Weibull-type Distribution Research Articles

Low-mass Stars: Their Protoplanetary Disk Lifetime Distribution

While most protoplanetary disks lose their gas within less than 10 Myr, individual disk lifetimes vary from <1 Myr to ≫20 Myr, with some disks existing for 40 Myr. Mean disk half-lifetimes hide this diversity; only a so-far nonexisting disk lifetime distribution could capture this fact. The benefit of a disk lifetime distribution would be twofold. First, it would provide a stringent test on disk evolution theories. Second, it could function as an input for planet formation models. Here, we derive such a disk lifetime distribution. We heuristically test different standard distribution forms for their ability to account for the observed disk fractions at certain ages. We here concentrate on the distribution for low-mass stars (spectral types M3.7–M6, M s ≈ 0.1–0.24 M ⊙) because disk lifetimes depend on stellar mass. A Weibull-type distribution (k = 1.78, λ = 9.15) describes the observational data if all stars have a disk at a cluster age t c = 0. However, a better match exists for lower initial disk fractions. For f(t=0) = 0.65, a Weibull distribution (k = 2.34, λ = 11.22) and a Gaussian distribution (σ = 9.52, μ = 9.52) fit the data similarly well. All distributions have in common that they are wide, and most disks are dissipated at ages >5 Myr. The next challenge is to quantitatively link the diversity of disk lifetimes to the diversity in planets.

A New Inverse Extended Weibull Distribution for Modelling Insurance Loss Data

This paper proposes a new three-parameter inverse Weibull-type distribution via a similar way of constructing a skew normal distribution. The additional shape parameter increases the distribution’s modelling capability as other three-parameter inverse Weibull-type distributions. The mathematical properties of this inverse extended Weibull distribution are studied and proven. In the empirical study of four insurance datasets from Australia, Denmark and the USA, we show that the proposed distribution performs very well compared with other three-parameter competitors.

Mechanical and critical current characteristics of high-strength BiSrCaCuO multi-filamentary tapes reinforced with thicker Ni-alloy laminations with various pre-tensions

We investigated the stress–strain relation and the applied strain dependence of the critical current of high-strength Bi2Sr2Ca2Cu3O (Bi2223) R&D tapes reinforced by thick Ni-based alloy sheets with various pre-tensions applied during the lamination process. We confirmed that the thickening of Ni-alloy laminations is useful for enhancing Young’s modulus compared with commercial tapes ( thick laminations). decreases gradually and reversibly with applied strains and then degrades irreversibly with larger tensile and compressive strains. The reversible strain regime tends to shift toward the tensile side with larger , indicating that application of large is effective in improving the reversible tensile strain limit. The observed can be reproduced well by model considering -linear critical current density, cross-sectional strain distribution, and Weibull-type distribution of filament damage. We clarified that different irreversible degradations with large tensile and compressive strains can be understood by differences in the dimensions and number of cracks and ruptures (tensile strains) and bucklings (compressive strains).

Processing and mechanical characterization of short carbon fiber-reinforced epoxy composites for material extrusion additive manufacturing

Fiber-reinforced polymer composites have been extensively utilized in recent years as feedstock materials for material extrusion additive manufacturing (AM) processes to improve strength, stiffness, and functionality of printed parts over unfilled printed polymers. However, the widespread adoption of AM of fiber-reinforced polymer composites requires a deeper understanding of the process-structure-property relationships in printed components, and such relationships are not well understood yet. Fiber length is critically important to the mechanical performance of short fiber composites, but very few studies to-date have focused on how the fiber length distribution (FLD) evolves during processing of composite feedstocks and how this evolution affects printing behavior and mechanical properties in 3D-printed composites. In this work, FLD is measured for carbon fiber reinforced epoxy composites over a wide range of ink compositions and shear mixing times, and the distributions are fit with a Weibull-type distribution function. The effects of FLD on the tradeoff between ink processability, ink rheology, printing behavior and mechanical properties are investigated. Furthermore, the effects of printing parameters (nozzle size and print speed) on mechanical anisotropy and fiber orientation distribution (FOD) in printed composites are explored. Mechanical properties of printed composites are characterized via 3 pt-flexural testing, and microstructure is investigated using optical and scanning electron microscopy (SEM), and x-ray computed tomography. Finally, the fitted Weibull parameters are fed into a composite model that incorporates FLD and FOD, and model predictions are found to be in excellent agreement with experimental observations.

