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Biomarkers are measured repeatedly in clinical studies until a pre-defined endpoint, such as death from certain causes, is reached. Such repeated measurements may present a dynamic process for understanding when to expect the study’s endpoint. Joint modelling is often employed to handle such a model. Typically, shared random effects are assumed to be common to both the longitudinal component and the study’s endpoint. These shared random effects usually assume homogeneous and follow a normal distribution. However, identifying homogeneous subgroups is important when the underlying population is heterogeneous. This issue has received little attention in the literature, particularly for multi-phase longitudinal responses. In this paper, we propose a joint modelling approach for longitudinal and survival models using a bent-cable mixed model for longitudinal measurements and a Weibull distribution for the survival component. We also incorporate a finite mixture of normal distribution assumptions to account for the unobserved heterogeneity in the shared random effects model. A Bayesian MCMC is developed for parameter estimation and inferences. The proposed method is evaluated using simulation studies and the Tehran Lipid and Glucose Study dataset. [ABSTRACT FROM AUTHOR]

Wind energy production has been a relevant topic of research for several years. Specifically, the estimation of wind energy potential has received important attention in different regions of the world. One of the main considerations for these estimations is based on the modeling of wind speed data based on probability density functions (PDF), given that once the best distribution for wind speed data is determined, the wind energy potential can be estimated. The objective of this paper is to investigate the wind speed and wind energy potential in Ciudad Juárez, México. To achieve this, three meteorological stations were installed in strategic open sites at a height of 10 meters within and on the edges of the city. Speed data were recorded for each meteorological station every ten minutes over a one-year period. The wind speed data were studied to define the best-fitting distribution, and different commercial wind turbines were considered to estimate the power curves for each location. With the characterized power curves, it was possible to estimate the potential energy production. In addition, wind shear was considered to estimate the energy production with wind turbines that have greater heights. The results show the importance of selecting the best distribution to estimate the wind energy potential of certain regions where measurements can be obtained from different locations. [ABSTRACT FROM AUTHOR]

Abstract.In the realm of mixed models, this article explores the estimation method for incorporating a mixture of distributions and addressing controlled heteroscedasticity. By relaxing the assumptions of normality and homoscedasticity, we introduce a more flexible approach to analyzing data. The article presents estimation techniques for variance components, estimable vectors, and cumulants, while also developing prediction intervals and prediction ellipsoids for future observations. A numerical example is employed to illustrate the method and compare it with traditional ANOVA and Bayesian estimation methods. The results demonstrate the superior flexibility and broader applicability of the proposed methods in diverse contexts. By extending the analysis beyond conventional assumptions, these approaches enhance the accuracy and robustness of statistical inference in mixed models. [ABSTRACT FROM AUTHOR]

AbstractRecently, considerable interest has been in deriving explicit expressions for moment of different truncated multivariate distributions. This is due to their usefulness in developing efficient EM-algorithm for the maximum likelihood estimation of the corresponding distributional parameters. One rich family of distributions is the multivariate normal mean-mixture distribution as it includes a number of important multivariate distributions as special cases. In this article, we derive exact expressions for the first two moments of doubly truncated multivariate normal mean-mixture distributions, from which results for various special cases are then deduced. [ABSTRACT FROM AUTHOR]

Abstract–Mixture models are widely employed in the analysis of heterogeneous data. However, existing approaches are based on the assumption that the observations in each component are normally distributed. The main objective of this article is to propose mixture models with Yeo-Johnson transformation to handle general heterogeneous data. Bayesian methods are developed for estimation and model comparison. The empirical performance of the proposed methodology is assessed through simulation studies. A real analysis of a data set derived from the National Longitudinal Survey of Youth 1997 is presented for illustration. [ABSTRACT FROM AUTHOR]

