General Weight Function Research Articles

Extended weighted generalized cumulative residual entropy for risk measurement

In this paper, we propose an extended variant of Weighted Generalized Cumulative Residual Entropy (WGCRE), its dynamic version, and a Weighted Generalized Cumulative Kerridge Inaccuracy measure by considering a general continuous non negative weight function. Several properties, including bounds and stochastic ordering, of these measures are obtained under appropriate assumptions. Furthermore, we introduce the empirical extended WGCRE as a non-parametric estimator, demonstrating its almost sure convergence to the proposed measure. A central limit theorem is established, examples are provided for illustrative purposes, and the stability of the empirical measure is examined. Finally, we demonstrate the practicality of this measure using real financial data for risk measurement.

Zone‐based failure risk assessment of fatigue crack growth caused by initial defects in powder turbine disc

AbstractIn this research, A zone‐based failure risk assessment (FRA) method of fatigue crack growth (FCG) caused by initial defects in the FGH96 alloy turbine disc is developed. Firstly, the initial defects distribution in the FGH96 alloy turbine disc is calculated based on the defect data. Subsequently, a probabilistic short FCG life model is established, taking into account the dispersion in grain size. Meanwhile, a probabilistic long FCG life model is established, incorporating the life dispersion factor. To calculate the Stress Intensity Factor (SIF) at any position of the disc, the general weight function method and the rectangular plate model are established. Finally, the zoning process is established, enabling a FRA that considers the FCG due to initial defects. The results indicate that the number of cycles corresponding to a 0.13% failure probability of the turbine disc is 7150, and the percentage of failures in each zone is analyzed.

General point load weight function for semi-elliptical surface cracks in round bars

Offshore mooring chains are continuously subjected to cyclic environmental loadings, resulting in fatigue cracks that always develop at the surface and grow across the circular cross-sections. To analyse the fatigue crack growth in a mooring chain based on a fracture mechanics-based approach, the evaluation of stress intensity factors (SIFs) of surface cracks in the chain segments becomes essential. A general point load weight function of SIFs for semi-elliptical surface cracks in straight and curved round bars was derived in this study. Parametric three-dimensional finite element (FE) models were developed to calculate the SIF solutions for determining the unknown parameters in the weight function. The analysis matrix covered wide ranges of aspect ratios 0.1 ≤ a/c ≤ 2.0, relative crack depths 0.03 ≤ a/D ≤ 0.6 and relative bar curvatures 0.0 ≤ D/Ri ≤ 1.5. The developed weight function was verified against the solutions available in the literature and FE results, indicating good accuracy. The proposed weight function can be used in SIF calculation and remaining fatigue life prediction of surface cracked round bar components under complex stress distributions, such as chain links.

Positive solution for a nonlocal problem with strong singular nonlinearity

Abstract In this article, we consider a nonlocal problem with a strong singular term and a general weight function. By using Ekeland’s variational principle, we prove a necessary and sufficient condition for the existence of a positive solution. Moreover, a method of algebraic analysis is used to deal with the multiplicity of solutions. Compared with the existing literature, our problems and results are novel.

MWS and FWS codes for coordinate-wise weight functions

A combinatorial problem concerning the maximum size of the (Hamming) weight set of an $$[n,k]_q$$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $$[n,k]_q$$ codes with the same weight set as $$\mathbb {F}_q^n$$ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily “short”, whereas MWS codes are necessarily “long”. For fixed k, q the values of n for which an $$[n,k]_q$$ -FWS code exists are completely determined, but the determination of the minimum length M(H, k, q) of an $$[n,k]_q$$ -MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on n for which an FWS code exists, and bounds on n for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $$M({\mathscr {L}},k,q)$$ (the minimum length of Lee MWS codes), and pose the determination of $$M({\mathscr {L}},k,q)$$ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

Markdown Optimization with Generalized Weighted Least Squares Estimation

Retailers increasingly apply price markdowns for their seasonal products. Efficiency of these markdown applications is driven by the accuracy of empirical models, especially toward the end of a selling season. In the literature, recent sales are recognized to be more important than older sales data for estimating the current period’s demand for a given markdown level. The importance difference between the weeks of a selling season is addressed by weighted least squares (WLS) method with continuous weight functions of time. This study suggests a generalization of the weight functions and a method for optimizing their shape and discretization parameters to stimulate the predictive accuracy of models. We find that addressing the importance difference of recent sales observations using our generalized weight functions improves the forecast accuracy by up to 20%, and most of the improvement stems from our weight discretization method.

