Mixture Weibull Model Research Articles

Comparing current and emerging practice models for the extrapolation of survival data: a simulation study and case-study

BackgroundEstimates of future survival can be a key evidence source when deciding if a medical treatment should be funded. Current practice is to use standard parametric models for generating extrapolations. Several emerging, more flexible, survival models are available which can provide improved within-sample fit. This study aimed to assess if these emerging practice models also provided improved extrapolations.MethodsBoth a simulation study and a case-study were used to assess the goodness of fit of five classes of survival model. These were: current practice models, Royston Parmar models (RPMs), Fractional polynomials (FPs), Generalised additive models (GAMs), and Dynamic survival models (DSMs). The simulation study used a mixture-Weibull model as the data-generating mechanism with varying lengths of follow-up and sample sizes. The case-study was long-term follow-up of a prostate cancer trial. For both studies, models were fit to an early data-cut of the data, and extrapolations compared to the known long-term follow-up.ResultsThe emerging practice models provided better within-sample fit than current practice models. For data-rich simulation scenarios (large sample sizes or long follow-up), the GAMs and DSMs provided improved extrapolations compared with current practice. Extrapolations from FPs were always very poor whilst those from RPMs were similar to current practice. With short follow-up all the models struggled to provide useful extrapolations. In the case-study all the models provided very similar estimates, but extrapolations were all poor as no model was able to capture a turning-point during the extrapolated period.ConclusionsGood within-sample fit does not guarantee good extrapolation performance. Both GAMs and DSMs may be considered as candidate extrapolation models in addition to current practice. Further research into when these flexible models are most useful, and the role of external evidence to improve extrapolations is required.

Mixture Models and the Krätzel Integral Transform

Continuous mixture Weibull models arise in many areas of sciences such as reliability studies, communications theory, etc. Due to its wide applicability, we introduce a class of continuous mixture Weibull models which is a combination of Weibull and generalized gamma distributions. Some characteristics of the distribution are obtained. It is seen that Krätzel integral enters into the model naturally, and then the model can be called as a Krätzel density. Applications of the density function related to fading channels and ultrasonic backscatter signals modeling are discussed. A real data analysis is given to illustrate the use of this distribution.

Modelling diameter distributions of two-cohort forest stands with various proportions of dominant species: A two-component mixture model approach

In recent years finite-mixture models have been employed to approximate and model empirical diameter at breast height (DBH) distributions. We used two-component mixtures of either the Weibull distribution or the gamma distribution for describing the DBH distributions of mixed-species, two-cohort forest stands, to analyse the relationships between the DBH components, age cohorts and dominant species, and to assess the significance of differences between the mixture distributions and the kernel density estimates. The data consisted of plots from the Świętokrzyski National Park (Central Poland) and areas close to and including the North Carolina section of the Great Smoky Mountains National Park (USA; southern Appalachians). The fit of the mixture Weibull model to empirical DBH distributions had a precision similar to that of the mixture gamma model, slightly less accurate estimate was obtained with the kernel density estimator. Generally, in the two-cohort, two-storied, multi-species stands in the southern Appalachians, the two-component DBH structure was associated with age cohort and dominant species. The 1st DBH component of the mixture model was associated with the 1st dominant species sp1 occurred in young age cohort (e.g., sweetgum, eastern hemlock); and to a lesser degree, the 2nd DBH component was associated with the 2nd dominant species sp2 occurred in old age cohort (e.g., loblolly pine, red maple). In two-cohort, partly multilayered, stands in the Świętokrzyski National Park, the DBH structure was usually associated with only age cohorts (two dominant species often occurred in both young and old age cohorts). When empirical DBH distributions representing stands of complex structure are approximated using mixture models, the convergence of the estimation process is often significantly dependent on the starting strategies. Depending on the number of DBHs measured, three methods for choosing the initial values are recommended: min.k/max.k, 0.5/1.5/mean, and multistart. For large samples (number of DBHs measured ⩾80) the multistage method is proposed – for the two-component mixture Weibull or gamma model select initial values using the min.k/max.k (for k=1,5,10) and 0.5/1.5/mean methods, run the numerical procedure for each method, and when no two solutions are the same, apply the multistart method also.