One-Parameter Weibull-Type Distribution, Its Relative Entropy with Respect to Weibull and a Fractional Two-Parameter Exponential Distribution

A new one-parameter distribution is presented with similar mathematical characteristics to the two parameter conventional Weibull. It has an estimator that only depends on the sample mean. The relative entropy with respect to the Weibull distribution is derived in order to examine the level of similarity between them. The performance of the new distribution is compared to the Weibull and in some cases the Gamma distribution using real data. In addition, the Exponential distribution is modified to include an extra parameter via a simple transformation using fractional mathematics. It will be shown that the modified version also exhibits Weibull characteristics for particular values of the second parameter.

Kernel regression with Weibull-type tails

We consider the estimation of the tail coefficient of a Weibull-type distribution in the presence of random covariates. The approach followed is non-parametric and consists of locally weighted estimation in narrow neighbourhoods in the covariate space. We introduce two families of estimators and study their asymptotic behaviour under some conditions on the conditional response distribution, the kernel function, the density function of the independent variables, and for appropriately chosen bandwidth and threshold parameters. We also introduce a Weissman-type estimator for estimating upper extreme conditional quantiles. The finite sample behaviour of the proposed estimators is examined with a simulation experiment. The practical applicability of the methodology is illustrated on a dataset of sea storm measurements.

Weibull-type speckle distributions as a result of saturation in stimulated scattering processes

AbstractDuring the propagation of an optically smoothed laser beam through a warm plasma the speckle field pattern and the corresponding speckle intensity distribution are modified in time and along the laser propagation direction. It is shown here that the laser–plasma interaction can change the character of speckle statistics from an initially exponential-type limit law to a Weibull-type law. The Weibull distribution is characterized by a power-law-type behavior in a limited interval of the random variable, which is, in the present case, the speckle intensity. The properties of the speckle distributions are studied using methods of extremal and order statistics. The scattering instability process (here stimulated Brillouin forward scattering) causing the change in speckle statistics has an onset behavior associated with a “critical gain” value, as pointed out in work by Rose and DuBois (1993b). The saturation of the instability process as a function of intensity explains the limited interval of the Weibull-type speckle distribution. The differences in the type of the speckle statistics are analyzed by using “excess over threshold” methods relying on the generalized Pareto distribution, which clearly brings to evidence the transition from an exponential type distribution to the Weibull-type distribution as a function of the instability gain value, that is, from the regime below critical gain to values above the critical gain.

On the surface wind speed probability density function over complex terrain

[1] The physical mechanisms determining the shape of the surface wind speed probability density function over complex terrain are investigated using observations from a dense mesoscale network and a high spatial resolution mesoscale simulation for the 1992 to 2005 period. Results indicate that the atmospheric stability plays a major role in controlling the shape of the wind speed distribution but its effects are strongly modulated by the contribution of different atmospheric scales of motion such as the mesoscale or synoptic scale. The local topographic features further modulate the relative contribution of each mechanism. As a consequence of the complicated interaction of these atmospheric processes the surface wind speed distribution can present a complicated shape that is not always expected to fit a unimodal Weibull type distribution over complex terrain regions.

Asymptotics for the Hirsch Index

Abstract. The last decade methods for quantifying the research output of individual researchers have become quite popular in academic policy making. The h-index (or Hirsch index) constitutes an interesting combined bibliometric volume/impact indicator that has attracted a lot of attention recently. It is now a common indicator, available for instance on the Web of Science. In this article, we establish the asymptotic normality of the empirical h-index. The rate of convergence is non-standard: , where f is the density of the citation distribution and n is the number of publications of a researcher. In case that the citations follow a Pareto-type respectively a Weibull-type distribution as defined in extreme value theory, our general result specializes well to results that are useful for practical purposes such as the construction of confidence intervals and pairwise comparisons for the h-index. A simulation study for the Pareto-type case shows that the asymptotic theory works well for moderate sample sizes already.

Analysis of non-steady-state distribution functions for grain growth and coarsening

The description of non-steady-state grain growth or precipitate coarsening using object radius distribution functions with multiple time-dependent parameters (distribution concept) appears promising. The present paper deals with the simplest case of non-steady-state distribution functions with two parameters–the first one scaling the object radius, the second determining the shape of the distribution function. The main question concerns the physical basis behind the evolution of these two parameters. The principle of maximum dissipation has proven to be a suitable tool to derive the evolution equations. Semi-analytical solutions for the evolving parameters of arbitrary two-parameter distribution functions can be developed. As examples, Kirkaldy- and Weibull-type distribution functions are investigated. It is shown that the parameters of the Kirkaldy distribution function are not independent and, thus, the general non-steady-state analysis fails. For a Weibull-type distribution function, nearly exact and simple analytical expressions for both parameters are presented and discussed for the grain growth and coarsening cases.

