Numerical analysis of EM estimation of mixture model parameters

Optimisation of distribution parameters is a very common problem. There are many sorts of distributions which can be used to model environment processes, biological functions or graphical data. However, it is common that parameters of those distribution may be, partially or completely unknown. Mixture models composed of a few distributions are easier to solve. In such a case simple estimation methods may be used to obtain results. Usually models are composed of several distributions. Those distributions may be of the same or different type. Such models are called mixture models. Finding their parameters may be complicated. Usually in such cases iterative methods need to be used. The paper gives a brief survey of algorithms designed for solving mixtures of distributions and problems connected with their usage.One of the most common method used to obtain mixture model parameters is Expectation-Maximization (EM) algorithm. EM is the iterative algorithm performing maximum likelihood estimation. The authors present the results of adjusting the Gaussian mixture models to the data. It is done with the usage of EM algorithm. The article gives advantages and disadvantages of EM algorithm. Improvements of EM applied in the case of large data are also presented. They help increase efficiency and decrease operation time of the algorithm. Another considered issue is the problem of optimal input parameters selection and its influence on the adjustment results. The authors also present algorithm performance observations

Outlier detection through mixtures with an improper component

The  paper  investigates the use of a finite  mixture model with an additional uniform density for outlier detection and robust estimation.  The main contribution of this paper lies in the  analysis of the properties of the improper component and the introduction of a modified EM algorithm  which, beyond providing the maximum likelihood estimates of the mixture parameters, endogenously provides a numerical value for the density of the uniform distribution used for the improper component. The mixing proportion of outliers may be known or unknown.  Applications to robust estimation and outlier detection will be discussed with particular attention to the normal mixture case

An analytical approach on parametric estimation of cure fraction based on weibull distribution using interval censored data.

In this article, we consider the Bounded Cumulative Hazard (BCH) model that is more appropriate than mixture cure model in case of cancer clinical trials when the population of interest contains long-term survivors or cured. We propose this cure rate model based on the Weibull distribution with interval censored data. Maximum likelihood estimation (MLE) method is proposed to estimate the parameters within the framework of expectation-maximization (EM) algorithm, Newton Raphson method also employed. The analysis showed that the cure fraction cannot be obtained analytically, but may be obtained from the numerical solution of the estimated equations. A simulation study is also provided for assessing the efficiency of the proposed estimation procedure

Model Selection for Gaussian Mixture Models

This paper is concerned with an important issue in finite mixture modelling, the selection of the number of mixing components. We propose a new penalized likelihood method for model selection of finite multivariate Gaussian mixture models. The proposed method is shown to be statistically consistent in determining of the number of components. A modified EM algorithm is developed to simultaneously select the number of components and to estimate the mixing weights, i.e. the mixing probabilities, and unknown parameters of Gaussian distributions. Simulations and a real data analysis are presented to illustrate the performance of the proposed method

S-estimation of hidden Markov models

A method for robust estimation of dynamic mixtures of multivariate distributions is proposed. The EM algorithm is modified by replacing the classical M-step with high breakdown S-estimation of location and scatter, performed by using the bisquare multivariate S-estimator. Estimates are obtained by solving a system of estimating equations that are characterized by component specific sets of weights, based on robust Mahalanobis-type distances. Convergence of the resulting algorithm is proved and its finite sample behavior is investigated by means of a brief simulation study and n application to a multivariate time series of daily returns for seven stock markets