Truncated Weibull Distribution Research Articles

An Approach to Remaining Life Estimation for Power Transformers with Biased Data

Abstract Power transformers are among the most important equipment in power systems. Estimating the remaining life of transformers is an important issue for energy companies because of the need for planning maintenance and capital expenditures. Traditional methods fail to provide accurate results mainly because data is incomplete, and there is a significant lack of records of equipment scrapped before 2015. Given this, the study provides a new approach to life estimation for power transformers based on survival analysis. First, the data was selected from only the transformers that were still operating after 2015. Then, a Bayesian survival model with a truncated Weibull distribution is developed to fit the biased data. Parameters are, at last, estimated through maximum likelihood estimation. The experimental results indicate that the proposed method performs well.

Unit upper truncated Weibull distribution with extension to 0 and 1 inflated model – Theory and applications

A two-parameter unit distribution and its regression model plus its extension to 0 and 1 inflation is introduced and studied. The distribution is called the unit upper truncated Weibull (UUTW) distribution, while the inflated variant is called the 0−1 inflated unit upper truncated Weibull (ZOIUUTW) distribution. The UUTW distribution has an increasing and a J-shaped hazard rate function. The parameters of the proposed models are estimated by the method of maximum likelihood estimation. For the UUTW distribution, two practical examples involving household expenditure and maximum flood level data are used to show its flexibility and the proposed distribution demonstrates better fit tendencies than some of the competing unit distributions. Application of the proposed regression model demonstrates adequate capability in describing the real data set with better modeling proficiency than the existing competing models. Then, for the ZOIUUTW distribution, the CD34+ data involving cancer patients are analyzed to show the flexibility of the model in characterizing inflation at both endpoints of the unit interval.

A new approach to project stand tables

Recent research studies have shown that an integrated system can be constructed to provide future predictions when the inputs are from plot summary, a tree list, or a diameter distribution. The current study addresses the missing link, allowing this integrated system to also accept inputs from a stand table. In this study, different existing techniques were employed to process data in stand table format. Trees in each diameter class were modeled with a truncated Weibull distribution. A tree list was generated by dividing each diameter class into 0.2-cm sub-classes, the midpoint diameter of which represented diameter of an individual tree. Projecting the stand table was equivalent to growing trees in the tree list. Future tree survival and diameter growth were predicted by use of an individual-tree model, which was derived from a whole-stand model. This new method outperformed three existing stand table projection models, based on two error indices.

A Note on the Right Truncated Weibull Distribution and the Minimum of Power Function Distributions

A Note on the Right Truncated Weibull Distribution and the Minimum of Power Function Distributions

An Algorithm to Determine the Truncated Weibull Parameters for Distribution of Throats and Pores in Random Network Models

In random network models, sizes for pores and throats are distributed according to a truncated Weibull distribution. As a result, parameters defining the shape of the distribution are critical for the characteristic of the network. In this paper, an algorithm to distribute pores and throats in random network was established to more representatively describe the topology of porous media. First, relations between Weibull parameters and the distribution of dimensionless throat sizes were studied and a series of standard curves were obtained. Then, by analyzing the capillary pressure curve of the core sample, frequency distribution histogram of throat sizes was obtained. All the sizes were transformed to dimensionless numbers ranged from 0 to 1. Curves of the core were compared to the standard curves, and truncated Weibull parameters could be determined according an inverse algorithm. Finally, aspect ratio and average length of throats were adjusted to simultaneously fit the porosity and the capillary pressure curves and the whole network was established. The predicted relative permeability curves were in good agreement with the experimental data of cores, indicating the validity of the algorithm.

