A unified Minorization-Maximization approach for estimation of general mixture models

The mixed distribution model is often used to extract information from heteroge-neous data and perform modeling analysis.When the density function of mixed distribution is complicated or the variable dimension is high,it usually brings challenges to the parameter es-timation of the mixed distribution model.The application of MM algorithm can avoid complex expectation calculations,and can also solve the problem of high-dimensional optimization by decomposing the objective function.In this paper,MM algorithm is applied to the parameter estimation problem of mixed distribution model.The method of assembly and decomposition is used to construct the substitute function with separable parameters,which avoids the problems of complex expectation calculations and the inversion of high-dimensional matrices.

A new method for evaluating the firing precision of multiple launch rocket system based on Bayesian theory

How to effectively evaluate the firing precision of weapon equipment at low cost is one of the core contents of improving the test level of weapon system.A new method to evaluate the firing precision of the MLRS considering the credibility of simulation system based on Bayesian theory is proposed in this paper.First of all,a comprehensive index system for the credibility of the simulation system of the firing precision of the MLRS is constructed combined with the group analytic hierarchy process.A modified method for determining the comprehensive weight of the index is established to improve the rationality of the index weight coefficients.The Bayesian posterior estimation formula of firing precision considering prior information is derived in the form of mixed prior distribution,and the rationality of prior information used in estimation model is discussed quantitatively.With the simulation tests,the different evaluation methods are compared to validate the effectiveness of the proposed method.Finally,the experimental results show that the effectiveness of estimation method for firing precision is improved by more than 25%.

Extreme value distributions of mixing two sequences with different MDA＇s

Suppose {Xi, i≥1} and {Yi, i≥1} are two independent sequences with distribution functions FX(x) and FY(x), respectively. Zi is the combination of Xi and Yi with a probability pn for each i with 1≤i≤n. The extreme value distribution ,n GZ(x) of this particular triangular array of the i.i.d. random variables Z1, , Z2, ,…, Zn n n ,nis discussed. We found a new form of the extreme value distribution ΛA(ρx)Λ(x)(0<ρ <1), which is not max-stable. It occurs if FX(x) and FY(x) belong to the same MDA(Λ). GZ(x) does not exist as mixture forms of the different types of extreme value distributions.

Global exponential stability of mixed discrete and distributively delayed cellular neural network

This paper concernes analysis for the global exponential stability of a class of recurrent neural networks with mixed discrete and distributed delays. It first proves the existence and uniqueness of the balance point, then by employing the Lyapunov-Krasovskii functional and Young inequality, it gives the sufficient condition of global exponential stability of cellular neural network with mixed discrete and distributed delays, in addition, the example is provided to illustrate the applicability of the result.

The New Mixed Generalized Erlang Distribution

In probability theory, the mixture distribution M has a density function for the collection of random variables and weighted by wi ≥ 0 and . These mixed distributions are used in various disciplines and aim to enrich the collection distribution to more parameters. A more general mixture is derived by Kadri and Halat, by proving the existence of such mixture by wi ∈ R, and maintaining . Kadri and Halat provided many examples and applications for such new mixed distributions. In this paper, we introduce a new mixed distribution of the Generalized Erlang distribution, which is derived from the Hypoexponential distribution. We characterize this new distribution by deriving simply closed expressions for the related functions of the probability density function, cumulative distribution function, moment generating function, reliability function, hazard function, and moments.

Analysis of College Students’ Test Scores Based on Two-Component Mixed Generalized Normal Distribution

In order to improve the fitting accuracy of college students’ test scores, this paper proposes two-component mixed generalized normal distribution, uses maximum likelihood estimation method and Expectation Conditional Maxinnization (ECM) algorithm to estimate parameters and conduct numerical simulation, and performs fitting analysis on the test scores of Linear Algebra and Advanced Mathematics of F University. The empirical results show that the two-component mixed generalized normal distribution is better than the commonly used two-component mixed normal distribution in fitting college students’ test data, and has good application value.

Bayesian Assessment for Reliability of Binomial Components Based on Information Fusion of Similar Products

A modified Bayesian reliability assessment method of binomial components was proposed by fusing prior information of similar products.The traditional Bayesian method usually directly used all the prior data,ignoring the differences between them,which might decrease the credibility level of reliability evaluation and result in data submergence.To solve the problem,a revised approach was derived to calculate groups of prior data's quantitative credibility,used for weighted data fusion.Then inheritance factor was introduced to build a mixed beta distribution to illustrate the innovation of new products.However,in many cases,inheritance factor was determined by Chi-square test that could not give out exact result with respect to rare failures.To make the model more precise,Barnard's exact test was suggested being used to calculate the inheritance factor.A numerical example is given to demonstrate that the modified method is successful and rational,while the classical method is too conservative and the traditional Bayesian method is too risky.

