Weibull Function Research Articles

How sampling scheme and species-area scale contribute to select the best statistical functions for upscaling plant species richness

Species–area curve is widely used to describe biological diversity across spatial scales. However, there are major ecological and statistical limitations in its application and inference, such as sampling methods, species-area scales, and statistical functions that must be considered in diversity assessment. This study aimed to explain relations among sampling methods, species-area curve scales and statistical functions in the Perk forested catchment, Zagros forests, western Iran. We studied plant diversity during the peak of the growing season, from early April to late May, 2019. The stratified random sampling was applied to collect species richness data on the same sized plots (4m2). The mean of data from noncontiguous plots was used to construct Scheiner's type IIIB curve. A comprehensive set of species-area models including five convex and three sigmoid models were tested. Akaike information criterion (AIC) and Bayesian information criterion (BIC) were used to choose the best-fitted function model. Furthermore, the Log Error of Extrapolation (LEE) was used to assess the reliability of extrapolation. Both AIC and BIC results suggested that the Lomolino and Cumulative Weibull functions as the best-fit curves for extrapolating species richness beyond sampling area. However, the LEE showed the superiority of the Rational function. This discrepancy between LEE and both AIC and BIC introduced models originates from the LEE limitation of application to tenfold of the sampling area. This study showed that the sampling scheme and the study scale are the two important factors in the shape of statistical function (convex or sigmoid) which is using in constructing species-area curves. In contrast to the prevailing use of power function, the results showed that in a broader scale like the landscape scale, the Lomolino and Cumulative Weibull functions are superior in estimating species richness in Zagros forests, western Iran.

Proposing a novel comprehensive evaluation model for the coal burst liability in underground coal mines considering uncertainty factors

Coal burst is a severe hazard that can result in fatalities and damage of facilities in underground coal mines. To address this issue, a robust unascertained combination model is proposed to study the coal burst hazard based on an updated database. Four assessment indexes are used in the model, which are the dynamic failure duration (DT), elastic energy index (WET), impact energy index (KE) and uniaxial compressive strength (RC). Four membership functions, including linear (L), parabolic (P), S and Weibull (W) functions, are proposed to measure the uncertainty level of individual index. The corresponding weights are determined through information entropy (EN), analysis hierarchy process (AHP) and synthetic weights (CW). Simultaneously, the classification criteria, including unascertained cluster (UC) and credible identification principle (CIP), are analyzed. The combination algorithm, consisting of P function, CW and CIP (P-CW-CIP), is selected as the optimal classification model in function of theory analysis and to train the samples. Ultimately, the established ensemble model is further validated through test samples with 100% accuracy. The results reveal that the hybrid model has a great potential in the coal burst hazard evaluation in underground coal mines.

Sensitivity analysis of a new model to predict the survival probability of artificial rock blocks upon dynamic impact

Rockfall fragmentation is a very complex phenomenon that is still poorly understood and modelled. Being able to adequately model fragmentation of impacting blocks, including change of shape, size, and energy after breakage, is essential to be able to predict realistic trajectories and design effective mitigation measures. In order to develop an accurate predictive model for rockfall fragmentation, it is necessary to better understand the fragmentation process and its likely outcomes. A novel model was recently proposed by the authors which can predict the survival probability (SP) of brittle spheres upon impact from the statistical distribution of material parameters, obtained by standard quasi-static tests (Brazilian tests and unconfined compression tests). The survival probability is described as a Weibull function whose two parameters (shape parameter-m-and scale parameter - critical kinetic energy) are predicted by the model. The model is based on theoretically-derived (from Hertzian contact theory) conversion factors used to transform the critical work required to fail disc samples in quasi-static indirect tension into the critical kinetic energy to cause failure of spheres at impact in vertical drop tests. This paper presents a sensitivity analysis on the parameters which influence the prediction of the critical kinetic energy.