We introduce a mixture of mixtures of bounded asymmetric generalized Gaussian and uniform distributions. Based on this framework, we propose model-based classification and model-based clustering algorithms. We develop an objective function for the minimum message length (MML) model selection criterion to discover the optimal number of clusters for the unsupervised approach of our proposed model. Given the crucial attention received by Explainable AI (XAI) in recent years, we introduce a method to interpret the predictions obtained from the proposed model in both learning settings by defining their boundaries in terms of the crucial features. Integrating Explainability within our proposed algorithm increases the credibility of the algorithm's predictions since it would be explainable to the user's perspective through simple If-Then statements using a small binary decision tree. In this paper, the proposed algorithm proves its reliability and superiority to several state-of-the-art machine learning algorithms within the following real-world applications: fault detection and diagnosis (FDD) in chillers, occupancy estimation and categorization of residential energy consumers. [ABSTRACT FROM AUTHOR]

Most empirical studies of complex networks return rich but noisy data, as they measure the network structure repeatedly but with substantial errors due to indirect measurements. In this article, we propose a novel framework, called the group-based binary mixture (GBM) modeling approach, to simultaneously conduct network reconstruction and community detection from such rich but noisy data. A generalized expectation-maximization (EM) algorithm is developed for computing the maximum likelihood estimates, and an information criterion is introduced to consistently select the number of communities. The strong consistency properties of the network reconstruction and community detection are established under some assumption on the Kullback-Leibler (KL) divergence, and in particular, we do not impose assumptions on the true network structure. It is shown that joint reconstruction with community detection has a synergistic effect, whereby actually detecting communities can improve the accuracy of the reconstruction. Finally, we illustrate the performance of the approach with numerical simulations and two real examples. for this article are available online. [ABSTRACT FROM AUTHOR]

Abstract Feature selection is a difficult but highly important preliminary step for bearings remaining useful life (RUL) estimation. To avoid the weights setting problem in hybrid metric, this work devotes to conduct feature selection by using a single metric. Due to noise and outliers, an existing feature selection metric, called monotonicity, used for estimating bearings RUL, requires data smoothing processing before adequate implementation. Such a smoothing process may remove significant part of meaningful information from data. To overcome this issue, a mixture distribution analysis-based feature selection metric is proposed. Moreover, based on this new metric, a feature selection approach for bearings RUL estimation is proposed. Numerical experiments benchmarking the proposed method and the existing metric monotonicity method on available real datasets highlight its effectiveness.

The receiver operating characteristics (ROC) analysis is commonly used in clinical settings to check the performance of a single threshold for distinguishing population-wise bimodal-distributed test results. However, for population-wise three-modal distributed test results, a single threshold ROC (stROC) analysis showed poor discriminative performance. The purpose of this study is to use a double-threshold ROC analysis for the three-modal distributed test results to provide better discriminative performance than the stROC analysis. A double-threshold receiver operating characteristic plot (dtROC) is constructed by replacing the single threshold with a double threshold. The sensitivity and specificity coordinates are chosen to maximize sensitivity for a given specificity value. Besides a simulation study assuming a mixture of lognormal, Poisson, and Weibull distributions, a clinical application is examined by a secondary data analysis of palpation test results of the C7 spinous process using the modified thorax–rib static technique. For the assumed mixture models, the discrimination performance of dtROC analysis outperforms the stROC analysis (area under ROC (AUROC) increased from 0.436 to 0.983 for lognormal distributed test results, 0.676 to 0.752 for the Poisson distribution, and 0.674 to 0.804 for Weibull distribution). [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach that utilizes the weighted harmonic mean in combination with multiple univariate cumulative distribution functions to create a fresh univariate cumulative distribution. In addition to an innovative expression, it exhibits direct ordering properties involving traditional mixtures of univariate cumulative distribution functions. After establishing the theoretical foundation, we leverage these findings to develop new families of trigonometric distributions. We highlight a special case that extends the exponentiated distribution's scope and demonstrates its superior performance in data fitting compared to fair competitors like the exponential, gamma, exponentiated exponential, and sine exponential distributions. In summary, our methodology offers a versatile framework for modeling and understanding a wide array of statistical phenomena. [ABSTRACT FROM AUTHOR]