Estimation of the parameters of a mixture Weibull model for analyze cure rate

Mixture models have been used to analyze clinical trials with potentially cured patients. Model parameters are estimated via an appropriate EM (Expectation Maximization) algorithm that perform the ML (Maximum Likelihood) in presence of missing data. The basic idea of the EM algorithm is to associate a complete data model to the incomplete structure that is observed in order to simplify the computation of maximum likelihood estimates. We will investigate estimators of the parameters of a mixture Weibull model for analyze cure rate of the breast cancer patient. The problem to estimated of parameters of a mixture Weibull model is solution of derivatives of loglikelihood expectation function is not close form. In this paper we introduce solution this problem with iteration support by Matlab program.

Two Improved Mixture Weibull Models for the Analysis of Wind Speed Data

AbstractIn this paper, the authors propose two improved mixture Weibull distribution models by adding one or two location parameters to the existing two-component mixture two-parameter Weibull distribution [MWbl(2, 2)] model. One improved model is the mixture two-parameter Weibull and three-parameter Weibull distribution [MWbl(2, 3)] model. The other improved model is the two-component mixture three-parameter Weibull distribution [MWbl(3, 3)] model. In contrast to existing literature, which has focused on the MWbl(2, 2) and the typical Weibull distribution models, the authors apply the MWbl(2, 3) model and MWbl(3, 3) model to fit the distribution of wind speed data with nearly zero percentages of null wind speed. The parameters of the two improved models are estimated by the maximum likelihood method in which the maximization problem is regarded as a nonlinear programming problem with only inequality constraints and is solved numerically by the interior-point method. The experimental results show that the mixture Weibull models proposed in this paper are more flexible than the existing models for the analysis of wind speed data in practice.

A Reliability Data Analysis Method Using Mixture Weibull Distribution Model

The weibull distribution plays a crucial role in reliability theory and life-testing experiments. Weibull mixtures are widely used to model lifetime and failure time data, since they exhibit a wide range of shapes for the failure rate function. In this paper, the failure data of crank rod system was analyzed by using mixture weibull distribution model. The distribution parameters of the mixture weibull distribution model were estimated by using maximum likelihood estimation and drawing method. The comparison of fitting degree of failure location between standard weibull distribution model and mixture weibull model was given. Results show that the fitting degree of the failure data in the mixture weibull distribution model is higher than that of the simple weibull distribution model, and it can more accurately described the failure distribution curve of the system in life cycle.

Reliability Assessment of Numerical Control Machine Tools Using Weibull Mixture Models

Weibull distribution is one of the most widely used model in reliability analysis of NC machine tools. In order to assess the reliability of NC machine tool, Generalized Expectation Maximization (GEM) algorithm is used to estimate parameters of mixture Weibull models. Akaike information criterion (AIC) and Bayesian information criterion (BIC) are used as comprehensive criteria to select the number of subpopulations of mixture models. The results show that mixture models are more suitable to assess reliability of NC machine tools than that of single model. Root mean square errors (RMSE) of mixture models reduce by 87.5% than that of single model. Reliability evaluation results of machine tool, such as MTBF etc., are given.

Diversity of patch structure in Central European forests: are tree diameter distributions in near‐natural multilayered Abies–Fagus stands heterogeneous?

