Weibull Type Research Articles

Inference on modified Weibull type distribution and its application to Competing risks data using MCMC

For lifetime data analysis, failure time analysis, or survival analysis, the Weibull distribution is commonly used due to its various hazard functions, which can be increasing, decreasing, or constant. We have extended the traditional twoparameter Weibull distribution to accommodate hazard functions that are increasing, decreasing, constant and bathtubshaped. Utilizing a competing risks approach, we applied this modified Weibull distribution to both simulated data and an observed mice dataset. Our findings indicate that the modified Weibull distribution provides a better fit to the mice dataset compared to the traditional Weibull distribution. We estimated the parameters of the modified Weibull distribution using Maximum likelihood estimation (MLE) and Bayesian methods. For MLE, we employed the Newton-Raphson numerical method, while for the Bayesian approach, we used the Metropolis-Hastings algorithm, an MCMC method. Additionally, we plotted hazard curves for both the simulated and mice datasets. The Kaplan-Meier survival curves were plotted along with the survival curve of the modified Weibull distribution.. KEYWORDS :Modified weibull distribution, Competing risks, MCMC, Information criterion, MLE.

Optimal Significance Levels and Sample Sizes for Signal Detection Methods Based on Non-constant Hazards.

Statistical methods for signal detection of adverse drug reactions (ADRs) in electronic health records (EHRs) need information about optimal significance levels and sample sizes to achieve sufficient power. Sauzet and Cornelius proposed tests for signal detection based on the hazard functions of Weibull type distributions (WSP tests) which use the time-to-event information available in EHRs. Optimal significance levels and sample sizes for the application of the WPS tests are derived. A simulation study was performed with a range of scenarios for sample size, rate of event due (ADRs), and not due to the drug and random time to ADR occurrence. Based on the area under the curve of the receiver operating characteristic graph, we obtain optimal significance levels of the different WSP tests for the implementation in a hypothesis free signal detection setting and approximate sample sizes required to reach a power of 80% or 90%. The dWSP-pPWSP (combination of double WSP and power WSP) test with a significance level of 0.004 was recommended. Sample sizes needed for a power of 80% were found to start at 60 events for an ADR rate equal to the background rate of 0.1. The number of events required for a background rate of 0.05 and an ADR rate equal to a 20% increase of the background rate was 900. Based on this study, it is recommended to use the dWSP-pWSP test combination for signal detection with a significance level of 0.004 when the same test is applied to all adverse events not depending on rates.

Comparative Performance of Point-to-Point Multiple Input Multiple Output System under Weibull and Rayleigh Fading Channels

Multiple-input multiple-output (MIMO) technologies use multiple antennas at the sender and the receiver to get the high data rate that the next-generation communication system needs. This paper compares MIMO systems using the Quadrature Phase Shift Keying (QPSK) modulation scheme for two-channel distributions of Rayleigh and Weibull types. The performance of the work is far with the Minimum Mean Square Equalizer (MMSE) in terms of Bit Error Rate (BER) for various antenna number situations on the transmitter and receiver sides. The MIMO module is carried out using MATLAB code. The channel noise will be a signal of random noise that is generated. To mitigate the impact of inter-channel interference, the MMSE approach employs inverse filtering at the receiver, and BER will be calculated. According to simulation results, the system's performance is primarily influenced by the number of antennas; it decreased as the number of antennas increased, and the BER of the Weibull channel decreased as the two-parameter value increased

Synthesis Weibull Stochastic Differential Equation: Properties and application

The Weibull distribution is suitable for modeling stochastic phenomena. However, most of the existing stochastic differential equations (SDEs) of Weibull type found in the literature do not follow the Weibull distribution, making them unsuitable for modeling phenomena that follow a Weibull random variable. To address this issue, this paper proposes a new SDE known as the Synthesis Weibull Stochastic Differential Equation (SWSDE) using the synthesis method. Some properties and numerical studies were carried out. The usefulness of the model is also demonstrated using Malawian wind speed data. The results show that the proposed SDE is suitable for predicting the data. The model also demonstrates a mean-reverting property, whereby the equilibrium is the solution of a differential equation in simulation studies. Additionally, the solutions of the model are found to always follow the Weibull distribution.

