Mixture Of Weibull Distributions Research Articles

Bayesian pairwise meta-analysis of time-to-event outcomes in the presence of non-proportional hazards: A simulation study of flexible parametric, piecewise exponential and fractional polynomial models.

Traditionally, meta-analysis of time-to-event outcomes reports a single pooled hazard ratio assuming proportional hazards (PH). For health technology assessment evaluations, hazard ratios are frequently extrapolated across a lifetime horizon. However, when treatment effects vary over time, an assumption of PH is not always valid. The Royston-Parmar (RP), piecewise exponential (PE), and fractional polynomial (FP) models can accommodate non-PH and provide plausible extrapolations of survival curves beyond observed data. Simulation study to assess and compare the performance of RP, PE, and FP models in a Bayesian framework estimating restricted mean survival time difference (RMSTD) at 50 years from a pairwise meta-analysis with evidence of non-PH. Individual patient data were generated from a mixture Weibull distribution. Twelve scenarios were considered varying the amount of follow-up data, number of trials in a meta-analysis, non-PH interaction coefficient, and prior distributions. Performance was assessed through bias and mean squared error. Models were applied to a metastatic breast cancer example. FP models performed best when the non-PH interaction coefficient was 0.2. RP models performed best in scenarios with complete follow-up data. PE models performed well on average across all scenarios. In the metastatic breast cancer example, RMSTD at 50-years ranged from -14.6 to 8.48 months. Synthesis of time-to-event outcomes and estimation of RMSTD in the presence of non-PH can be challenging and computationally intensive. Different approaches make different assumptions regarding extrapolation and sensitivity analyses varying key assumptions are essential to check the robustness of conclusions to different assumptions for the underlying survival function.

Reliability model for key components of urban rail transit train based on improved hunter-prey optimization

The reliability of key components of urban rail transit (URT) plays an important role in the maintenance plans of URT. It is necessary to establish the reliability model of URT trains. In the current research, the reliability model has a limited scope of application and fails to accurately depict the reliability of key components in URT trains. To solve the above problem, a multi-peak type mixture Weibull distribution model is established using several three-parameter Weibull distributions based on fault modes of components sourced from historical lifetime data. Due to the complexity of this model, parameter estimation is challenging. For this purpose, an improved hunter-prey optimization (IHPO) was proposed to improve parameter estimation accuracy. Firstly, an improved Hénon chaos map was introduced to improve the distribution of the initial population. Secondly, the Lévy flight was introduced to increase the probability of the individual spreading to the whole range at the late stage. Lastly, a nonlinear balance factor was proposed to enhance the algorithm’s global search capability. The simulation experiment was carried out with examples of the balanced pressing wheel and the wheelset. The IHPO algorithm-based parameter estimation method shows the highest R-square with values of 0.996 and 0.999, respectively, and the lowest root mean square error with values of 0.019 and 0.008, respectively. The simulation results demonstrate that the stability and optimization of the HPO are improved, and the multi-peak mixture Weibull distribution model based on the IHPO can accurately depict URT trains’ reliability.

Reliability evaluation of components with multiple failure modes based on mixture Weibull distribution using expectation maximization algorithm

Reliability evaluation of components with multiple failure modes based on mixture Weibull distribution using expectation maximization algorithm

Brazilian wind energy generation potential using mixtures of Weibull distributions

As concerns about the greenhouse effect and the resulting increase in carbon dioxide levels in the atmosphere continue to mount, there is an increasing need to curtail the use of fossil fuels for energy generation, especially given the ever-growing energy demands of the population. In this work, we applied the Weibull–Weibull distribution mixture to adjust the wind speed series of all 575 weather stations at six different heights for the 12 months of the year in the period from 01-01-2000 to 03-31-2022 measured in the hourly interval. We present the main results for the month with the lowest and the month with the highest incidence of wind speed in Brazil for the height of 120 m, which are the months of May and September, respectively. In estimating the Weibull–Weibull parameters, the Expectation–Maximization algorithm was used. Using the results obtained at each meteorological station and the Inverse Distance Weighting method, we predicted wind energy generation at points without information about wind speed. We observed that the Weibull–Weibull distribution mixture proved excellent for estimating wind energy density distribution over large areas We were able to provide a suitable fit for the different regions of Brazil. The findings indicate that the model employed has the potential to serve as a strong contender for adoption in other nations, enabling the integration of alternative energy sources in a complementary fashion. This measure could prove instrumental in mitigating the use of environmentally hazardous energy sources.

