Mixture Of Probability Distributions Research Articles

A Rényi-type quasimetric with random interference detection

AbstractThis paper introduces a new dissimilarity measure between two discrete and finite probability distributions. The followed approach is grounded jointly on mixtures of probability distributions and an optimization procedure. We discuss the clear interpretation of the constitutive elements of the measure under an information-theoretical perspective by also highlighting its connections with the Rényi divergence of infinite order. Moreover, we show how the measure describes the inefficiency in assuming that a given probability distribution coincides with a benchmark one by giving formal writing of the random interference between the considered probability distributions. We explore the properties of the considered tool, which are in line with those defining the concept of quasimetric—i.e. a divergence for which the triangular inequality is satisfied. As a possible usage of the introduced device, an application to rare events is illustrated. This application shows that our measure may be suitable in cases where the accuracy of the small probabilities is a relevant matter.

Statistical analysis of wind power density using different mixture probability distribution functions in coastal region of Andhra Pradesh

Statistical analysis of wind power density using different mixture probability distribution functions in coastal region of Andhra Pradesh

A Python package based on robust statistical analysis for serial crystallography data processing.

The term robustness in statistics refers to methods that are generally insensitive to deviations from model assumptions. In other words, robust methods are able to preserve their accuracy even when the data do not perfectly fit the statistical models. Robust statistical analyses are particularly effective when analysing mixtures of probability distributions. Therefore, these methods enable the discretization of X-ray serial crystallography data into two probability distributions: a group comprising true data points (for example the background intensities) and another group comprising outliers (for example Bragg peaks or bad pixels on an X-ray detector). These characteristics of robust statistical analysis are beneficial for the ever-increasing volume of serial crystallography (SX) data sets produced at synchrotron and X-ray free-electron laser (XFEL) sources. The key advantage of the use of robust statistics for some applications in SX data analysis is that it requires minimal parameter tuning because of its insensitivity to the input parameters. In this paper, a software package called Robust Gaussian Fitting library (RGFlib) is introduced that is based on the concept of robust statistics. Two methods are presented based on the concept of robust statistics and RGFlib for two SX data-analysis tasks: (i) a robust peak-finding algorithm and (ii) an automated robust method to detect bad pixels on X-ray pixel detectors.

A Novel Multiobjective Formulation for Optimal Wind Speed Modeling via a Mixture Probability Density Function

Over the past decades, the mathematical formulation of wind turbines (WTs) has been handled using different methodologies to model the probabilistic nature via different distribution functions. Many recently published articles have applied either the wind speed or the obtained active power from the WT on various probabilistic curves, such as Weibull, log-normal, and Gamma. In this work, the wind speed was modeled at five different locations in Egypt via a novel mixture probability distribution function (MPDF) that included four well-known distribution functions used to imitate the probabilistic nature of wind speed. Moreover, a decision-making multiple objective formulation was developed to optimally fit the MPDF with a minimum root mean square error (RMSE) and ensure reliable fitting by two other effective indices. Two methodologies, namely, equal and variable class widths, were investigated to model the density of wind speed and obtain a more realistic model for the tested wind speed profiles. The results showed the effectiveness of the proposed MPDF model as the RMSE was effectively minimized using multiobjective particle swarm optimization (MOPSO), showing nearly 10% improvement compared to the nondominated sorting genetic algorithm (NSGA-II).

Data Typology Using Technologies for Separating Mixtures of Probability Distributions

The authors studied heterogeneous samples (for a number of statistical populations) taken from finite mixtures of probability distribu- tions, which reflect a number of socio-economic characteristics of the Russian society, using the information resources of the Federal State Statistics Service; in this case the number of mixing distributions (components), as well as their corresponding weights and parameters, may be unknown. In terms of content, the problem of separation (decomposition) of mixtures can be reduced to estimating unknown parameters of mixing distributions and their weights.The article considers and discusses methods for solving this problem, their advantages and disadvantages, conditions and areas of application. In the study the decomposition of mixtures of probability distributions on the population of subjects of the RussianFederation was carried out according to three interrelated statistical indicators – the unemployment rate, the poverty rate, the level of violence. These indicators can be considered as statistical measure of achieving sustainable development goals at the regional level and, at the same time, measures of socio-economic «health» of territory. Typologies of subjects of the Russian Federation were carried out according to the listed indicators. A cross-comparison of the results of the obtained typological groupings was performed, which also made it possible to identify the subjects of the Russian Federation that differ in both negative and positive trends in the context of the indicators under consideration.The authors underline that the results of the study can be used by the authorities to develop specific measures for the socio-economic development of Russia and its regions.

