Parameters Of Lognormal Distribution Research Articles

Change point detection in length-biased lognormal distribution

In this paper, we develop two procedures for identifying the dynamic trends in the parameters of length-biased lognormal distribution based on likelihood ratio and modified information criterion. These methods mainly consider the test of the existence of the change point and provide the maximum likelihood estimation of the change point when the change point exists. In addition, the asymptotic distribution of test statistic based on likelihood ratio is derived as an extreme value distribution, while the asymptotic distribution of test statistic based on modified information criterion is derived as a chi-square distribution. And the consistency of parameter estimation is proved. Simulations are conducted to study the performance of the proposed method in terms of power, coverage probabilities and average sizes of confidence sets. The proposed methods are applied to a real data to illustrate the detecting procedures.

Intermittency of Gravity Wave Potential Energy Generated by Mountains Revealed from COSMIC-2 Observations

The intermittency of gravity wave potential energy (GWPE) in the upper troposphere and stratosphere was investigated using the Constellation Observing System for Meteorology, Ionosphere, and Climate-2 (COSMIC-2) temperature data over three typical mountains (Tibetan Plateau, Rocky Mountains, and Andes). These typical mountains have high sea level elevations but different land–sea contrast. The probability density function (PDF) of GWPE has the independent variable of GWPE and dependent variable of occurrence probability of GWPE over a region. Our analysis showed that the PDFs of GWPE over these three mountains roughly followed lognormal distributions in all heights and months. But, the key parameters (mean value and standard deviation) of lognormal distribution varied with heights and months. Above each mountain, the two key parameters exhibited similar temporal and spatial distributions. They had the largest values around the tropopause region, smaller values in the lower stratosphere (~20–30 km), and larger values in the upper stratosphere (~35–45 km). The intermittency of GWs is represented as the ratio of the GWPE at 50th percentile to the GWPE at 90th percentile. The weakest intermittency was at ~20–30 km (above the zonal mean winds of zero) over the Tibetan Plateau and Rocky Mountains in all months and over the Andes from November to March, respectively. Generally, the weakest intermittency (~0.4) occurred in the region where the key parameters were the smallest around summer. The key parameters of lognormal distribution were dominated by annual variation over the Andes throughout the height range, 8–50 km. However, the semiannual variations are also significant in the lower stratosphere over the Tibetan Plateau and Rocky Mountains. The seasonal variations in the intermittency were not as obvious as those of the key parameters. The lognormal distributions and the intermittencies derived here provide an observational constraint on the tunable parameters in GW parameterization schemes.

Optimization of behaviour strategies within the simulation model of a multi-agent socio-economic system

This article presents a new simulation model of a multi-agent socio-economic system (MA-SES), in which individual, including interproduct interactions are implemented. Within the MA-SES, the models of agent-producers and agents-consumers behaviour with their objective functions are studied. In particular, producers follow their own production strategies by choosing the moments for introducing new fixed assets and labour resources. Consumers participate in barter and monetary deals by interacting with other agents. The states of readiness of agents-producers to introduce new fixed assets and labour resources, as well as the states of readiness of agents-consumers to complete barter and monetary deals are set up for each time moment using lognormal distributions with given characteristics, which are the control parameters of the system. Important bi-objective optimisation problems are formulated for ensembles of agents-producers and agents-consumers. A new parallel hybrid genetic algorithm (MORCGA-MOPSO) was developed, in particular, providing the possibility of seeking the Pareto-optimal solutions for maximising the average (over an ensemble of agents) profit and the total number of agent-producers’ buyers, as well as maximising the average utility and monetary savings of agents-consumers. The parameters of log-normal distributions that determine the states of interacting agents that make individual decisions are computed. The features of the behavioural strategies of producers and consumers are determined, which make possible to achieve an improvement in the values of the objective functions through controlling the dynamics of the input of production resources and choosing the preferred types of interproduct interactions, in particular, barter, monetary etc.

