DOCSLIB.ORG

The EM Algorithm in Multivariate Gaussian Mixture Models Using Anderson Acceleration

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The EM Algorithm in Multivariate Gaussian Mixture Models Using Anderson Acceleration The EM Algorithm in Multivariate Gaussian Mixture Models using Anderson Acceleration by Joshua H. Plasse A Project Report Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Master of Science in Applied Mathematics by May 2013 APPROVED: Dr. Homer F. Walker, Thesis Advisor Dr. Bogdan Vernescu, Department Head Abstract Over the years analysts have used the EM algorithm to obtain maximum likelihood estimates from incomplete data for various models. The general algorithm admits several appealing properties such as strong global convergence; however, the rate of convergence is linear which in some cases may be unacceptably slow. This work is primarily concerned with applying Anderson acceleration to the EM algorithm for Gaussian mixture models (GMM) in hopes of alleviating slow convergence. As preamble we provide a review of maximum likelihood estimation and derive the EM algorithm in detail. The iterates that correspond to the GMM are then formulated and examples are provided. These examples show how faster convergence is experienced when the data are well separated, whereas much slower convergence is seen whenever the sample is poorly separated. The Anderson acceleration method is then presented, and its connection to the EM algorithm is discussed. The work is then concluded by applying Anderson acceleration to the EM algorithm which results in reducing the number of iterations required to obtain convergence. i Acknowledgments I would first like to thank my advisor and project mentor Dr. Homer Walker. His guidance throughout the duration of this exposition proved invaluable. I also would like to express my gratitude to Professor Mayer Humi and Professor Marcus Sarkis for instilling a strong background in various areas of numerical analysis. I would also like to thank Dr. Joseph Fehribach for taking his time to conduct an independent study. Secondly, I would like to express my thanks to Kyle Dunn, Marisa Zemsky, William Sanguinet and Dan Brady, who helped vectorize MATLAB code, produce aesthetically appealing graphics and help alleviate problems in LATEX. I would also like to acknowledge Mike Malone and Sia Najafi who provided me with a powerful computer that was used extensively in the computation of large scale data sets. Lastly, I would like to thank my mother, father, Briannah and Alannah for all the love and support that they have given more over the span of the last twenty four years. To Caitlin, I express my utmost gratitude for the patience and encouragement you have shown me which has helped me get where I am today. It is to them I dedicate this project. This work was supported in part through U.S. Department of Energy Office of Science Grant DE- SC0004880 to Worcester Polytechnic Institute. ii Contents Abstract i Acknowledgments ii List of Figures iv List of Tables v Introduction 1 Maximum Likelihood Estimation.....................................2 Calculating an MLE Analytically..................................2 The EM Algorithm 5 General EM.................................................5 Convergence Properties..........................................6 EM Algorithm for Mixture Models....................................7 EM on Normal Mixtures.......................................9 Examples.................................................. 12 Well-Separated Case (WSC)..................................... 12 Poorly-Separated Case (PSC).................................... 17 Observations............................................. 23 Anderson Acceleration 24 Fixed-Point Iteration............................................ 24 Acceleration Algorithm.......................................... 24 Example................................................... 27 Conclusions 30 Future Work................................................ 30 Bibliography 31 Appendix A 32 Appendix B 44 iii List of Figures 1 Scatter Plot of our Random Sample (WSC)............................ 12 2 After 3 Iterations (WSC)...................................... 13 3 After 5 Iterations (WSC)...................................... 13 4 After 15 Iterations (WSC)...................................... 14 5 Mean Errors - log jjµi+1 − µijj1 (WSC).............................. 14 6 Proportion Errors - log jjαi+1 − αijj1 (WSC)........................... 14 7 Covariance Errors - log jjΣi+1 − Σijj1 (WSC)........................... 