Genetic least square estimation approach to wind power curve modelling and wind power prediction

Wind power curve (WPC) is an important index of wind turbines, and it plays an important role in wind power prediction and condition monitoring of wind turbines. Motivated by model parameter estimation of logistic function in WPC modelling, aimed at the problem of selecting initial value of model parameter estimation and local optimum result, based on the combination of genetic algorithm and least square estimation method, a genetic least square estimation (GLSE) method of parameter estimation is proposed, and the global optimum estimation result can be obtained. Six evaluation indices including the root mean square error, the coefficient of determination R2, the mean absolute error, the mean absolute percentage error, the improved Akaike information criterion and the Bayesian information criterion are used to select the optimal power curve model in the different candidate models, and avoid the model’s over-fitting. Finally, to predict the annual energy production and output power of wind turbines, a two-component Weibull mixture distribution wind speed model and five-parameter logistic function power curve model are applied in a wind farm of Jiangsu Province, China. The results show that the GLSE approach proposed in this paper is feasible and effective in WPC modelling and wind power prediction, which can improve the accuracy of model parameter estimation, and five-parameter logistic function can be preferred compared with high-order polynomial and four-parameter logistic function when the fitting accuracy is close.

Wind power potential over India using the ERA5 reanalysis

India’s growing installed capacity for electricity generation faces demands from surging domestic consumption, national energy security and obligations under climate change accords. It is exigent to increase focus on renewables as their share in net electricity generated may reach 22 % by 2030, against the goal of 40 % from non-fossil fuels by 2030 as pledged under the Paris Agreement. Wind energy is preferable at higher levels of renewables due to favourable capacity factors. Prior investigations necessary to develop wind farms are better achieved through reanalysis datasets than satellite-derived wind products, which are temporarily coarse and limited to off-shore regions. In this study, we estimate wind power potential at 100 m level over the Indian domain using data from the ERA5 reanalysis for 1979–2018 through the Weibull mixture distribution. Our findings indicate that India’s mainland (off-shore) regions have a wind power potential of nearly 7.2 TW (18.4 TW), considering just 3 % of the available area. Further, up to 53 TW is available for exploitation over sparsely vegetated areas, which extend up to 25 % of India’s mainland. Future efforts for exploiting wind energy should focus on the availability of wind resources and their proximity to energy-hungry regions.

Too much information: Why CDCL solvers need to forget learned clauses.

Conflict-driven clause learning (CDCL) is a remarkably successful paradigm for solving the satisfiability problem of propositional logic. Instead of a simple depth-first backtracking approach, this kind of solver learns the reason behind occurring conflicts in the form of additional clauses. However, despite the enormous success of CDCL solvers, there is still only a limited understanding of what influences the performance of these solvers in what way. Considering different measures, this paper demonstrates, quite surprisingly, that clause learning (without being able to get rid of some clauses) can not only help the solver but can oftentimes deteriorate the solution process dramatically. By conducting extensive empirical analysis, we furthermore find that the runtime distributions of CDCL solvers are multimodal. This multimodality can be seen as a reason for the deterioration phenomenon described above. Simultaneously, it also gives an indication of why clause learning in combination with clause deletion is virtually the de facto standard of SAT solving, in spite of this phenomenon. As a final contribution, we show that Weibull mixture distributions can accurately describe the multimodal distributions. Thus, adding new clauses to a base instance has an inherent effect of making runtimes long-tailed. This insight provides an explanation as to why the technique of forgetting clauses is useful in CDCL solvers apart from the optimization of unit propagation speed.

