithms based on linear algebra and their strengths and weaknesses.

Corrigendum to “Eigenvector method, consistency test and inconsistency repairing for an incomplete fuzzy preference relation” [Applied Mathematical Modelling 37(2013) 5171–5183

Corrigendum to “Eigenvector method, consistency test and inconsistency repairing for an incomplete fuzzy preference relation” [Applied Mathematical Modelling 37(2013) 5171–5183

Incomplete pairwise comparison matrices: Ranking top women tennis players

The method of pairwise comparisons is frequently applied for ranking purposes. This article aims to rank top women tennis players based on their win/lose ratios. Incomplete pairwise comparison matrices (PCMs) were constructed from data obtained from the WTA (Women’s Tennis Association) homepage. The database contains head-to-head results from the period between 1973 and 2022 for 28 players who had the position No. 1 in the official rankings of WTA. The weight vector was calculated from the incomplete PCM with the logarithmic least squares method and the eigenvector method. The results are not surprising: Serena Williams, Steffi Graf, and Martina Navratilova stand in the first three positions, and Martina Hingis, Kim Clijsters, and Justine Henin follow them. We also tested the frequently used probability-based Bradley-Terry method and found high rank-correlation values. Using graph representations, the results gave us a deeper insight into the properties of incomplete PCMs. Special attention was given to the nontransitive triads. A data modification was necessary to remove ties in order to apply the commonly used tests. The results indicate that ordinally nontransitive triads are not significant in the data we analysed.

Assessment of key regulatory genes and identification of possible drug targets for Leprosy (Hansen’s disease) using network-based approach

ABSTRACT Leprosy is a major health concern and continues to be a source of fear and stigma among people worldwide. Despite remarkable achievements in the treatment, understanding of pathogenesis and transmission, epidemiology of leprosy still remains inadequate. The prolonged incubation period, slow rates of occurrence in those exposed and deceptive clinical presentation pose challenges to develop reliable strategies to stop transmission. Hence, there is a need for improved diagnostics and therapies to prevent mortality caused by leprosy. The objectives of this study are to identify significant genes from protein-protein interactions (PPIs) network of leprosy and to choose the most effective therapeutic targets. Fifty genes related with leprosy were discovered by literature mining. These genes were used to construct a primary network. Leading Eigen Vector method was used to break down the primary network into various sub-networks or communities. It was found that the primary network was divided into many sub-networks at the 6 levels. Seed genes were traced at each level till key regulatory genes were identified. Three seed genes, namely, GNAI3, NOTCH1, and HIF1A, were able to make their way till the final motif stage. These genes along with their interacting partners were considered key regulators of the leprosy network. This study provides leprosy-associated key genes which can lead to improved diagnosis and therapies for leprosy patients.

Do They Align? Congruence Between Patient Preferences of People Living with Cognitive Impairments and Physicians' Judgements for Person-Centered Care: An Analytic Hierarchy Process Study.

Person-centered care (PCC) requires knowledge about patient preferences. Among people living with cognitive impairments (PlwCI), evidence on quantitative, choice-based preferences, which allow to quantify, weigh, and rank care elements, is limited. Furthermore, data on the congruence of patient preferences with physicians' judgements for PCC are missing. Such information is expected to support the implementation of PCC; state-of-the-art medical care aligned with patients' preferences. To elicit patient preferences and physicians' judgements for PCC and their congruence. Data from the mixed-methods PreDemCare study, including a cross-sectional, paper-and-pencil, interviewer-assisted analytic hierarchy process (AHP) survey conducted with n = 50 community-dwelling PlwCI and n = 25 physicians. Individual AHP weights (preferences/judgements) were calculated with the principal eigenvector method and aggregated per group by aggregation of individual priorities mode. Individual consistency ratios (CRs) were calculated and aggregated per group. Group differences in preferences/judgements were investigated descriptively by means and standard deviations (SDs) of AHP weights, resulting ranks, and boxplots. Additionally, differences between groups were investigated with independent paired t-test/Mann Whitney U-test. Sensitivity of AHP results was tested by inclusion/exclusion of inconsistent respondents, with an accepted threshold at CR≤0.3 for patients, and CR≤0.2 for physicians, due to better cognitive fitness of the latter group. Patient preferences and physicians' judgements did not differ significantly, except for the criterion Memory Exercises (AHP weights (mean (SD)): 0.135 (0.066) versus 0.099 (0.068), p = 0.01). We did not see rank-reversals of criteria after exclusion of inconsistent participants. Mean CR for patients at the criteria level was 0.261, and 0.181 for physicians. Physicians' judgements in our setting aligned well with patients' preferences. Our findings may be used to guide the implementation of preference-based PCC.