Feature selection is a difficult but highly important preliminary step for bearings remaining useful life (RUL) estimation. To avoid the weights setting problem in hybrid metric, this work devotes to conduct feature selection by using a single metric. Due to noise and outliers, an existing feature selection metric, called monotonicity, used for estimating bearings RUL, requires data smoothing processing before adequate implementation. Such a smoothing process may remove significant part of meaningful information from data. To overcome this issue, a mixture distribution analysis-based feature selection metric is proposed. Moreover, based on this new metric, a feature selection approach for bearings RUL estimation is proposed. Numerical experiments benchmarking the proposed method and the existing metric monotonicity method on available real datasets highlight its effectiveness.Article Highlights: A feature selection method for bearings remaining useful life estimation Retrieval the underlying trend of vibration signal Address the drawbacks of smoothing methods that remove meaningful information from data [ABSTRACT FROM AUTHOR]

How much genome differences between species reflect neutral or adaptive evolution is a central question in evolutionary genomics. In humans and other mammals, the presence of adaptive versus neutral genomic evolution has proven particularly difficult to quantify. The difficulty notably stems from the highly heterogeneous organization of mammalian genomes at multiple levels (functional sequence density, recombination, etc.) which complicates the interpretation and distinction of adaptive versus neutral evolution signals. In this study, we introduce mixture density regressions (MDRs) for the study of the determinants of recent adaptation in the human genome. MDRs provide a flexible regression model based on multiple Gaussian distributions. We use MDRs to model the association between recent selection signals and multiple genomic factors likely to affect the occurrence/detection of positive selection, if the latter was present in the first place to generate these associations. We find that an MDR model with two Gaussian distributions provides an excellent fit to the genome-wide distribution of a common sweep summary statistic (integrated haplotype score), with one of the two distributions likely enriched in positive selection. We further find several factors associated with signals of recent adaptation, including the recombination rate, the density of regulatory elements in immune cells, GC content, gene expression in immune cells, the density of mammal-wide conserved elements, and the distance to the nearest virus-interacting gene. These results support the presence of strong positive selection in recent human evolution and highlight MDRs as a powerful tool to make sense of signals of recent genomic adaptation. [ABSTRACT FROM AUTHOR]

Abstract Background The problem of dealing with misreported data is very common in a wide range of contexts for different reasons. The current situation caused by the Covid-19 worldwide pandemic is a clear example, where the data provided by official sources were not always reliable due to data collection issues and to the high proportion of asymptomatic cases. In this work, a flexible framework is proposed, with the objective of quantifying the severity of misreporting in a time series and reconstructing the most likely evolution of the process. Methods The performance of Bayesian Synthetic Likelihood to estimate the parameters of a model based on AutoRegressive Conditional Heteroskedastic time series capable of dealing with misreported information and to reconstruct the most likely evolution of the phenomenon is assessed through a comprehensive simulation study and illustrated by reconstructing the weekly Covid-19 incidence in each Spanish Autonomous Community. Results Only around 51% of the Covid-19 cases in the period 2020/02/23–2022/02/27 were reported in Spain, showing relevant differences in the severity of underreporting across the regions. Conclusions The proposed methodology provides public health decision-makers with a valuable tool in order to improve the assessment of a disease evolution under different scenarios.

We estimate the distribution of COVID-19 mortality (measured as daily deaths) from the start of the pandemic until July 31st, 2022, for six European countries and the USA. We use the Pareto, the stretched exponential, the log-normal and the log-logistic distributions as well as mixtures of the log-normal and log-logistic distributions. The main results are that the Pareto does not describe well the data and that mixture distributions tend to offer a very good fit to the data. We also compute Value-at-Risk measures as well as mortality probabilities with our estimates. We also discuss the implications of our results and findings from the point of view of public health planning and modelling.