Forecasting Monthly Wind Energy Using an Alternative Machine Training Method with Curve Fitting and Temporal Error Extraction Algorithm

The aim of this research was to forecast monthly wind energy based on wind speed measurements that have been logged over a one-year period. The curve type fitting of five similar probability distribution functions (PDF, pdf), namely Weibull, Exponential, Rayleigh, Gamma, and Lognormal, were investigated for selecting the best machine learning (ML) trained ones since it is not always possible to choose one unique distribution function for describing all wind speed regimes. An ML procedural algorithm was proposed using a monthly forecast-error extraction method, in which the annual model is tested for each month, with the temporal errors between target and measured values being extracted. The error pattern of wind speed was analyzed with different error estimation methods, such as average, moving average, trend, and trained prediction, for adjusting the intended following month’s forecast. Consequently, an energy analysis was performed with effects due to probable variations in the selected Lognormal distribution parameters, according to their joint Gaussian probability function. Error estimation of the implemented method was carried out to predict its accuracy. A comparison procedure was performed and was found to be in line with the conducted Markov series analysis.

Comparative fire-following-earthquake performance of code-compliant low-rise base-isolated structures

Seismic base isolation is an effective way of protecting structures against seismic loads and is very effective in terms of both collapse mitigation of structures and protection of non-structural elements under severe ground shaking. In this study, the structural demand of base-isolated three- and four-story steel moment-resisting frames is determined in case of a fire event followed by an earthquake, and compared with the results of fixed-base frames, which have similar geometry, load, and seismic hazard for a location in California. Four compartments are selected as possible locations for fire events in each building, and beams and columns in those compartments are exposed to a representative temperature increase in time, which includes a cooling phase as well. Maximum, minimum, and residual axial force, and moment demands on elements of the fire compartments, and drift demand of structural frames on the first and second floor, where the fire is assumed to occur, are determined and compared. Results are given in terms of parameters of three-parameter log-normal distribution, hence fragility curves can be constructed for each response considered in the study. Seismic isolation is effective in reducing both maximum and residual drift demand of the frames, and axial force in the beam element for each compartment considered. Fixed-base frames have 20% more maximum axial load on beams. Beam and column elements in the four-story configuration are under relatively more moments in case of a fire, while the performance of three-story frames depends on the location of the assumed fire.

Bayesian interval estimations for the mean of delta-three parameter lognormal distribution with application to heavy rainfall data.

Flash flooding is caused by heavy rainfall that frequently occurs during a tropical storm, and the Thai population has been subjected to this problem for a long time. The key to solving this problem by planning and taking action to protect the population and infrastructure is the motivation behind this study. The average weekly rainfall in northern Thailand during Tropical Storm Wipha are approximated using interval estimations for the mean of a delta-three parameter lognormal distribution. Our proposed methods are Bayesian confidence intervals-based noninformative (NI) priors (equal-tailed and highest posterior density (HPD) intervals based on NI1 and NI2 priors). Our numerical evaluation shows that the HPD-NI1 prior was closer to the nominal confidence level and possessed the narrowest expected length when the variance was small-to-medium for a large threshold. The efficacy of the methods was illustrated by applying them to weekly natural rainfall data in northern Thailand to examine their abilities to indicate flooding occurrence.

Influence of raindrop size distribution variations on the rain specific attenuation coefficients

The raindrop size distribution is one of main term to evaluate microwave attenuation due to rainfall. In this paper, the lognormal distribution is introduced to represent the raindrop-size distribution based on the measurement performed at the Chungnam National University. Since the distribution parameters are strongly scattered around the fitted value, we developed a new statistical model for the lognormal distribution parameters. Based on the new statistical model for raindrop size distribution parameters we have estimated influence of raindrop size distribution variations on the rain specific attenuation coefficients.

Quasi-Static Variation of Power-Law and Log-Normal Distributions of Urban Population.

We analytically derived and confirmed by empirical data the following three relations from the quasi-time-reversal symmetry, Gibrat’s law, and the non-Gibrat’s property observed in the urban population data of France. The first is the relation between the time variation of the power law and the quasi-time-reversal symmetry in the large-scale range of a system that changes quasi-statically. The second is the relation between the time variation of the log-normal distribution and the quasi-time-reversal symmetry in the mid-scale range. The third is the relation among the parameters of log-normal distribution, non-Gibrat’s property, and quasi-time-reversal symmetry.

A regression model for extreme events and the presence of bimodality with application to energy generation data

The application of the theory of extreme values has been growing due to increasing interest in extreme natural events. Many articles on extreme values in data modelling consider unimodal data. This work introduces an appropriate regression for extreme values to detect the presence of bimodality by means of systematic components of two parameters of the odd log-logistic log-normal distribution. The global influence is addressed to verify the model robustness and to find possible influential points. Quantile residuals are proposed to detect distribution deficiencies and outliers in the new regression. A real dataset from the electricity generation area is analysed, namely the Santo Antônio Hydroelectric Plant in the state of Rondônia (Brazil), to illustrate the potential of the new regression. The main results indicate that the proposed regression can identify changes in the means and variability of the power generation between extreme events, that is, between the months of June and December.