Form of Distribution of Dendro/Morphometric Variables for Brazilian Pine in Southern Brazil

The form of distribution found for the dendro/morphometric variables determines the structure, stability, productivity of forest stands, being a tool to propose silvicultural interventions, management, conservation of species, and dynamics of this environment. Thus, this study evaluates, using probability density functions (pdf), the form of distribution of these variables for araucaria in five sites in southern Brazil, aiming to establish the dynamics and identify the existence of a standard—or the lack thereof—to propose the need for silvicultural interventions to conserve the species and the future forest structure. The Normal, Log-Normal, Weibull and Gamma probability density functions were tested. Results show no significant changes in the shape and dimension in the forest structure dynamics, but a period of stability in the pattern of dendro/morphometric values, resulting from the stagnation of the values of the variables, non-intervention in the forest, relationship with the site, density, competition, and position of the tree in the forest stratum, which compromises the future structure of this forest typology. The study proves that the distribution probability of the variables can be used in management for species conservation and future structure development, as this influences the growth dynamics and processes, resource availability, and the stability, diversity, vitality, and productivity of the species.

Utilization of high-speed imaging for atmospheric scintillation studies.

Atmospheric scintillation studies have been traditionally undertaken utilizing nonimaging detection. When imaging devices are used, they typically detect the resultant signal at the receiver plane. Here, a high-speed camera has been utilized in atmospheric scintillation field trials, imaging a laser source (i.e., imaging the object plane) over a near ground path length of 1.5 km. The statistical nature of the acquired atmospheric scintillation data is characterized using a range of probability density functions. The exponentiated Weibull function was found to best describe the nature of scintillation over the broadest range of a scintillation index typical of atmospheric scintillation. A preliminary investigation into the relationship between the fit variables of three of the better-performing probability density functions and the scintillation index is presented, along with suggestions for future use of digital cameras in atmospheric scintillation studies.

Exploring Wind Energy Potential as a Driver of Sustainable Development in the Southern Coasts of Iran: The Importance of Wind Speed Statistical Distribution Model

Wind energy as a clean and inexhaustible source of renewable energy can be a key element of sustainable development that decreases dependence of countries on fossil fuels. Therefore, implementing accurate and comprehensive feasibility studies in countries with a high level of consumption of traditional energy resources is vital; an approach encouraged and supported by green funds and climate change action. It is also crucial to helping spur economic and sustainable growth of these countries. In this regard, this study aims at accurate evaluation of onshore wind energy potential in seven coastal cities in the south of Iran. Six Probability Distribution Functions (PDFs) were examined over representative stations. It was deduced that the Weibull function, which is the most used PDF in similar studies, was only applicable to one station. Here, Gamma distribution offered the best fit for three stations and for the other ones, Generalized Extreme Value (GEV) performed better. Considering the ranking of six examined PDFs and the simplicity of Gamma, it was identified as the effective function in the southern coasts of Iran bearing in mind the geographic distribution of stations. Moreover, six wind energy converter power curve functions contributed to investigating the capacity factor. It is found that, using only one function could cause under- or over-estimation. Then, stations were classified based on the National Renewable Energy Laboratory system. Last but not least, examining a range of wind energy converters enabled scholars to extend this study into practice and prioritize the development of stations considering budget limits.

A reliability prediction methodology based on improved radial basis function neural network (RBFNN): a condensate light crude oil stabilization facility as case study

In this article, a risk analysis was carried out on condensate stabilisation facility. The condensate or light crude oil produced from oil and gas facilities can be highly volatile i.e. the liquid can vaporise very easily, forming a flammable atmosphere, thereby making it difficult or unsafe for storage in atmospheric tanks. Due to its volatility, a high vapour pressure can result in seal damage to floating roof tanks with the potential to result in roof collapse and/or loss of containment of the stored liquid to atmosphere (on top of the roof). The Radial Basis Function Neural Network (RBFNN) proposed method herein is based on clustering of input space vectors and computing weights of Euclidian distances; histogram equalisation within each cluster will determine the centre and width of each receptive field for the estimation of the risk parameters. The RBFNN is designed to estimate the quality of performing consistently well (Reliability) to minimise any risk in the tank. The obtained results demonstrate the high performances of the proposed method where the deviation between Weibull function and the predicted value using RBFNN is less than 0.5%.