A Mixture of Weibull Distribution and its Inverse: Properties and Estimation

A Mixture of Weibull Distribution and its Inverse: Properties and Estimation

Mixture probability distribution functions using novel metaheuristic method in wind speed modeling

Wind speed modeling is essential for predicting wind energy potential and properly utilizing wind power. The efficiency of the chosen probability distribution function (PDF) to describe the measured wind speed frequency distribution determines the quality of the wind speed assessment. The original probability distribution functions are insufficient in wind speed modeling as they have a significant error between real wind speed frequency distribution and the estimated distribution curve, while the mixture probability distributions can enhance the fitting of wind speed frequency distribution and reduce this error. The objective of this work is to model wind speed characteristics using two and three components mixture probability distribution functions generated from the combination of original Weibull, Gamma, and Inverse Gaussian distribution functions. To examine the fitness of the probability distribution functions, statistical errors such as the root mean square error (RMSE), Chi-square error (X2), and coefficient of Correlation (R2) are used as judging criteria. Results indicate that the two-component mixture Weibull distribution gives the best fitting among the original and the other two-component mixture PDFs. Moreover, the results show that the three-component mixture Weibull distribution gives the best fitting among all distributions proposed in this work. The mixture distribution functions used in this study have a higher level of complexity than the original distribution in estimating parameters as they have a higher number of optimization variables that need an artificial intelligence optimization method. The novel metaheuristic Aquila Optimizer (AO) method is used to estimate the parameters of proposed original and mixture PDFs in this work.

Distribution of shear fragments classified by morphology

In studies on metal shell under internal explosive loading, unlike brittle metals, ductile metals usually exhibit shear fracture. Existing literature mostly focuses on slip fracture caused by unidirectional shear fracture, in fact, the shell fractures in the two directions of maximum shear stress, making morphology of the fragments diversified, considering fragment morphology in the distribution may achieve a better description. In previous studies, Poisson mixtures and Weibull mixtures model are only used for fragment distribution statistics as a functional form, and parameters (weight coefficient and distribution modulus) in the function lack physical process description. In this paper, an experimental study on the fragmentation of 50SiMnVB steel shell under TNT internal explosive loading was carried out. Through the observation and analysis of recovered fragments, the shear fragments with cross-sections of parallelogram, right-angled trapezoid, and isosceles right-angled triangle are divided into three types: A, B, and C. According to “A-type fragments created by the parallel through-the-thickness fractures, B-type and C-type fragments created by interaction of intersecting fractures,” the morphological parameters of fragments in the distribution model were derived, including fragment size (width, length, thickness, mass, etc.), shape parameters, and weight coefficient. Weibull mixture distribution model is used to describe the distribution of the total fragments, whereas Weibull distribution is used to describe the distribution of each type of fragment. Through the comparison of theoretical calculations and experimental data, the applicability of the theoretical model in the field of “shear fracture of ductile steel shell under internal explosive loading” is verified.

OP236 Evidence Synthesis Of Time-To-Event Outcomes In The Presence Of Non-Proportional Hazards