Reliability Estimation of Aviation Pyrotechnics System Based on Mixed Beta Distribution

For evaluating reliability of an aviation pyrotechnics system,a Bayesian reliability estimation method utilizing reliability information of a system and its units was put forward. Firstly,an inheritance factor was determined by chi-square goodness of fit test.Then the mixed prior distribution was obtained based on the inheritance factor. Finally, the density function of posterior distribution was obtained and used to assess the reliability of system.According to the new method, the reliability of an aviation pyrotechnics system was evaluated to reach 0. 989 6 at the confidence level of 0. 90. To reach the reliability index,the required numbers of trials of system and its units were given. It is instructional to apply the new method on the reliability estimation of aviation pyrotechnics systems.

Experimental study by online measurement of the precipitation of nickel hydroxide: Effects of operating conditions

The objective of this work is using the online measurement method to study the process of precipitation of nickel hydroxide in a single-feed semi-batch stirred reactor with an internal diameter ofD = 240mm. The effects of impeller speed, impeller type, impeller diameter and feed location on the mean particle size d43 and particle size distribution （PSD） were investigated, d43 and PSD were measured online using a Malvern Insitec Liquid Pro- cess Sizer every 20 s. It was found that d43 varied between 13 kwh and 26 lain under different operating conditions, and it decreased with increasing impeller diameter. When feeding at the off-bottom distance of D/2 under lower impeller speeds, d43 was significantly smaller than that at D/3. PSDs were slightly influenced by operating conditions.

ON THE INCREMENTS DISTRIBUTION OF STOCK PRICES

In this paper,the models of increment distributions of stock price are constructed with two approaches. The first approach is based on limit theorems of random summation. The second approach is based on the statistical analysis of the increment distribution of the logarithms of stock prices.

More general results on mixed extreme value distributions

The sequences {Zi,n, 1≤i≤n}, n≥1 are multi-nomial distribution among i.i.d, random variables {X1,i, i≥1}, {X2,i, i≥1 } {Xm,i, i≥1 }. The extreme value distribution Gz（x） of this particular triangular array of i.i,d, random variables Z1,n, Z2 n,...,Zn,n is discussed. A new type of not max-stable extreme value distributions which are Fréchet mixture, Gumbel mixture and Weibull mixture has been found if Fj,…… Fm belong to the same MDA. Whether mixtures of different types of extreme value distributions exist or not and the more general case are discussed in this paper. We found that Gz（x） does not exist as mixture forms of the different types of extreme value distributions after we investigated all cases.

A class of not max-stable extreme value distributions

The sequences {Zi , 1≤i≤n}, n≥1 have multi-nomial distribution among i.i.d. random variables {X1, , i≥1}, {X2, , ,n i i i≥1}, …, {Xm , i≥1}. The extreme value distribution GZ(x) of this particular triangular array of i.i.d. random variables Z1, , Z2, , …, ,i n n r ?1 Zn is discussed in this paper. We found a new type of not max-stable extreme value distributions, i) GZ (x) = ,n ∏Φα Ai(x)×Φαr (x); i i=1 r ?1 r?1 ii) GZ (x) = ∏Ψα Ai(x)×Ψαr (x); iii) GZ (x) = ∏Λ Ai(λix)×Λ(x), r≥2, 0<α1≤α2≤…≤αr and λi∈(0,1] for i, 1≤i≤r?1 which occur if i i=1 i=1 Fj, …, Fm belong to the same MDA.

Mixtures of Distributions and Volatility

Using Monte Carlo methods we generate time series with the following features： a） series with distributions that are the mix of two normal distributions with different variances, b） series that satisfy volatility models, c） series that satisfy an AR（1） model but with contaminated errors that follow the same distribution as the mixes given in a） and d） series that follow the same distribution as the mixes given in a） but with conditional heterocedasticity. From the analysis we see that it is difficult to identify in practical situations the real generating process of the series. In fact, the processes that come from distribution mixes have many similar characteristics to the ones that satisfy the volatility scheme. We use the corresponding theoretical considerations and also the usual tools in the identifying process of any time series; that is, series graphs, histograms, the corresponding sampling distributions, correlograms and partial correlograms.

Parameter Estimation of Mixed Weibull Distributions Using Cuckoo Search

The lifetime data of products with multiple failure modes which are collected from life testing are often fitted by the mixed Weibull distributions. Since the mixed Weibull distributions contain no less than five parameters,the parameter estimation is difficult and inaccurate. In order to enhance the accuracy,a new method of parameter estimation based on Cuckoo search( CS) is proposed. An optimization model for the mixed Weibull distribution is formulated by minimizing the residual sum of squares. The optimal parameters are searched via CS algorithm. In the case study,the lifetime data come from the life testing of diesel injectors and are fitted by the twocomponent Weibull mixture. Regarding the maximum absolute error and the accumulative absolute error between estimated and observed values as the accuracy index of parameter estimation,the results of four parameter estimation methods that the graphic estimation method,the nonlinear least square method,the optimization method based on particle swarm optimization( PSO) and the proposed method are compared. The result shows that the proposed method is more efficient and more accurate than the other three methods.