EFFECT OF HYDROGEN PEROXIDE CONCENTRATION AND pH ON INACTIVATION KINETICS OF ESCHERICHIA COLI

ABSTRACT The inactivation kinetics of Escherichia coli ATCC 35218 in hydrogen peroxide solutions at different concentrations (0–3.00% w/v) and pH values (3.0–7.2) at 25C was studied. Effectiveness of H2O2 solutions against E. coli ATCC 35218 appeared to improve at both higher H2O2 concentrations and more acidic pH values. As the H2O2 level was lowered, the influence of the pH of the solution on the rate of E. coli destruction was more notorious. Most of the inactivation semilogarithmic curves were not linear. Inactivation kinetics was modeled by using a Weibull type distribution of resistances. Quadratic polynomial models were applied to express the effects of H2O2 concentration and pH on the exposure time needed to reduce the microbial population by 5 log.PRACTICAL APPLICATIONSThis study gives useful information to predict the effect of H2O2 treatment on Escherichia coli survival as a basis for understanding the only effects of hydrogen peroxide concentration and pH levels, without considering other factors involved in sanitization such as produce surface characteristics, microorganism attachment features and presence of organic material other than the bacteria.Further similar studies using different E. coli strains and other pathogenic bacteria, biofilms and fresh produce would allow to optimize hydrogen peroxide treatments, avoiding or minimizing adverse effects on quality of fresh produce.

Asymptotics for the Hirsch Index

The last decade methods for quantifying the research output of individual researchers have become quite popular in academic policy making. The h- index (Hirsch, 2005) constitutes an interesting quality measure that has attracted a lot of attention recently. It is now a standard measure available for instance on the Web of Science. In this paper we establish the asymptotic normality of the empirical h-index. The rate of convergence is non-standard: ph=(1 nf(h)), where f is the density of the citation distribution and n the number of publications of a researcher. In case that the citations follow a Pareto-type or a Weibull-type distribution as defined in extreme value theory, our general result nicely specializes to results that are useful for constructing confidence intervals for the h-index.

Direct Tensile Performance of UHPCC Element Based on Damage Mechanics

Direct uniaxial tension test of ultra high performance cementitious composites I shape specimens have been investigated in this paper. A nonlinear analytical model based on continuum damage mechanics is developed to characterize tensile stress-strain constitutive response of UHPCC. Basic governing equations of damage evolution and material constitutive relation are established considering random damage which conforms to a modified Weibull type distribution proposed in this paper. Calculation suggests that Weibull distribution can describe damage evolution of UHPCC and predict the constitutive relation and damage evolution equation.

Determination of the adhesion energy of MEMS structures by applying Weibull-type distribution function

A novel test method to determine the adhesion energy of various MEMS surfaces in contact is presented. The method is based on an energy equilibrium state consideration between mechanical deformation energy and adhesion energy. The adhesion energy of the measurement structures is determined by means of pull-off tests and the results are modelled using a Weibull-type distribution function for stiction. In contrast to previous test methods, based on the same physical principle, our new method not only allows to characterize various materials but also various surface geometries. For a contact pair of a circular silicon disc of 200 mm in diameter and of a silicon substrate both covered with native oxide we measured under atmospheric conditions the average adhesion energy as to be 0.13 mJ/m 2. The low value of adhesion energy is due to the curvature of one contact face.

Extended Weibull type distribution and finite mixture of distributions

An extended form of Weibull distribution is suggested which has two shape parameters ( m and δ ). Introduction of another shape parameter δ helps to express the extended Weibull distribution not only as an exact form of a mixture of distributions under certain conditions, but also provides extra flexibility to the density function over positive range. The shape of density function of the extended Weibull type distribution for various values of the parameters is shown which may be of some interest to Bayesians. Certain statistical properties such as hazard rate function, mean residual function, r th moment are defined explicitly. The proposed extended Weibull distribution is used to derive an exact form of two, three and k -component mixture of distributions. With the help of a real data set, the usefulness of mixture Weibull type distribution is illustrated by using Markov Chain Monte Carlo (MCMC), Gibbs sampling approach.