15 8 Applying K-Means to our Sample (WSC)............................. 15 9 Mean and Proportion Errors using K-Means (WSC)....................... 16 10 Covariance Errors using K-Means (WSC)............................. 16 11 Scatter Plot of our Random Sample (PSC)............................ 17 12 After 50 Iterations (PSC)...................................... 18 13 After 100 Iterations (PSC)...................................... 18 14 After 250 Iterations (PSC)...................................... 19 15 Mean Errors - log jjµi+1 − µijj1 (PSC)............................... 20 16 Proportion Errors - log jjαi+1 − αijj1 (PSC)............................ 20 17 Covariance Errors - log jjΣi+1 − Σijj1 (PSC)........................... 20 18 Applying K-Means to our Sample (PSC).............................. 21 19 After 25 Iterations using K-Means................................. 22 20 After 150 Iterations using K-Means................................. 22 21 Plots of Performance With and Without Acceleration...................... 28 iv List of Tables 1 True Parameter Values (WSC)................................... 12 2 Final Approximated Parameter Values (WSC).......................... 15 3 Initial Parameter Guesses Using K-Means (WSC)......................... 16 4 True Parameter Values (PSC).................................... 17 5 Approximated Parameter Values (PSC).............................. 19 6 Initial Parameter Guesses using K-Means (PSC)......................... 21 7 Without Acceleration........................................ 29 8 With Acceleration.......................................... 29 v Introduction Of interest is the estimation of parameters in a mixture model where all underlying components are multivariate Gaussian distributions of dimension at least two. To be precise, throughout this exposition the model used will be a Gaussian mixture model (GMM) that represents a population composed of m 2 Z+ subpopulations. To accompany our model we also assume that we have an unlabeled random sample X = (x1; x2; :::; xN ) which was obtained in an independent and identically distributed (iid) fashion. An unlabeled sample in this sense means that for any xk 2 X ; k = 1; :::; N, the true subpopulation to which xk belongs is not known. The goal is to make accurate statistical inferences on properties of the subpopulations using only the unlabeled sample X . As will be seen in the sequel, maximum likelihood estimation via the EM algorithm is a powerful tool in obtaining accurate parameter estimates for the GMM as well as various other models. Mathematically we define the GMM as m X p(xjΦ) = αipi(xjφi); (1) i=1 T d where x = (x1; :::; xd) 2R , φi = (µi; Σi), Φ = (α1; :::; αm; φ1; :::; φm) 2 Ω and each pi is a d-dimensional multivariate Gaussian distribution given by, 1 −1=2(x−µ )T Σ−1(x−µ ) p (xjφ ) = e i i i : (2) i i d=2 1=2 (2π) (det Σi) d th Similar to the univariate Gaussian distribution, µi 2 R represents the mean vector for the i subpop- th ulation; whereas, Σi is the d × d symmetric positive definite covariance matrix that corresponds to the i subpopulation. The collection of αi's are known as the model's mixture proportions, i.e. each αi represents th the probability that a randomly selected xk 2 X was generated from the i subpopulation. Since each αi is a probability, it follows that αi 2 [0; 1] for all i and the αi's are constrained to sum to one. Due to this constraint, it follows that p(xjΦ) is a probability density function (pdf) since m X 1 = αi i=1 m X Z = αi pi(xjφi)dx d i=1 R m Z X = αipi(xjφi)dx d R i=1 Z = p(xjΦ)dx: d R 1 where pdf stands for probability density function. Now that we have defined the model we seek numerical methods that will allow one to compute accurate estimates for Φ. Since ultimately our goal is to introduce, derive and accelerate the EM algorithm for (1), which is used for maximum likelihood estimation, we provide the reader with a review of the method. Maximum Likelihood Estimation The notations that will be developed reflects those that are used in Casella and Berger [2]. First we suppose that we have an n-tuple of random vectors X = (X1;X2; :::; Xn) that was generated in an iid fashion. We also assume that the distribution of X depends on a fixed unknown parameter θ = (θ1; :::; θk) that takes its value in the parameter space Θ. Thus for any Xi 2 X the individual probability density (mass) iid function (pdf (pmf)) for Xi will be denoted by Xi ∼ f(xijθ) for i = 1; :::; n. We may now take advantage of our sample being iid by defining our joint pdf (pmf) for X as n Y f(xjθ) = f(xijθ1; :::; θk): (3) i=1 Here x = (x1; :::; xn) denotes the observed values of X1:; ; ; :Xn. To understand how to compute a maximum likelihood estimate (MLE) for a given distribution, one must be familiar with the concept of likelihood functions, which we define below. Definition 1. Let f(xjθ) denote the joint pdf (pmf) of our random sample X = (X1; :::; Xn) and be defined as in (3). Then given the observed values X = x, the likelihood function is defined as n Y L(θjx) = L(θ1; ...θkjx1; :::; xn) = f(xjθ) = f(xijθ1; :::; θk): (4) i=1 The reader should be aware