The Mixture of Probability Distribution Functions for Wind and Photovoltaic Power Systems Using a Metaheuristic Method

The rising use of renewable energy sources, particularly those that are weather-dependent like wind and solar energy, has increased the uncertainty of supply in these power systems. In order to obtain considerably more accurate results in the analysis of power systems, such as in the planning and operation, it is necessary to tackle the stochastic nature of these sources. Operators require adequate techniques and procedures to mitigate the negative consequences of the stochastic behavior of renewable energy generators. Thus, this paper presents a modification of the original probability distribution functions (PDFs) where the original PDFs are insufficient for wind speed and solar irradiance modeling because they have a significant error between the real data frequency distribution and the estimated distribution curve. This modification is using a mixture of probability distributions, which can improve the fitting of data and reduce this error. The main aim of this paper is to model wind speed and solar irradiance behaviors using a two-component and a three-component mixture of PDFs generated from the integration of the original Weibull, Lognormal, Gamma, and Inverse-Gaussian PDFs. Three statistical errors are used to test the efficiency of the proposed original and mixture PDFs, which are the root mean square error (RMSE), the coefficient of correlation (R2), and the Chi-square error (X2). The results show that the mixture of PDFs gives better fitting criteria for wind speed and solar irradiance frequency distributions than the original PDFs. The parameters of the original and the mixture of PDFs are calculated using the innovative metaheuristic Mayfly algorithm (MA). The three-component mixture of PDFs lowered the RMSE by about 73% and was 17% more than the best original and the two-component mixture distributions.

Binomial-Geometric Mixture and Its Applications

A mixture distribution is a combination of two or more probability distributions; it can be obtained from different distribution families or the same distribution families with different parameters. The underlying distributions may be discrete or continuous, so the resulting mixture probability distribution function should be a mass or density function. In the last few years, there has been great interest in the problem of developing a mixture distribution based on the binomial distribution. This paper uses the probability generating function method to develop a new two-parameter discrete distribution called a binomial-geometric (BG) distribution, a mixture of binomial distribution with the number of trials (parameter <img src=image/13428702_01.gif>) taken after a geometric distribution. The quantile function, moments, moment generating function, Shannon entropy, order statistics, stress-strength reliability and simulating the random sample are some of the statistical highlights of the BG distribution that are explored. The model's parameters are estimated using the maximum likelihood method. To examine the performance of the accuracy of point estimates for BG distribution parameters, the Monte Carlo simulation is performed with different scenarios. Finally, the BG distribution is fitted to two real lifetime count data sets from the medical field. As a result, the proposed BG distribution is an overdispersed right-skewed and can accommodate a constant hazard rate function. The proposed BG distribution is appropriate for modelling the overdispersed right-skewed real-life count data sets and it can be an alternative to the negative binomial and geometric distributions.

On the Stability of Mixtures of Probability Distributions to Perturbations of Mixing Distributions

On the Stability of Mixtures of Probability Distributions to Perturbations of Mixing Distributions

A probabilistic neural network for uncertainty prediction with applications to manufacturing process monitoring

Modeling the uncertainty from data is an essential quest in the learning of neural network models but has not been well addressed. A probabilistic neural network with Gaussian-mixture distributed parameters is developed in this work to provide an efficient and high-fidelity solution for learning multimodal uncertainties in neural networks. An adaptive Gaussian mixture scheme is adopted to refine the Gaussian mixture probability distributions and ensure the fidelity of uncertainty propagation in both linear and nonlinear transformations through the network. As its predictive distribution can be inferred analytically, this probabilistic network can be trained efficiently using a backpropagation method based on gradient descent. The proposed network not only achieves a state-of-the-art performance when benchmarked on a series of public datasets but also improves the accuracy and uncertainty quantification quality in two manufacturing process monitoring schemes. In a tool wear monitoring scheme for machining, it reduced the root mean square error (RMSE) by 44% and narrowed the confidence intervals of tool wear prediction by 35% compared to a neuro-fuzzy model. In a porosity monitoring system for additive manufacturing, the proposed network improved the porosity detection accuracy by 2% to 93.6% and quantified confidence intervals that were not available in conventional deep learning models. All these successes prove that the proposed probabilistic neural network can be a promising solution to address practical problems subject to significant uncertainties.

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.