Calculating the efficacy of heat sterilization processes

The isothermal semi-logarithmic survival curves of certain bacterial spores, C. botulinum and B. sporothermodurans among them, are non-linear. Hence, the methods to calculate the efficacy of processes to destroy them need to be revised. These spores’ survival curves could be described by a power law model, which is based on the assumption that the spores’ heat resistances have a Weibull type distribution, with a practically temperature independent shape factor. The temperature dependence of the ‘rate parameter’ of the power law model (related to the reciprocal of the distribution’s scale factor) could be described by a log logistic or a discontinuous linear model. The survival characteristics of the two spores are described in terms of the power law model’s exponents and the log logistic and discontinuous linear models’ two parameters; the temperature level where the inactivation accelerates and the rate at which it rises with temperature at the lethal range. These three parameters, together with the temperature profiles were used to simulate the outcome of different heat processes by solving, numerically, a rate based survival model. The resulting survival curves could then be compared and the processes’ lethality assessed in terms of the final survival ratio that they had produced. The method to calculate the survival curves is applicable to thermal processes having either a continuous or a discontinuous temperature profile. The same survival model could also be used to estimate the spores’ survival parameters directly from the non-isothermal survival curves in simulated inactivation data to which a scatter had been added.

SURVIVAL OF LISTERIA INNOCUA IN APPLE JUICE AS AFFECTED BY VANILLIN OR POTASSIUM SORBATE

ABSTRACT Survival of stationary phase Listeria innocua (as surrogate microorganism for L. monocytogenes) inoculated in apple juice (pH 3.3 or 3.8) supplemented with vanillin (1,500 ppm or 3,000 ppm) or potassium sorbate (500 ppm or 1,000 ppm) and stored at room temperature was studied. L. innocua survived in apple juice without the preservatives at pH 3.3 or 3.8, with minimal population reductions. In the juices with the incorporation of potassium sorbate or vanillin, L. innocua behavior depended on the pH value, the type of antimicrobial and its concentration. At pH 3.3, the presence of vanillin (3,000 ppm) or potassium sorbate (1,000 ppm or 500 ppm) decreased L. innocua counts, with population reductions ranging from 4 to 5 log cycles after a 4 h – 8 h exposure at 30C. However, at pH 3.8, L. innocua showed sensitivity only to 3,000 ppm vanillin. Survival curves were successfully fitted using a Weibull type distribution of resistances. The results suggest that the use of potassium sorbate or vanillin could prevent the survival of L. innocua in contaminated unpasteurized and pasteurized apple juice. Vanillin, a natural antimicrobial, would be particularly suitable as an antilisterial additive for less acidic apple juice.

GENERALIZED GAMMA TYPE DISTRIBUTION AND ITS HAZARD RATE FUNCTION

A new generalized gamma distribution is defined involving a parameter δ = λ − 1; λ ≥ 0 in the Kobayashi's (1991) function Γλ(m,n). The parameter δ will relax the restriction on the parameter λ > 0 in all probability distributions using Kobayashi's (1991) type functions. The hazard rate function of this distribution has the property of monotonicity and that of bathtub. The trend of density function and statistical properties are studied for generalized gamma distribution. With the help of a real life data set the fitting of the distribution is shown. A new Weibull type distribution is also considered and the hazard rate function of this distribution also has both the properties namely bathtub and monotonicity.

Bubble size distribution and coarsening of aqueous foams

This paper characterises the bubble size distribution of aqueous foams, produced by a compressed-air foam generator. The time evolution of bubble size distribution in aqueous foams is experimentally measured using a CCD video camera. A computer model, which predicts the change in bubble size distribution with time, is used to simulate the coarsening and disproportionation of aqueous foams. As an extension of previous computer simulations, our model incorporates the variation in liquid fraction during the foam aging process, thereby enabling the simulation of both wet and dry foams. It is found that a Weibull-type distribution best approximates the narrow bubble-size distribution produced by a compressed-air foam generator. The modelled predictions show good agreement with the experimental measurements and a sensitivity analysis indicates a significant dependence of the model on drainage (hence on the thickness of the lamellae), the Henry constant, the gas diffusivity and the surface tension of the foam solution.

Size effect on tensile strength of carbon fibers

Tensile strength of carbon fibers exhibits statistical Weibull type distribution and significant size dependence. In the present work, ten types of polyacrylonitrile based and mesophase-pitch based carbon fiber monofilaments were tested for two or three different gauge lengths. Size effect in both axial and radial directions were analyzed based on the two parameters Weibull statistics. It was found that the size effect in axial direction was almost similar for all fibers tested. This result suggests that the tensile strength obtained for a certain gauge length is a meaningful measure as a representative strength of the fiber strength. In radial direction, the size effect of the tensile strength was larger than that in axial direction. The tensile strength of the carbon fibers seemed to have unisotropic statistical characteristics. Size dependence in diameter was numerically simulated with an assumption of unisotropic distribution of Reynolds-Sharp type defects.