Censored expectation maximization algorithm for mixtures: Application to intertrade waiting times

In a previous analysis the problem induced by “zero-inflation” in time series data (caused by high frequency trading in the electronic order book) was handled by left-truncating the waiting times between consecutive limit orders . We demonstrated, using rigorous statistical methods, that the truncated Weibull distribution describes the corresponding stochastic dynamics for the entire range of inter-arrival limit order waiting times, except for a region close to zero. However, since the truncated Weibull distribution was not able to describe the prodigious “zero-inflated” probability mass in the neighbourhood of zero (making up approximately 50% of the data for limit orders), it became clear that the entire probability distribution must be a mixture distribution of which the Weibull distribution is a significant part. To investigate this idea, we use a “censored expectation–maximization algorithm” to analyse the intertrade waiting times data for four selected stocks trading on the London Stock Exchange. The intertrade waiting times usually have a much lower percentage of zero inflation, typically around 2.5%. Making use of this new method and testing various mixture models, we show that the desired mixture consists of the Weibull distribution with the universal shape parameter of β≃0.57 plus an additional exponential distribution. This is the same value for the shape parameter found already in our previous study. The “1 exponential + 1 Weibull” mixture describes the intertrade waiting times extremely well at all time scales. While the Weibull component dominates in the transition and tail regions the exponential distribution explains the “zero-inflated” excess mass.

New Probability Distributions in Astrophysics: V. The Truncated Weibull Distribution

We demonstrate that certain astrophysical distributions can be modelled with the truncated Weibull distribution, which can lead to some insights: in particular, we report the average value, the rth moment, the variance, the median, the mode, the generation of random numbers, and the evaluation of the two parameters with maximum likelihood estimators. The first application of the Weibull distribution is the initial mass function for stars. The magnitude version of the Weibull distribution is applied to the luminosity function for the Sloan Digital Sky Survey (SDSS) galaxies and to the photometric maximum of the 2MASS Redshift Survey (2MRS) galaxies. The truncated Weibull luminosity function allows us to model the average value of the absolute magnitude as a function of the redshift for the 2MRS galaxies.

Reference priors via [formula omitted]-divergence for a certain non-regular model in the presence of a nuisance parameter

This paper presents reference priors for non-regular model whose support depends on an unknown parameter. A multi-parameter family which includes both regular and non-regular structures is considered. The resulting priors are obtained by asymptotically maximizing the expected α-divergence between the prior and the corresponding posterior distribution. Some examples of reference priors for typical multi-parameter non-regular distributions such as the location-scale family of distributions and the truncated Weibull distribution are also given.

Does the Inclusion of Truncation point in a Finite Mixture Model Improve Diameter Distribution Estimation of Degraded Stand?

ABSTRACT This study proposed an alternative approach that could be used to model irregular diameter distribution with fixed measurement threshold. Sixteen plots of 0.0625 ha size from a degraded teak (Tectona grandis L. f.) stands were used as case study, in which only trees with diameter ≥ 10 cm were considered. Four methodologies were evaluated: method 1 used the Weibull distribution neglecting the truncation point and irregularities; method 2 utilized truncated Weibull distribution; method 3 used two components 2-parameter Weibull in a mixture model, and in method 4, two components truncated Weibull in a mixture model was used. The models were assessed with different indices including Kolmogorov-Smirnov and Anderson-Darling statistics. Relative rank sum of the models was analyzed with one-way analysis of variance. The result showed that model developed with method 4 had the smallest fit indices and relative rank which was significantly different from models obtained with other methods. It also had the smallest root mean square error (5.146 m3 ha−1) and critical error (1.362 m3 ha−1) for estimating stand volume in diameter classes. Thus, incorporating truncated Weibull in a finite mixture is the best strategy for modeling diameter distribution of degraded teak stands with fixed measurement threshold.