Load more

Recommended publications
  • LECTURE 13 Mixture Models and Latent Space Models
    LECTURE 13 Mixture models and latent space models We now consider models that have extra unobserved variables. The variables are called latent variables or state variables and the general name for these models are state space models.A classic example from genetics/evolution going back to 1894 is whether the carapace of crabs come from one normal or from a mixture of two normal distributions. We will start with a common example of a latent space model, mixture models. 13.1. Mixture models 13.1.1. Gaussian Mixture Models (GMM) Mixture models make use of latent variables to model different parameters for different groups (or clusters) of data points. For a point xi, let the cluster to which that point belongs be labeled zi; where zi is latent, or unobserved. In this example (though it can be extended to other likelihoods) we will assume our observable features xi to be distributed as a Gaussian, so the mean and variance will be cluster- specific, chosen based on the cluster that point xi is associated with. However, in practice, almost any distribution can be chosen for the observable features. With d-dimensional Gaussian data xi, our model is: zi j π ∼ Mult(π) xi j zi = k ∼ ND(µk; Σk); where π is a point on a K-dimensional simplex, so π 2 RK obeys the following PK properties: k=1 πk = 1 and 8k 2 f1; 2; ::; Kg : πk 2 [0; 1]. The variables π are known as the mixture proportions and the cluster-specific distributions are called th the mixture components.
    [Show full text]
  • START HERE: Instructions Mixtures of Exponential Family
    Machine Learning 10-701 Mar 30, 2015 http://alex.smola.org/teaching/10-701-15 due on Apr 20, 2015 Carnegie Mellon University Homework 9 Solutions START HERE: Instructions • Thanks a lot to John A.W.B. Constanzo for providing and allowing to use the latex source files for quick preparation of the HW solution. • The homework is due at 9:00am on April 20, 2015. Anything that is received after that time will not be considered. • Answers to every theory questions will be also submitted electronically on Autolab (PDF: Latex or handwritten and scanned). Make sure you prepare the answers to each question separately. • Collaboration on solving the homework is allowed (after you have thought about the problems on your own). However, when you do collaborate, you should list your collaborators! You might also have gotten some inspiration from resources (books or online etc...). This might be OK only after you have tried to solve the problem, and couldn’t. In such a case, you should cite your resources. • If you do collaborate with someone or use a book or website, you are expected to write up your solution independently. That is, close the book and all of your notes before starting to write up your solution. • Latex source of this homework: http://alex.smola.org/teaching/10-701-15/homework/ hw9_latex.zip. Mixtures of Exponential Family In this problem we will study approximate inference on a general Bayesian Mixture Model. In particular, we will derive both Expectation-Maximization (EM) algorithm and Gibbs Sampling for the mixture model. A typical
    [Show full text]
  • Robust Mixture Modelling Using the T Distribution
    Statistics and Computing (2000) 10, 339–348 Robust mixture modelling using the t distribution D. PEEL and G. J. MCLACHLAN Department of Mathematics, University of Queensland, St. Lucia, Queensland 4072, Australia [email protected] Normal mixture models are being increasingly used to model the distributions of a wide variety of random phenomena and to cluster sets of continuous multivariate data. However, for a set of data containing a group or groups of observations with longer than normal tails or atypical observations, the use of normal components may unduly affect the fit of the mixture model. In this paper, we consider a more robust approach by modelling the data by a mixture of t distributions. The use of