In this paper, we present a two-component Weibull mixture model. An important property is that this new model accommodates bimodality, which can appear in data representing phenomena in some heterogeneous populations. We provide statistical properties, such as the quantile function and moments. Additionally, the expectation-maximization (EM) algorithm is used to find maximum-likelihood estimates of the model parameters. Further, a Monte Carlo study is carried out to evaluate the performance of the estimators on finite samples. The new model's relevance is shown with an application referring to the vote proportion for the Brazilian presidential elections runoff in 2018. The proportion of votes is an important measure in analyzing electoral data. Since it is a variable limited to the unitary interval, unit distributions should be considered to analyze its probabilistic behavior. Thus, the introduced model is suitable for describing the characteristics detected in these data, such as the asymmetric behavior, bimodality, and the unit interval as support. In the application, the superiority of the proposed model is verified when comparing the fit with the two-component beta mixture models. [ABSTRACT FROM AUTHOR]

There currently exist a variety of statistical methods for modeling bivariate extremes. However, when the dependence between variables is driven by more than one latent process, these methods are likely to fail to give reliable inferences. We consider situations in which the observed dependence at extreme levels is a mixture of a possibly unknown number of much simpler bivariate distributions. For such structures, we demonstrate the limitations of existing methods and propose two new methods: an extension of the Heffernan–Tawn conditional extreme value model to allow for mixtures and an extremal quantile-regression approach. The two methods are examined in a simulation study and then applied to oceanographic data. Finally, we discuss extensions including a subasymptotic version of the proposed model, which has the potential to give more efficient results by incorporating data that are less extreme. Both new methods outperform existing approaches when mixtures are present. [ABSTRACT FROM AUTHOR]

In this paper, a new generalization of one parameter Lindely distribution is proposed. The new distribution is a mixture distribution of Gamma distributions with fixed scale parameter and variable shape parameter. The distribution is called 'GOLD Distribution' as it is a generalization for several distributions such as exponential, Lindely, Sujatha, Amarendra, Devya and Shambhu distributions. The probability density and cumulative density functions are derived. Also, the statistical properties of the GOLD distribution are discussed. Parameter estimation using the maximum likelihood and the method of moments are given. Moreover, an illustration of the usefulness of the GOLD distribution in survival data analysis is discussed based on a real lifetime data. [ABSTRACT FROM AUTHOR]

Background: The problem of dealing with misreported data is very common in a wide range of contexts for different reasons. The current situation caused by the Covid-19 worldwide pandemic is a clear example, where the data provided by official sources were not always reliable due to data collection issues and to the high proportion of asymptomatic cases. In this work, a flexible framework is proposed, with the objective of quantifying the severity of misreporting in a time series and reconstructing the most likely evolution of the process. Methods: The performance of Bayesian Synthetic Likelihood to estimate the parameters of a model based on AutoRegressive Conditional Heteroskedastic time series capable of dealing with misreported information and to reconstruct the most likely evolution of the phenomenon is assessed through a comprehensive simulation study and illustrated by reconstructing the weekly Covid-19 incidence in each Spanish Autonomous Community. Results: Only around 51% of the Covid-19 cases in the period 2020/02/23–2022/02/27 were reported in Spain, showing relevant differences in the severity of underreporting across the regions. Conclusions: The proposed methodology provides public health decision-makers with a valuable tool in order to improve the assessment of a disease evolution under different scenarios. [ABSTRACT FROM AUTHOR]

The Shanker distribution, a one-parameter lifetime distribution with an increasing hazard rate function, is recommended by Shanker for modelling lifespan data. In this study, we examine the theoretical and practical implications of 2-component mixture of Shanker model (2-CMSM). A significant feature of proposed model's hazard rate function is that it has rising, decreasing, and upside-down bathtub forms. We investigate the statistical characteristics of a mixed model, such as the probability-generating function, the factorial-moment-generating function, cumulants, the characteristic function, the Mills ratio, the mean residual life, and the mean time to failure. There is a graphic representation of density, mean, hazard rate functions, coefficient of variation, skewness, and kurtosis. Our final approach is to estimate the parameters of the mixture model using appropriate approaches such as maximum likelihood, least squares, and weighted least squares. Using a simulation analysis, we examined how the estimates behaved graphically. The simulation results demonstrated that, in the majority of cases, the maximum likelihood estimates have the smallest mean square errors among all other estimates. Finally, we observed that when the sample size rises, the precision measures decrease for all of the estimation techniques, indicating that all of the estimation approaches are consistent. Through two real data analyses, the suggested model's validity and adaptability are contrasted with those of other models, including the mixture of the exponential distributions and the Lindley distributions. [ABSTRACT FROM AUTHOR]