Probabilistic modeling of fatigue damage in asphalt mixture

Fatigue cracking is considered as one of the major distress in asphalt pavements. Environmental factors like aging and moisture intrusion further complicate the cracking process. In this study, the effect of two levels of aging and three levels of moisture conditioning on the fatigue damage of asphalt mixtures was determined using a probabilistic approach. The fracture energy, the stiffness modulus, and the fatigue life of four different asphalt mixtures were determined at 15 °C and 25 °C. It was noticed that the stiffness modulus and fatigue life of asphalt mixtures increased with an increase in the level of aging and reduced with an increase in moisture conditioning cycles. At three freeze–thaw cycles, the reduction in fatigue life of asphalt mixtures was much higher compared to the stiffness modulus of the asphalt mixtures. It was noticed that moisture had a higher negative impact on the fatigue life of asphalt mixtures at 15 °C as compared to 25 °C. Both stiffness modulus and fatigue life of asphalt mixtures reduced with the increase in test temperature. The fatigue damage of unconditioned and moisture conditioned asphalt mixtures followed a lognormal distribution. The lognormal distribution parameters were used to determine the probability of fatigue damage at each loading cycle. It was noticed that a moisture conditioned asphalt mixture had a higher probability of fatigue damage compared to unconditioned asphalt mixture at different aging condition and test temperature. Asphalt mixtures subjected to three freeze thaw cycles had a higher probability of fatigue damage compared to one freeze thaw cycle. Further, the probability of fatigue damage was higher in the case of long term aged asphalt mixtures when compared to short term aged asphalt mixtures at a lower temperature of 15 °C. The damage based fatigue life prediction model had a good coefficient of determination for different conditions evaluated in this study. A Monte Carlo simulation with one million iterations was used to simulate the fatigue life of the damage based fatigue life prediction model. It was observed that a moisture susceptible asphalt mixture might not show an increase in fatigue life with an increase in the aging level.

An application of Particle Swarm Optimization (PSO) algorithm with daily precipitation data in Campina Grande, Paraíba, Brazil

We study the daily precipitation in the municipality of Campina Grande, estimating the parameters of Gamma, Log-Normal, and Weibull distributions. To evaluate the parameter estimators, we compared the Particle Swarm Optimization (PSO) versus Maximum Likelihood Estimation (MLE) to analyze and understand the behaviour of the daily precipitation in Campina Grande. In most cases, our results show evidence that the PSO algorithm is an efficient and robust technique. Notwithstanding, the algorithm also presents an efficient parameter estimation due to its fast convergence.

Corrected Maximum Likelihood Estimations of the Lognormal Distribution Parameters

As a result of asymmetry in practical problems, the Lognormal distribution is more suitable for data modeling in biological and economic fields than the normal distribution, while biases of maximum likelihood estimators are regular of the order O ( n − 1 ) , especially in small samples. It is of necessity to derive logical expressions for the biases of the first-order and nearly consistent estimators by bias correction techniques. Two methods are adopted in this article. One is the Cox-Snell method. The other is the resampling method known as parametric Bootstrap. They can improve maximum likelihood estimators performance and correct biases of the Lognormal distribution parameters. Through Monte Carlo simulations, we obtain average root mean squared error and bias, which are two important indexes to compare the effect of different methods. The numerical results reveal that for small and medium-sized samples, the performance of analytical bias correction estimation is superior than bootstrap estimation and classical maximum likelihood estimation. Finally, an example is given based on the actual data.

Single cycle supply chain coordination model for fuzzy stochastic demand of perishable items

Market demand of the perishable bakery products are volatile. Hence, demand of such items are random variable. The most appealing demand distribution of such items is lognormal demand distribution. Using historical data, the parameters of lognormal distribution are obtained. But, there is a problem of overestimation or underestimation of the parameters. Fuzzy triangular numbers are used to overcome such problems. Single period (newsboy) inventory model is used to obtain an optimal order quantity. The retailers and manufacturers profit is obtained by using the supply chain coordination model. At the end of day, all the remaining items rotted. Hence, there is no gain from unsold items. Also, the some items are defective at the time of purchase. Thus, loss sharing strategy is proposed between manufacturers and retailers. All of these aspects are incorporated in the supply chain coordination model. The data is collected from Koregaon Park, Pune market, India to illustrate the methodology.

