Fourth Central Moment Research Articles

Comparison of Moments for Classical‐, Quasi‐, and Convolution‐Fickian Dispersion of a Conservative Tracer

Transport moments up to fourth order for convolution‐, quasi‐, and classical‐Fickian dispersion of a conservative tracer are presented and compared. Convolution‐Fickian yields a non‐Gaussian distribution while quasi‐ and classical‐Fickian result in Gaussian type distributions regardless of the distribution of a stationary velocity field. The first and second central spatial moments are the same for convolution‐ and quasi‐Fickian dispersion, while the third and fourth central moments differ.

An Eulerian scheme for the second‐order approximation of subsurface transport moments

The moments of a conservative tracer cloud migrating in a mean uniform flow field are estimated using an operator approximation scheme; results are presented for the second, third, and fourth central moments in the mean flow direction. It is assumed that the spatially variable flow field, and therefore the tracer migration problem itself, is amenable to a probabilistic description; the effects of local dispersion on cloud migration are neglected in this study. Variation in the flow field is assumed to be the result of spatial variation in the hydraulic conductivity; spatial variation in porosity is assumed negligible. The operator approximation scheme, as implemented in this study, is second‐order correct, which requires a second‐order correct approximation of the velocity field correlation structure. Because estimation of the velocity correlation structure is decidedly the most difficult aspect of second‐order analysis, an ad hoc extension of the imperfectly stratified approximation developed earlier is implemented for this purpose. The first‐order approximation resulting from the operator expansion scheme is equivalent to small perturbation Eulerian results presented earlier (Naff, 1990, 1992). The infinite‐order approximation resulting from this scheme is equivalent to the exponential operator results obtained by Van Kampen (1976).

Use of higher order signal moments and high speed digital sampling technique for neutron flux measurements

The second (conventional variance or Campbell signal), the third, and the modified fourth order central signal moments associated with the amplified and filtered currents from two electrodes of an ex-core neutron sensitive fission detector were measured versus the reactor power of the 1-MW TRIGA reactor in Mexico City. Two channels of a high-speed (400-MHz) multiplexing data sampler and an analog-to-digital converter with 12-b resolution and 1-Mword buffer memory were used. The data were further retrieved into a PC, and estimates for autocorrelation and cross-correlation moments up to the fifth order, coherence, skewness, excess, etc., quantities were calculated offline. Five-mode operation of the detector was achieved, including conventional counting rates and currents in agreement with theory and the authors' previous results with analog techniques. The signals are proportional to the neutron flux and reactor power in some flux ranges. The suppression of background noise is improved and the lower limit of the measurement range is extended as the order of moment is increased, in agreement with theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Single frequency laser probing of velocity component correlations and transport properties of Ba+ drifting in Ar

Velocity distributions for Ba+ ions drifting in argon under the influence of an external electric field are measured at directions of 0°, 45°, and 90° with respect to the electric field using single frequency laser-induced fluorescence probing. Values for the reduced mobility, translational temperatures (second central moments), skewness (third central moment), and excess (fourth central moment) are presented as a function of field strength up to E/N values of 201 Td, which corresponds to a mean Ba+/Ar center-of-mass collision energy of 0.371±0.038 eV. Maxima are observed in both the reduced mobility, 2.40±0.05 cm2 V−1 s−1 at 160 Td, and in the skewness, 0.95±0.03 at 119 Td. The dimensionless skewness parameter characterizes the asymmetry of the velocity distributions and is the ratio of the cube root of the third central moment to the square root of the second central moment. A comparison of the moments of the measured 45° velocity distribution to a distribution synthesized without correlation from the 0° and 90° distributions shows that a positive correlation exists between velocity components parallel and perpendicular to the electric field. This is the first experimental verification of velocity component correlation in drifting ions.

Robustness to nonnormality of the Durbin-Watson test for autocorrelation

Abstract This study investigates the robustness to nonnormality of the null distribution of the Durbin-Watson test for autocorrelation in regression errors. The first four moments of the null distribution are derived to the order O ( 1 n 3 ) when the regression errors are nonnormal, with n the sample size. It is found that nonnormality has an insignificant effect on the mean and the fourth central moment of the distribution. The variance tends to be deflated (inflated) if the error distribution is long-tailed (short-tailed). The third central moment is reduced if the error distribution is skewed (left or right). Both skewness and kurtosis of the distribution are affected by nonnormality, but these effects are negligible in large samples. It seems the test is relatively robust for moderate nonnormality and moderately large sample size. These effects are also found to have insignificant influence from the regressors.