The fundamental assumption of flood frequency analysis is that flood samples are generated by the same flood generation mechanism (FGM). However, flood events are usually triggered by the interaction of meteorological factors and watershed properties, which results in different FMGs. To solve this problem, researchers have put forward traditional two-component mixture distributions (TCMD-T) without clearly linking each component distribution to an explicit FGM. In order to improve the physical meaning of mixture distributions in seasonal snow-covered areas, the ratio of rainfall to flood volume (referred to as rainfall–flood ratio, RF) method was used to classify distinct FGMs. Thus, the weighting coefficient of each component distribution was determined in advance in the rainfall–flood ratio based TCMD (TCMD-RF). TCMD-RF model was applied to 34 basins in Norway. The results showed that flood types can be clearly divided into rain-on-snow-induced flood, snowmelt-induced flood and rainfall-induced flood. Moreover, the design flood and associated uncertainties were also estimated. It is found that TCMD-RF model can reduce the uncertainties of design flood by 20% compared with TCMD-T. The superiority of TCMD-RF is attributed to its clear classification of FGMs, thus determining the weighting coefficients without optimization and simplifying the parameter estimation procedure of mixture distributions. [ABSTRACT FROM AUTHOR]

The multiple stream process involves equipment with several modules producing the same items, but with respective random variables' subpopulations. This particularity demands specific control charts for the statistical processes monitoring. Among the initiatives proposed are the finite mixture control chart and T2-Hotelling control chart, but there are no comparative studies between them. This article aims to study and compare the proposed use of a finite mixture control chart with the T2-Hotelling control chart for monitoring the multiple stream process (MSP). The modeling and simulation were performed using actual data from the dietary sector to develop an illustrative case. The results show that the T2-Hotelling charts' performance was superior in detecting special causes and generating false alarms about the process and supplementary to having less mathematical difficulty in implementing. However, the finite mixture chart is a more recently used approach, considering the literature's occurrence, compared with the T2-Hotelling chart. Furthermore, the finite mixture chart considering all streams and the variability between them in just one control chart is more straightforward in terms of sampling, but more difficult in mathematical terms for its application. In conclusion, the finite mixture and T2-Hotelling charts are suitable for solving MSP, monitoring with no losing information on each stream variability, and not masking effects from special causes in the process. [ABSTRACT FROM AUTHOR]

We estimate the distribution of COVID-19 mortality (measured as daily deaths) from the start of the pandemic until July 31st, 2022, for six European countries and the USA. We use the Pareto, the stretched exponential, the log-normal and the log-logistic distributions as well as mixtures of the log-normal and log-logistic distributions. The main results are that the Pareto does not describe well the data and that mixture distributions tend to offer a very good fit to the data. We also compute Value-at-Risk measures as well as mortality probabilities with our estimates. We also discuss the implications of our results and findings from the point of view of public health planning and modelling. [ABSTRACT FROM AUTHOR]

We study some properties of the failure rate for a mixture of Erlang distributions. In particular, we discuss mixtures of Erlang distributions with the same scale parameter and we describe the behavior of the failure rate. Some remarks are also given for an infinite mixture of Erlangs, in particular with regard to stochastic order properties. [ABSTRACT FROM AUTHOR]

This paper studies the prediction problem of future order statistics based on a mixture of exponential distributions. We suggest different scenarios to construct predictive intervals and point predictions for the future observations in two cases. In the first case, we assume fixed sample size, while in the second case, the sample size is assumed to be positive integer–valued random variable independent of the observations. [ABSTRACT FROM AUTHOR]

In this paper, by using a new method, we derive the trapezoidal beta (TB) distribution and its properties. The TB distribution is a mixture model, generalizes both the beta and rectangular beta distributions, and allows one to describe bounded data with heavy right and/or left tails. In relation to the two-parameter beta distribution, we add two additional parameters which have an intuitive interpretation. The four TB parameters are estimated with the expectation-maximization algorithm. We conduct a simulation study to evaluate performance of the TB distribution. An application with real data is carried out, which includes a comparison among the beta, rectangular beta and TB distributions indicating that the TB one describes these data better.