the ECM algorithm to fit this t mixture model is described and examples of its use are given in the context of clustering multivariate data in the presence of atypical observations in the form of background noise. Keywords: finite mixture models, normal components, multivariate t components, maximum like- lihood, EM algorithm, cluster analysis 1. Introduction tailed alternative to the normal distribution. Hence it provides a more robust approach to the fitting of normal mixture mod- Finite mixtures of distributions have provided a mathematical- els, as observations that are atypical of a component are given based approach to the statistical modelling of a wide variety of reduced weight in the calculation of its parameters. Also, the random phenomena; see, for example, Everitt and Hand (1981), use of t components gives less extreme estimates of the pos- Titterington, Smith and Makov (1985), McLachlan and Basford terior probabilities of component membership of the mixture (1988), Lindsay (1995), and B¨ohning (1999).
    [Show full text]
  • Lecture 16: Mixture Models
    Lecture 16: Mixture models Roger Grosse and Nitish Srivastava 1 Learning goals • Know what generative process is assumed in a mixture model, and what sort of data it is intended to model • Be able to perform posterior inference in a mixture model, in particular { compute the posterior distribution over the latent variable { compute the posterior predictive distribution • Be able to learn the parameters of a mixture model using the Expectation-Maximization (E-M) algorithm 2 Unsupervised learning So far in this course, we've focused on supervised learning, where we assumed we had a set of training examples labeled with the correct output of the algorithm. We're going to shift focus now to unsupervised learning, where we don't know the right answer ahead of time. Instead, we have a collection of data, and we want to find interesting patterns in the data. Here are some examples: • The neural probabilistic language model you implemented for Assignment 1 was a good example of unsupervised learning for two reasons: { One of the goals was to model the distribution over sentences, so that you could tell how \good" a sentence is based on its probability. This is unsupervised, since the goal is to model a distribution rather than to predict a target. This distri- bution might be used in the context of a speech recognition system, where you want to combine the evidence (the acoustic signal) with a prior over sentences. { The model learned word embeddings, which you could later analyze and visu- alize. The embeddings weren't given as a supervised signal to the algorithm, i.e.
    [Show full text]
  • Linear Mixture Model Applied to Amazonian Vegetation Classification
    Remote Sensing of Environment 87 (2003) 456–469 www.elsevier.com/locate/rse Linear mixture model applied to Amazonian vegetation classification Dengsheng Lua,*, Emilio Morana,b, Mateus Batistellab,c a Center for the Study of Institutions, Population, and Environmental Change (CIPEC) Indiana University, 408 N. Indiana Ave., Bloomington, IN, 47408, USA b Anthropological Center for Training and Research on Global Environmental Change (ACT), Indiana University, Bloomington, IN, USA c Brazilian Agricultural Research Corporation, EMBRAPA Satellite Monitoring Campinas, Sa˜o Paulo, Brazil Received 7 September 2001; received in revised form 11 June 2002; accepted 11 June 2002 Abstract Many research projects require accurate delineation of different secondary succession (SS) stages over large regions/subregions of the Amazon basin. However, the complexity of vegetation stand structure, abundant vegetation species, and the smooth transition between different SS stages make vegetation classification difficult when using traditional approaches such as the maximum likelihood classifier (MLC). Most of the time, classification distinguishes only between forest/non-forest. It has been difficult to accurately distinguish stages of SS. In this paper, a linear mixture model (LMM) approach is applied to classify successional and mature forests using Thematic Mapper (TM) imagery in the Rondoˆnia region of the Brazilian Amazon. Three endmembers (i.e., shade, soil, and green vegetation or GV) were identified based on the image itself and a constrained least-squares solution was used to unmix the image. This study indicates that the LMM approach is a promising method for distinguishing successional and mature forests in the Amazon basin using TM data. It improved vegetation classification accuracy over that of the MLC.