Downscaling fire weather extremes from historical and projected climate models

An important aspect of predicting future wildland fire risk is estimating fire weather—weather conducive to the ignition and propagation of fire—under realistic climate change scenarios. Because the majority of area burned occurs on a few days of extreme fire weather, this task should be able to resolve fire weather extremes. In this paper, we present a statistical downscaling procedure based on distribution based scaling (DBS) to bias correct the Fire Weather Index (FWI), part of the Canadian Forest Fire Danger Rating System, as calculated from modeled climate data. Our study area is western Canada (British Columbia and Alberta) and we consider both an historical control period (1990–2000) and three future time periods (2020–2030, 2050–2060, and 2080–2090). The historical data used to calibrate the DBS procedure comprises weather station data and weather from the North American Regional Reanalysis (NARR), whereas the future climate projections are the output of three regional climate models, corresponding to different model parameterizations and downscaled from the NCAR Community Earth System Model under the RCP 8.5 scenario. By fitting a truncated Weibull distribution to observed and modeled FWI values, our method is able to reproduce historical extremes in fire weather indices as determined by the distribution of annual potential spread days, which are defined as days with FWI values greater than 19. Moreover, by calibrating the DBS procedure with gridded reanalysis data as well as station observations, we are able to project future spread day distributions over the entire study area. The results of this study show the DBS procedure leads to a greater number of projected annual spread days at most locations compared with estimates using the uncorrected model output, and that all three RCM models show positive increases in potential annual spread days for the 2050–2060 and 2080–2090 time periods.

Study of cut-set distributions in the fuzzy reliability evaluation models

The performances of the cut-set distributions in the fuzzy reliability evaluation are studied based on cut-set method. Firstly, a theorem is proved to indicate the convergence defect of the model with the three commonly used cut-set distributions, including uniform distribution, linear distribution and truncated normal distribution. Secondly, a general method is proposed to construct a new family of cut-set distributions named intrinsic cut-set distributions, and three specific intrinsic cut-set distributions are obtained based on this method, including modified truncated normal distribution, truncated lognormal distribution and truncated Weibull distribution. Thirdly, numerical examples are carried out to verify the above theoretical results. It is shown that, compared with the three commonly used cut-set distributions, the proposed intrinsic cut-set distributions make the evaluation more stable and the fuzzy reliability model achieve good convergence at the boundary cases, which could effectively improve the evaluation accuracy and broaden the application of the model. Finally, some recommendations are given to show how to choose a suitable cut-set distribution in practice.

20 - 1-Day Acceptance and Commitment Training workshop for the prevention of persistent pain and prolonged opioid use

The performances of the cut-set distributions in the fuzzy reliability evaluation are studied based on cut-set method. Firstly, a theorem is proved to indicate the convergence defect of the model with the three commonly used cut-set distributions, including uniform distribution, linear distribution and truncated normal distribution. Secondly, a general method is proposed to construct a new family of cut-set distributions named intrinsic cut-set distributions, and three specific intrinsic cut-set distributions are obtained based on this method, including modified truncated normal distribution, truncated lognormal distribution and truncated Weibull distribution. Thirdly, numerical examples are carried out to verify the above theoretical results. It is shown that, compared with the three commonly used cut-set distributions, the proposed intrinsic cut-set distributions make the evaluation more stable and the fuzzy reliability model achieve good convergence at the boundary cases, which could effectively improve the evaluation accuracy and broaden the application of the model. Finally, some recommendations are given to show how to choose a suitable cut-set distribution in practice.

Factors Influencing the Incubation of an Infectious Form of Creutzfeldt-Jakob Disease

The French epidemics of iatrogenic Creutzfeldt-Jakob disease after growth hormone (GH) treatment provide an opportunity to understand factors governing the inter-human transmission of prions. The present analysis relying on truncated Weibull distribution supports a relationship between host genetics, dose of the at-risk GH, age at treatment onset, and duration of the incubation period.