Study of the modified Weibull function to analyze reliability in engineering components that fail and are repairable

An engineering component is physically inoperative when it fails or has conditions that prevent it from operating and has a functional failure. The useful or operating times and the non-operating or downtimes are collected to determine the reliability or the probability of operation of the component during a period. The use of distributions is the most accepted method for determining component reliability. Among the main distributions used are the exponential, Log Normal, Weibull, etc. The Weibull distribution is widely used to estimate the reliability of a component, understood as the physical capacity of the element to function according to operating parameters. A study of three models based on Weibull modifications was conducted to determine the reliability of repairable components subject to maintenance activities. Two models represent the modified Weibull function with two parameters and an additional model that represents the Weibull function transmuted with failure rate function that fits the Davies curve. The characteristics, the physical interpretation, the applications, and the feasibility of using each model in analysis of reliability in elements and repairable in engineering are presented.

Effect of Carburizing and Nitriding on Fatigue Properties of 18Cr2Ni4WA Steel in Very High Cycle Fatigue Regime

The work aims to study the influence of carburizing and nitriding on fatigue properties of 18Cr2Ni4WA high strength steel in very high cycle fatigue regime. Very high cycle fatigue tests were carried out on 18Cr2Ni4WA Steel after carburizing and nitriding respectively. The micro morphology of fatigue fracture was observed by scanning electron microscope, the failure mode and failure mechanism were discussed. The relationship between fatigue life and defect size, FGA size, fish eye size of fracture was analyzed. The characteristic size of defects is evaluated by Gumbel, Weibull and GEV distribution functions, and a modified Akiniwa fatigue life prediction model considering the relationship between FGA size and inclusion size was established. The results showed that, nitriding and carburizing treatment improve the surface fatigue limit of the steel. The fatigue life decreases with the increase of internal defect size and FGA size. After carburizing and nitriding treatment, the internal fatigue strength of the specimen decreases slightly. When the failure probability is 99%, the internal defect sizes of nitrided specimens calculated by Weibull, Gumbel and GEV distributions are 141.5 μm, 148.4 μm and 211.7 μm respectively. The calculated internal defect sizes of carburized specimens are 47 μm, 67.8 μm and 40 μm respectively. Compared with the experimental data, the fatigue strength predicted by GEV is the most appropriate. carburizing and nitriding treatment can improve the surface fatigue strength of 18Cr2Ni4WA steel, but slightly reduce the internal fatigue strength. The prediction result of the new model is conservative when the failure probability is 99%, which is suitable for engineering application.

An adapted Weibull function for agricultural applications

The Weibull function is applied extensively in the life sciences and engineering but underused in agriculture. The function was consequently adapted to include parameters and metrics that increase its utility for characterizing agricultural processes. The parameters included initial and final dependent variables (Y0 and YF, respectively), initial independent variable (x0), a scale constant (k), and a shape constant (c). The primary metrics included mode, integral average, domain, skewness, and kurtosis. Nested within the Weibull function are the Mitscherlich and Rayleigh functions where c is fixed at 1 and 2, respectively. At least one of the three models provided an excellent fit to six example agricultural datasets, as evidenced by large adjusted coefficient of determination (RA2 ≥ 0.9266), small normalized mean bias error (MBEN ≤ 1.49%), and small normalized standard error of regression (SERN ≤ 8.08%). The Mitscherlich function provided the most probable (PX) representation of corn (Zea mays L.) yield (PM = 87.2%); Rayleigh was most probable for soil organic carbon depth profile (PR = 96.4%); and Weibull was most probable for corn seedling emergence (PW = 100%), nitrous oxide emissions (PW = 100%), nitrogen mineralization (PW = 58.4%), and soil water desorption (PW = 100%). The Weibull fit to the desorption data was also equivalent to those of the well-established van Genuchten and Groenevelt–Grant desorption models. It was concluded that the adapted Weibull function has good potential for widespread and informative application to agricultural data and processes.