IntroductionSynthesis of clinical effectiveness is a well-established component of health technology assessment (HTA) combining data from multiple trials to obtain an overall pooled estimate of clinical effectiveness, which may inform an associated economic evaluation. Time-to-event outcomes are often synthesized using effect measures from Cox proportional hazards models assuming a constant hazard ratio over time. However, where treatment effects vary over time an assumption of proportional hazards is not always valid. Several methods have been proposed for synthesizing time-to-event outcomes in the presence of non-proportional hazards. However, guidance on choosing between these methods and the implications for HTA is lacking.MethodsWe applied five methods for estimating treatment effects from time-to-event outcomes, which relax the proportional hazards assumption to a network of melanoma trials, reporting overall survival: restricted mean survival time, an accelerated failure time generalized gamma model, piecewise exponential, fractional polynomial and Royston-Parmar models. We conducted a simulation study to compare these five methods. Simulated individual patient data was generated from a mixture Weibull distribution assuming a treatment-time interaction. Each simulated meta-analysis consisted of five trials with varying numbers of patients and length of follow-up across trials. For each model fitted to each dataset, we calculated the restricted mean survival time at the end of observed follow-up and following extrapolation to a 20-year time horizon.ResultsAll models fitted the melanoma data reasonably well with some variation in the treatment rankings and differences in the survival curves. The simulation study demonstrated the potential for different conclusions from different modelling approaches.ConclusionsThe restricted mean survival time, generalized gamma, piecewise exponential, fractional polynomial and Royston-Parmar models can all accommodate non-proportional hazards and differing lengths of trial follow-up within an evidence synthesis of time-to-event outcomes. Further work is needed in this area to extend the simulation study to the network meta-analysis setting and provide guidance on the key considerations for informing model choice for the purposes of HTA.

Learning Interpretable Mixture of Weibull Distributions—Exploratory Analysis of How Economic Development Influences the Incidence of COVID-19 Deaths

This paper presents an algorithm for learning local Weibull models, whose operating regions are represented by fuzzy rules. The applicability of the proposed method is demonstrated in estimating the mortality rate of the COVID-19 pandemic. The reproducible results show that there is a significant difference between mortality rates of countries due to their economic situation, urbanization, and the state of the health sector. The proposed method is compared with the semi-parametric Cox proportional hazard regression method. The distribution functions of these two methods are close to each other, so the proposed method can estimate efficiently.

Bayesian Approximation for a Mixture Weibull and Lomax Distributions based on Progressive Type-II Censoring Scheme

Bayesian Approximation for a Mixture Weibull and Lomax Distributions based on Progressive Type-II Censoring Scheme

Modeling in Forestry Using Mixture Models Fitted to Grouped and Ungrouped Data

The creation and maintenance of complex forest structures has become an important forestry objective. Complex forest structures, often expressed in multimodal shapes of tree size/diameter (DBH) distributions, are challenging to model. Mixture probability density functions of two- or three-component gamma, log-normal, and Weibull mixture models offer a solution and can additionally provide insights into forest dynamics. Model parameters can be efficiently estimated with the maximum likelihood (ML) approach using iterative methods such as the Newton-Raphson (NR) algorithm. However, the NR algorithm is sensitive to the choice of initial values and does not always converge. As an alternative, we explored the use of the iterative expectation-maximization (EM) algorithm for estimating parameters of the aforementioned mixture models because it always converges to ML estimators. Since forestry data frequently occur both in grouped (classified) and ungrouped (raw) forms, the EM algorithm was applied to explore the goodness-of-fit of the gamma, log-normal, and Weibull mixture distributions in three sample plots that exhibited irregular, multimodal, highly skewed, and heavy-tailed DBH distributions where some size classes were empty. The EM-based goodness-of-fit was further compared against a nonparametric kernel-based density estimation (NK) model and the recently popularized gamma-shaped mixture (GSM) models using the ungrouped data. In this example application, the EM algorithm provided well-fitting two- or three-component mixture models for all three model families. The number of components of the best-fitting models differed among the three sample plots (but not among model families) and the mixture models of the log-normal and gamma families provided a better fit than the Weibull distribution for grouped and ungrouped data. For ungrouped data, both log-normal and gamma mixture distributions outperformed the GSM model and, with the exception of the multimodal diameter distribution, also the NK model. The EM algorithm appears to be a promising tool for modeling complex forest structures.