Bayesian hierarchical modelling of sea-level extremes in the Finnish coastal region

Abstract. Occurrence probabilities of extreme sea levels required in coastal planning, e.g. for calculating design floods, have been traditionally estimated individually at each tide-gauge location. However, these estimates include uncertainties, as sea-level observations typically have only a small number of extreme cases such as annual maxima. Moreover, exact information on sea-level extremes between the tide-gauge locations and incorporation of dependencies between the adjacent stations is often lacking in the analysis. In this study, we use Bayesian hierarchical modelling to estimate return levels of annual maxima of short-term sea-level variations related to storm surges in the Finnish coastal region. We use the generalised extreme value (GEV) distribution as the basis and compare three hierarchical model structures of different complexity against tide-gauge-specific fits. The hierarchical model structures allow us to share information on annual maximum sea levels between the neighbouring stations and also provide a natural way to estimate uncertainties in the theoretical estimates. The results show that compared to the tide-gauge-specific fits, the hierarchical models, which pool information across the tide gauges, provide narrower uncertainty ranges for both the posterior parameter estimates and the corresponding return levels in most locations. The estimated shape parameter of the GEV model is systematically negative for the hierarchical models, which indicates a Weibull type of behaviour for the extremes along the Finnish coast. The negative shape parameter also allows us to calculate the theoretical upper limit for the annual maximum sea levels on the Finnish coast. Depending on the tide gauge and hierarchical model considered, the median value of the theoretical upper limit was 47–73 cm higher than the highest observed sea level.

A comparative study to choose the appropriate growth model to forecast COVID-19 cases in Iraq

COVID-19 infection cases forecasting is a process of estimating future values based on historical data which is playing an important role in health decision making in various fields. Daily infection cases of COVID-19 can be considered as a time series represent the growth of the number of infected persons in a population. Consequently, the growth models may be used to forecast any population growth such as population of infected people with the Covid-19 virus. The popular models of growth such as logistic, log-logistic, Gompertz, Weibull and Richards models are useful to describe the growth of many phenomena like an epidemic and the spread of the number of infected people. The main objective of this paper is to choose a successful growth model after comparing these models to make good use of the current data on COVID-19 in Iraq to better understand the spread of this disease and to forecast the future daily infection cases. AIC, BIC and other goodness of fit criteria and daily infection cases in Iraq for the period from 1st Jan. 2021 until 30th April 2021 were used to compare these models and choose the successful model. The results of fitting these model show that the appropriate models are Weibull type 1 and log-logistic with five parameters models, and the predicted numbers of infected cases are near the actual numbers of infected cases.

Experimental analysis and numerical simulation of wave overtopping on a fixed vertical cylinder under regular waves

Wave overtopping phenomenon affects relatively narrow offshore marine structures different from shoreline linear structures, where there is not defined a precise prediction methodology as it is the case of the behaviour at long coastal defences. In the present study a combined experimental and numerical approach has been followed to obtain an empirical relation that represents the relative overtopping discharge over a fixed vertical cylinder exposed to non-impulsive wave conditions. The phenomenon follows a Weibull type dependence on the relative freeboard in a similar way as the case of vertical walls but reporting a decreasing overtopping rate at higher freeboards. In addition, a direct linear relationship between the relative mean flow thickness computed at the centre of the circular crest of the cylinder and the relative overtopping discharge has been observed. This methodology may be used as an indirect cost-effective method to characterize experimentally the wave overtopping phenomenon in cylindrical structures of full-scale prototypes without the need of accumulating and characterising huge amounts of overtopped water volumes. The present study contains a systematic analysis of the dispersion obtained in the experimental and computational results to evaluate the performance attributed to the proposed empirical expressions. • Overtopping has been characterized in a fixed vertical cylinder and compared with the behaviour of long shoreline defences. • A RANS-VOF Eulerian numerical model has been validated experimentally with a physical scaled model in a wave flume. • A new Weibull type expression predicts the overtopping discharge in a vertical cylinder for non-impulsive wave conditions. • The flow thickness measured at the crest has been proposed to determine the mean overtopping discharge over the structure.

