Canonical transformations of skew-normal variates

Conditions are given for linear functions of skew-normal random vectors to maximize skewness and kurtosis. As a direct implication, several measures of their multivariate skewness and kurtosis are shown to be equivalent. An estimator of the shape parameter with good statistical properties is also considered. These results are strictly related to canonical forms of skew-normal distributions and linear transformations to normality.     

A novel statistical tolerance analysis method for assembled parts

Tolerance analysis of an assembly is an important issue for mechanical design. Among various tolerance analysis methods, statistical analysis is the most commonly employed method. However, the conventional statistical tolerance method is often based on the normal distribution. It fails to predict the resultant tolerance of an assembly of parts with non-normal distributions. In this paper, a novel method based on statistical moments is proposed. Tolerance distributions of parts are first transferred into statistical moments that are then used for computing tolerance stack-up. The computed moments, particularly the variance, the skewness and the kurtosis, are then mapped back to probability distributions in order to calculate the resultant tolerance of the assembly. The proposed method can be used to analyse the resultant tolerance specification for non-normal distributions with different skewness and kurtosis. Simulated results showed that tail coefficients of different distributions with the same kurtosis are close to each other for normalised probabilities between ?3 and 3. That is, the tail coefficients of a statistical distribution can be predicted by the coefficients of skewness and kurtosis. Two examples are illustrated in the paper to demonstrate the proposed method. The predicted resultant tolerances of the two examples are only 0.5% and 1.5% differences compared with that by the Monte Carlo simulation for 1,000,000 samples. The proposed method is much faster in computation with higher accuracy than conventional statistical tolerance methods. The merit of the proposed method is that the computation is fast and comparatively accurate for both symmetrical and unsymmetrical distributions, particularly when the required probability is between ±2σ and ±3σ.     

The beta exponential distribution

The exponential distribution is perhaps the most widely applied statistical distribution for problems in reliability. In this note, we introduce a generalization—referred to as the beta exponential distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the beta exponential distribution. We derive expressions for the moment generating function, characteristic function, the first four moments, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, the distribution of sums and ratios, and the asymptotic distribution of the extreme order statistics. We also discuss simulation issues, estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix. We hope that this generalization will attract wider applicability in reliability.     

涡旋光束通过非高斯随机粗糙面的场分布特性

基于角谱衍射理论,利用Johnson传递系统数值模拟非高斯粗糙面,研究了拉盖尔-高斯涡旋光束通过随机非高斯粗糙表面的场分布特性。在分析了非高斯粗糙面方向自相关长度、峰度、偏斜以及均方根粗糙度对涡旋光束场分布影响的基础上,研究了涡旋光束通过随机粗糙表面后光束光强分布变化时的均方根粗糙度取值范围,并通过实验,将实验数据与仿真结果进行了对比分析。结果表明:当非高斯粗糙面方向相关长度为20 mm,偏斜为0.001,峰度为6,均方根粗糙度大于0.12 mm时,拉盖尔-高斯光束透过随机表面的光强分布不再保持空心分布,对应的相位奇点消失。     

Incoherent superposition of off-axis polychromatic Hermite-Gaussian modes

We study a statistical ensemble of multimode laser beams. Each beam is made up of an incoherent superposition of off-axis polychromatic Hermite-Gaussian modes. We obtain analytic expressions for the squared beam radius, the waist position, the Rayleigh range, the skewness parameter, the kurtosis parameter, and the squared beam-propagation factor. We demonstrate that the squared beam radius has a quadratic dependence on the distance from the waist plane. The skewness parameter may be different from zero in the near-field zone, but it tends to zero in the far-field zone. The kurtosis parameter in the far-field zone coincides with the kurtosis parameter of the incoherent superposition of on-axis modes.     