Estimation of travel time distributions for urban roads using GPS trajectories of vehicles: a case of Athens, Greece

Investigating travel time distribution and associated variability is important for a variety of transport planning, traffic management, and control projects. Studies that investigated travel time distribution tend to be limited to explore changes in characteristics of distribution with respect to space and time of day. Given the availability of big dataset that contains seven different types of vehicle trajectories in the city of Athens for around 56,000 trips which are traversing on more than 1.8 million road links, this study presents the detailed investigation of travel time distribution in different spatiotemporal settings. The study considered four different types of urban roads and six time intervals along with consideration of weekdays and weekends. The empirical investigation employed Kruskal-Wallis, Chi-square, and Kolmogorov-Smirnov tests to fit travel time data into seven unimodal statistical distributions that are found in the literature to describe travel time distribution. It is found that lognormal distribution outperformed other distribution, and all of the considered categories of travel time data are well-fitted to this distribution. Additionally, parameters of lognormal distribution for different categories of travel time data are not significantly different from each other, which led to the conclusion that travel time distribution is roughly independent of space and time, which is in agreement with a few earlier studies that are limited in their scope especially in relation with availability of data. With this important finding, this study estimate values of travel time variability for different classes of individuals employing a standard approach that requires time of day independent standardized distribution of travel time. It is estimated that for Athens population value of travel time variability is approximately half of the value of travel time. This is useful to carry out cost-benefit analyses for mobility-related projects in Athens, Greece.

Fatigue crack growth assessment method subject to model uncertainty

Fatigue crack growth (FCG) is an important degradation process of many critical mechanical equipment. Probabilistic FCG models are often used to account for the variability among FCG process conditions. In the well-known model of Yang and Manning, a deterministic FCG model is randomized by multiplying the crack growth rate with a random multiplier assumed to obey a lognormal distribution and unknown parameters are jointly estimated through Maximum likelihood estimation. By so doing, the modeling error due to inappropriate choice of the deterministic FCG model and that due to unsuitable assignment of the probability distribution of the random multiplier cannot be distinguished. Besides, the model uncertainty of the random multiplier is not explicitly considered. In this paper, a two-step least-square estimation method is proposed, which estimates the unknown parameters in the deterministic FCG model at first, and generates a sample set for the estimation of the random multiplier considering model uncertainty by way of Bayesian model selection. In Bayesian model selection, three types of Bayes factor are considered to select the appropriate candidate model and a simulation experiment is carried out to guide their selection. The effectiveness and feasibility of the proposed method are illustrated through two case studies using the real FCG datasets.

Modeling of probability in obtaining intensity-duration-frequency relationships of rainfall occurrence for Pelotas, RS, Brazil

ABSTRACT Based on historical series, for each locality, equations can characterize the relationship between intensity, duration and frequency of rainfall occurrence. The objective of this study was to present two equations that can describe the occurrence of intense rainfall in Pelotas, RS state, over the period 1982-2015. The two equations were denominated conventional and hybrid, depending on the probabilistic model used. Following the conventional methodology, the parameters of Normal, Log-Normal, Gumbel and Gamma probability distributions were adjusted by the maximum likelihood method for return periods of 2, 5, 10, 20, 25, 50 and 100 years. The maximum intensity values for the hybrid equation were obtained using the empirical model of Weibull, considering return periods of 2, 5, 10, 20 and 25 years. On the other hand, the same theoretical distributions used in the conventional equation were applied to return periods of 50 and 100 years. The Kolmogorov-Smirnov test was used to select the best fitting distribution for the data. In order to verify the information acquired through the Weibull empirical model in comparison to the theoretical distributions, the t-test was applied to the angular coefficients. Significant differences were not verified between the values of maximum rainfall intensities obtained using the two methodologies, for the pre-established durations and return periods. Thus, considering the maximum rainfall intensities values (durations of 5-1440 min) and return periods of 2-100 years in the municipality of Pelotas, RS, Brazil, both the hybrid and the conventional intense rainfall equations can be used.

