Parameter Weibull Distribution Research Articles

Age-related changes in dynamic corneal backscatter observed in Scheimpflug imaging.

Corneal visualization Scheimpflug technology (Corvis ST) has the potential to indirectly provide information on corneal structure via the analysis of statistical properties of the backscatter. The aim of this work was to ascertain whether there are age-related changes in the dynamics of corneal backscatter during an air-puffed induced corneal deformation. Retrospective data from Corvis ST measurements of 151 young subjects (19-30 years) and 82 older subjects (50-87 years) were considered. Each measurement consisted of 140 frames (sampling frequency: 4330 fps). For every frame the cornea was first segmented, then regions of interest, encompassing temporal, central and nasal parts of cornea were selected, to which the parameters of Weibull distribution (scale and shape) were fitted, leading to time series of the estimated parameters. Apparent differences were found between the parameters of Weibull distribution between the two considered groups that manifest themselves mostly in the nasal region of the cornea. However, those differences cannot be attributed to the age alone. For this, a normalization method is proposed that leads to a much better separation between the groups in all considered regions. Clinical Relevance- The parameters of the corneal backscat-ter are widely used to asses corneal clarity (so-called corneal densitometry). Recently the parameters of Weibull distribution fitted to the corneal backscatter data have been used to support diagnosis of keratoconus. This work contributes to the assessment of corneal clarity by identifying the apparent age-related differences in those parameters when dynamic raw data is considered highlighting the need for such parameters to be appropriately normalized. Further it shown that the shape parameter of Weibull distribution unlike the scale parameter carries that information already for the raw non-normalized data.

Using two and three-parameter Weibull statistical model for predicting the loading rate effect on low-temperature fracture toughness of asphalt concrete with the ENDB specimen

Loading rate can affect significantly the load bearing capacity and cracking resistance of asphalt mixtures that are basically visco-elastic materials. On the other hand, the scatter of mechanical test results obtained from the asphalt mixtures (as multi ingredients and randomly distributed heterogeneous materials) would be high. Therefore, in this research the effect of loading rate on mode I fracture toughness of a fixed hot mix asphalt (HMA) mixture is investigated experimentally using several test replicates and the obtained data are analyzed using two and three parameter Weibull statistical models. The Edge Notch Disc Bend (ENDB) specimen, (that is a circular disc containing an edge crack along the diameter and loaded by three point bending) is utilized for conducting the KIc experiments under three loading rates of 0.5, 5 and 10 mm/min. For each loading rate, twenty five ENDB specimens are tested at a fixed low temperature of −12 °C and the probability of failure and Weibull distribution curves are obtained. It is shown that both two and three parameter Weibull models can be utilized successfully for predicting the mode I fracture behavior of tested asphalt material in the considered range for the loading rate. By increasing the loading rate from 0.5 to 10 mm/min, the mean KIc values and other Weibull parameters are increased. The Weibull distribution parameters obtained experimentally from the loading rates of 5 and 10 mm/min, are also predicted in terms of the Weibull parameters of the ENDB specimen tested at loading rate of 0.5 mm/min considered as a reference case.

Correlation analysis of three-parameter Weibull distribution parameters with wind energy characteristics in a semi-urban environment

Suburban areas are rich in wind resources, but wind energy characteristics are complicated by the strong turbulence disturbance of buildings. Current work on wind resource assessment is focused on the fitting of the wind speed distribution function and the measurement of wind energy abundance. The mathematical relationship between the distribution function and the main wind energy characteristics under suburban wind field conditions needs to be studied.In this study, a ZephIR 300 LiDAR wind measurement system is built in a suburban area, and wind speed data samples from heights of 11–199 m are obtained. The functional relationship between parameters is derived through a comparative analysis, which reveals that the variation of shape parameter k in the range of 1.05–1.25 causes abrupt changes in wind power density (WPD). The positive proportional relationship between the scale parameter and WPD and its correlation with height are also obtained. Turbulence intensity (TI) decrease as the scale parameter increases from 1.18 to 2.50 and fluctuates within 10.54% as the scale parameter c continues to increase from 2.50 to 5.58. The scale parameter corresponding to the minimum value of TI at each height is provided.