聚脲材料动态压缩力学行为的数值模拟研究

以聚脲材料动态压缩力学特性为研究对象,提出了考虑动态弹性模量、动态强度因子和动态切线模量的简化三直线弹塑性本构模型;基于ANSYS/LS-DYNA有限元分析软件,建立了低、中、高不同应变率下聚脲材料压缩有限元模型,并与实验结果进行对比分析。结果表明：动态弹性模量增大因子、动态强度因子和动态切线模量因子随应变率增加而有规律的增大,均和应变率的对数呈双线性关系。在中低应变率下,三者的线性关系斜率都比较平缓;在高应变率下,三者的线性关系斜率都比较陡,且弹性模量动态增大因子的斜率比动态强度因子的更大,而第二阶段动态切线模量因子的斜率与动态强度因子的基本一致,但第三阶段动态切线模量因子的斜率是动态强度因子的2.3倍左右,说明高应变率下聚脲材料的后期应力强化效应更加显著。聚脲材料的简化三直线弹塑性本构模型可以在ANSYS/LS-DYNA有限元软件中较好地实现。该文建立的有限元模型能较为准确地模拟聚脲材料压缩实验,进一步验证了简化弹塑性本构模型在不同应变率压缩加载下的有效性。该研究可以为聚脲涂覆加固防护结构有限元模型提供材料模型参数依据。     

Limit analysis of the statistics of quasi-steady non-linear aerodynamic forces for small turbulence intensities

This paper focuses on the estimation of statistical characteristics of a specific quasi-steady wind loading used in buffeting analyses. In this loading, the intrinsic non-linearity of aerodynamic coefficients is considered and approached by a polynomial expression of any a priori chosen degree. As rigorous developments of the statistical moments would result in impractical formulations, we suggest to consider the smallness of the turbulence intensities to construct, by means of a limit analysis, approximate expressions of the raw moments of aerodynamic forces. From these expressions, approximate cumulants and associated dimensionless characteristics, such as skewness and excess coefficients, are derived. The accuracy of the proposed analytical relations is assessed by comparison with Monte Carlo simulations, and the relevance of the sophisticated non-linear loading under consideration is compared to more traditional models.     

Controlling first four moments for robust optimization

The paper addresses the solution of robust moment-based optimization problems after a multipoint reformulation. The first four moments are considered (i.e. mean, variance, skewness and kurtosis) going beyond classical engineering optimization based on the control of the mean and variance . In particular, the impact on the design of a control of the third and fourth moments are discussed. The multipoint formulation leads to discrete expressions for the moments. linking moment-based and multipoint optimizations. The linearity of the sums in the discrete moments permits an easy evaluation of their gradients with respect to the design variables. Optimal sampling issues are analyzed and a procedure is proposed to quantify the confidence level on the robustness of the design. The proposed formulation is fully parallel and the time-to-solution is comparable to single-point situations. It is applied to three problems: an analytical least-square minimization problem, a shape optimization problem with a reduced-order model, and a full aircraft shape optimization robust over a range of transverse winds.     

Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics

The emissivity of two-dimensional anisotropic rough sea surfaces with non-Gaussian statistics is investigated. The emissivity derivation is of importance for retrieval of the sea-surface temperature or equivalent temperature of a rough sea surface by infrared thermal imaging. The well-known Cox-Munk slope probability-density function, considered non-Gaussian, is used for the emissivity derivation, in which the skewness and the kurtosis (related to the third- and fourth-order statistics, respectively) are included. The shadowing effect, which is significant for grazing angles, is also taken into account. The geometric optics approximation is assumed to be valid, which means that the rough surface is modeled as a collection of facets reflecting locally the light in the specular direction. In addition, multiple reflections are ignored. Numerical results of the emissivity are presented for Gaussian and non-Gaussian statistics, for moderate wind speeds, for near-infrared wavelengths, for emission angles ranging from 0 degrees (nadir) to 90 degrees (horizon), and according to the wind direction. In addition, the emissivity is compared with both measurements and a Monte Carlo ray-tracing method.     

Elastic microcracked bodies with random properties

The stochastic properties of a ‘dense’ distribution of microcracks in an elastic body are analyzed by using a multifield continuum model describing the influence of the microcracks on the gross mechanical behavior of the body. Numerical examples are presented: the strain localization phenomena that have been already found in deterministic bodies are confirmed, and similar patterns are shown to exist also for the stochastic moments of the displacements. In particular, the patterns in the portraits of skewness and kurtosis become stronger when the correlation of the distance between neighboring microcracks increases. Such a distance is considered as a uni-variate non-Gaussian random field. Strain localization is an indicator toward the irreversible growth and coalescence of microcracks.     

