Analysis of random laser scattering pulse signals with lognormal distribution

The statistical distribution of natural phenomena is of great significance in studying the laws of nature. In order to study the statistical characteristics of a random pulse signal, a random process model is proposed theoretically for better studying of the random law of measured results. Moreover, a simple random pulse signal generation and testing system is designed for studying the counting distributions of three typical objects including particles suspended in the air, standard particles, and background noise. Both normal and lognormal distribution fittings are used for analyzing the experimental results and testified by chi-square distribution fit test and correlation coefficient for comparison. In addition, the statistical laws of three typical objects and the relations between them are discussed in detail. The relation is also the non-integral dimension fractal relation of statistical distributions of different random laser scattering pulse signal groups.

Two Probability Plots of the Three-Parameter Lognormal Distribution

The two-parameter lognormal distribution is a variant of the normal distribution and the three-parameter lognormal distribution is an extension of the two-parameter lognormal distribution by introducing a location parameter. The Q-Q plot of the three-parameter lognormal distribution is widely used. To obtain the Q-Q plot one needs to iteratively try different values of the shape parameter and subjectively judge the linearity of the Q-Q plot. In this paper,a mathematical method was proposed to determine the value of the shape parameter so as to simplify the generation of the Q-Q plot. Then a new probability plot was proposed,which was more easily obtained and provided more accurate parameter estimates than the Q-Q plot. These are illustrated by three realworld examples.

Acceptance sampling plan of accelerated life testing for lognormal distribution under time-censoring

Abstract Lognormal distribution is commonly used in engineering. It is also a life distribution of important research values. For long-life products follow this distribution, it is necessary to apply accelerated testing techniques to product demonstration. This paper describes the development of accelerated life testing sampling plans （ALSPs） for lognormal distribution under time-censoring conditions. ALSPs take both producer and consumer risks into account, and they can be designed to work whether acceleration factor （AF） is known or unknown. When AF is known, lift testing is assumed to be conducted under accelerated conditions with time-censoring. The producer and con- sumer risks are satisfied, and the size of test sample and the size of acceptance number arc opti- mized. Then sensitivity analyses are conducted. When AF is unknown, two or more predetermined levels of accelerated stress are used. The sample sizes and sample proportion allo- cated to each stress level are optimized. The acceptance constant that satisfies producer and consumer risk is obtdned by minimizing the generalized asymptotic variance of the test statistics. Finally, the properties of the two ALSPs （one for known-AF conditions and one for unknown AF conditions） are investigated to show that the proposed method is corrcct and usablc through numerical examples.

Identifying the failure mechanism in accelerated life tests by two-parameter lognormal distributions

The failure mechanism stimulated by accelerated stress in the degradation may be different from that under normal conditions, which would lead to invalid accelerated life tests. To solve the problem, we study the re- lation between the Arrhenius equation and the lognormal distribution in the degradation process. Two relationships of the lognormal distribution parameters must be satisfied in the conclusion of the unaltered failure mechanism, the first is that the logarithmic standard deviations must be equivalent at different temperature levels, and the second is that the ratio of the differences between logarithmic means must be equal to the ratio of the differences between reciprocals of temperature. The logarithm of distribution lines must simultaneously have the same slope and regular interval lines. We studied the degradation of thick-film resistors in MCM by accelerated stress at four temperature levels （390, 400, 410 and 420 K）, and the result agreed well with our method.

The Lognormal Distribution and Quantum Monte Carlo Data

Quantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis cannot be based on simple Gaussian assumptions.To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them.This method entails reconstructing the probability distribution of a set of data,with given mean and variance,that has been assumed to be lognormal prior to undergoing a blocking or renormalization transformation.In doing so,we perform a numerical evaluation of the renormalized sum of lognormal random variables.This technique is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate the ground state energy or dominant eigenvalue of a Hamiltonian matrix.

Species Abundance in a Forest Community in South China：A Case of Poisson Lognormal Distribution

Case studies on Poisson lognormal distribution of species abundance have been rare, especially in forest communities. We propose a numerical method to fit the Poisson lognormal to the species abundance data at an evergreen mixed forest in the Dinghushan Biosphere Reserve, South China. Plants in the tree, shrub and herb layers in 25 quadrats of 20 m× 20 m, 5 m× 5 m, and 1 m× 1 m were surveyed. Results indicated that: (i) for each layer, the observed species abundance with a similarly small median, mode, and a variance larger than the mean was reverse J-shaped and followed well the zero-truncated Poisson lognormal; (ii) the coefficient of variation, skewness and kurtosis of abundance, and two Poisson lognormal parameters (& and μ) for shrub layer were closer to those for the herb layer than those for the tree layer; and (iii) from the tree to the shrub to the herb layer, the α and the coefficient of variation decreased, whereas diversity increased. We suggest that: (i) the species abundance distributions in the three layers reflects the overall community characteristics; (ii) the Poisson lognormal can describe the species abundance distribution in diverse communities with a few abundant species but many rare species; and (iii) 1/α should be an alternative measure of diversity.