An Inventory Model with Three Parameter Weibull Distribution for Deteriorating Items under Inflation

This study explores a predetermined inventory system for deteriorating items with a Weibull distribution, taking into consideration nonlinear demand and trade credit under inflation and allowing for shortages and partial backlogs. The reason for this study is to find the appropriate order and replenishment policy to actually reduce relevant costs. The model is built under two scenarios; case 1: \(\left(0 \leq M \leq t_{2}\right)\) . The consumer will be charged interest on the outstanding debt if he does not pay the provider by time M. and case 2: \(\left(0 \leq t_{1} \leq M\right)\) if the consumer gets to sell all of his commodity as well as earns interest on the revenue until the account is settled. To demonstrate the use and performance of the model, numerical approach and sensitivity analysis are actually given. As a result, this model will assist retailer in determining the optimal replenishment cycles in a variety of situations, as well as provide an innovative management insight that will aid the industry reduce relevant cost.

Wind speed Weibull distribution and wind energy potential of Chandpur, Bangladesh

Weibull distribution goes particularly well with wind speed data and is necessary to assess wind energy harnessing potential at a site. For the best estimation of the statistical distribution parameters at Chandpur (23.21116° N; 90.64237° E), located at the southern part of Bangladesh, six models: graphical method, method of moments, power density method, Justus method, Lysen method and maximum likelihood method have been analyzed for 18.8 m, 40.2 m and 59.9 m above ground level (AGL) using measured wind speed data for 2014-2017. Out of the six models, Justus method has showed the best result for estimation of Weibull parameters at all heights AGL with average RMSE of 0.0082 and R2 of 0.9650. Further more to evaluate the wind energy potential at modern wind turbine hub height of 100 m, firstly the wind shear exponent at the site (the average value is 0.37 considering the 10 m elevation of the site above sea level) is estimated from the wind data of 3 heights; secondly average wind speed (4.88 m/s), wind power density (108.8 Wm-2), Weibull distribution parameters (shape factor 2.29 and scale factor 5.33 m/s) and wind turbine power curve related wind speeds (cut in 2.28 m/s, rated 9.33 m/s and cut out 21.03 m/s)have been extrapolated at the hub height. The average wind speed and wind power density values show that the class of the site is 1 in the worldwide accepted wind power classification, based on long term practical wind farm installation experiences; the low cut in speed (<3 m/s) shows that suitable wind turbines for the site are commercially unavailable for the site and turbines with 3 m/s cut in speed have average annual capacity factor of 16.62 %, less than the commercially viable capacity factor >25 %. The wind power class (< class 3) and capacity factor (<25 %) strictly reveals that Chandpur is not suitable for wind energy exploration in near future until a high technical improvement of wind turbines and reduction of relevant costs. DUJASE Vol. 6 (2) 99-108, 2021 (July)

A New Transmuted Weibull Distribution: Properties and Application

This paper proposes a new three parameter Weibull distribution obtained using a new Transmutation technique namely New Transmuted Weibull distribution. A comprehensive account of some of the mathematical properties of new model are derived. Entropy estimation and parameter estimation is also carried out using different methods. Finally, it will be shown that the analytical results are applicable to model real data.

On transmuted skewed Laplace distribution and its application

The aim of this study is to add a shape parameter into the existing 3-parameter skewed Laplace distribution using the method of transmutation and apply the proposed distribution on two life simulated data generated using R-statistical software. The parameters of the proposed distribution are obtained using the method of maximum likelihood estimation. The fitness of the proposed distribution was compared with that of existing skewed Laplace distribution and four parameter Weibull distribution using measures of fitness criteria, and the results obtained showed that the proposed model can be used effectively to model real life situations than the Skewed Laplace Distribution. The results also showed that the Four Parameter Weibull Distribution which is a competing model performed better than the Skewed Laplace Distribution and Transmuted Skewed Laplace Distribution.