Assessment of failure probabilities and the allowable size of defects in structural elements using the criteria of fracture mechanics

The possibility of assessing the safe size of the defects of metal continuity using risk criteria are considered. Such defects occur at all stages of the life cycle of structures. Assessment of their hazard and determination of their allowable size becomes important when the defects can lead to brittle or quasi-brittle fracture. In this case, models of linear and nonlinear fracture mechanics are used. In these models, defects are considered as internal elliptical or surface semi-elliptical cracks. The stochastic variety of shapes, sizes, locations, and orientations of defects has a significant effect on the destruction mechanisms. In this regard, the relevant probabilistic problem of assessing the permissible size of defects according to the criteria of the risk of destruction comes to the fore. We consider a general approach to assessing hazard of defects by risk criteria. Two settings of the probabilistic problem of risk assessment based on of one- and two-parameter fracture criteria are presented. The risk function in the form of the probability of fracture according to a given criterion is used as the main calculated characteristic. The expression of the risk function based on one-parameter criteria of fracture mechanics is presented. The main attention is paid to the probabilistic model based on the two-parameter fracture criterion by E. M. Morozov. This criterion provides ample opportunities for analyzing fracture mechanisms with a change in the size of defects. An expression for the risk function based on the family of two-dimensional probability distributions of Lu – Bhattacharya of the Weibull type is obtained. It is shown that the correlations of the fracture mechanisms can significantly affect the values of the fracture probabilities, and, consequently, the allowable size of defects.

Current Trends from Safety to Security Engineering

Safety Engineering mitigates normal events. Its contemporary equivalence, for extreme disasters, demands the deployment of Statistics of Extremes, Gumbel [1]. Civil Engineering research has addressed extreme natural and industrial disasters by employing extreme statistical distributions of Gumbel and Weibull types, respectively.

Estimation of the Parameters of Type-II Discrete Weibull Distribution Under Type-I Censoring

Maximum likelihood and proportion estimators of the parameters of the discrete Weibull type II distribution with type I censored data are discussed. A simulation study is performed to generate data from this distribution for suggested values of its parameters and to get the Maximum likelihood estimates of the parameters numerically. The method of proportions suggested by Khan et al. (1989) is also used to estimate the model's parameters. Numerical examples are used to perform a comparison study between the two method results according the values of the estimates and their corresponding mean squared errors.

The effect of size and stress state on the strength of architectural glass. Experiments versus theory

The multi-scale approach in the statistical evaluation of the macroscopic strength of annealed glass for architectural applications indicates a close correlation with the population of surface microcracks present in the glass sheets. The theory predicts a dependence of the strength capacity on the size of the glass element and on the state of stress acting on it. To investigate this effect, an experimental campaign has been conducted on samples of three different sizes, tested in a ring-on-ring apparatus. Data were re-normalized according to statistics of the Weibull type and re-scaled to account for the phenomenon of static fatigue. For small- and medium-size specimens, the results are in agreement with the prediction of Weibull statistical theory, according to which the larger the specimen, the greater the probability of finding cracks of critical size and, consequently, the lower the macroscopic strength on a statistical basis. On the other hand, quite surprisingly, large samples show an opposite trend. Since the state of stress is not uniform and equibiaxial in the large specimens, on the contrary to what happens in the small and medium size ones, this finding suggest a revision of classical approach based on linear elastic fracture mechanics and its correlation with the weakest-link-in-the-chain rationale in the evaluation of the mechanical properties of brittle materials such as glass.