In this paper, we present a two-component Weibull mixture model. An important property is that this new model accommodates bimodality, which can appear in data representing phenomena in some heterogeneous populations. We provide statistical properties, such as the quantile function and moments. Additionally, the expectation-maximization (EM) algorithm is used to find maximum-likelihood estimates of the model parameters. Further, a Monte Carlo study is carried out to evaluate the performance of the estimators on finite samples. The new model’s relevance is shown with an application referring to the vote proportion for the Brazilian presidential elections runoff in 2018. The proportion of votes is an important measure in analyzing electoral data. Since it is a variable limited to the unitary interval, unit distributions should be considered to analyze its probabilistic behavior. Thus, the introduced model is suitable for describing the characteristics detected in these data, such as the asymmetric behavior, bimodality, and the unit interval as support. In the application, the superiority of the proposed model is verified when comparing the fit with the two-component beta mixture models.

This work investigates aspects of the global sensitivity analysis of computer codes when alternative plausible distributions for the model inputs are available to the analyst. Analysts may decide to explore results under each distribution or to aggregate the distributions, assigning, for instance, a mixture. In the first case, we lose uniqueness of the sensitivity measures, and in the second case, we lose independence even if the model inputs are independent under each of the assigned distributions. Removing the unique distribution assumption impacts the mathematical properties at the basis of variance‐based sensitivity analysis and has consequences on result interpretation as well. We analyze in detail the technical aspects. From this investigation, we derive corresponding recommendations for the risk analyst. We show that an approach based on the generalized functional ANOVA expansion remains theoretically grounded in the presence of a mixture distribution. Numerically, we base the construction of the generalized function ANOVA effects on the diffeomorphic modulation under observable response preserving homotopy regression. Our application addresses the calculation of variance‐based sensitivity measures for the well‐known Nordhaus' DICE model, when its inputs are assigned a mixture distribution. A discussion of implications for the risk analyst and future research perspectives closes the work. [ABSTRACT FROM AUTHOR]

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. • Inter-arrival and inter-trade waiting times can be explained by mixture distributions for all time scales. • Mixture distributions are necessary to explain small time scales (i.e. high frequency trading). • Waiting times approximately > 50 milliseconds can be explained well with a Weibull distribution. • Expectation–maximization algorithm provides a reliable platform to study mixture distributions. [ABSTRACT FROM AUTHOR]

This work investigates aspects of the global sensitivity analysis of computer codes when alternative plausible distributions for the model inputs are available to the analyst. Analysts may decide to explore results under each distribution or to aggregate the distributions, assigning, for instance, a mixture. In the first case, we lose uniqueness of the sensitivity measures, and in the second case, we lose independence even if the model inputs are independent under each of the assigned distributions. Removing the unique distribution assumption impacts the mathematical properties at the basis of variance-based sensitivity analysis and has consequences on result interpretation as well. We analyze in detail the technical aspects. From this investigation, we derive corresponding recommendations for the risk analyst. We show that an approach based on the generalized functional ANOVA expansion remains theoretically grounded in the presence of a mixture distribution. Numerically, we base the construction of the generalized function ANOVA effects on the diffeomorphic modulation under observable response preserving homotopy regression. Our application addresses the calculation of variance-based sensitivity measures for the well-known Nordhaus' DICE model, when its inputs are assigned a mixture distribution. A discussion of implications for the risk analyst and future research perspectives closes the work., (© 2021 The Authors. Risk Analysis published by Wiley Periodicals LLC on behalf of Society for Risk Analysis.)