Polydispersity effects on the magnetization of diluted ferrofluids:a lognormal analysis

Based on a lognormal particle size distribution, this paper makes a model analysis on the polydispersity effects on the magnetization behaviour of diluted ferrofluids. Using a modified Langevin relationship for the lognormal dispersion, it first performs reduced calculations without material parameters. From the results, it is extrapolated that for the ferrofluid of lognormal polydispersion, in comparison with the corresponding monodispersion, the saturation magnetization is enhanced higher by the particle size distribution. It also indicates that in an equivalent magnetic field, the lognormally polydispersed ferrofluid is magnetically saturated faster than the corresponding monodispersion. Along the theoretical extrapolations, the polydispersity effects are evaluated for a typical ferrofluid of magnetite, with a dispersity of σ = 0.20. The results indicate that the lognormal polydispersity leads to a slight increase of the saturation magnetization, but a noticeable increase of the speed to reach the saturation value in an equivalent magnetic field.

Fracture Statistics Dominated by Hot-Cutting Defect and Deviation from Weibull Distribution

This paper characterizes the rarely noticeable hot-cutting defect and statistically models the fracture governed by this new type defect. The morphology of the defect on fired ceramic is examined and quantitatively featured through comparing the fracture strength governed by intrinsic defect and hot-cutting defect. Weibull distribution is utilized to fit the observed strength data and chi-square goodness-of-fit test is conducted to analyze the deviation. Kernel density estimation is introduced to explore the underlying strength distribution dominated by hot-cutting defect. Based on the shape information from kernel density estimating,gamma and lognormal distribution are compared and the hot-cutting defect governing fracture statistics is finally confirmed by chisquare test. Results show that the newly-defined hot-cutting defect is more dangerous than the intrinsic defect and the priori Weibull distribution fails to describe the fracture statistic governed by the edge defect while the lognormal with a slightly right skewed shape fits it very well.

Exact Statistical Distribution and Correlation of Human Height and Weight: Analysis and Experimental Confirmation

The statistical relationship between human height and weight is of especial importance to clinical medicine, epidemiology, and the biology of human development. Yet, after more than a century of anthropometric measurements and analyses, there has been no consensus on this relationship. The purpose of this article is to provide a definitive statistical distribution function from which all desired statistics (probabilities, moments, and correlation functions) can be determined. The statistical analysis reported in this article provides strong evidence that height and weight in a diverse population of healthy adults constitute correlated bivariate lognormal random variables. This conclusion is supported by a battery of independent tests comparing empirical values of 1) probability density patterns, 2) linear and higher order correlation coefficients, 3) statistical and hyperstatistics moments up to 6th order, and 4) distance correlation (dCor) values to corresponding theoretical quantities: 1) predicted by the lognormal distribution and 2) simulated by use of appropriate random number generators. Furthermore, calculation of the conditional expectation of weight, given height, yields a theoretical power law that specifies conditions under which body mass index (BMI) can be a valid proxy of obesity. The consistency of the empirical data from a large, diverse anthropometric survey partitioned by gender with the predictions of a correlated bivariate lognormal distribution was found to be so extensive and close as to suggest that this outcome is not coincidental or approximate, but may be a consequence of some underlying biophysical mechanism.

Seismic loss assessment of RC high-rise buildings designed according to Eurocode 8

A probabilistic seismic loss assessment of RC high-rise(RCHR)buildings designed according to Eurocode 8 and located in the Southern Euro-Mediterranean zone is presented herein.The loss assessment methodology is based on a comprehensive simulation approach which takes into account ground motion(GM)uncertainty,and the random effects in seismic demand,as well as in predicting the damage states(DSs).The methodology is implemented on three RCHR buildings of 20-story,30-story and 40-story with a core wall structural system.The loss functions described by a cumulative lognormal probability distribution are obtained for two intensity levels for a large set of simulations(NLTHAs)based on 60 GM records with a wide range of magnitude(M),distance to source(R)and different site soil conditions(SS).The losses expressed in percent of building replacement cost for RCHR buildings are obtained.In the estimation of losses,both structural(S)and nonstructural(NS)damage for four DSs are considered.The effect of different GM characteristics(M,R and SS)on the obtained losses are investigated.Finally,the estimated performance of the RCHR buildings are checked to ensure that they fulfill limit state requirements according to Eurocode 8.

SAMPLING INSPECTION OF RELIABILITY IN (LOG)NORMAL CASE WITH TYPE I CENSORING

This article proposes a statistical method for working out reliability sampling plans under Type I censored sample for items whose failure times have either normal or lognormal distributions. The quality statistic is a method of moments estimator of a monotonous function of the unreliability. An approach of choosing a truncation time is recommended. The sample size and acceptability constant are approximately determined by using the Cornish-Fisher expansion for quantiles of distribution. Simulation results show that the method given in this article is feasible.