Wind energy potential assessment based on wind speed, its direction and power data

Based on wind speed, direction and power data, an assessment method of wind energy potential using finite mixture statistical distributions is proposed. Considering the correlation existing and the effect between wind speed and direction, the angular-linear modeling approach is adopted to construct the joint probability density function of wind speed and direction. For modeling the distribution of wind power density and estimating model parameters of null or low wind speed and multimodal wind speed data, based on expectation–maximization algorithm, a two-component three-parameter Weibull mixture distribution is chosen as wind speed model, and a von Mises mixture distribution with nine components and six components are selected as the models of wind direction and the correlation circular variable between wind speed and direction, respectively. A comprehensive technique of model selection, which includes Akaike information criterion, Bayesian information criterion, the coefficient of determination R2 and root mean squared error, is used to select the optimal model in all candidate models. The proposed method is applied to averaged 10-min field monitoring wind data and compared with the other estimation methods and judged by the values of R2 and root mean squared error, histogram plot and wind rose diagram. The results show that the proposed method is effective and the area under study is not suitable for wide wind turbine applications, and the estimated wind energy potential would be inaccuracy without considering the influence of wind direction.

Mixture Weibull probabilistic model in Wind Turbine Power Analysis

Wind energy is renewable and environment friendly. It is pollution free source of energy when compared with fossil fuels that pollute the lower layer of atmosphere. A new method depending on the parameters of three parameter mixture Weibull distribution is presented in this paper for calculating the output energy. Parameter estimation is much critical for the application of statistical model and is a challenging problem precisely for a Weibull distribution with more than two parameters. A study on estimation of parameters using maximum likelihood method is discussed. An expression to estimate the Capacity factor is formulated and the average power produced from a turbine is computed using the three parameter mixture Weibull distribution.

Mathematical representation of Wind Turbine Power reliability

Nowadays wind resource assessment utilize new advanced technologies and appropriate analytical methods which are applied to estimate how much wind as a fossil free fuel will be available from a wind farm over the period of its performance. The utmost important piece of information is determining the expected generation of energy from the power plant and ultimately how much cost effective it will be. In our study we have proposed a new method, the improved mixture Weibull distribution produced from the combination of two and three parameter Weibull distribution with six parameters which include shape, scale, location parameter and the weight component or the mixing parameter. The basic properties of the improved mixture Weibull distribution and the estimation of parameters using maximum likelihood method are discussed. The estimated parameters are used to derive a mathematical model to compute the capacity factor and wind power density.

L-Moments Estimations for Mixture of Weibull Distributions

Mixture of Weibull distributions has wide application in modeling of heterogeneous data sets. The parameter estimation is one of the most important problems related to mixture of Weibull distributions. In this paper, we propose a L-moment estimation method for mixture of two Weibull distributions. The proposed method is compared with maximum likelihood estimation (MLE) method according to the bias, the mean absolute error, the mean total error and completion time of the algorithm (time) by simulation study. Also, applications to real data sets are given to show the flexibility and potentiality of the proposed estimation method. The comparison shows that, the proposed method is better than MLE method.

L-Moments Estimations for Mixture of Weibull Distributions

L-Moments Estimations for Mixture of Weibull Distributions

Wind Speed and Power Density Analyses Based on Mixture Weibull and Maximum Entropy Distributions

Wind resource is important part of the utilization of renewable energy. To effectively estimate the wind energy potential for a given area, a variety of probability density functions (pdf) have been available in literature. In this paper, the bimodal mixture Weibull function (WW) and the probability function derived with maximum entropy principle (MEP) will be used and com- pared with the conventional Weibull function. Wind speed data measured at three wind farms experiencing different climatic environments in Taiwan are selected as sample data to test their performance. Judgment criterions include four kinds of statistical errors, i.e. the max error in Kolmogorov-Smirnov test, Chi-square error, root mean square error and relative error of wind potential energy. The results show that the proposed WW and MEP pdfs describe wind charac- terizations better than the conventional Weibull pdf, irrespective of wind speed and wind power density data, particularly for a location where wind regime presents two humps on it. For wind speed distributions, the WW pdf describes best according to the Kolmogorov-Smirnov test; while for wind power density, the MEP pdf outperforms the others.