Copula-Based Approach to Construct a Joint Probabilistic Model of Earthquakes and Strong Winds

Current structural design codes usually treat multiple hazards separately, and probabilistic backbones are rare for extreme hazard combinations, e.g., earthquake and strong wind, which may cause unforeseen damage to engineering structures exposed to multiple extreme hazards during their lifecycles. This study presents an innovative copula-based approach to construct the joint cumulative distribution function (JCDF) of the peak ground acceleration (PGA) and strong wind speed ([Formula: see text]). Six commonly used Archimedean copulas are applied to bond the JCDF with the corresponding marginal cumulative distribution functions (MCDFs) of PGA and [Formula: see text]. A total of 76 low-probability-high-consequence extreme events with a simultaneously occurring earthquake and strong wind are abstracted from data recorded from 1971–2017 in Dali Prefecture, China. The statistical analysis results show that the Frechet and truncated Weibull distributions are the optimal expressions for the marginal distributions of PGA and [Formula: see text], respectively, while the Joe Archimedean copula can yield good JCDF estimation. Monte Carlo simulation is employed to establish a target dependent multihazard database that can be used for the performance-based design of engineering structures against multiple natural hazards. A high-rise building is used to study the performance under the multihazard of an earthquake and strong wind. The results show that the maximum inter-story drift ratio of the building under multiple hazards increases by 14.4–21.3% compared with the structural response induced by an earthquake alone.

Multiple imputation approach for interval-censored time to HIV RNA viral rebound within a mixed effects Cox model.

We present a method to fit a mixed effects Cox model with interval-censored data. Our proposal is based on a multiple imputation approach that uses the truncated Weibull distribution to replace the interval-censored data by imputed survival times and then uses established mixed effects Cox methods for right-censored data. Interval-censored data were encountered in a database corresponding to a recompilation of retrospective data from eight analytical treatment interruption (ATI) studies in 158 human immunodeficiency virus (HIV) positive combination antiretroviral treatment (cART) suppressed individuals. The main variable of interest is the time to viral rebound, which is defined as the increase of serum viral load (VL) to detectable levels in a patient with previously undetectable VL, as a consequence of the interruption of cART. Another aspect of interest of the analysis is to consider the fact that the data come from different studies based on different grounds and that we have several assessments on the same patient. In order to handle this extra variability, we frame the problem into a mixed effects Cox model that considers a random intercept per subject as well as correlated random intercept and slope for pre-cART VL per study. Our procedure has been implemented in R using two packages: truncdist and coxme, and can be applied to any data set that presents both interval-censored survival times and a grouped data structure that could be treated as a random effect in a regression model. The properties of the parameter estimators obtained with our proposed method are addressed through a simulation study.

COMPARISON OF DIABETIC NEPHROPATHY ONSET TIME OF TWO GROUPS WITH LEFT TRUNCATED AND RIGHT CENSORED DATA

Abstract The present paper is concerned with the comparison of the nephropathy onset time of type-2 diabetic patients, grouped on the basis of gender and age at the time of diabetes diagnosis. Diabetic Nephropathy (DN) onset time is assumed to follow Weibull distribution with fixed left truncation. The likelihood ratio test is applied on uncensored cases and Thoman and Bain two sample tests is applied with generated left truncated Weibull distributions. To avoid the model validity issues for left truncated and right censored data (LTRC), the nonparametric approach, suggested by Kaplan and Meier, is used to compare the survival function of two groups over different time periods. Another method based on median survival time of the pooled group is applied to compare the survival function of two groups with LTRC data. The major advantage of developing methods for comparing the nephropathy onset times of DM patients is that the expected DN onset time of new DM patients can be predicted depending on the patient group.

Bonus-Malus System with the Claim Frequency Distribution is Geometric and the Severity Distribution is Truncated Weibull

Bonus-Malus system is said to be optimal if it is financially balanced for insurance companies and fair for policyholders. Previous research about Bonus-Malus system concern with the determination of the risk premium which applied to all of the severity that guaranteed by the insurance company. In fact, not all of the severity that proposed by policyholder may be covered by insurance company. When the insurance company sets a maximum bound of the severity incurred, so it is necessary to modify the model of the severity distribution into the severity bound distribution. In this paper, optimal Bonus-Malus system is compound of claim frequency component has geometric distribution and severity component has truncated Weibull distribution is discussed. The number of claims considered to follow a Poisson distribution, and the expected number λ is exponentially distributed, so the number of claims has a geometric distribution. The severity with a given parameter θ is considered to have a truncated exponential distribution is modelled using the Levy distribution, so the severity have a truncated Weibull distribution.