The distribution of velocities of a concentric sphere in a dilute dispersion of spheres sedimenting in a spherical container

We consider the motion of identical spheres in an incompressible fluid which is bounded by a spherical container wall. The test sphere is momentarily concentric with the container and the centres of the other spheres are distributed in the accessible region according to a Poisson process. We derive the third and fourth polyadic central moments of velocity and examine their components. These moments, together with the mean and covariance which are known from earlier work, provide a good description of the theoretical initial distribution. Estimates of the hard-sphere equilibrium distribution are obtained from computer simulations using a fixed number of hard spheres. Simulated values agreed closely with the theoretical, indicating that the Poisson assumption, which greatly simplifies the analysis, has little effect on the results. In small containers, the vertical component of velocity is skewed downward and is platykurtic. In large containers, the distributions of all components approach normality.

X-ray Diffraction Study of Layer Shift in Chlorites by Fourth Moment

An expression connecting the fourth central moment of an X-ray diffraction line profile with the particle size and stacking disorders of the type of layer displacement in the specimen studied, has been derived on the supposition that the line broadening is due to particle size as well as stacking disorders of the above type. It has been shown that from a single reflection, the defect parameter can be easily calculated and the method can be utilised as widely as the method of second moment (i.e. variance) in the study of disorders in layer silicate clay minerals.

Uncertainty analysis of fault trees with statistically correlated failure data

An exposition is made of the method of moments for the uncertainty analysis of fault trees with statistically correlated failure data. Exact formulas for the mean and variance of the top-event probability, as well as accurate formulas for its third and fourth central moments, are derived. An approximate analysis of partial correlation is also given. Two test problems are studied, the first of which demonstrates a good agreement between the present method and the Monte Carlo method for a case of total correlation, and also reveals that the effect of correlation can be quite significant. The second test problem studies the effect of partial correlation for different parameters of the basic events.

Mixtures, myths and kurtosis

Kurtosis, usually as measured by the standardised fourth central moment, has been examined on a number of occasions by observing the effect of contaminating the distribution, that is, mixing in another distribution. However, superficial treatment can lead, and indeed has led, to misunderstandings. This paper considers, firstly for a symmetric distribution contaminated at two points symmetrically placed around its centre and then for a mixture of two continuous symmetric distributions, the behaviour of three measures of kurtosis. This is done in general and not just as the mixing proportion tends to zero as in the influence function approach. It is seen that when both scale and kurtosis change, the latter is not necessarily intuitive. It is also illustrated that parameter interpretation in terms of distributional properties such as shape can be misleading without the use of the appropriate distributional partial ordering

New studies of X-ray diffraction line profiles from aggregates of distorted crystallites I

The fourth central moment of an X-ray diffraction profile from an aggregate of distorted crystallites has been expressed by Mitra (1964a) as a function of the crystallite size, strain and strain gradients in the specimen. While the usual methods of line profile analysis yield information regarding either the apparent strain or the rms strain, the present study provides additional information regarding strain distribution in the form of strain derivatives and rms displacements of atoms over a given distancet in the direction of study. The strain parameters like 〈ee′〉, 〈ee″〉 have been obtained from fourth moment of the strain profile against range plots. The strain parameters thus obtained have subsequently been used to determine the rms displacements of the atoms. Alloys of copper and zinc at different stages of cold working and annealing have been studied by this method. The results have been discussed in the light of dislocation distribution, polygonisation and grain growth as well as distortion waves in the distorted crystals.

On Testing Equality of Means of Correlated Variates with Incomplete Data

AbstractA statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. Expressions for the second and fourth central moments of the statistic are obtained. These moments are used to approximate the distribution of the statistic by a Student's t distribution under the null hypothesis. The powers of the test are computed and compared with those of the conventional paired t and the other known statistics.

The Number of Observed Classes from a Multiple Hypergeometric Distribution

Abstract The distribution of the number of classes observed, when sampling without replacement from a finite population comprised of k classes, is derived and factorial moments are given. The mean and up to the fourth central moment are presented. A simple application to an ecological problem illustrates the computations, and other potential applications are discussed.

Estimation of variance of sample variance in sample random sampling

As it is known, the general m.v.u estimator of the variance of sample variance in simple random sampling is given by a linear combination with fixed coefficients of the fourth central moment and the square of the variance observed in the sample. In this paper, the best unbiased estimator is determined when the coefficient are allowed to depend on the moments of the standardized population. Explicit formulas are given for sapling with repetition. The results are illustrated on ten theoretical populations of different shape. By means of a simulation exercise it is investigated what happens when in the best estimator population moment, assumed unknown, are substituted by the corresponding sample moments. In the appendix other results concerning the fourth central moment and the square of variance are given.

