Mixture model

About: Mixture model is a research topic. Over the lifetime, 18155 publications have been published within this topic receiving 588317 citations.

A gentle tutorial of the em algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models

Abstract: We describe the maximum-likelihood parameter estimation problem and how the ExpectationMaximization (EM) algorithm can be used for its solution. We first describe the abstract form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the parameters of a mixture of Gaussian densities, and 2) finding the parameters of a hidden Markov model (HMM) (i.e., the Baum-Welch algorithm) for both discrete and Gaussian mixture observation models. We derive the update equations in fairly explicit detail but we do not prove any convergence properties. We try to emphasize intuition rather than mathematical rigor.

Point Set Registration: Coherent Point Drift

Abstract: Point set registration is a key component in many computer vision tasks. The goal of point set registration is to assign correspondences between two sets of points and to recover the transformation that maps one point set to the other. Multiple factors, including an unknown nonrigid spatial transformation, large dimensionality of point set, noise, and outliers, make the point set registration a challenging problem. We introduce a probabilistic method, called the Coherent Point Drift (CPD) algorithm, for both rigid and nonrigid point set registration. We consider the alignment of two point sets as a probability density estimation problem. We fit the Gaussian mixture model (GMM) centroids (representing the first point set) to the data (the second point set) by maximizing the likelihood. We force the GMM centroids to move coherently as a group to preserve the topological structure of the point sets. In the rigid case, we impose the coherence constraint by reparameterization of GMM centroid locations with rigid parameters and derive a closed form solution of the maximization step of the EM algorithm in arbitrary dimensions. In the nonrigid case, we impose the coherence constraint by regularizing the displacement field and using the variational calculus to derive the optimal transformation. We also introduce a fast algorithm that reduces the method computation complexity to linear. We test the CPD algorithm for both rigid and nonrigid transformations in the presence of noise, outliers, and missing points, where CPD shows accurate results and outperforms current state-of-the-art methods.

Hierarchical mixtures of experts and the EM algorithm

Abstract: We present a tree-structured architecture for supervised learning. The statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models (GLIM's). Learning is treated as a maximum likelihood problem; in particular, we present an Expectation-Maximization (EM) algorithm for adjusting the parameters of the architecture. We also develop an on-line learning algorithm in which the parameters are updated incrementally. Comparative simulation results are presented in the robot dynamics domain.

Mixture models : inference and applications to clustering

Abstract: General Introduction Introduction History of Mixture Models Background to the General Classification Problem Mixture Likelihood Approach to Clustering Identifiability Likelihood Estimation for Mixture Models via EM Algorithm Start Values for EMm Algorithm Properties of Likelihood Estimators for Mixture Models Information Matrix for Mixture Models Tests for the Number of Components in a Mixture Partial Classification of the Data Classification Likelihood Approach to Clustering Mixture Models with Normal Components Likelihood Estimation for a Mixture of Normal Distribution Normal Homoscedastic Components Asymptotic Relative Efficiency of the Mixture Likelihood Approach Expected and Observed Information Matrices Assessment of Normality for Component Distributions: Partially Classified Data Assessment of Typicality: Partially Classified Data Assessment of Normality and Typicality: Unclassified Data Robust Estimation for Mixture Models Applications of Mixture Models to Two-Way Data Sets Introduction Clustering of Hemophilia Data Outliers in Darwin's Data Clustering of Rare Events Latent Classes of Teaching Styles Estimation of Mixing Proportions Introduction Likelihood Estimation Discriminant Analysis Estimator Asymptotic Relative Efficiency of Discriminant Analysis Estimator Moment Estimators Minimum Distance Estimators Case Study Homogeneity of Mixing Proportions Assessing the Performance of the Mixture Likelihood Approach to Clustering Introduction Estimators of the Allocation Rates Bias Correction of the Estimated Allocation Rates Estimated Allocation Rates of Hemophilia Data Estimated Allocation Rates for Simulated Data Other Methods of Bias Corrections Bias Correction for Estimated Posterior Probabilities Partitioning of Treatment Means in ANOVA Introduction Clustering of Treatment Means by the Mixture Likelihood Approach Fitting of a Normal Mixture Model to a RCBD with Random Block Effects Some Other Methods of Partitioning Treatment Means Example 1 Example 2 Example 3 Example 4 Mixture Likelihood Approach to the Clustering of Three-Way Data Introduction Fitting a Normal Mixture Model to Three-Way Data Clustering of Soybean Data Multidimensional Scaling Approach to the Analysis of Soybean Data References Appendix

Sample Splitting and Threshold Estimation

Abstract: .Threshold models have a wide variety of applications in economics. Direct applications include models of separating and multiple equilibria. Other applications include empirical sample splitting when the sample split is based on a continuously-distributed variable such as firm size. In addition, threshold models may be used as a parsimonious strategy for nonparametric function estimation. For example, the threshold autoregressive model .TAR is popular in the nonlinear time series literature. Threshold models also emerge as special cases of more complex statistical frameworks, such as mixture models, switching models, Markov switching models, and smooth transition threshold models. It may be important to understand the statistical properties of threshold models as a preliminary step in the development of statistical tools to handle these more complicated structures. Despite the large number of potential applications, the statistical theory of threshold estimation is undeveloped. It is known that threshold estimates are super-consistent, but a distribution theory useful for testing and inference has yet to be provided. This paper develops a statistical theory for threshold estimation in the regression context. We allow for either cross-section or time series observations. Least squares estimation of the regression parameters is considered. An asymptotic distribution theory . for the regression estimates the threshold and the regression slopes is developed. It is found that the distribution of the threshold estimate is nonstandard. A method to construct asymptotic confidence intervals is developed by inverting the likelihood ratio statistic. It is shown that this yields asymptotically conservative confidence regions. Monte Carlo simulations are presented to assess the accuracy of the asymptotic approximations. The empirical relevance of the theory is illustrated through an application to the multiple . equilibria growth model of Durlauf and Johnson 1995 .