Quantitative spectrophotometry of scattering media via frequency-domain and constant-intensity measurements

The attenuation of light by scattering and absorbing media is nonlinearly dependent upon the absorption coefficient, since detected light has experienced many different flight times. The frequency response of such a sample to modulated light is also nonlinear. We derive an expression for the attenuation that includes both the absorption coefficient and the modulation frequency. Its form is a power series whose coefficients are the cumulants of the temporal point-spread function (TPSF). Recasting this expression in terms of intensity leads to a similar expression with the cumulants replaced by the moments of the TPSF. A means of exploiting this relationship to produce estimates of the absolute concentration of the absorbing species is suggested.     

Density Matrix Elements and Moments for Generalized Gaussian State Fields

Abstract

A single-mode radiation field with Gaussian Wigner functions (Gaussian state field) is studied. The single-mode Gaussian-field generating function for both the density matrix elements and the moments of the creation and annihilation operators is calculated. It is shown that the matrix elements as well as the moments of different orders can be expressed in terms of Hermitian polynomials of two variables. Furthermore, the cumulants of the Gaussian state field are calculated. Finally, we generalize the Gaussian state field and compute the corresponding moments and density matrix elements.     

Stochastic response moments for linear systems

State space moment analysis is developed as a practical tool for investigating the response of a linear system subjected to stochastic excitation. General formulations are presented to show that the method can be used to evaluate response moments, or cumulants, of any order for both stationary and nonstationary response. The limitation is that the excitation of the linear system must be a generalized white noise called a delta-correlated process. This generalization of the Lyapunov method for finding response covariances gives a comparable matrix method for finding the higher order moments which are often important in predicting failure due to first-passage or fatigue. The technique used here involves rewriting the mth order tensor of mth order cumulants into a minimum length vector, making use of all inherent symmetry, in order to minimize the size of the resulting matrix. Easily implemented algorithms are presented for finding the terms in this matrix. General relationships are also given relating the eigenvalues and eigenvectors of this matrix for mth order cumulants to those of a much smaller matrix. This eigen solution is needed for evaluating nonstationary response cumulants, and the given relationships provide a particularly efficient method for evaluating the eigenvalues. The method is illustrated by evaluating the 35 fourth cumulants of nonstationary response for a class of two-degree-of-freedom oscillators.     

Non-Gaussian stochastic response of nonlinear compliant platforms

A moment function method is presented to estimate the stochastic response of compliant offshore platforms with nonlinearity in stiffness based on non-Gaussian closure. For guyed towers with clump weight, the nonlinearity in stiffness is of the softening type. The random wave loading is expressed in terms of a rational spectrum, making the system Markovian. Using Ito's rule for stochastic differentiation, differential equations for moments up to the fourth order are developed. The system of equations is closed by considering the fifth and sixth cumulants to be zero. For stationary response, differential equations become algebraic equations. The moments are obtained by solving the system of nonlinear algebraic equations. It is observed that the Gaussian closure method is inadequate for defining the complete probabilistic characteristics of the response.     

Trivariate characteristics of intensity fluctuations for heavily saturated optical systems

Trivariate cumulants of intensity fluctuations have been computed starting from a trivariate intensity probability distribution function, which rests on the assumption that the variation of intensity has a maximum entropy distribution with the constraint that the total intensity is constant. The assumption holds for optical systems such as a thin, long, mirrorless gas laser amplifier where under heavy gain saturation the total output approaches a constant intensity, although intensity of any mode fluctuates rapidly over the average intensity. The relations between trivariate cumulants and central moments that were needed for the computation of trivariate cumulants were derived. The results of the computation show that the cumulants have characteristic values that depend on the number of interacting modes in the system. The cumulant values approach zero when the number of modes is infinite, as expected. The results will be useful for comparison with the experimental triavariate statistics of heavily saturated optical systems such as the output from a thin, long, bidirectional gas laser amplifier.     

Using partial probability weighted moments and partial maximum entropy to estimate quantiles from censored samples

The maximum entropy principle constrained by probability weighted moments is an useful technique for unbiasedly and efficiently estimating the quantile function of a random variable from a sample of complete observations. However, censored or incomplete data are often encountered in engineering reliability and lifetime distribution analysis. This paper presents a new distribution free method for the estimation of the quantile function of a non-negative random variable using a censored sample of data, which is based on the principle of partial maximum entropy (MaxEnt) in which partial probability weighted moments (PPWMs) are used as constraints. Numerical results and practical examples presented in the paper confirm the accuracy and efficiency of the proposed partial MaxEnt quantile function estimation method for censored samples.     

Extreme quantile estimation using order statistics with minimum cross-entropy principle

The paper presents a general approach to the estimation of the quantile function of a non-negative random variable using the principle of minimum cross-entropy (CrossEnt) subject to constraints specified in terms of expectations of order statistics estimated from observed data.Traditionally CrossEnt is used for estimating the probability density function under specified moment constraints. In such analyses, consideration of higher order moments is important for accurate modelling of the distribution tail. Since the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain, the use of CrossEnt quantile estimates in extreme value analysis is fairly limited.The present paper is an attempt to overcome this problem via the use of probability weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting a PWM as the moment of quantile function, the paper derives an analytical form of quantile function using the CrossEnt principle. Monte Carlo simulations are performed to assess the accuracy of CrossEnt quantile estimates obtained from small samples.     

