Unbiased estimates of the Weibull parameters by the linear regression method

For sample sizes from 5 to 100, the bias of the scale parameter was investigated for probability estimators, P = (i − a)/(n + b), which yield unbiased estimates of the shape parameter. A class of unbiased estimators for both the shape and scale parameters was developed for each sample size. In addition, the percentage points of the distribution of unbiased estimate of the shape parameter were determined for all sample sizes. The distribution of the scale parameter was found to be normal by using the Anderson-Darling goodness-of-fit test. How the results can be used to establish confidence intervals on both the shape and scale parameters are demonstrated in the paper.     

On estimating percentiles of the Weibull distribution by the linear regression method

Probability estimators developed previously by the authors have been used to obtain unbiased estimates of the Weibull parameters by the linear regression method. Using these unbiased estimators, percentiles of the Weibull distribution have been estimated. Since these percentiles are determined from the estimated parameters, they also have distributions and subsequently are determined for five sample sizes. Analysis has shown that the distributions of these estimated percentiles are neither normal, lognormal, three-parameter Weibull nor three-parameter log-Weibull. A new methodology to estimate the percentile with a specified level of confidence has been introduced. The step-by-step use of the methodology is demonstrated by examples in this paper.     

Influence of heat treatments on microstructural parameters and mechanical properties of P92 steel

Abstract

The microstructural parameters (dislocation density, martensite lath width, precipitate diameters, and volume fractions) have been measured for the 9%Cr steel P92 (NF616) after different heat treatments. The austenitising temperatures were 970, 1070, and 1145°C and the tempering temperatures 715, 775, and 835°C. Increasing the austenitising temperature led to an increase in the austenite grain size and in the martensite lath width, but no significant effect on the tensile properties at 20, 600, and 650°C was observed. The creep strength was, however, reduced by tempering at 835°C due to rapid recovery of the martensitic structure with a sharp decrease in dislocation density. The lowest creep strength was found for the P92 steel subjected to a heat treatment that produced a fully ferritic microstructure; the secondary creep rate was four orders of magnitude higher than that of the steel in the usual martensitic condition.     

Estimation of mixed Weibull distribution parameters using the SCEM-UA algorithm: Application and comparison with MLE in automotive reliability analysis

The shuffled complex-evolution metropolis algorithm (SCEM-UA) is used to estimate mixed Weibull distribution parameters in automotive reliability analysis. The results are compared with maximum likelihood estimation (MLE) results. The comparison shows that, in the examples given, SCEM-UA can deliver more accurate results than MLE overall.     

A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

Reliability analysis of Weibull multicomponent system with stress-dependent parameters from accelerated life data

In this paper, reliability estimation of multicomponent system under a multilevel accelerated life testing. When the lifetime of components follows Weibull distribution, the problem of point and interval estimates are discussed from different perspectives. Under a general life-stress assumption that there are multiple nonconstant and stress-dependent scale and shape parameters, the maximum likelihood estimates of unknown parameters along with associated existence and uniqueness are established. Approximate confidence intervals are constructed correspondingly via expected Fisher information matrix. Furthermore, some pivotal quantities are constructed and alternative generalized point and interval estimates are also proposed for comparison. In addition, predictive intervals for the lifetime of the multicomponent system are discussed under classical and generalized pivotal approaches, respectively. The results show that the proposed generalized estimates are superior to the conventional likelihood approach in terms of the accuracy. A real data example is carried out to illustrate the implementations of the proposed methods.     

Estimation of the Parameters of a Mixture of Exponential and Weibull Distributions with Progressive Censoring

A method is proposed for estimating the parameters of a mixture of exponential and Weibull distributions using censored samples. Preliminary estimates obtained by graphical analysis are refined by the method of maximum likelihood. The efficiency of the method is confirmed by the results of a statistical modeling.     

Generalized linear regression analysis of association of universal helmet laws with motorcyclist fatality rates

This study evaluates the association of universal helmet laws with U.S. motorcyclist fatality rates from 1993 through 2002 using climate measures as statistical controls for motorcycling activity via quasi-maximum likelihood generalized linear regression analyses. Results revealed that motorcyclist fatalities and injuries are strongly associated with normalized heating degree days and precipitation inches, and that universal helmet laws are associated with lower motorcyclist fatality rates when these climate measures, and their interaction, are statistically controlled. This study shows that climate measures have considerable promise as indirect measures (proxies) of motorcycling activity in generalized linear regression studies.     

Statistical monitoring of multivariate multiple linear regression profiles in phase I with calibration application

In some statistical process control applications, there are some correlated quality characteristics which can be modeled as linear functions of some explanatory variables. We refer to this structure as multivariate multiple linear regression profiles. When the correlation structure between quality characteristics is ignored and profiles are monitored separately then misleading results could be expected. Hence, developing methods to account for this multivariate structure is required. In this paper, we specifically focus on phase I monitoring of multivariate multiple linear regression profiles and develop four methods for this purpose. The performance of the developed methods is compared through simulation studies in terms of probability of a signal. In addition, a diagnostic scheme to find the out‐of‐control samples is developed. Finally, the application of the proposed methods is illustrated using a calibration application at the National Aeronautics and Space Administration (NASA) Langley Research Center. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation of regression coefficients subject to exact linear restrictions when some observations are missing and quadratic error balanced loss function is used

This article considers a linear regression model when a set of exact linear restrictions binding the coefficients is available and some observations on the study variable are missing. Estimators for the vectors of regression coefficients are presented and their superiority properties with respect to the criteria of the variance covariance matrix and the risk under balanced loss functions are analyzed.     