Stochastic seismic response of multi-layered soil with random layer heights

This paper deals with the effect of layer height randomness on the seismic response of a layered soil. These parameters are assumed to be lognormal random variables. The analysis is carried out via Monte Carlo simulations coupled with the stiffness matrix method. A parametric study is conducted to derive the stochastic behavior of the peak ground acceleration and its response spectrum,the transfer function and the amplification factors. The input soil characteristics correspond to a site in Mexico City and the input seismic accelerations correspond to the Loma Prieta earthquake. It is found that the layer height heterogeneity causes a widening of the frequency content and a slight increase in the fundamental frequency of the soil profile,indicating that the resonance phenomenon is a concern for a large number of structures. Variation of the layer height randomness acts as a variation of the incident angle,i.e.,a decrease of the amplitude and a shift of the resonant frequencies.

Copula joint function and its application in probability seismic hazard analysis

A mature mathematical technique called copula joint function is introduced in this paper, which is commonly used in the financial risk analysis to estimate uncertainty. The joint function is generalized to the n-dimensional Frank’s copula. In addition, we adopt two attenuation models proposed by YU and Boore et al, respectively, and construct a two-dimensional copula joint probabilistic function as an example to illustrate the uncertainty treatment at low probability. The results show that copula joint function gives us a better prediction of peak ground motion than that resultant from the simple linear weight technique which is commonly used in the traditional logic-tree treatment of model uncertainties. In light of widespread application in the risk analysis from financial investment to insurance assessment, we believe that the copula-based technique will have a potential application in the seismic hazard analysis.

Constant Stress Accelerated Life Tests for Vacuum Fluorescent Display Based on Least Square Method

To estimate the life of vacuum fluorescent display （VFD） more accurately and reduce test time and cost, four constant stress accelerated life tests （CSALTs） were conducted on an accelerated life test model. In the model, statistical analysis of test data is achieved by applying lognormal function to describe the life distribution, and least square method （LSM） to calculate the mean value and the standard deviation of logarithm. As a result, the accelerated life equation was obtained, and then a self-developed software was developed to predict the VFD life. The data analysis results demonstrate that the VFD life submits to lognormal distribution, that the accelerated model meets the linear Arrhenius equation, and that the precise accelerated parameter makes it possible to acquire the life information of VFD within one month.

The Perturbed Compound Poisson Risk Model with Proportional Investment

In this paper, the insurance company considers venture capital and risk-free investment in a constant proportion. The surplus process is perturbed by diffusion. At first, the integro-differential equations satisfied by the expected discounted dividend payments and the Gerber-Shiu function are derived. Then, the approximate solutions of the integro-differential equations are obtained through the sinc method. Finally, the numerical examples are given when the claim sizes follow different distributions. Furthermore, the errors between the explicit solution and the numerical solution are discussed in a special case.

Fatigue behavior of direct laser deposited Ti-6.5Al-2Zr-1Mo-1V titanium alloy and its life distribution model

The present work aims to investigate the fatigue behavior of Direct Laser Deposition(DLD) Ti-6.5 Al-2 Zr-1 Mo-1 V titanium alloy under constant amplitude stress. 22 pieces of DLD Ti-6.5 Al-2 Zr-1 Mo-1 V titanium alloy standard cylinder specimens were tested under a stress level of 800 MPa with a stress ratio of 0.06. Fatigue fractography and fatigue life data were obtained.Through the fracture surface analysis, the specimens were divided into two categories in accordance with the location of crack initiation and defect types. Comparison of fatigue life and behavior between two specimen types was given, which was followed by a discussion about the impact of defect type, size and position on the fatigue life of the specimen. The fatigue test results also show a large variation of fatigue life. To illustrate the statistical characteristics of the fatigue life, probabilistic analysis was performed, and a novel bimodal lognormal model was established. The model has a good fit with the experimental data and can reduce the scatter of the fatigue life significantly.

Estimation of load and resistance factors using the thirdmoment method based on the 3P-Iognormal distribution

Load and resistance factors are generally obtained using the first order reliability method(FORM)in which the design point should be determined and derivative-based iterations used.In this article,the thirdmoment reliability index,based on the three-parameter lognormal(3P-lognormal)distribution,is investigated.A simple method based on the third-moment method for estimating load and resistance factors is then proposed,and a simple formula for the target mean resistance is also presented to avoid iterative computations.Unlike the currently used method,the proposed method can be used to determine load and resistance factors,even when the probability density functions(PDFs)of the basic random variables are not available.Moreover,the proposed method does not require the iterative computation of derivatives or any design points.Thus,the method provides a more convenient and effective way to estimate load and resistance factors in practical engineering applications.Numerical examples are presented to demonstrate the advantages of the proposed third moment method for determining load and resistance factors.