Transmuted Topp-Leone Weibull Lifetime Distribution: Statistical Properties and Different Method of Estimation

In this work we focus on proposing a new lifetime Weibull type model called the transmuted Topp-Leone Weibull and studying its properties. We derive some new bivariate and multivariate transmuted Topp-Leone Weibull versions using “Farlie Gumbel Morgenstern (FGM) Copula”, “modified FGM Copula”, “Clayton Copula” and “Renyi's entropy Copula”. The estimation of its unknown parameters is carried out by considering different method of estimation. The statistical performances of all methods are studied by two real data sets and a numerical Monte Carlo simulation. The Cramer-Von Mises method is the best method for modeling the carbon fibers data. The maximum likelihood method is the best method for modeling the Leukemia data, however all other methods performed well.

Bayesian Estimation of the Parameters of Discrete Weibull Type (I) Distribution

Bayesian estimation of the continuous Weibull distribution parameters was studied by Ahmad and Ahmad (2013) under the assumption of knowing the shape parameter. Bayesian estimates are considered here of the parameters of the discrete Weibull Type I [DW(I)] distribution and are obtained under two different assumptions: when the shape parameter is known, and when both parameters are independent random variables. A Mathcad program is performed to simulate data from the DW(I) distribution considering different values of the parameters and different sample sizes, and to obtain Bayesian parameter estimates. The resulted estimates are compared to the ML and proportion estimates obtained by Khan et al. (1989).

Effect of biaxial stress state on seismic fragility of concrete gravity dams

Dams are important structures for management of water supply for irrigation or drinking, flood control, and electricity generation. In seismic regions, the structural safety of concrete gravity dams is important due to the high potential of life and economic loss if they fail. Therefore, the seismic analysis of existing dams in seismically active regions is crucial for predicting responses of dams to ground motions. In this paper, earthquake response of concrete gravity dams is investigated using the finite element (FE) method. The FE model accounts for dam-water-foundation rock interaction by considering compressible water, flexible foundation effects, and absorptive reservoir bottom materials. Several uncertainties regarding structural attributes of the dam and external actions are considered to obtain the fragility curves of the dam-water-foundation rock system. The structural uncertainties are sampled using the Latin Hypercube Sampling method. The Pine Flat Dam in the Central Valley of Fresno County, California, is selected to demonstrate the methodology for several limit states. The fragility curves for base sliding, and excessive deformation limit states are obtained by performing non-linear time history analyses. Tensile cracking including the complex state of stress that occurs in dams was also considered. Normal, Log-Normal and Weibull distribution types are considered as possible fits for fragility curves. It was found that the effect of the minimum principal stress on tensile strength is insignificant. It is also found that the probability of failure of tensile cracking is higher than that for base sliding of the dam. Furthermore, the loss of reservoir control is unlikely for a moderate earthquake.

Bivariate Software Reliability Growth Models under Budget Constraint for Development Management

Software reliability growth is observed by investing not only the testing-time but also the testing-effort in the testing-phase of software development process. If the testing-time (testing-effort) is reduced to some extent, it is possible to observe the software reliability growth by investing the amount of testing-effort (testing-time) which can compensate the insufficiency of the testing-time (testing-effort). However, most of the existing software reliability growth models (SRGMs) are constructed as univariate models and the substitutability between the testing-time and testing-effort is not considered. Additionally, it is necessary to remove many faults efficiently within the budget. In this paper, we develop bivariate Weibull type SRGMs under budget constraint based on the Cobb-Douglas type and CES (constant elasticity of substitution) type testing-time functions. Simultaneously, we evaluate the substitutability between the testing-time and testing-effort factors which are software reliability growth factors. Finally, we conduct the sensitivity analysis and show numerical examples by using actual data sets.