Optimal distributed generation and battery energy storage units integration in distribution systems considering power generation uncertainty

This paper proposes an application of the recent metaheuristic rider optimization algorithm (ROA) for determining the optimal size and location of renewable energy sources (RES) including wind turbine (WT), photovoltaic (PV), and biomass-based Distributed Generation (DG) units in distribution systems (DS). The main objective function is to minimize the total power and energy losses. Power loss-sensitivity factor (PLSF) is used with the ROA to determine the suitable candidate buses and accelerate the solution process. The Weibull and Beta probability distribution functions (PDF) are employed to characterize the variability of wind speed and solar radiation, respectively. The high penetration of intermittent renewable resource together with demand variations has introduced many challenges to distribution systems such as power fluctuations, voltage rise, high losses, and low voltage stability, therefore battery energy storage (BES) and dispatchable Biomass are considered to smooth out the fluctuations and improve supply continuity. The standard 33 and 69-bus test systems are used to verify the effectiveness of the proposed technique compared with other well-known optimization techniques. The results show that the developed approach accelerates to the near-optimal solution seamlessly, and in steady convergence characteristics compared with other techniques.

A python program to model and analyze wind speed data

A python program has been developed to analyze wind distributions using the Weibull density function. A two-parameter Weibull function is frequently used to model and assess wind potential and wind distribution. This python program finds first Weibull parameters from the recorded wind data by five different methods, namely, Empirical Method(EPM), Method of Moment (MoM), Energy Pattern Factor Method (EPFM), Maximum Likelihood Method (MLM), Modified Maximum Likelihood Method (MMLM), the parameters are then used to find theoretically fitted pdfs. The program is implemented on wind distribution of two cities of Pakistan (Chakri and Sadiq Abad). The program-generated pdfs were plotted with the histogram of recorded data, the fitting was excellent. To check the validity of the fitted pdfs, statistical errors Root Mean Square (RMSE), MeanAbsolute Percent Error (MAPE), Mean Absolute Error (MABE), and Chi-square statistic are calculated. In all cases,these statistical errors are well below the acceptance range. Both pictorial results and numerical values of statistical errors indicate the performance of the python program to analyze wind speed data

Characterizing Energetic Dependence of Low-Energy Neutron-Induced SEU and MCU and Its Influence on Estimation of Terrestrial SER in 65-nm Bulk SRAM

Characterizing low-energy neutrons (<; 10 MeV)-induced single-event upsets (SEUs) and multiple cells upsets (MCUs) is essential to validate the current standard for terrestrial soft error rate (SER) and investigate its enhancement. Following our preliminary analysis on the contribution of the low-energy neutrons to the total terrestrial SER at the nominal operating voltage of 1.0 V by Liao et al. (2020), this article newly presents and analyzes the data measured at the low operating voltage of 0.4 V. The dependence of SEU cross section on the neutron energy is similar between the operating voltages of 0.4 and 1.0 V, including onset energy of around 6 MeV. The existence of MCUs at 4.1-MeV neutrons was also confirmed at both the operating voltages. Based on the measurement, we approximate the dependence of SEU and MCU cross sections as Weibull functions of the neutron energy. The terrestrial SER of SEUs and MCUs was calculated by folding the Weibull function and the flux spectrum. The calculated result indicates that the SER originating from the low-energy neutrons is less than 6% in the terrestrial environment at New York and Tokyo City. We confirm that disregarding the flux of neutrons below 10 MeV in the acceleration factor calculation at accelerated neutron tests, which follows the current standard defined in JESD89, could give a reasonable SER estimation accuracy for both SEUs and MCUs. On the other hand, for covering the beams having an extremely high proportion of low-energy neutrons, considering the flux of neutrons above 6 MeV would be an option for better SER estimation.

The Impact of Voltage Stability Constraint L-Index on Power System Optimization Base on Interior Point Algorithm by Considering the Integration of Renewable Energy