PSO-KS Algorithm for Fitting Lognormal Distribution: Simulation and Empirical Implementation to Women’s Age at First Marriage Data

Lognormal distribution plays an essential role in the distribution modeling of right-skewed data in many areas. In social sciences, for instance, it can be used to model women’s age at first marriage pattern, a key indicator in studying fertility patterns. Distribution fitting is a fundamental and essential stage of data modeling before doing advancing the analysis. Kolmogorov-Smirnov (KS) distance is applicable as maximum goodness-of-fit (GOF) estimators for distribution parameters. Minimizing KS distance is optimization problem. Particle swarm optimization (PSO) algorithm is a general optimizer that can handle various optimization problems. This study assesses the characteristics of minimum KS distance estimator for lognormal distribution parameters. KS distance estimators were obtained via optimization using the PSO algorithm, so the combination of these is called the PSO-KS algorithm. We conducted a simulation to assess the performance of PSO-KS, Maximum Likelihood (MLE), Method of Moment (MME). The bias and mean square error (MSE) of point estimators were used in simulation to assess the characteristics of estimators. Meanwhile, MSE of distribution fitting, KS distance, and log-likelihood value were used to evaluate the GOF characteristics. Moreover, we demonstrated the performance of the algorithm by implementing it to women’s age at first marriage data in Indonesia. The results show that based on the bias and MSE properties, the PSO-KS point estimators yield similar characteristics with MLE, but better than MME. From the GOF perspective, PSO-KS outperforms in MSE of distribution fitting and KS distance, but not in log-likelihood value. We also observed these patterns in the women’s age at first marriage data. The contributions of this study are two-fold, first to assess the PSO-KS algorithm in the lognormal distribution case. Second, it implements the algorithm on women’s age at first marriage data, which has broad social, economic, and public health implications.

Performance analysis of dual-media cooperative communication based on wireless and power line under hybrid fading

Wireless communication and power line communication are extensively applied in various fields, such as household Internet of things. For cooperative communication of amplify–forward relay using wireless access and power line transmission, the hybrid model of universal Nakagami wireless fading and lognormal power line fading was used in this study. A comparative analysis of similarities between Gamma and lognormal distributions and lognormal distribution characteristics of relay link signal-to-noise ratio was also carried out. Given the deficiencies of lognormal variable add approximation method, lognormal distribution parameters combined computing method based on moment generation function was proposed. Theoretical formulas of outage probability and bit error rate after amplify–forward relay and maximal-ratio combining was derived on the basis of moment generation function of the total output signal-to-noise ratio of the system. Effectiveness and reliability of the algorithm and theoretical formula were verified through simulation, and influence rules of hybrid channel fading and power distribution on the system performance were analyzed.

Droplet size prediction model based on the upper limit log-normal distribution function in venturi scrubber

Abstract Droplet size and distribution are important parameters determining venturi scrubber performance. In this paper, we proposed physical models for a maximum stable droplet size prediction and upper limit log-normal (ULLN) distribution parameters. For the proposed maximum stable droplet size prediction model, a Eulerian-Lagrangian framework and a Reitz-Diwakar breakup model are solved simultaneously using CFD calculations to reflect the effect of multistage breakup and droplet acceleration. Then, two ULLN distribution parameters are suggested through best fitting the previously published experimental data. Results show that the proposed approach provides better predictions of maximum stable droplet diameter and Sauter mean diameter compared to existing simple empirical correlations including Boll, Nukiyama and Tanasawa. For more practical purpose, we developed the simple, one dimensional (1-D) calculation of Sauter mean diameter.

Spreadsheet for evaluation of global risks in conformity assessment of a multicomponent material or object

A user-friendly MS-Excel spreadsheet is developed for evaluation of global consumer's and producer's risks in conformity assessment of chemical composition of a multicomponent material or object, when up to four component concentrations are under control. These risks are probabilities of incorrect conformity decisions related to a material batch (lot or similar) randomly drawn from a statistical population of such batches. The probabilities characterize the material quality globally, allowing the prediction of false decisions on conformity of a future batch, based on the future measurement results. The spreadsheet program evaluates risks using Monte Carlo simulations. As input data, the following need to be provided to the software: parameters of normal or lognormal distribution of actual (‘true’) values of the component concentrations (prior distribution); parameters of the distribution of measurements results at the actual value of the component concentration (likelihood function); and correlation matrices for couples of the actual components' concentrations under control and also for corresponding measurement results. The spreadsheet is validated by comparison of the risk estimates with those calculated in R programing environment by numerical integration of the relevant analytical formulae. The developed Excel file and a demonstration videos of its use are available as electronic supplementary material.