    [Show full text]
  • Graph Laplacian Mixture Model Hermina Petric Maretic and Pascal Frossard
    1 Graph Laplacian mixture model Hermina Petric Maretic and Pascal Frossard Abstract—Graph learning methods have recently been receiv- of signals that are an unknown combination of data with ing increasing interest as means to infer structure in datasets. different structures. Namely, we propose a generative model Most of the recent approaches focus on different relationships for data represented as a mixture of signals naturally living on between a graph and data sample distributions, mostly in settings where all available data relate to the same graph. This is, a collection of different graphs. As it is often the case with however, not always the case, as data is often available in mixed data, the separation of these signals into clusters is assumed form, yielding the need for methods that are able to cope with to be unknown. We thus propose an algorithm that will jointly mixture data and learn multiple graphs. We propose a novel cluster the signals and infer multiple graph structures, one for generative model that represents a collection of distinct data each of the clusters. Our method assumes a general signal which naturally live on different graphs. We assume the mapping of data to graphs is not known and investigate the problem of model, and offers a framework for multiple graph inference jointly clustering a set of data and learning a graph for each of that can be directly used with a number of state-of-the- the clusters. Experiments demonstrate promising performance in art network inference algorithms, harnessing their particular data clustering and multiple graph inference, and show desirable benefits.
    [Show full text]
  • Comparing Unsupervised Clustering Algorithms to Locate Uncommon User Behavior in Public Travel Data
    Comparing unsupervised clustering algorithms to locate uncommon user behavior in public travel data A comparison between the K-Means and Gaussian Mixture Model algorithms MAIN FIELD OF STUDY: Computer Science AUTHORS: Adam Håkansson, Anton Andrésen SUPERVISOR: Beril Sirmacek JÖNKÖPING 2020 juli This thesis was conducted at Jönköping school of engineering in Jönköping within [see field of study on the previous page]. The authors themselves are responsible for opinions, conclusions and results. Examiner: Maria Riveiro Supervisor: Beril Sirmacek Scope: 15 hp (bachelor’s degree) Date: 2020-06-01 Mail Address: Visiting address: Phone: Box 1026 Gjuterigatan 5 036-10 10 00 (vx) 551 11 Jönköping Abstract Clustering machine learning algorithms have existed for a long time and there are a multitude of variations of them available to implement. Each of them has its advantages and disadvantages, which makes it challenging to select one for a particular problem and application. This study focuses on comparing two algorithms, the K-Means and Gaussian Mixture Model algorithms for outlier detection within public travel data from the travel planning mobile application MobiTime1[1]. The purpose of this study was to compare the two algorithms against each other, to identify differences between their outlier detection results. The comparisons were mainly done by comparing the differences in number of outliers located for each model, with respect to outlier threshold and number of clusters. The study found that the algorithms have large differences regarding their capabilities of detecting outliers. These differences heavily depend on the type of data that is used, but one major difference that was found was that K-Means was more restrictive then Gaussian Mixture Model when it comes to classifying data points as outliers.