Constructing Bootstrap Confidence Intervals of Process Capability Indices for a Three Parameter Weibull Distribution

Statistical Quality Control (SQC) technique is used in investigating the quality improvement features in a manufacturing process. One of the important tools in SQC is the process capability indices (PCIs), for measuring and comparing the characteristics of a productionprocess to engineering specifications. This article evaluates the performance of the PCIs for a three parameter Weibull distribution which is commonly employed in almost all fields of studies, such as reliability, stability and breakage data. In this article, three different techniques of constructing bootstrap confidence intervals (BCIs) of PCIs are investigated using simulations for the three parameterWeibull distribution. The three different techniques considered are percentile bootstrap, bias-corrected percentile bootstrap and normal bootstrap techniques. By using simulation, the authors investigate the average widths of the BCI of each of the three different techniques. It is found that the average widths of the bias-corrected percentile bootstrap technique are shorter than that of the other two techniques.

Experimental and statistical investigation on the wear and hardness behaviour of multiwalled carbon nanotubes reinforced copper nanocomposites

In this experimental work the two parameter Weibull distribution statistical method is used to investigate the effect of fabrication process on the wear and hardness properties of multiwalled carbon nanotubes reinforced copper nanocomposites (MWCNTs/Cu). The MWCNTs/Cu nanocomposites are fabricated by the powder metallurgy processing route in two ways (i) MWCNTs/Cu nanocomposites without hot pressing (NHP-MWCNTs/Cu) and (ii) MWCNTs/Cu nanocomposites with hot pressing (HP-MWCNTs/Cu). The MWCNTs doping weight percentage (wt.%) include 0.0 (Pure copper), 0.5, 1.0 and 1.5. The MWCNTs/Cu nanocomposites are subjected to hardness and wear tests using micro-Vickers hardness tester and rotating type pin-on-disc tribometer, respectively. From the experimental results it is found that the MWNTs/Cu nanocomposites have higher hardness and lower wear rate than the pure Cu. Further, the HP-MWCNTs/Cu nanocomposites showed higher hardness and lower wear rate than the NHP-MWCNTs/C nanocomposites. From statistical analysis it is found that the HP-MWCTs/Cu nanocomposites showed more reliable and repeatable experimental data than the NHP-MWCNTs/Cu nanocomposites. Statistical analysis showed that the effectiveness of MWCNTs doping on the wear rate of Cu is more effective at lower sliding speed. Finally, it is observed form the statistical analysis that both theoretical and experimental results of hardness and wear tests showed good correlation.

An Iterative Method for Parameter Estimation of the Three-Parameter Weibull Distribution Based on a Small Sample Size with a Fixed Shape Parameter

The three-parameter Weibull distribution is popular in reliability analysis. However, in the context of a small sample size, the issue of parameter estimation of the three-parameter Weibull distribution is difficult to address. If only a small sample size is available, a reasonable method is to empirically determine the shape parameter, and the emphasis is placed on location parameter estimation. This paper presents a theoretical model to establish the relationship between the location parameter, the minimal sample value, and the sample size. Moreover, an iterative method is proposed to estimate the Weibull location parameter and scale parameter with an empirical value for the shape parameter. Compared with the maximum likelihood method (MLE), the Weibull plot with the sample correlation coefficient and TL-moment, the proposed method can more effectively estimate the parameters of the three-parameter Weibull distribution.

Sequential Bayesian inference for Weibull distribution parameters with initial hyperparameter optimization for system reliability estimation

In this study, we investigated the sequential Bayesian inference of the Weibull distribution parameters of the components of a system the remaining useful life of which can be estimated using lifetime estimation or survival function of certain types of components. Failure observations are collected sequentially and analyzed as censored data. Here, we propose a sequential Bayesian inference with optimized prior distribution (SBOPD) to provide an effective way to estimate the parameters of the Weibull distribution of the components. In this framework, the Bayesian inference of the parameters of the Weibull distribution is updated by a certain number of new failure observations. Two measures are used to estimate more accurately and effectively in the preliminary stages of the system life cycle; the posterior distribution of the previous update is used as the prior information of the new update, and the initial prior distribution is optimized using prior knowledge of the parameters. With optimized hyperparameters for the initial prior distribution, parameter estimation with SBOPD provided a better estimate at the preliminary stages of the system life than conventional methods.