Capacity of unsignalized intersection with mixed distribution headway

In order to describe the time-headway distribution more precisely in urban traffic network,the mixed distribution model was introduced which has been widely used in mathematical statistics,and a capacity model of unsignalized intersections was obtained based on gap acceptance theory.The new model is suitable for absolute and limited priority controlled conditions and can be regarded as a more general form which handles simple headway distributions including lognormal distribution,negative exponential distribution and shifted negative exponential distribution.Through analyses of the main influencing factors in this model,the proportion of free flowing and the standard variance of gaps between any two continuous following vehicles are high sensitivity with the capacity when major stream volume is low.Besides,the capacity is affected deeply by the mean value of following vehicle gaps when major stream value is fixed and the proportion of free flowing is small.At last,the observed minor stream capacity is obtained by the survey date in Changchun city,and the average relative error between the theoretical capacity proposed in this paper is 13.73%,meanwhile the accuracy increases by 16.68% compared with the theoretical value when major stream obeys shifted negative exponential distribution.

Clustering-based maintainability demonstration for complex systems with a mixed maintenance time distribution

During maintainability demonstration,the maintenance time for complex systems consisting of mixed technologies generally conforms to a mixture distribution.However existing maintainability standards and guidance do not explain explicitly how to deal with this situation.This paper develops a comprehensive maintainability demonstration method for complex systems with a mixed maintenance time distribution.First of all,a K-means algorithm and an expectation-maximization(EM)algorithm are used to partition the maintenance time data for all possible clusters.The Bayesian information criterion(BIC)is then used to choose the optimal model.After this,the clustering results for equipment are obtained according to their degree of membership.The degree of similarity for the maintainability of different kinds of equipment is then determined using the projection method.By using a Bootstrap method,the prior distribution is obtained from the maintenance time data for the most similar equipment.Then,a test method based on Bayesian theory is outlined for the maintainability demonstration.Finally,the viability of the proposed approach is illustrated by means of an example.

Gamma-Generalized Inverse Gaussian Class of Distributions

Gamma distribution nests exponential, chi-squared and Erlang distributions;while generalized Inverse Gaussian distribution nests quite a number of distributions. The aim of this paper is to construct a gamma mixture using Generalized inverse Gaussian mixing distribution. The rth moment of the mixture is obtained via the rth moment of the mixing distribution. Special cases and limiting cases of the mixture are deduced.

Robust Variance Components Estimation in the PERG Mixed Distributions of Empirical Variances—PEROBVC Method

A mixed distribution of empirical variances, composed of two distributions the basic and contaminating ones, and referred to as PERG mixed distribution of empirical variances, is considered. In the paper a robust inverse problem solution is given, namely a (new) robust method for estimation of variances of both distributions—PEROBVC Method, as well as the estimates for the numbers of observations for both distributions and, in this way also the estimate of contamination degree.

New method to identify optimal discontinuity set number of rock tunnel excavation face orientation based on Fisher mixed evaluation

Discontinuity is critical for strength,deformability,and permeability of rock mass.Set information is one of the essential discontinuity characteristics and is usually accessed by orientation grouping.Traditional methods of identifying optimal discontinuity set numbers are usually achieved by clustering validity indexes,which mainly relies on the aggregation and dispersion of clusters and leads to the inaccuracy and instability of evaluation.This paper proposes a new method of Fisher mixed evaluation(FME)to identify optimal group numbers of rock mass discontinuity orientation.In FME,orientation distribution is regarded as the superposition of Fisher mixed distributions.Optimal grouping results are identified by considering the fitting accuracy of Fisher mixed distributions,the probability monopoly and central location significance of independent Fisher centers.A Halley-Expectation-Maximization(EM)algorithm is derived to achieve an automatic fitting of Fisher mixed distribution.Three real rock discontinuity models combined with three orientation clustering algorithms are adopted for discontinuity grouping.Four clustering validity indexes are used to automatically identify optimal group numbers for comparison.The results show that FME is more accurate and robust than the other clustering validity indexes in optimal discontinuity group number identification for different rock models and orientation clustering algorithms.

Uncertainty modeling of wind power frequency regulation potential considering distributed characteristics of forecast errors

Large-scale integration of wind power generation decreases the equivalent inertia of a power system, and thus makes frequency stability control challenging. However, given the irregular, nonlinear, and non-stationary characteristics of wind power, significant challenges arise in making wind power generation participate in system frequency regulation. Hence, it is important to explore wind power frequency regulation potential and its uncertainty. This paper proposes an innovative uncertainty modeling method based on mixed skew generalized error distribution for wind power frequency regulation potential. The mapping relationship between wind speed and the associated frequency regulation potential is established, and key parameters of the wind turbine model are identified to predict the wind power frequency regulation potential. Furthermore, the prediction error distribution of the frequency regulation potential is obtained from the mixed skew model. Because of the characteristics of error partition, the error distribution model and predicted values at different wind speed sections are summarized to generate the uncertainty interval of wind power frequency regulation potential. Numerical experiments demonstrate that the proposed model outperforms other state-of-the-art contrastive models in terms of the refined degree of fitting error distribution characteristics. The proposed model only requires the wind speed prediction sequence to accurately model the uncertainty interval. This should be of great significance for rationally optimizing system frequency regulation resources and reducing redundant backup.