Quadrature as applied to computer models for robust design: theoretical and empirical assessment

This paper develops theoretical foundations for extending Gauss–Hermite quadrature to robust design with computer experiments. When the proposed method is applied with m noise variables, the method requires 4m + 1 function evaluations. For situations in which the polynomial response is separable, this paper proves that the method gives exact transmitted variance if the response is a fourth-order separable polynomial response. It is also proven that the relative error mean and variance of the method decrease with the dimensionality m if the response is separable. To further assess the proposed method, a probability model based on the effect hierarchy principle is used to generate sets of polynomial response functions. For typical populations of problems, it is shown that the proposed method has less than 5% error in 90% of cases. Simulations of five engineering systems were developed and, given parametric alternatives within each case study, a total of 12 case studies were conducted. A comparison is made between the cumulative density function for the hierarchical probability models and a corresponding distribution function for case studies. The data from the case-based evaluations are generally consistent with the results from the model-based evaluation.

A copula-based approach to joint modelling of multiple longitudinal responses with multimodal structures

This article introduces a flexible modelling strategy to extend the familiar mixed-effects models for analysing longitudinal responses in the multivariate setting. By initiating a flexible multivariate multimodal distribution, this strategy relaxes the imposed normality assumption of related random-effects. We use copulas to construct a multimodal form of elliptical distributions. It can deal with the multimodality of responses and the non-linearity of dependence structure. Moreover, the proposed model can flexibly accommodate clustered subject-effects for multiple longitudinal measurements. It is much useful when several subpopulations exist but cannot be directly identifiable. Since the implied marginal distribution is not in the closed form, to approximate the associated likelihood functions, we suggest a computational methodology based on the Gauss–Hermite quadrature that consequently enables us to implement standard optimization techniques. We conduct a simulation study to highlight the main properties of the theoretical part and make a comparison with regular mixture distributions. Results confirm that the new strategy deserves to receive attention in practice. We illustrate the usefulness of our model by the analysis of a real-life dataset taken from a low back pain study.

ESTIMASI PARAMETER MODEL PROBIT PADA DATA PANEL MENGGUNAKAN OPTIMASI BFGS

Salah satu model yang dapat menjelaskan pola hubungan antara variabel dependen yang bersifat kategorik dengan variabel independen adalah regresi probit. Dalam regresi probit, variabel independen dapat bersifat kategorik atau kontinu. Regresi probit mrnggunakan fungsi link dari distribusi normal standar. Jika pemodelan regresi probit melibatkan data silang dan deret waktu, disebut model probit data panel. Estimasi parameter model probit data panel random effect menggunakan maximum likelihood estimation (MLE) dengan pendekatan Gauss Hermite Quadrature. Proses iterasi menggunakan metode BFGS. Metode ini digunakan untuk mendapatkan hasil estimasi parameter yang closed form.

Quadrature Kalman filter–based state of charge estimation for lithium-ion battery

This article presents an approach to lithium-ion battery state of charge estimation based on the quadrature Kalman filter. Among the existing state of charge estimation approaches, the extended Kalman filter–based state of charge and unscented filter–based state of charge algorithms are influenced by the linearization or the solution of sigma points. The proposed quadrature Kalman filter–based state of charge algorithm avoids these problems. Specifically, the battery system equations are built based on the second-order resistance–capacitance equivalent circuit model, and the parameters are identified according to the hybrid pulse power characterization discharging test. Then, the quadrature points and corresponding weights are defined by the Gauss–Hermite quadrature rule, and the Kronecker tensor product is adopted to solve the points of multivariate. In addition, the stability of quadrature Kalman filter–based state of charge is verified. Finally, the simulation is carried out under the discharging and urban dynamometer driving schedule condition, which demonstrates that the quadrature Kalman filter–based state of charge algorithm has a better performance compared with extended Kalman filter–based state of charge and unscented filter–based state of charge.

Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack

In this study, singular integral solutions were studied to investigate scattering of Rayleigh waves by subsurface cracks. Defining a wave scattering model by objects, such as cracks, still can be quite a challenge. The model’s analytical solution uses five different numerical integration methods: (1) the Gauss–Legendre quadrature, (2) the Gauss–Chebyshev quadrature, (3) the Gauss–Jacobi quadrature, (4) the Gauss–Hermite quadrature and (5) the Gauss–Laguerre quadrature. The study also provides an efficient dynamic finite element analysis to demonstrate the viability of the wave scattering model with an optimized model configuration for wave separation. The obtained analytical solutions are verified with displacement variation curves from the computational simulation by defining the correlation of the results. A novel, verified model, is proposed to provide variations in the backward and forward scattered surface wave displacements calculated by different frequencies and geometrical crack parameters. The analytical model can be solved by the Gauss–Legendre quadrature method, which shows the significantly correlated displacement variation with the FE simulation result. Ultimately, the reliable analytic model can provide an efficient approach to solving the parametric relationship of wave scattering.

A note on the accuracy of adaptive Gauss–Hermite quadrature

Summary Numerical quadrature methods are needed for many models in order to approximate integrals in the likelihood function. In this note, we correct the error rate given by Liu & Pierce (1994) for integrals approximated with adaptive Gauss–Hermite quadrature and show that the approximation is less accurate than previously thought. We discuss the relationship between the error rates of adaptive Gauss–Hermite quadrature and Laplace approximation, and provide a theoretical explanation of simulation results obtained in previous studies regarding the accuracy of adaptive Gauss–Hermite quadrature.

An Approximation for BER of Optical Wireless System under Weak Atmospheric Turbulence using Point Estimate

Obtaining analytically closed-form expression for bit error rate (BER) of an optical wireless communication (OWC) system under weak atmospheric turbulence is extremely difficult and intractable. In this paper, a simple closed-form approximate expression for average BER of OWC system under weak turbulence condition is obtained using point estimate method. The proposed expression is used to compute the BER for on-off keying (OOK) and sub carrier intensity modulated (SIM) binary phase shift keying (BPSK) modulation schemes. Results obtained using proposed expression are compared with those obtained using Gauss-Hermite quadrature approximation and Monte-Carlo simulations.