A multi-objective programming approach to Weibull parameter estimation

Weibull distribution is widely used in various areas such as life tables, failure rates, and definition of wind speed distribution. Therefore, parameter estimation for the Weibull distribution is an important problem in many real data applications. The least square (LS), the weighted least square (WLS) and the maximum likelihood (ML) are the most popular methods for the parameter estimation in the Weibull distribution. In this study, based on the LS, WLS and ML estimation methods, a multi-objective programming approach is proposed for the parameter estimation of two-parameter Weibull distribution. This new approach evaluates together LS, WLS and ML methods in the estimation process. NSGA-II method, which is a multi-objective heuristic optimization method, is used to solve the proposed multi-objective estimation model. To evaluate the applicability and performance of the proposed approach, a detailed Monte Carlo simulation study based on deficiency criteria and a real data application are designed. The results illustrated that the proposed multi-objective programming approach provides quite accurate parameter estimates for the two parameter Weibull distribution with respect to deficiency criteria.

Comperative Analysis of Weibull Parameter Estimation Using HPSOBA and Standard PSO and Classical Bat Algorithms

The paper benevolences a hybrid metaheuristic optimization algorithm developed for the computation of Weibull probability and cumulative density functions (PDF) and (CDF) respectively. The algorithm is designed through espousing protruding topographies wich are used in the Bat algorithm (BA) and in the Particle Swarm Optimization (PSO) and is used to analyze the figure and measure constraints of Weibull Distribution (WD). The established algorithm is also verified on target test functions. The results and computational time are compared with the results obtained through the implementation of standard PSO and BA. The computed values of the parameters of WD are also verified through the calculation of, index of agreement (AI), mean absolute percentage error (MAPE) coefficient of determination (R2) and root mean square error (RMSE). The results of parameter estimation are also compared with standard PSO and BA for the mentioned region. The comparative analysis of the results of benchmark functions and Weibull parameter estimation reveal the robustness and verstality of the developed algorithm.

Analytical Solution of two Model Equations for the Variation of Capital Market Prices

This paper investigated different methods for estimation of parameters of Weibull distribution, using Mean Square Error (MSE) as a criterion for selecting the best model. The result showed that Method of Moments outstripped other methods. In the same vain, the estimated results were logically extended to form a matrix that would help in predicting different commodity price processes, and the result obtained by exploring the properties of the principal component analysis solution and results showed the level of proportion accounted by first Principal Component Analysis (PCA). However, the eigenvectors describe the direction of the stock market prices in terms of changes in short-run and long-run respectively.

G-R Curves Based on Weibull Distribution

China Earthquake time intervals have been calculated and passed the hypothesis test based on Weibull distribution. Comparative analyses have been carried out with the China Earthquake Catalogue according to different time intervals and different zones. From the results, we can see that the shape parameters are enlarged with the year’s extrapolation. b value curves and Amira images of the global plate subduction zones have been drawn, and the parameters of Weibull Distribution have been calculated. The relationship diagrams of b values and the parameters of Weibull Distribution of the whole subduction zones have been drawn.

Machine learning‐based statistical downscaling of wind resource maps using multi‐resolution topographical data

AbstractThe need for high‐resolution wind resource maps is increasing with the increase in the supply and development of wind power. Many physical downscaling models have been developed and applied to make these maps. However, as the existing models require extensive computations and time, statistical models with higher efficiency are being studied. Statistical models such as regression and machine learning models can quickly calculate wind resource maps, but they have a problem of low accuracy. This study proposes a machine learning model with new topography‐derived variables to interpret the physical characteristics of the wind. As the shape of topography, which was unable to be interpreted in previous studies, can be considered with new derived variables, a significant performance improvement was identified. The analysis was conducted using 1 km Weather Research and Forecasting (WRF) results and ERA5 reanalysis data from South Korea. Two Weibull distribution parameter maps were calculated and used as input and output data. Three collections of derived variables were devised and compared. Therefore, the multi‐resolution topography data showed the highest improvements with approximately 15% reduction in root mean square error (RMSE) for both the linear regression and machine learning models. In particular, the land area showed a decrease of 20%. The best proposed models showed an RMSE of 7% and 8% for two Weibull parameters. The results are expected to serve as a reference for continuing research and utilization of statistical models.