Application of the method of moments to X-ray diffraction line profiles from paracrystals: Native cellulose fibres

The `microparacrystallite size' and `distortion parameter' of delignified ramie, hemp and jute have been determined – with the assumption that these materials are paracrystalline in nature – with the second and fourth central moments of X-ray diffraction line profiles. A new method of curvature correction has been developed. Background as well as non-additivity corrections have also been accounted for. Theories of determining the `microparacrystallite size' and the `distortion parameter' from single reflections independently from each of these two central moments have been developed, and described. Determination of these two parameters for several directions in the samples studied have been based on the above work. It has been shown that, in conformity with the recent findings of Hosemann &amp; Balta Calleja [Ber. Bunsenges. Phys. Chem. (1980), 84, 91], the larger the paracrystalline distortion the smaller is the `microparacrystallite size'.

A Methodology for obtaining the probability density function of the present worth of probabilistic cash flow profiles

Abstract A new approach is presented for analyzing probabilistic cash flow profiles. Integral transform theory is used to obtain a complete analytic characterization of the probability density function of the net present worth of such cash flow profiles. This represents an extension of the current techniques of risk analysis. The methodology includes the usual evaluation and use of the expected value and the second, third and fourth central moments of the density of present worth. However, these descriptive measures do not always provide sufficient information for a complete managerial analysis. The method presented here enables management to fully differentiate between competing investment alternatives using existing methods, such as stochastic dominance, which are based on a knowledge of the form of the associated probability density functions. Simple formulae for evaluating moments of any or all orders are given. Illustrative examples are included to accompany the analytic development for several repre...

Stochastic Breakeven Analysis

This paper presents a practical approach for studying the profit probability distribution defined in stochastic breakeven analysis. After presenting a procedure for calculating the mean and the second to fourth central moments of the profit distribution, the several important uses of these four moments are discussed. A major application of the four moments is in the fitting of a Pearson's curve. Using a numerical example in conjunction with a set of published tables, we then demonstrate the simplicity of our proposed approach as well as the resultant high accuracy in estimating probabilities of various profit levels.

Experimental characterization of elution profiles in gas chromatography using central statistical moments : Study of the relationship between these moments and mass transfer kinetics in the column

A high-precision gas chromatograph under computer control, with digital data acquisition of the chromatographic signal, was used to measure the central statistical moments of chromatographic zones. It is shown that for fairly symmetrical peaks, the moments are determined with a better accuracy, even at a low signal-to-noise ratio, from a least-squares fit of the Gram-Charlier series on the experimental elution profiles. The Gram-Charlier series is a good model for representing the peak elution profile provided that the skew coefficient is small. The second central moment derived from the Gram-Charlier model was studied as a function of the carrier gas velocity, using the usual HETP concept. The third and the fourth central moments were found to follow similar relationships to the carrier gas velocity. For strongly tailing peaks, it is shown that the moments are not sufficient for characterizing the peak shape. This tailing has a kinetic origin and results from slow mass transfer in the stationary phase. The time constant of the exponential decay adjusted on the peak tail gives a good indication of the desorption time of the molecule from the most active sites of the packing.

Estimating Moments of Universe Scores and Associated Standard Errors in Multiple Matrix Sampling for All Item-Scoring Procedures

Described herein are formulas and computational procedures for estimating the mean and second through fourth central moments of universe scores through multiple matrix sampling. Additionally, procedures are given for approximating the standard error associated with each estimate. All procedures are applicable when items are scored either dichotomously or polychotomously.

The dependence of shell model state densities on angular momentum

The centroid and second through fourth central moments of shell model state densities are shown to have a very simple dependence on the projection of the angular momentum along the z-axis. This simple dependence follows from the fact that the single-particle angular momenta add statistically for a large number of nucleons.

Some properties of an estimator for the variance of a normal distribution

is the multiple of the sample variance with the minimum mean-squared error (m.s.e.) in estimating variance of a normal distribution (for example, see Kendall and Stuart [3]). Let XI, Xs,..-, X~ be a random sample from a population with mean /~ and variance a s, and let/~s=t~Ja 4 be a coefficient of kurtosis where p~ is the fourth central moment. Under the situation that /~ is known as an a priori information, Singh et al. [5] proposed the estimator