Impact of Autonomous and Human-Driven Trucks on Flexible Pavement Design

Truck platooning with autonomous and connected vehicles has several advantages compared with traditional trucking. Platooning improves overall road safety and reduces fuel consumption by up to 15%. This is a result of the advanced control systems and high steering accuracy of autonomous vehicles. These advanced control systems present an opportunity to pavement design engineers, as the lateral positions of the vehicles could be altered to create less damaging loading scenarios. This study introduces an expected response framework to quantify the impact of lateral position on pavement performance. Using the expected response framework, any mixture of human-driven and autonomous vehicles can be analyzed by characterizing lane position as a mixture probability distribution instead of point loads. Pavement damage can subsequently be computed by using the expectation of the responses. This approach requires little computational effort and is easily incorporated in any mechanistic–empirical design or optimization framework. The approach is illustrated by analyzing four flexible pavement sections using the expected response framework. Compared with human-driven trucks, optimized lateral position could decrease pavement damage by 40%. Channelized traffic on the other hand could increase pavement damage by 60%. A simplified approach is introduced alongside a reliability analysis and fragility curves for various pavement structures. Distributed traffic was found to have the lowest probability of failure among all traffic scenarios.

Probability Models with Discrete and Continuous Parts

In mathematical statistics courses, students learn that the quadratic function is minimized when x is the mean of the random variable X, and that the graphs of this function for any two distributions of X are simply translates of each other. We focus on the problem of minimizing the function defined by in the context of mixtures of probability distributions of the discrete, absolutely continuous, and singular continuous types. This problem is important, for example, in Bayesian statistics, when one attempts to compute the decision function, which minimizes the expected risk with respect to an absolute error loss function. Although the literature considers this problem, it does so only under restrictive conditions on the distribution of the random variable X, by, for example, assuming that the corresponding cumulative distribution function is discrete or absolutely continuous. By using Riemann-Stieltjes integration, we prove a theorem, which solves this minimization problem under completely general conditions on the distribution of X. We also illustrate our result by presenting examples involving mixtures of distributions of the discrete and absolutely continuous types, and for the Cantor distribution, in which case the cumulative distribution function is singular continuous. Finally, we prove a theorem that evaluates the function y(x) when X has the Cantor distribution.

Analysis of Decoherence in Linear and Cyclic Quantum Walks

We theoretically analyze the case of noisy Quantum walks (QWs) by introducing four qubit decoherence models into the coin degree of freedom of linear and cyclic QWs. These models include flipping channels (bit flip, phase flip and bit-phase flip), depolarizing channel, phase damping channel and generalized amplitude damping channel. Explicit expressions for the probability distribution of QWs on a line and on a cyclic path are derived under localized and delocalized initial states. We show that QWs which begin from a delocalized state generate mixture probability distributions, which could give rise to useful algorithmic applications related to data encoding schemes. Specifically, we show how the combination of delocalzed initial states and decoherence can be used for computing the binomial transform of a given set of numbers. However, the sensitivity of QWs to noisy environments may negatively affect various other applications based on QWs.

Data reduction for serial crystallography using a robust peak finder.

A peak-finding algorithm for serial crystallography (SX) data analysis based on the principle of 'robust statistics' has been developed. Methods which are statistically robust are generally more insensitive to any departures from model assumptions and are particularly effective when analysing mixtures of probability distributions. For example, these methods enable the discretization of data into a group comprising inliers (i.e. the background noise) and another group comprising outliers (i.e. Bragg peaks). Our robust statistics algorithm has two key advantages, which are demonstrated through testing using multiple SX data sets. First, it is relatively insensitive to the exact value of the input parameters and hence requires minimal optimization. This is critical for the algorithm to be able to run unsupervised, allowing for automated selection or 'vetoing' of SX diffraction data. Secondly, the processing of individual diffraction patterns can be easily parallelized. This means that it can analyse data from multiple detector modules simultaneously, making it ideally suited to real-time data processing. These characteristics mean that the robust peak finder (RPF) algorithm will be particularly beneficial for the new class of MHz X-ray free-electron laser sources, which generate large amounts of data in a short period of time.

Representing an alarm system for processes with variables with mixture probability distribution

Representing an alarm system for processes with variables with mixture probability distribution

The similarity about the probability distributions of variables in the Bayesian regression model and application

The linear regression model as well as the time series model is applied in many fields, in which the mean of the dependent variable is one function of the mean of the independent variables. However, to consider the regression model following in the Classical Statistics (the Frequent Statistics), it means that the parameters are the constants, in many situations, the regression model does not describe the fluctuation of both the dependent variable and the independent variables. Therefore, we need to modify the parameters following the random variable form, not the constant form, like as the regression in Bayesian Statistics. The other side, when the parameters considered as the random variables, computations in the regression model becomes very complex, because we need to compute the product of the probability distributions. So, we must evaluate about to vary of the variables' probability distributions not only the normal distribution, the Student distribution t, the Poisson distribution, the binomial distribution… In this paper, we estimated the dependent variable's probability distribution form through the simple Bayesian regression model in cases having many the probability distribution forms of the independent variable. In addition, we apply the results to real stock price data, proving that the most appropriate probability distribution with the data is a mixture of probability distributions, not a single normal distribution.