Estimation of Particle Size Distribution Based on Observed Weights of Groups of Particles

Particulate matter is collected on some sampling medium. The particles of interest are collected by some means, and weighed as a group. Weights are observed for a number of replicate samples. These observed weights may include background contribution from particles existing on the sampling medium prior to its use. Given these data, plus similar date to establish the background, the problem is to estimate the average number of particles per sample, and their weight distribution. The estimation is accomplished by equating sample moments to population moments. The first four moments of the population are found for an arbitrary weight distribution function possessing finite first four moments. Specific estimates are found in the event the weight distribution is exponential in form, and approximate sampling variances of these estimates are derived. A numerical example is included.     

The Percentile Points of Distributions Having Known Cumulants

In an earlier study of the uses of moments and cumulants in the specification of statistical distributions, the authors developed explicit asymptotic expansions, expressing any desired percentile point of such distributions in terms of known cumulants.

The general formulae are now presented as far as the sixth adjustment, based on the eighth cumulant, and also numerical tables showing the coefficients of all terms for ten chosen levels of significance over the range 0.5 to 0.0005 (single tail), together with the first five Ilermite polynomials and tables for the common tests of significance, χ², t and z. at the same levels.     

Probabilistic model of multiple light scattering based on rigorous computation of the first and the second moments of photon coordinates

We consider a concise method based on recurrent relations that permit rigorous computing of the first and the second moments of the components of the vector locating a randomly walking photon in an infinite homogeneous light-scattering medium. On assumption that the components obey a three-dimensional Gaussian distribution a probability density for the photon locations at the Nth scattering event can readily be written down and the light-intensity distribution in the medium may be calculated. The results from theoretical analyses are compared with the solution of a light-diffusion equation and with results of Monte Carlo simulations and show a better fit with simulated data than the diffusion approximation.     

Probabilistic crack trajectory analysis by a dimension reduction method

This paper proposes a novel analysis method of stochastic crack trajectory based on a dimension reduction approach. The developed method allows efficiently estimating the statistical moments, probability density function and cumulative distribution function of the crack trajectory for cracked elastic structures considering the randomness of the loads, material properties and crack geometries. First, the traditional dimension reduction method is extended to calculate the first four moments of the crack trajectory, in which the responses are eigenvectors rather than scalars. Then the probability density function and cumulative distribution function of the crack trajectory can be obtained using the maximum entropy principle constrained by the calculated moments. Finally, the simulation of the crack propagation paths is realized by using the scaled boundary finite element method. The proposed method is well validated by four numerical examples performed on varied cracked structures. It is demonstrated that this method outperforms the Monte Carlo simulation in terms of computational efficiency, and in the meanwhile, it has an acceptable computational accuracy.     

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.     

Wigner distribution moments in fractional Fourier transform systems

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.     

基于分位数函数和Bootstrap分布的不确定度评定研究

最大信息熵方法是基于概率分布评定测量不确度的主要方法之一。其所依赖的高阶矩需要较大样本的测量数据,而校准/检测实验室的测量一般为小样本,故用最大熵方法评定小样本测量不确定度缺乏一定的可靠性。提出了基于分位数函数和概率权重矩作为约束条件的最大信息熵不确定度评定法,把矩的计算从高次降为一次,并结合遗传算法求解概率分布,用Bootstrap分布估计扩展不确定度和包含区间,解决了由分位数区间估计分布不对称所致的复杂计算问题。     

Joint statistics of combined first- and second-order random processes

Two-term Volterra series are often used to describe non-linear random processes in subject areas as diverse as communication theory and the dynamics of offshore structures. It is well known that the characteristic function, the moments and the cumulants of such processes can be calculated analytically and that the probability density function can be calculated accurately and efficiently. This paper considers the joint statistics between two such random processes by deriving: (i) an exact expression for the joint characteristic function; (ii) an efficient means for calculating the joint moments; (iii) an ‘exact’ numerical means for calculating the joint probability density function (jpdf). For the special case of a combined first- and second-order process and a pure first-order process it is shown that it is possible to derive analytical expressions for the characteristic function and to calculate the jpdf accurately and efficiently using saddle-point integration. In addition to the above, the maximum entropy principle is used to calculate the jpdf.     

Relationships Among Moments of Order Statistics in Samples from Two Related Populations

It is shown that the moments of order statistics in samples drawn from a population symmetric about zero can be expressed in terms of the moments of order statistics in samples drawn from the population obtained by folding the symmetric population at zero. Also, odd moments of the largest order statistic in samples of even sizes and even moments of the largest order statistic in samples of odd sizes drawn from the folded population can be expressed purely in terms of the moments of the order statistics in samples drawn from the symmetric population. The cumulative round off error involved in numerical evaluation of the moments of order statistics from the symmetric population, using a table of the moments of the order statistics from the folded population, is not serious except for the mixed moments in which case, a bound on the maximum error is available. An application is also considered.     

累积量算法在PCS中的颗粒粒径反演

介绍了应用光子相关光谱法测量纳米颗粒粒径的基本理论,该理论利用了悬浮于介质中的散射体的布朗运动所引起的散射光强涨落现象.详细分析了利用累积量算法反演光子相关光谱法测量中的颗粒粒径.针对已经标定的PMMA颗粒,利用累积量算法进行反演,对反演结果进行分析.累积量算法比较简单,计算快速,计算结果重复性好,对单分散颗粒系以及窄分布的颗粒系能给出较准确的平均粒径以及与颗粒粒径分布有关的一些参数.     

单自由度非线性随机参数系统的可靠性分析

阐述了具有随机参数的单自由度非线性振动系统的可靠性方法，使用四阶矩技术确定了系统响应和状态函数的前四阶矩，应用Ｅｄｇｗｏｒｔｈ级数把未知响应和状态函数的概率分布展开成标准正态分布的表达式，从而获得了系统的可靠度。