An Intelligent Gray Prediction Model Based on Fuzzy Theory

In order to improve the forecasting effect of the gray prediction model, this paper combines the fuzzy theory to construct the gray prediction model and explores its forecasting accuracy. Moreover, this paper uses the entropy weight method to obtain the objective weight to correct the subjective weight, which makes the weight calculation more reasonable. In view of the uncertainty of the control signal of the research object, this paper introduces the gray system theory to conduct cluster analysis on the fire control computer and mainly introduces the general whitening weight function. Furthermore, this paper adopts the center point mixed with a triangular whitening weight function to carry out gray clustering according to the difficulty of defining the gray class boundary and gives the solution steps to obtain the intelligent gray prediction model. Finally, this paper verifies that the intelligent gray prediction model based on fuzzy theory has a good effect through experiments, which can effectively improve the prediction effect of the intelligent prediction model.

Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.

Evaluation and Numerical Simulation of Music Education Informationization Based on the Local Linear Regression Method

The study of this theoretical problem enables sparse or dense functional data, including educational information evaluation data. The choice of different weights is subjected to principal component analysis. The evaluation of music education informatization level mainly evaluates the status quo of music education informatization development, provides a basis for formulating and adjusting music education informatization development policies, and provides support for educational decision-making, to promote the sustainable and balanced development of music education informatization. The evaluation of music education informatization has become the key promotion work of music education informatization at this stage. This paper studies the convergence rate of functional principal components based on the local linear method under general weighting conditions. First, we introduce the related research on the estimation of mean and covariance function under general weighting. Secondly, for principal functional components under general weighting, namely, eigenvalues and eigen functions, the text gives the corresponding estimated values and derives its strong uniform convergence rate. Finally, the convergence rate was verified by simulation research. The estimation methods and conclusions of this article enrich the research of functional linear regression models and will help analyze the complex and changeable problems encountered in the application of music education information.

Determination of complex stress intensity factors for interface cracks in bi-material specimens subjected to ununiform stresses

This paper presents the calculation of complex stress intensity factors (K) for interface cracks in commonly used bi-material specimens subjected to arbitrary ununiform stress fields. First, detailed finite element analyses were carried out for cracks in centre-cracked plate (CCP) and single edge-cracked plate (SECP) bi-material specimens of different material combinations under non-uniform stress distributions. The complex SIFs were determined for the loading cases of uniform, linear, parabolic or cubic stress distributions applied to the crack faces in both normal and shear directions. Then, general mathematical forms of the two complex crack face weight functions for interface cracks, one for normal loading, and one for shear loading, have been formulated by taking advantages of the knowledge of specific analytical complex weight functions available in the literature, and certain properties of the complex weight functions in general. The existence of the general functional forms simplifies the determination of these weight functions for specific crack geometrical configurations and bi-material combinations. The determination of a specific weight function is thus reduced to the determination of the unknown parameters of the generalized weight function functional forms. It was demonstrated these unknown parameters can be determined from reference stress intensity factors. This method was used to derive the complex weight functions for interface cracks in bi-material CCP and SECP specimens. The derived closed-form weight functions were then validated against complex stress intensity factors solutions for several non-linear normal and shear stress distributions.