Variation of Elastic Energy Shows Reliable Signal of Upcoming Catastrophic Failure

We consider the Equal-Load-Sharing Fiber Bundle Model as a model for composite materials under stress and derive elastic energy and damage energy as a function of strain. With gradual increase of stress (or strain) the bundle approaches a catastrophic failure point where the elastic energy is always larger than the damage energy. We observe that elastic energy has a maximum that appears after the catastrophic failure point is passed, i.e., in the unstable phase of the system. However, the slope of elastic energy {\it vs.\/} strain curve has a maximum which always appears before the catastrophic failure point and therefore this can be used as a reliable signal of upcoming catastrophic failure. We study this behavior analytically for power-law type and Weibull type distributions of fiber thresholds and compare the results with numerical simulations on a single bundle with large number of fibers.

Bayesian approach to extreme-value statistics based on conditional maximum-entropy method

Recently, the conditional maximum-entropy method (abbreviated as C-MaxEnt) has been proposed for selecting priors in Bayesian statistics in a very simple way. Here, it is examined for extreme-value statistics. For the Weibull type as an explicit example, it is shown how C-MaxEnt can give rise to a prior satisfying Jeffreys’ rule.

Effect of high pressure carbon dioxide on tomato juice: Inactivation kinetics of pectin methylesterase and polygalacturonase and determination of other quality parameters

Tomato juice, Lycopersicon esculentum cv. Canario, has been treated by high pressure carbon dioxide (HPCD) as non-thermal preservation treatment. The inactivation kinetics for pectinmethylesterase (PME) and polygalacturonase (PG) were determined at different pressures (8.5–20 MPa) and temperatures (35–55 °C). At the highest operating pressure and temperature essayed in this work, it was found that PME could be almost completely inactivated, whereas PG resulted to be more HPCD resistant at the working conditions. PME enzyme inactivation curves were properly described by a Weibull type model, while the fractional conversion model was the most appropriate for the PG with a sharp initial decrease in activity. On the contrary, high pressure processing (HPP) led to a nearly complete inactivation of PG while PME was very resistant at 600 MPa. It was also found that HPCD treatment led to a smaller particle size distribution of tomato juice.

Development and Validation of Height-Diameter Models for Cupressus lusitanica in Gergeda Forest, Ethiopia

This study was conducted to fit and evaluate ten existing nonlinear height diameter functions for Cupressus lusitanica in Gergeda forest Ethiopia. A total of 260 trees were measured for their diameter at breast height (D) and height using destructive sampling methods. This data were randomly split in to two datasets for model development (50%) and for model validation (50%). Considering hetroscadasticity of variance, all functions were fitted using weighted nonlinear least square regression. To evaluate the performance of each model, five fit statistics-such as Coefficient of determination (R2), root mean square error (RMSE), bias (Ē), absolute mean deviation (AMD), and coefficient of variation (CV%) were used. Among all the models tested, the Weibull type function of the form H = 1.3 + a (1-exp (-bDc)) + ɛ was observed to give the best fit based on the model’s goodness of fit and predictive ability. Therefore, this model with three parameters has been conformed to provide reliable estimate of total tree height for Cupressus lusitanica in Gergeda forest, Ethiopia.

Development of a sequential alkalic salt and dilute acid pretreatment for enhanced sugar recovery from corn cobs

This study presents a sequential sodium phosphate dodecahydrate (Na3PO4·12H2O) and sulfuric acid (H2SO4) pretreatment to enhance sugar release from corn cobs. The effects of Na3PO4·12H2O concentration, H2SO4 concentration and solid to liquid ratio on the reducing sugar yield were investigated. The developed model had a coefficient of determination value (R2) of 0.98. Maximum reducing sugar yield of 0.99 ± 0.01 g/g was obtained with 12.70% Na3PO4·12H2O, 1.04% H2SO4 and 14.49% solid to liquid ratio. A 9-fold increase in the sugar yield was observed compared to previous reports on corn cobs. An intelligent model was developed to determine functional relationships between the input and output parameters. Reducing sugar yield was majorly dependent on Na3PO4·12H2O concentration and fits a Weibull type of relationship. Sequential pretreatment of lignocellulosic material with alkalic salt and dilute acid significantly enhanced sugar recovery and demonstrated high efficiency for microbial production of biofuels and bioproducts.