Evolution of arbitrary moments of radiant intensity distribution for partially coherent general beams in atmospheric turbulence

The evolution law of arbitrary order moments of the Wigner distribution function, which can be applied to the different spatial power spectra, is obtained for partially coherent general beams propagating in atmospheric turbulence using the extended Huygens–Fresnel principle. A coupling coefficient of radiant intensity distribution (RID) in turbulence is introduced. Analytical expressions of the evolution of the first five-order moments, kurtosis parameter, coupling coefficient of RID for general beams in turbulence are derived, and the formulas are applied to Airy beams. Results show that there exist two types for general beams in turbulence. A larger value of kurtosis parameter for Airy beams also reveals that coupling effect due to turbulence is stronger. Both theoretical analysis and numerical results show that the maximum value of kurtosis parameter for an Airy beam in turbulence is independent of turbulence strength parameter and is only determined by inner scale of turbulence. Relative angular spread, kurtosis and coupling coefficient are less influenced by turbulence for Airy beams with a smaller decay factor and a smaller initial width of the first lobe.     

Scanning-laser glint measurements of sea-surface slope statistics

A scanning-laser glint meter designed for field measurements of sea-surface slope statistics is described. A narrow laser beam is scanned in a line, and specular reflections (glints) are counted in bins according to their slope angle. From normalized glint histograms, moments to the fourth order are calculated, and slope probability density functions are approximated with a Gram-Charlier expansion. Field measurements with this instrument show good agreement with previous results when the stability (essentially air-sea temperature difference) is near neutral (zero). Under conditions of negative stability (warm ocean), both the mean-square slope and the probability density function kurtosis increase.     

水下航行器空化噪声偏度和峰度分析

空化是水下高速航行器在进行总体设计时需要考虑的重要因素。对空化产生的噪声特性进行研究有助于水下航行器航行性能和目标检测能力的提高。文章首先介绍了空化噪声的形成机理和数学模型,然后基于实测数据对空化噪声相应的统计特性进行了分析,计算了典型的单个声脉冲形式的空化噪声的概率分布、偏度和峰度,对比分析了无空化发生与空化发展比较充分条件下实测空化噪声的偏度和峰度特性。结果表明,在空化充分发展时水下航行器空化噪声具有明显的非高斯特性,其偏度和峰度值明显大于理想高斯噪声或者实际的海洋环境噪声。有利于提高空化条件下水下航行器对目标辐射噪声检测的准确性以及检测系统的环境适应能力。     

An alternative time-domain index for condition monitoring of rolling element bearings—A comparison study

Statistical moments have been widely used for detection and diagnosis of rolling element bearing damage. Among them, Kurtosis and Honarvar third moment S_r are the major parameters. In this paper a new statistical moment, from the viewpoint of Rényi entropy, is derived, which is shown to be as effective as kurtosis and S_r. Comprehensive comparisons of kurtosis, S_r and this moment are performed, and the results from simulations and experiments show the new moment has a better overall performance than kurtosis and S_r. On the one hand, this moment behaves much like kurtosis but is less susceptible to spurious vibrations, which is considered to be one of the main shortcomings of higher statistical moments including kurtosis. On the other hand, from the viewpoint of sensitivity to incipient faults, which is the major drawback of lower statistical moments including S_r, the new moment is superior to S_r. Moreover, the sensitivity of this new moment to changes of bearing speed and load is also less than kurtosis and is close to that of S_r.     

Characterization of optical fields with quantized orbital angular momentum by invariants of higher order moments of radial coordinates

We show that the skewness and kurtosis parameters of optical fields with quantized orbital angular momenta (OAM) and integer topological charge, which depend on the propagation distance only through normalized transverse coordinates, remain invariant at propagation through axially symmetric first-order optical systems, if defined in terms of higher-order moments of the radial coordinate. The values of these parameters, which characterize the shape of optical fields, depend on the type of OAM beams (Gaussian, Laguerre–Gauss or spiral phase plates in far-field) and the topological charge. As a result, the skewness and kurtosis can be used to identify the type of OAM beam and the absolute value of the topological charge for Gaussian and Laguerre–Gauss vortices encountered in most applications.     