Leader–Follower Flocking for Discrete-Time Cucker–Smale Models With Lossy Links and General Weight Functions

In this article, we investigate leader–follower flocking for the Cucker–Smale model with lossy links and general weight functions. Here, the loss phenomenon of the control packages from the controller to the actuator is characterized by a Bernoulli stochastic variable and the edge weights of the interaction network are determined by a general update rule based on the distance between agents. A method based on the products of substochastic matrices is developed to analyze the flocking behavior with lossy links. By means of this method, a sufficient condition which depends on the agents’ initial states, the topology structure, the successful information transmission rate and the weight function is established. Compared with the previous Cucker–Smale flocking results, which are only applicable to positive and decreasing weight functions with specific forms, our result is more general and can be applied to arbitrary positive and decreasing weight functions with nonzero lower bounds. Finally, our result is illustrated through a simulation example.

Addendum to "Dimensional Reduction for 6D Vibrotactile Display".

In our recent article, we found a general quadratic weighting function to predict how a six-dimensional (6D) vibrotactile stimulus rendered at the haptic interaction point (HIP) of a kinesthetic haptic interface, with the stylus held in a precision pen-hold grasp, is mapped to an equivalent one-dimensional (1D) stimulus that is normalized by the detection threshold. However, in that work we did not constrain the weighting function to be positive semidefinite, and as a result, the model will not generalize well to all future 6D inputs. In this addendum, we reconsider the original data set, incorporating the positive-semidefinite constraint. We find that as few as four independent parameters are required to describe the coupling, and the model has one fewer coupling term than originally proposed. We also describe the process of fitting a positive-semidefinite function to a new stylus at some specific excitation frequency.

On General Exponential Weight Functions and Variation Phenomenon

General weighted exponential distributions including modified exponential ones are widely used with great ability in statistical applications, particularly in reliability. In this paper, we investigate full exponential weight functions and their extensions from any nonnegative continuous reference weighted distribution. Several properties and their connections with the recent variation phenomenon are then established. In particular, characterizations, weightening operations and dual distributions are set forward. Illustrative examples and concluding remarks are extensively discussed.

On a generalized energy conservation/dissipation time finite element method for Hamiltonian mechanics

Energy conserving and dissipative algorithm designs in Hamilton’s canonical equations via Petrov–Galerkin time finite element methodology are proposed in this paper that provide new avenues with high-order convergence rate, improved solution accuracy, and controllable numerical dissipation. Lagrange quadratic shape function in time with flexible interpolation points are considered to approximate the solution over a time interval. Instead of specifying the weight functions, two algorithmic parameters, namely, a principal root (ρq∞) and a spurious root (ρp∞), are introduced to formulate a generalized weight function, which enables us to introduce controllable numerical dissipation with respect to displacement and momenta while preserving high-order convergence, and features with improved solution accuracy. A family of third-order accurate time finite element algorithms with controllable dissipation and improved solution accuracy is presented in both the homogeneous and non-homogeneous dynamic problems; and via setting (ρq∞,ρp∞)=(1,1), this third-order family of algorithms directly leads to a new family of fourth-order accurate non-dissipative algorithms in general homogeneous problems; and the fourth-order accuracy is also preserved in non-homogeneous problems when the third-order time derivative of the external excitation has the order of O(Δtn)(n≤1). Numerical examples are performed to demonstrate the pros/cons for the conserving properties of various schemes in the proposed Petrov–Galerkin time finite element of algorithms.

A phaseless inverse problem for electrodynamic equations in the dispersible medium

ABSTRACT For electrodynamic equations related to non-conducting dispersible medium we consider the inverse problem of recovering two variable coefficients from a given phaseless information of solutions to the equations. One of these coefficients is the permittivity while the second one characterizes the time dispersion of the medium. We suppose that unknown coefficients differ from given constants inside of a compact domain Ω. A plane electromagnetic wave going in the direction ν from infinity fall down on this domain and modulus of the electric strength is measured on a part of the boundary of Ω for all . The inverse problem consists in determining unknown functions from this information. We reduce the inverse problem to two problems: (1) the inverse kinematic problem for recovering the refractive index and (2) the integral geometry problem for recovering the second coefficient related to the dispersion. An uniqueness theorem for the first problem is stated on the base of known results. The second problem differs from have studied by the more general weight function and it is still open. Then we demonstrate that under some natural assumption the weight function uniformly close to 1. Replacing the weight function by 1, we obtain the integral geometry problem for which the uniqueness theorem and stability estimate are established and some numerical algorithms are proposed.