Monitoring a bi-attribute high-quality process using a mixture probability distribution

High-quality processes with very low defect rates are usually modeled using a Poisson distribution. Considering any correlation between the number of defects on a unit and the fraction of nonconforming units, a bi-attribute process is introduced and the relationship between the two features is discussed in this research. A generalization of the Poisson distribution which is called k-inflated Poisson (KIP) distribution is derived to model these types of processes. Based on the KIP distribution, a two-parameter distribution, a sampling method is suggested for inspection of a high-quality process. The joint distribution of the bi-attribute high-quality process is also introduced and the correlation between its attributes is discussed. The results of numerical examples provided some perspective to the model and supported the theoretical findings that there is no significant correlation between the two features. Then a charting procedure is suggested for monitoring the two parameters of the KIP distribution.

Identification of Homo sapiens cancer classes based on fusion of hidden gene features.

Classification of Homo sapiens cancer genes in molecular level is a challenging research issue as they are extremely pseudo random in nature. Signature gene features need to be exposed to distinctly identify the gene class. Tree-structured filter bank is chosen to perform feature extraction and dimension reduction of the genes. Extracted gene features are fused using Gaussian mixture probability distribution function and identify different cancer classes depending on amount of correlation and exploiting maximum likelihood function. The algorithm is tested on 161 sample gene data of 7 different cancer classes. Sensitivity, specificity, accuracy, precision and F-score are used as metrics to judge the performance of the system and ROC is plotted in comparison with existing electrical network model based classifier. The proposed classifier can identify more than stated number of cancer classes which is a major limitation of the existing electrical network based method. The proposed algorithm is validated by comparing the results with other seven existing image processing based methods.

Probability density function for wave elevation based on Gaussian mixture models

In this article, Gaussian mixture models are used to estimate the probability density function of wave elevation in the context of the second-order random wave theory. Two approaches are used to construct the Gaussian mixture probability distribution. One is the moment estimate in which the unknown parameters are determined by matching the moments of Gaussian mixture model with those of the real wave process. The other is the maximum likelihood estimation in which the expectation–maximization (EM) algorithm is used to determine the parameters in statistical models. The proposed Gaussian mixture distribution is favorably validated by using Monte Carlo simulations in comparison with other theoretical distribution models. Numerical results reveal a clear dependence of the probability distribution structure on the wave steepness and the spectral shape. Finally, three sets of observation data are applied to further confirm the accuracy and efficiency of Gaussian mixture model.

Probabilistic Mixtures in Measurements of Interterritorial Differentiation

The article summarizes research results of the study on the problem of assessing the differentiation level of socio-economic development of territorial units of the Russian Federation. The authors propose an approach to measuring differentiation using mixtures of probability distributions.This technique was developed and tested on real data that allows one to determine the presence or absence of interregional differentiation. The research hypothesis, that interterritorial differentiation is estimated by a specific statistical indicator selected based on a content, qualitative analysis, served as a theoretical platform of this methodology. Differentiation is practically absent if the entire statistical population is described by a single law of probability distribution. If the statistical population is described by a mixture of probability distributions, then one should expect the presence of a significant level of differentiation by the considered indicator.In mathematical statistics, the problem of separating a mixture of probability distributions (estimating parameters of distribution densities and weighting coefficients) is traditionally solved using several similar methods. For example, the expectation-maximization (EM) algorithm, median modifications of the EM-algorithm, SEM-algorithm, taking into account the specifics of the selected object (constituent entities of the Russian Federation a small sample). To solve this problem, the authors used the SEM algorithm. As the information base of the empirical study, official statistics were used (open data from the Federal State Statistics Service).The typologies of the constituent entities of the Russian Federation were identified based on two characteristics within 2005-2017-time interval. The first one being the level of violence (using the “homicide rate” indicator the number of homicides and attempted murders per 100000 population). And second average per capita income, which made it possible, among other things, to additionally test the hypothesis of the traditional use of differentiation trends in the level of violence as an indicator of economic inequality. According to the authors, the results of this study can be used as instrumental and informational support for managerial decisions aimed at regulating the differentiation of Russian regions by the level of violence and economic inequality.