Renewable energy (RE) generation including wind and solar farms has experienced significant growth due to the challenge of climate and energy crisis. High penetration of RE affects system stability and generation cost optimization. In this paper, voltage stability L-index is proposed as the constraint of the primal-dual interior point method for optimal power flow by considering the integration of wind and photovoltaic cell power on system. L-index is used in this study to indicate the voltage security while renewable energy is prefixed into power system to build a new (REL-OPF) formulation that can consider the voltage stability margin and generation cost of the power grid. Weibull and Beta probability density functions are applied to determine the outputs of wind and photovoltaic power generation respectively. This model has the advantage that there is no need to calculate the critical point and could obtain the demanded voltage security level by adjusting the upper limits of indicator L constraints. The optimal solution can concurrently satisfy both the voltage security and the economical demand for power systems. The overall problem is simulated under coding written in MATLAB environment. The results obtained from regulating the upper limit of voltage stability constraint on IEEE30-bus has shown that the attention of RE generations into the system leads the fuel cost objective to decrease randomly as well as enlarge the distance of stability to a critical point with faster convergence.

Prediction of sulfamerazine and sulfamethazine solubility in some cosolvent mixtures using non-ideal solution models

Introduction: Experimental Solubilities of sulfamerazine (SMR) and sulfamethazine (SMT) in some (methanol + water), (ethanol + water) and (1-propanol + water) cosolvent mixtures were collected from the literature at five temperatures from 293.15 to 313.15 K. Methodology: The results were analyzed with the van’t Hoff, Apelblat modified, Buchowski-Ksiazaczak λh, van’t Hoff-Yaws model, twoparameter Weibull function model. It was determined that the models that best describe the solubility of these sulfonamides in (alcohol + water) mixtures were thevan’t Hoff and Apelblat models, obtaining correlation indices greater than 0.99 in all cases. Results: The results obtained with the Modified Apelblat equation presents ahigh correlation index for the solubility of SMR and SMT in cosolvent mixtures, followed by the van’t Hoff-Yaws model that presents a high fit of the estimated data with respect to the theoretical ones. According to the two parameter Weibull function model, the solubility of SMR and SMT in the co-solvent mixtures shows important deviations from ideality, which is consistent with the literature. The results are discussed in terms of the solute-solvent interactions that occur in this system.

RAPID AND SIMPLE DETERMINATION OF IBUPROFEN AND CAFFEINE IN FIXED-DOSE COMBINATION FORMULATIONS: APPLICATION TO DISSOLUTION STUDIES

Objective: To develop a UV-derivative spectrophotometric method with zero-crossing determinations for the simultaneous quantification of ibuprofen (IBU) and caffeine (CAF) in fixed-dose combination formulations (soft gelatin capsules). The proposed method was validated, and it was applied to determine the in vitro dissolution performance of IBU and CAF from a commercial formulation.&#x0D; Methods: The method is based on the use of the second-derivatives of the zero-order spectra and measurement at zero-crossing wavelengths. Linearity, accuracy, precision, stability, and influence of the filter were evaluated. Dissolution profiles of IBU and CAF were obtained with the USP Apparatus 2 at 100 rpm and 900 ml of 0.1 M phosphate buffer pH 7.4 as dissolution medium. Dissolution samples were treated with the proposed UV-derivative method and results were compared with data previously published.&#x0D; Results: The zero-crossing points for the determination of IBU and CAF were found at 235.6 nm and 218.8 nm, respectively. The method was linear in the range of 7.5-15 µg/ml for IBU and 5-25 µg/ml for CAF (R2&gt;0.999, *P&lt;0.05). The precision and accuracy of the method were within acceptable criteria (CV&lt;0.99% and recovery 97.97% for IBU and CV&lt;1.76% and recovery 99.05% for CAF). Fiberglass filters were the best option to filter samples and stability of all drugs was adequate when solutions were stored at 25 °C during 24 h. Dissolution of IBU and CAF at 60 min was 99-100% with dissolution profiles of sigmoidal S-shape. Weibull function and Logistic were the best-fit models that describe the in vitro dissolution performance of both drugs.&#x0D; Conclusion: The proposed UV-derivative method allows the simultaneous determination of IBU and CAF in fixed-dose combination formulations. The method generates reliable information that can be compared with published data. The proposed UV-derivative method is rapid and simple and can be easily adopted for routine analysis of IBU and CAF.