Truncated Weibull Distribution Functions and Moments

The Weibull distribution is widely used in natural sciences and engineering, especially in material reliability. For one is often confronted to censored data sample in these fields, the truncated Weibull distribution functions and moments can be of great interest. McEwen and Parresol (1991) became the main reference over the years as they exhibit the moments expressions of the truncated distribution for both the two-parameter and three-parameter Weibull random variables. Unfortunately, a mistake has been made when writing the doubly-truncated three-parameter Weibull probability distribution function and it naturally led to wrong results. In this article, we aim at correcting this error and giving explicit mathematical demonstrations of how moments of truncated Weibull distributions can be computed. In order to be as clear as possible we give the explicit expressions and proofs for the two-parameter Weibull one-sided and doubly-truncated moments as well as for the three-parameter Weibull distribution, although every result can be derived from the doubly-truncated three-parameter Weibull distribution moments. Besides, an R code with a function computing these moments is given in the last section.

Intra- and inter-specific differences in crown architecture in Chinese subtropical mixed-species forests

Absorption of photosynthetically active radiation (APAR) is fundamental for tree growth and is strongly influenced by crown architecture. The aim of this study was to quantify the intra- and inter-specific variability in crown architecture in monospecific and mixed-species subtropical Chinese forests. A total of 68 trees, including Castanopsis eyrei, Castanopsis sclerophylla, Cunninghamia lanceolata, Cyclobalanopsis glauca and Liquidambar formosana were destructively sampled and their crown architectures were quantified in terms of the vertical distribution of live branch diameter, individual branch leaf area, leaf area and leaf-area density. The vertical distributions were fitted by a two-parameter right truncated Weibull distribution. Inter-specific variability was assessed using ANCOVA and post hoc Tukey tests and intra-specific variability was assessed by fitting linear and linear mixed effect models. The peak in the vertical distribution of leaf area was highest for the least shade tolerant species, L. formosana (relative depth into the crown of 0.5), intermediate for C. sclerophylla (0.55), C. glauca (0.55) and C. lanceolata (0.6) and lowest for C. eyrei (0.75). For all species, the vertical distribution of leaf area was influenced by tree size except for C. glauca. For L. formosana and C. lanceolata, the distribution of leaf area or branch sizes shifted upwards as tree diameter increased, possibly to overtop neighbouring trees. In contrast, as C. eyrei stem diameter increased, the vertical distribution of mean branch diameter shifted downwards, indicating that larger C. eyrei trees invested a higher proportion of their crown growth into their lower crown when compared to smaller trees. The vertical distribution of leaf-area density varied between species but not within a given species. Crown architectures were not influenced by stand density (basal area) or the species composition of the plot. This intra-specific consistency is useful for modelling light in forests. This study shows that there is a significant inter-specific variability in the crown architectures of the co-occurring species in these subtropical forests. There is also significant intra-specific variability related to tree size and this relationship varies between species. This crown architectural variability and its effect on stand structure are likely to influence the light absorption of these stands.

Recurrence relations between moments of progressive type-II right censored order statistics from doubly truncated Weibull distribution

In this paper, we establish some new recurrence relations for the single and product moments of progressively Type-II right censored order statistics from the doubly truncated Weibull distribution. These relations generalize the results given by Balakrishnan et al. (Statistics 35:537–556, 2001) for the progressively Type-II right censored order statistics for standard exponential and right truncated exponential distributions. These results, also generalize the corresponding results for usual order statistics due to Khan et al. (Naval Res Logist Q 32:693, 1985, J Statist Plann Inference 8:175–183, 1983) for the doubly truncated Weibull distribution. In the last section, some deduction and particular cases are also discussed.