Prediction of band gap of transition metal sulfide with Janus structure by deep learning atomic feature representation method

With the development of artificial intelligence, machine learning (ML) is more and more widely used in material computing. To apply ML to the prediction of material properties, the first thing to do is to obtain effective material feature representation. In this paper, an atomic feature representation method is used to study a low-dimensional, densely distributed atomic eigenvector, which is applied to the band gap prediction in material design. According to the types and numbers of atoms in the chemical formula of material, the Transformer Encoder is used as a model structure, and a large number of material chemical formula data are trained to extract the features of the training elements. Through the clustering analysis of the atomic feature vectors of the main group elements, it is found that the element features can be used to distinguish the element categories. The Principal Component Analysis of the atomic eigenvector of the main group element shows that the projection of the atomic eigenvector on the first principal component reflects the outermost electron number corresponding to the element. It illustrates the effectiveness of atomic eigenvector extracted by using the transformer model. Subsequently, the atomic feature representation method is used to represent the material characteristics. Three ML methods named Random Forest (RF), Kernel Ridge Regression (KRR) and Support Vector Regression (SVR) are used to predict the band gap of the two-dimensional transition metal chalcogenide compound MXY (M represents transition metal, X and Y refer to the different chalcogenide elements) with Janus structure. The hyperparameters of ML model are determined by searching for parameters. To obtain stable results, the ML model is tested by 5-fold cross-validation. The results obtained from the three ML models show that the average absolute error of the prediction using atomic feature vectors based on deep learning is smaller than that obtained from the traditional Magpie method and the Atom2Vec method. For the atomic eigenvector method proposed in this paper, the prediction accuracy of the KRR model is better than that of the results obtained from the Magpie method and Atom2Vec method. It shows that the atomic feature vector proposed in this paper has a certain correlation between the features, and is a low-dimensional and densely distributed feature vector. Visual analysis and numerical experiments of material property prediction show that the atomic feature representation method based on deep learning extraction proposed in this paper can effectively characterize the material features and can be applied to the tasks of material band gap prediction.

A new algorithm for computing priority vector of pairwise comparisons matrix with fuzzy elements

Applying the Analytic Hierarchy Process (AHP) in a decision making (DM) problem, fuzzy elements are appropriate whenever the decision maker is uncertain about the value of his/her evaluation of the relative importance of the elements in question, i.e., criteria and/or alternatives. The method, often called the fuzzy AHP, is also used when aggregating crisp pairwise comparisons of a group of decision makers in the group DM problem. In this paper, the DM problem is formulated in a general setting using pairwise comparisons matrices with elements from an Abelian linearly ordered group (alo-group). Such an approach enables extensions of traditional multiplicative, additive, or fuzzy approaches. Here, we propose some desirable properties (consistency, coherency, and intensity) of priority vectors, and derive sufficient conditions for the existence of priority vectors with those properties. In general, the most popular methods for deriving the priority vector – the Eigenvector Method and the Geometric Mean Method – do not always provide priority vectors having these desirable properties. Here, we formulate a new solution algorithm for deriving the priority vector based on a specific optimization problem satisfying the desirable properties under appropriate assumptions. Two illustrating examples of the algorithm are presented and discussed.

A joint DOA and polarization estimation method based on the conformal polarization sensitive array from the sparse reconstruction perspective

The conformal polarization sensitive array (CPSA) is formed by placing some vector sensors on the conformal array and it has a wide range of practical application in direction of arrival (DOA) and polarization parameters estimation. However, due to the diversity of each sensor’s direction, the performance of the conventional parameters estimation methods based on the CPSA would decrease greatly, especially under the low signal to noise ratio (SNR) and limited snapshots. In order to solve this problem, a unified framework and sparse reconstruction perspective for joint DOA and polarization estimation based on CPSA is proposed in this paper. Specifically, the array received signal model of the CPSA is formulated first and the two-dimensional spatial sparsity of the incident signals is then exploited. Subsequently, after employing the singular value decomposition method to reduce the dimension of array output matrix, the variational sparse Bayesian learning and orthogonal matching pursuit methods are utilized to solve the source DOA estimation, respectively. Finally, the polarization parameters are obtained by the minimum eigenvector method. Simulation results demonstrate that the novel approaches can provide improved estimation accuracy and resolution with low SNR and limited snapshots.