Evaluation of weibull parameter estimators for wind speed of Jumla, Nepal

Weibull Probability Distribution Function (PDF) is widely used across world for estimation of wind power. Weibull function is a two parameter probability distribution function. The methods employed for the evaluation of these two parameters are critical for the efficient use of Weibull PDF. In the present study, three different Weibull PDF parameter estimators have been evaluated. For this purpose, the daily averaged wind speed data of Jumla Station, Nepal for period of 10 year (2004 – 2014: 2012 excluded) is studied. The parameter estimator evaluated in this study are Method of Moments (MoM), Least Square Error Method (LSEM) and Power Density Method (PDM). It has been found that Method of Moments (MoM) is the best estimator for evaluating Weibull Parameters.

Forecasting Hospitalizations Due to COVID-19 in South Dakota, USA.

Anticipating the number of hospital beds needed for patients with COVID-19 remains a challenge. Early efforts to predict hospital bed needs focused on deriving predictions from SIR models, largely at the level of countries, provinces, or states. In the USA, these models rely on data reported by state health agencies. However, predicting disease and hospitalization dynamics at the state level is complicated by geographic variation in disease parameters. In addition, it is difficult to make forecasts early in a pandemic due to minimal data. Bayesian approaches that allow models to be specified with informed prior information from areas that have already completed a disease curve can serve as prior estimates for areas that are beginning their curve. Here, a Bayesian non-linear regression (Weibull function) was used to forecast cumulative and active COVID-19 hospitalizations for SD, USA, based on data available up to 2020-07-22. As expected, early forecasts were dominated by prior information, which was derived from New York City. Importantly, hospitalization trends differed within South Dakota due to early peaks in an urban area, followed by later peaks in rural areas of the state. Combining these trends led to altered forecasts with relevant policy implications.Supplementary InformationThe online version contains supplementary material available at 10.1007/s41666-021-00094-8.

Evaluation methods for estimation of Weibull parameters used in Monte Carlo simulations for safety analysis of pressure vessels

Abstract The test data for static burst strength and load cycle fatigue strength of pressure vessels can often be well described by Gaussian normal or Weibull distribution functions. There are various approaches which can be used to determine the parameters of the Weibull distribution function; however, the performance of these methods is uncertain. In this study, six methods are evaluated by using the criterion of OSL (observed significance level) from Anderson-Darling (AD) goodness of Fit (GoF), These are: a) the norm-log based method, b) least squares regression, c) weighted least squares regression, d) a linear approach based on good linear unbiased estimators, e) maximum likelihood estimation and f) method of moments estimation. In addition, various approaches of ranking function are considered. The results show that there are no outperforming methods which can be identified clearly, primarily due to the limitation of the small sample size of the test data used for Weibull analysis. This randomness resulting from the sampling is further investigated by using Monte Carlo simulations, concluding that the sample size of the experimental data is more crucial than the exact method used to derive Weibull parameters. Finally, a recommendation is made to consider the uncertainties of the limitations due to the small size for pressure vessel testing and also for general material testing.

Evaluating diameter distribution series of small-leaved lime (Tilia cordata Mill.) in forest stands

BackgroundThe paper provides studies on the structure of planted small-leaved lime (Tilia cordata Mill.) in the conditions of the Bashkir Cis-Urals. This work aimed to analyze their assortment by diameters and compile appropriate tables. This is the first study of lime for this region. The results of the study are based on data from 69 temporary sample plots. Stands are represented by trees of 11–79 years old, not affected by thinning. They belong to the I–III growth classes and the most common goutweed forests.ResultsIt was found, that small-leaved limes have a specific structure—coefficient of diameter variation in stands decreases with a higher average diameter, reaching 26 and 16% at 28 cm with thin and dense initial planting, respectively. The variability of tree diameters is related to the average stand diameter and is due to the initial density of the grown plantations. Correlations between the coefficients of asymmetry and excess with age and the average diameter was revealed.ConclusionsThe verification of theoretically calculated frequencies of distribution series to empirically observed frequencies showed a discrepancy in 29 and 19% of the total number, respectively. For the remaining series of distributions, the scale and shape parameters of the Weibull function were modeled using the average, minimum and maximum diameters, standard deviation, coefficients of asymmetry and excess. This made it possible to develop stand tables for small-leaved limes depending on the average diameter.