Nonexistence for hyperbolic problems on Riemannian manifolds1

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

Gravity sparse inversion using the interior-point method and a general model weighting function

This study presents an optimized gravity-sparse inversion method. The proposed method minimizes the global objective function using interior-point method for boundary constraints and a general weighting function comprising the depth, compactness, and kernel weighting functions of the density models. For the compactness weighting function, practical experiments demonstrate that the recovered model becomes more compact with an increasing value for the relative exponential factor β. However, if no appropriate boundary-constraint method is applied, the inversion results cannot be controlled within the designated constraint bounds when β needs to be set to a large value to obtain compact inversion results. The interior-point method allows the use of a larger β to obtain more compact inversion results without violating the boundary constraints. Additionally, models in close proximity can more clearly be recognized using this method. To improve the computational efficiency and obtain a more accurate regularization parameter, the preconditioned conjugate gradient and L-curve, or line search methods, were also applied. The proposed method was applied for three synthetic examples: two positive bodies adjacent to each other at different depths inverted using noise-free gravity anomaly data, three bodies (positive or negative) at different depths inverted using noise-free or contaminated gravity anomaly data, and three bodies (positive or negative) characterized by a certain dip angle inverted using contaminated gravity anomaly data. This method was also applied for the inversion of a Woodlawn sulfide body, Missouri iron ore body, and granitoid rock body in the Rio Maria region in the state of Para, Brazil. In all six test cases, larger β values were used and the density models were recovered with sharper boundaries within the designated bounds.

A Technique for Seamless Forecast Construction and Validation from Weather to Monthly Time Scales

Abstract Seamless prediction means bridging discrete short-term weather forecasts valid at a specific time and time-averaged forecasts at longer periods. Subseasonal predictions span this time range and must contend with this transition. Seamless forecasts and seamless validation methods go hand-in-hand. Time-averaged forecasts often feature a verification window that widens in time with growing forecast leads. Ideally, a smooth transition across daily to monthly time scales would provide true seamlessness—a generalized approach is presented here to accomplish this. We discuss prior attempts to achieve this transition with individual weighting functions before presenting the two-parameter Hill equation as a general weighting function to blend discrete and time-averaged forecasts, achieving seamlessness. The Hill equation can be tuned to specify the lead time at which the discrete forecast loses dominance to time-averaged forecasts, as well as the swiftness of the transition with lead time. For this application, discrete forecasts are defined at any lead time using a Kronecker delta weighting, and any time-averaged weighting approach can be used at longer leads. Time-averaged weighting functions whose averaging window widens with lead time are used. Example applications are shown for deterministic and ensemble forecasts and validation and a variety of validation metrics, along with sensitivities to parameter choices and a discussion of caveats. This technique aims to counterbalance the natural increase in uncertainty with forecast lead. It is not meant to construct forecasts with the highest skill, but to construct forecasts with the highest utility across time scales from weather to subseasonal in a single seamless product.

Generalized weight function selection criteria for the compressive sensing based robust DOA estimation methods

The performances of the conventional direction-of-arrival (DOA) estimation algorithms degrade significantly when some sensors in the array are miscalibrated. Inspired by a maximum correntropy criterion (MCC) based robust DOA estimation algorithm, this paper proposes a series of weight function selection criteria for the compressive sensing (CS) based DOA estimation algorithm to suppress the effect of the sensor miscalibrations under the half-quadratic (HQ) optimization framework. Based on the analysis of how the MCC works under the HQ framework, four weight function selection criteria are proposed to diminish the contribution of the observations received by the miscalibrated sensors. The analytical relationship between constraint function and weight function is then derived, based on which another selection criterion is supplemented to ensure that the selected weight function can be applied to the HQ framework. A generalized framework for the CS based robust DOA estimation methods is thus established, where the specific form of the constraint functions does not need to be known. Examples are presented to verify the effectiveness and generality of the proposed selection criteria.