A Method of Estimating the Process Capability Index from the First Four Moments of Non‐normal Data

A method is presented to estimate the process capability index (PCI) for a set of non‐normal data from its first four moments. It is assumed that these four moments, i.e. mean, standard deviation, skewness, and kurtosis, are suitable to approximately characterize the data distribution properties. The probability density function of non‐normal data is expressed in Chebyshev–Hermite polynomials up to tenth order from the first four moments. An effective range, defined as the value for which a pre‐determined percentage of data falls within the range, is solved numerically from the derived cumulative distribution function. The PCI with a specified limit is hence obtained from the effective range. Compared with some other existing methods, the present method gives a more accurate PCI estimation and shows less sensitivity to sample size. A simple algebraic equation for the effective range, derived from the least‐square fitting to the numerically solved results, is also proposed for PCI estimation. Copyright © 2004 John Wiley & Sons, Ltd.     

块石空间分布对土石混合体边坡稳定性的影响

土石混合体是自然界中分布广泛的一种地质材料, 受组分多样性, 块石粒径大小不一和块石空间分布位置随机等因素的影响, 其物理力学性质十分复杂。在土石混合体边坡的稳定性分析中, 以往大多将其简化为均匀地质体结构而不考虑块石的影响。该文采用有限元强度折减法, 对土体和四种不同类型块石分布的土石混合体边坡分别在自重状态(准静态)和打桩作用(动态)下的稳定性进行了分析, 探讨了块石空间分布对剪切带和边坡稳定性的影响。计算结果表明, 土石混合体边坡内的剪切带具有明显的绕石效应, 位于土石混合体边坡内部不同位置处的块石对边坡稳定性产生的影响不同。     

Sensitivity analysis using anchored ANOVA expansion and high‐order moments computation

An anchored analysis of variance (ANOVA) method is proposed in this paper to decompose the statistical moments. Compared to the standard ANOVA with mutually orthogonal component functions, the anchored ANOVA, with an arbitrary choice of the anchor point, loses the orthogonality if employing the same measure. However, an advantage of the anchored ANOVA consists in the considerably reduced number of deterministic solver's computations, which renders the uncertainty quantification of real engineering problems much easier. Different from existing methods, the covariance decomposition of the output variance is used in this work to take account of the interactions between non‐orthogonal components, yielding an exact variance expansion and thus, with a suitable numerical integration method, provides a strategy that converges. This convergence is verified by studying academic tests. In particular, the sensitivity problem of existing methods to the choice of anchor point is analyzed via the Ishigami case, and we point out that covariance decomposition survives from this issue. Also, with a truncated anchored ANOVA expansion, numerical results prove that the proposed approach is less sensitive to the anchor point. The covariance‐based sensitivity indices (SI) are also used, compared to the variance‐based SI. Furthermore, we emphasize that the covariance decomposition can be generalized in a straightforward way to decompose higher‐order moments. For academic problems, results show the method converges to exact solution regarding both the skewness and kurtosis. Finally, the proposed method is applied on a realistic case, that is, estimating the chemical reactions uncertainties in a hypersonic flow around a space vehicle during an atmospheric reentry. Copyright © 2015 John Wiley & Sons, Ltd.     

Hot entanglement in a simple dynamical model

Abstract

The theory of balanced homodyne and heterodyne detection is developed for inputs in which the signal field is in an arbitrary quantum state and the local-oscillator field is in a highly excited coherent state. Exact expressions are derived for the photocount moment-generating functions in the special case of a coherent signal. For more general signals, the first two moments of the photocount probability distribution are determined. The moments are evaluated for the examples of a coherent signal with a chaotic noise component, and for squeezed light derived from a degenerate and from a non-degenerate parametric amplifier. The corresponding moments for direct detection are obtained so that comparisons can be made. The Kelley-Kleiner photocount distribution formula is adapted to balanced detection schemes. Light beams are characterized throughout by their energy fluxes, and the theory accordingly describes steady-state experiments.     

An efficient point estimate method for probabilistic analysis

A new and efficient point estimate method is developed to calculate the statistical moments of a random quantity, Z, that is a function of n random variables, X. The method is an extension of Rosenblueth's two-point concentration method. The method uses m × n concentrations matching up to the first m × n non-crossed moments of each random variable and crossed second order moments of the random variables. The kth moment of Z is calculated by weighting the value of Z to the power of k evaluated at n × m locations. Simple to use formulas are provided for two special cases of the method, i.e. 2n-concentration scheme and 2n + 1-concentration scheme. This 2n-concentration scheme considers the skewness of probability density function. The 2n + 1-concentration scheme considers the skewness and kurtosis of probabilility density function. The correlations between the random variables are considered by using a rotational transformation based on the eigenvector of covariance matrix. Illustrative examples are presented.