    [Show full text]
  • Semiparametric Estimation in the Normal Variance-Mean Mixture Model∗
    (September 26, 2018) Semiparametric estimation in the normal variance-mean mixture model∗ Denis Belomestny University of Duisburg-Essen Thea-Leymann-Str. 9, 45127 Essen, Germany and Laboratory of Stochastic Analysis and its Applications National Research University Higher School of Economics Shabolovka, 26, 119049 Moscow, Russia e-mail: [email protected] Vladimir Panov Laboratory of Stochastic Analysis and its Applications National Research University Higher School of Economics Shabolovka, 26, 119049 Moscow, Russia e-mail: [email protected] Abstract: In this paper we study the problem of statistical inference on the parameters of the semiparametric variance-mean mixtures. This class of mixtures has recently become rather popular in statistical and financial modelling. We design a semiparametric estimation procedure that first es- timates the mean of the underlying normal distribution and then recovers nonparametrically the density of the corresponding mixing distribution. We illustrate the performance of our procedure on simulated and real data. Keywords and phrases: variance-mean mixture model, semiparametric inference, Mellin transform, generalized hyperbolic distribution. Contents 1 Introduction and set-up . .2 2 Estimation of µ ...............................3 3 Estimation of G with known µ ......................4 4 Estimation of G with unknown µ .....................7 5 Numerical example . .8 6 Real data example . 10 7 Proofs . 11 arXiv:1705.07578v1 [stat.OT] 22 May 2017 7.1 Proof of Theorem 2.1......................... 11 7.2 Proof of Theorem 3.1......................... 13 7.3 Proof of Theorem 4.1......................... 17 References . 20 ∗ This work has been funded by the Russian Academic Excellence Project \5-100". 1 D.Belomestny and V.Panov/Estimation in the variance-mean mixture model 2 1.
    [Show full text]
  • Structure of a Mixture Model
    In statistics, a mixture model is a probabilistic model for representing the presence of sub-populations within an overall population, without requiring that an observed data-set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population-identity information. Some ways of implementing mixture models involve steps that do attribute postulated sub-population-identities to individual observations (or weights towards such sub-populations), in which case these can be regarded as a types of unsupervised learning or clustering procedures. However not all inference procedures involve such steps. Mixture models should not be confused with model for compositional data, i.e., data whose components are constrained to sum to a constant value (1, 100%, etc.). Structure of a mixture model General mixture model A typical finite-dimensional mixture model is a hierarchical model consisting of the following components: • N random variables corresponding to observations, each assumed to be distributed according to a mixture of K components, with each component belonging to the same parametric family of distributions but with different parameters • N corresponding random latent variables specifying the identity of the mixture component of each observation, each distributed according to a K-dimensional categorical distribution • A set of K mixture weights, each of which is a probability (a real number between 0 and 1), all of which sum to 1 • A set of K parameters, each specifying the parameter of the corresponding mixture component.
    [Show full text]
  • Student's T Distribution Based Estimation of Distribution
    1 Student’s t Distribution based Estimation of Distribution Algorithms for Derivative-free Global Optimization ∗Bin Liu Member, IEEE, Shi Cheng Member, IEEE, Yuhui Shi Fellow, IEEE Abstract In this paper, we are concerned with a branch of evolutionary algorithms termed estimation of distribution (EDA), which has been successfully used to tackle derivative-free global optimization problems. For existent EDA algorithms, it is a common practice to use a Gaussian distribution or a mixture of Gaussian components to represent the statistical property of available promising solutions found so far. Observing that the Student’s t distribution has heavier and longer tails than the Gaussian, which may be beneficial for exploring the solution space, we propose a novel EDA algorithm termed ESTDA, in which the Student’s t distribution, rather than Gaussian, is employed. To address hard multimodal and deceptive problems, we extend ESTDA further by substituting a single Student’s t distribution with a mixture of Student’s t distributions. The resulting algorithm is named as estimation of mixture of Student’s t distribution algorithm (EMSTDA). Both ESTDA and EMSTDA are evaluated through extensive and in-depth numerical experiments using over a dozen of benchmark objective functions. Empirical results demonstrate that the proposed algorithms provide remarkably better performance than their Gaussian counterparts. Index Terms derivative-free global optimization, EDA, estimation of distribution, Expectation-Maximization, mixture model, Student’s t distribution I. INTRODUCTION The estimation of distribution algorithm (EDA) is an evolutionary computation (EC) paradigm for tackling derivative-free global optimization problems [1]. EDAs have gained great success and attracted increasing attentions during the last decade arXiv:1608.03757v2 [cs.NE] 25 Nov 2016 [2].