用三线法标定车载摄像机的外部参数

根据摄像机透视投影原理,三条互不重合的平行线在图像平面上具有相同的消失点和不同的斜率。按照针孔成像模型,经过数学推导可以建立摄像机的外部参数与消失点和成像斜率之间的解析表达式,无需经过迭代和优化计算。实现过程只需要地面上有三条平行线,然后通过手工或自动的方式确定出这三条直线成像的交点和斜率,就可以标定出摄像机的外部参数。实验表明,该方法可以在不同环境下进行车载摄像机标定,方位角和俯仰角的标定误差小于百分之一弧度。     

Comparative analysis of linear and non-linear method of estimating the sorption isotherm parameters for malachite green onto activated carbon

The experimental equilibrium data of malachite green onto activated carbon were fitted to the Freundlich, Langmuir and Redlich-Peterson isotherms by linear and non-linear method. A comparison between linear and non-linear of estimating the isotherm parameters was discussed. The four different linearized form of Langmuir isotherm were also discussed. The results confirmed that the non-linear method as a better way to obtain isotherm parameters. The best fitting isotherm was Langmuir and Redlich-Peterson isotherm. Redlich-Peterson is a special case of Langmuir when the Redlich-Peterson isotherm constant g was unity.     

基于部分可辨识振动系统的载荷恢复方法

在复模态分析和参数辨识的基础上,对部分可辨识振动系统的载荷识别问题进行了研究,提出了基于部分可辨识复模态参数的载荷恢复方法。恢复模型存在幅值模糊和相位模糊,不存在排列模糊。数值仿真验证了该方法的有效性,并讨论了提高恢复精度的方法。     

Estimation of high‐temperature fracture parameters for small punch specimen with a surface crack

An outline of a newly proposed methodology for evaluating creep crack growth (CCG) parameters using cracked small‐punch (SP) specimens is explained. Three‐dimensional finite element analyses were performed to calculate the stress intensity factor along the crack front for a surface crack formed at the centre of a SP specimen. Effects of crack ratio, (a/t); crack aspect ratio, (a/c); and thickness of the specimen, (t), on the fracture parameters were studied. It was observed that the minimum variation of K‐value along the crack front can be achieved when a/c was 0.50 except the location very near the intersection of the crack and free surface. This condition is similar to the case of constant K‐values along the crack front of the conventional compact tension specimen. Thus, it can be argued that the SP specimen with a surface crack is a suitable specimen geometry for CCG testing. The proposed CCG test method was found to be practically applicable for the crack geometry of 0.10 to 0.30 of a/t with constant aspect ratio of 0.50. An estimation of the K and C_t‐parameter under the small scale creep condition was derived. Future work for further development of the suggested CCG testing is discussed.     

Reliability of systems with randomly varying parameters by the path integration method

The paper considers a first passage time reliability problem for systems subjected to multiplicative and additive white noises. For numerical calculations of the reliability function and the first passage time the path integration method is properly adapted and used. Some results of numerical calculations are compared to approximate analytical results, obtained by the stochastic averaging method.     

Probabilistic response of linear structures equipped with nonlinear damper devices (PIS method)

Passive control introducing energy absorbing devices into the structure has received considerable attention in recent years. Unfortunately the constitutive law of viscous fluid dampers is highly nonlinear, and even supposing that the structure behaves linearly, the whole system has inherent nonlinear properties. Usually the analysis is performed by a stochastic linearization technique (SLT) determining a linear system equivalent to the nonlinear one, in a statistical sense. In this paper the effect of the non-Gaussianity of the response due to the inherent nonlinearity of the damper device will be studied in detail via the Path Integral Solution (PIS) method. A systematic study is conducted showing that for a very wide range of parameters the SLT gives satisfactory results in terms of variance of displacement and velocity but not in terms of joint Probability Density Function (PDF). It has also been shown that at steady state the two processes, displacement and velocity, may be considered as independent ones.     

计算区间特征值的Rayleigh商法

摘要：为提高区间特征值的计算效率,首先分析了区间结构矩阵特征值的设计变量的单调性问题,随后采用区间因子法给出了基于Rayleigh商计算结构广义特征值的一种快速算法。当诸参数的变化区间相对其均值的离差较小时,区间结构矩阵特征值的计算精度主要由各区间因子的差值大小所决定;且当各区间因子的差值为零时,误差亦为零。最后给出的两个算例表明：相对于全局优化算法和区间离散法,该方法具有计算效率高、误差小的特点,具有一定的工程实用价值。     

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.     

A stochastic collocation method for large classes of mechanical problems with uncertain parameters

The paper presents a self-contained and didactic approach to the stochastic collocation method. The method relies on the Lagrange polynomials and the Gauss quadrature rule. It is presented for large classes of mechanical problems, i.e. static problems, dynamic problems and spectral problems. After a general presentation of each of them, examples and results are provided. Numerical results show the high rate of convergence of the proposed method.     

Probabilistic analysis of truss structures with uncertain parameters (virtual distortion method approach)

A new approach for probabilistic characterization of linear elastic redundant trusses with uncertainty on the various members subjected to deterministic loads acting on the nodes of the structure is presented. The method is based on the simple observation that variations of structural parameters are equivalent to superimposed strains on a reference structure depending on the axial forces on the elastic modulus of the original structure as well as on the uncertainty (virtual distortion method approach). Superposition principle may be applied to separate contribution to mechanical response due to external loads and parameter variations. Statically determinate trusses dealt with the proposed method yields explicit analytic solution in terms of displacements while redundant trusses have been studied by means of an asymptotic expansion exhibiting explicit dependence on parameter fluctuations. Probabilistic characterization of the response may then be obtained both for statically determinate and statically indeterminate stochastic trusses.