AbstractNear‐natural multilayered Abiesalba Mill.–Fagus sylvatica L. forests form structural mosaics and consist of patches in different developmental stages and phases. Knowledge of the diversity of patch structure and adequate methods to describe the diameter structure is essential for modeling forest dynamics. The hypotheses tested in the study are that near‐natural multilayered stands are structurally heterogeneous (i.e., tree diameter (DBH) distributions of these stands are heterogeneous) and, that in these forests the finite‐mixture models are suitable for modeling the empirical DBH distributions. Diversity of patch structure was studied based on data collected from 33 sample plots. In multilayered stands, four groups of empirical DBH distributions were distinguished using hierarchical cluster analysis (HCA) and correspondence analysis (CA). Stands investigated are structurally heterogeneous; 27% multilayered stands showed the slightly rotated sigmoid (SRS) type of empirical DBH distribution, 34% the distinctly rotated sigmoid (DRS) type, 18% the bimodal M‐shaped (BMS) type, and 21% the unimodal highly skewed (UHS) type. The gamma distribution, the two‐component mixture gamma model, and the two‐component mixture Weibull model were more flexible for the SRS type of DBH distributions. The average p‐values (Chi‐square test) for these theoretical distributions were 0.4712, 0.4718, and 0.4660, respectively. The two‐component mixture gamma model and the two‐component mixture Weibull model were a good choice for modeling the DRS, BMS, and UHS types of DBH distributions. The average p‐values (Chi‐square test) for these models ranged from 0.2684 to 0.4854. In near‐natural multilayered Abies–Fagus forest patches of different DBH distributions occur together. The empirical DBH distributions in these stands are characterized by irregular and complicated shapes and therefore are best approximated by finite‐mixture models.

Portfolio Credit Risk Valuation: An Empirical Analysis Using a Mixture-Weibull Implied Hazard Rate Density

Through empirical analysis I show that portfolio credit risk can be valued in a reduced-form framework using hazard rates that are mixture-Weibull distributed. Each mixing density has its own degree of default correlation and probability of observing high default rates. The relative contributions of each density is determined by the mixing weights. How to imply the parameters of the mixture-Weibull distribution by calibrating a CDO valuation model to market spreads is shown. Using two and three mixing densities, I fit the model to daily 5-year CDO spreads for data spanning 2004 to 2008, and I consider CDOs written on both the North American and European investment grade reference portfolios. For the North American CDOs, I report root-mean-squared-errors of 2.009 and 0.713 basis points for the two and three mixture-Weibull models, respectively, while for the European CDOs I report root-mean-squared-errors of 0.953 and 0.220 basis points, respectively. In considering the model during periods of financial distress there is evidence that the dispersion of the implied hazard rate density is increasing in systematic default risk.

On the recent papers by Nagode et al.

The recent work Professor Nagode and colleagues is discussed with respect to the use of mixture Weibull models.

Fitting irregular diameter distributions of forest stands by Weibull, modified Weibull, and mixture Weibull models

Irregular diameter frequency distributions of forest stands include multimodal structure of mixed-species stands, highly skewed and highly irregular shapes of uneven-aged stands, and rotated sigmoid form of old-growth stands. In this study, a traditional two-parameter Weibull model, a modified two-parameter Weibull model, and a finite mixture of two-parameter Weibull models were used to fit four artificial example plots. The model fitting and comparison results indicate that the mixture Weibull model is more flexible to fit various irregular diameter distributions, while the traditional Weibull model fails in every case to adequately describe these frequency distributions. The modified Weibull model is a good choice for fitting the “rotated-sigmoid” diameter distribution of an uneven-aged old-growth stand. However, it may not be sufficient when a diameter frequency distribution is multimodal or highly irregular in shape.

The relationship between purchase regularity and propensity to accelerate

Identifying random and regular-buying household segments and relating them to demographic and shopping characteristics has been the focus of many marketing studies. Missing from the marketing literature, however, is a study that relates purchase regularity to marketing mix sensitivities. Such study could provide substantive implications since it would explore a practical dimension of a segmentation scheme based on purchase regularity. In this article, we investigate the relationship between purchase regularity and propensity to accelerate through the use of a mixture Weibull model of purchase timing. Applying this perspective to purchase timing data on four product categories (ketchup, sugar, bathroom tissue, margarine), we show that in the frequently purchased categories of bathroom tissue and margarine, random buyers do not exhibit any propensity to accelerate while regular buyers do. In the occasionally purchased categories of ketchup and sugar on the other hand, random buyers exhibit at least as much propensity to accelerate their purchases as regular buyers do. Our rationale for these results is based on information-theoretic arguments suggesting that propensity to accelerate depends on the frequency at which a product category is purchased.