Calculation of Weibull distribution parameters at low wind speed and performance analysis

In this study, the compatibility of the real wind energy potential to the estimated wind energy potential by Weibull distribution function (WDF) of a region with low average wind speed potential was examined. The main purpose of this study is to examine the performance of six different methods used to find the coefficients of the WDF and to determine the best-performing method for the selected region. In this study 7 years' worth of hourly wind speed data obtained from the general directorate of meteorology of the Adana region in Turkey were used. The root mean square error statistical indicator was used to compare the efficiency of all methods used. Another main purpose of this study is to observe how the performance of the methods used changes over the years. The obtained results showed that the performances of the methods used exhibited slight changes over the years, but when evaluated in general, it was observed that all methods showed acceptable performance. Based on the obtained results, when the 7-year data are evaluated in this selected region, it can be said that the moment method shows the best performance.

Techno-economic analysis to develop the first wind farm in the Egyptian western desert at Elkharga Oasis

Abstract During the past few years, Egypt has increased the share of electricity generation from renewable energy sources to achieve the renewable-energy strategy. The Egyptian Ministry of Electricity and Renewable Energy has set an ambitious target to increase the share of renewable energy among the total energy mix to be 20% by 2022 and 42% by 2035. This target will be achieved using several approved policies such as feed-in tariffs, build–own–operate and independent power producers under a merchant scheme. This paper presents a full analysis of wind-potential characteristics at Elkharga Oasis in Egypt based on an actual wind-measurement campaign taken by a met mast at two height levels of 10 and 25 m, respectively. The measurements show an average annual wind speed of 5.72 m/s at 10-m height and 6.53 m/s at >25-m height. The estimated wind speed, however, is 8.38 m/s at 120 m, which is technically convenient to generate electricity from wind energy. In this paper, the wind potential in the area is assessed using the statistical method of the Weibull probability density function. The different methods to estimate the Weibull distribution parameters are presented and the relevant accuracy is verified based on the root mean square error. A techno-economic assessment and selection of the optimal wind-turbine size with ideal micro-sitings are performed using the software package windPRO. Different cases of study for three typical wind-turbine models with various rotor diameters, power capacity, layout configuration and annual yield are assessed at the site. The selection of the most convenient wind turbine is determined based on the IEC 61400 standard criteria and the turbine that produces the lowest levelized cost of electricity.

C-Vine Copula 기반의 풍력 발전 투입을 위한 확률론적 조류 계산

Owing to the increase in renewable energy, the power system is exposed to more variability and uncertainty, which increases the difficulty in analyzing the power systems. Thus, an accurate power system analysis considering the stochastic characteristics of renewable energy is required. In this paper, we propose a Monte-Carlo based probabilistic load flow method that takes into account the statistical dependencies between the power generation from the multiple wind sites. To this end, the vine copula method was employed to capture their dependencies and the marginal distribution of the power generation from the multiple wind sites is modeled using a 3-parameter Weibull distribution. In addition, we suggested the chance-constrained optimization problem for properly allocating a target wind power capacity to the buses. From the numerical experiment conducted on the IEEE 39-bus system, it has been demonstrated that the proposed method can more accurately identify the potential risk with the modeling of the dependencies between the power generation from the multiple wind sites as compared to the deterministic approach or probabilistic load flow based on the random sampling method.

IGBT lifetime model considering composite failure modes

As a new generation of power devices, IGBT devices are widely used in fields such as aerospace, power transmission and transformation. Failures happen to the modules under different mechanisms and temperature conditions, but the current practice rarely considers the interaction of multiple failure mechanisms in the lifetime model, leading to an over optimistic lifetime prediction. With the analysis of the IGBT failure mechanism, this paper studies the IGBT's overall failure probability distribution assuming the failure of bonding wire and solder layer follows Weibull distribution. By exploring the impacts of the Weibull parameters, the relationship between the lifetime of each failed component and the IGBT's overall lifetime is quantified with the fitted coupling coefficients. The analyzing results show that when the bonding wire and the solder layer have similar lifetime, the IGBT's whole lifetime will be reduced, and the reduction degree is affected by the shape parameters of Weibull distribution, up to 30%. The IGBT lifetime model considering coupling effect is then established, which is verified by the experiments. The experimental results prove the existence of coupling effect, as well as the superiority of the model in lifetime prediction, whose error is reduced by 4.1% compared with that without coupling effect.