    [Show full text]
  • The Infinite Gaussian Mixture Model
    in Advances in Neural Information Processing Systems 12 S.A. Solla, T.K. Leen and K.-R. Muller¨ (eds.), pp. 554–560, MIT Press (2000) The Infinite Gaussian Mixture Model Carl Edward Rasmussen Department of Mathematical Modelling Technical University of Denmark Building 321, DK-2800 Kongens Lyngby, Denmark [email protected] http://bayes.imm.dtu.dk Abstract In a Bayesian mixture model it is not necessary a priori to limit the num- ber of components to be finite. In this paper an infinite Gaussian mixture model is presented which neatly sidesteps the difficult problem of find- ing the “right” number of mixture components. Inference in the model is done using an efficient parameter-free Markov Chain that relies entirely on Gibbs sampling. 1 Introduction One of the major advantages in the Bayesian methodology is that “overfitting” is avoided; thus the difficult task of adjusting model complexity vanishes. For neural networks, this was demonstrated by Neal [1996] whose work on infinite networks led to the reinvention and popularisation of Gaussian Process models [Williams & Rasmussen, 1996]. In this paper a Markov Chain Monte Carlo (MCMC) implementation of a hierarchical infinite Gaussian mixture model is presented. Perhaps surprisingly, inference in such models is possible using finite amounts of computation. Similar models are known in statistics as Dirichlet Process mixture models and go back to Ferguson [1973] and Antoniak [1974]. Usually, expositions start from the Dirichlet process itself [West et al, 1994]; here we derive the model as the limiting case of the well- known finite mixtures. Bayesian methods for mixtures with an unknown (finite) number of components have been explored by Richardson & Green [1997], whose methods are not easily extended to multivariate observations.
    [Show full text]
  • Outlier Detection and Robust Mixture Modeling Using Nonconvex Penalized Likelihood
    Journal of Statistical Planning and Inference 164 (2015) 27–38 Contents lists available at ScienceDirect Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi Outlier detection and robust mixture modeling using nonconvex penalized likelihood Chun Yu a, Kun Chen b, Weixin Yao c,∗ a School of Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China b Department of Statistics, University of Connecticut, Storrs, CT 06269, United States c Department of Statistics, University of California, Riverside, CA 92521, United States article info a b s t r a c t Article history: Finite mixture models are widely used in a variety of statistical applications. However, Received 29 April 2014 the classical normal mixture model with maximum likelihood estimation is prone to the Received in revised form 9 March 2015 presence of only a few severe outliers. We propose a robust mixture modeling approach Accepted 10 March 2015 using a mean-shift formulation coupled with nonconvex sparsity-inducing penalization, Available online 19 March 2015 to conduct simultaneous outlier detection and robust parameter estimation. An efficient iterative thresholding-embedded EM algorithm is developed to maximize the penalized Keywords: log-likelihood. The efficacy of our proposed approach is demonstrated via simulation EM algorithm Mixture models studies and a real application on Acidity data analysis. Outlier detection ' 2015 Elsevier B.V. All rights reserved. Penalized likelihood 1. Introduction Nowadays finite mixture distributions are increasingly important in modeling a variety of random phenomena (see Everitt and Hand, 1981; Titterington et al., 1985; McLachlan and Basford, 1988; Lindsay, 1995; Böhning, 1999). The m-component finite normal mixture distribution has probability density m I D X I 2 f .y θ/ πiφ.y µi; σi /; (1.1) iD1 T 2 where θ D .π1; µ1; σ1I ::: I πm; µm; σm/ collects all the unknown parameters, φ.·; µ, σ / denotes the density function 2 Pm D of N(µ, σ /, and πj is the proportion of jth subpopulation with jD1 πj 1.
    [Show full text]