DOCSLIB.ORG

Testing for Outliers from a Mixture Distribution When Some Data Are Missing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Testing for Outliers from a Mixture Distribution When Some Data Are Missing TESTING FOR OUTLIERS FROM A MIXTURE DISTRIBUTION WHEN SOME DATA ARE MISSING WayneA. Woodward* and StephanR Sain** * SouthernMethodist University ** University of Coloradoat Denver ABSTRACT We considerthe problemof multivariateoutlier testingfrom a populationfrom which a training sampleis available.We assumethat a new observationis obtained,and we test whetherthe new observationis from the populationof the training sample. Problemsof this sort arisein a numberof applicationsincluding nuclear monitoring, biometrics (including fingerprint and handwriting identification), and medical diagnosis. In many casesit is reasonableto model the population of the training sample using a mixture- of-normalsmodel (e.g. whenthe observationscome ftom a variety of sourcesor the data are substantially non-normal). In this paper we consider a modified likelihood ratio test / that is applicableto the casein which: (a) the training datafollow a mixture-of-normals distnoution, (b) all labelsin the training sampleare missing, (c) someof the observation vectorsin the training samplehave missing information, and (d) the numberof componentsin the mixture is unknown. The approachoften usedin practiceto handlethe fact that someof the data vectorshave missing observations is to perform the test basedonly on the datavectors with full data. Whenlarge amounts of data are missing,use of this strategymay leadto lossof valuableinformatio~ especiallyin the caseof smalltrajning sampleswhich, for example,is often the casein the nuclearmonitoring settingmentioned previously. An alternativeprocedure is to incorporateall n of the datavectors using the EM algorithmto handlethe missingdata. We use simulationsand examplesto comparethe useof the EM algorithmon the entire data set with the useof only the completedata vectors. Key Words: EM Algorithm, Mixture Model, Missing Data, Outlier Detection 1 1. Introduction We considerthe problemof testinga new data valueto determinewhether it shouldbe consideredan outlier from a distributionfor which we havea training sample, i.e."outlier testing." Fis~ Gray, and McCartor (1996) and Taylor andHartse (1997) haveused a likehl1oodratio test for detectingoutliers ftom a multivariatenormal (MVN) distn"butionfit to the training data when no datawere missing. Theseauthors applied the test to the problemof detectingseismic signals of undergroundnuclear explosions when a training sampleof non-nuclearseismic events is available. Our focus in this paperwill be the casein which the training dataare modeledas a mi:xtureof normals. A mixture model is an obviouschoice for a wide variety of settings. For example,in the seismicsetting discussed above, the populationof non-nuclear observationsin a particularregion may consistof observationsfrom a variety of sources suchas earthquakesand mming explosions,and ~ring typesof earthquakes. Additionally, in the areaof~cal diagnosis,benign tumors may be ofseveral types, etc. The flexibility of the mixture-of-normalsmodel alsomakes it usefulfor modelingnon- nom1aJityeven ifdisting uishablecomponents are not present.The training datawill be considereda sampleof size n ftom a mixture distn"butionwhose density is givenby m j(z) = E~ji(Zj Pi, Ei) i) i=1 wherem is the numberof componentsin the mixture,fi(Z; J.J.i,Ei) is the MVN density with meanvector /.I, and covariancematri:x E, associatedwith the ith component,the Ai' 1, . , m are the mixing proportions, and :z;is a d..djmen..~Qna1vector of variables. Letting the training samplebe denotedby Xl, ..., Xn and the new observation (whosedistribution is unknown) by Xu, then we wish to test the hypotheses 2 Ho : Xu E n Hl: x.~n wheren denotesthe populationof the training data. We considerthe casein which datamay be missingin the training data. In the case of a mixture model,there are at leastthree different waysin which "data" maybe missing: (a) missing labels (b) unknownnumber of components ( c) missingdata in the datavectors A "tabel" is saidto be known for a given observationif it is known to which componentin the mixture that observationbelongs. Wang,W~ard, Gray, Wiechecki,and Sain 1997) developed a modified likelihood ratio test for the case in which some but not all of the labels may be missmg. The authorsassumed that the numberof componen~ m, is known and that thereis no missingdata in the datavectors. The likehl1oodfimction under Ho (i.e. under the assumptionthat Xu E Ho) is denotedby Lo( 9) where9 is an unknown vector-valuedparameter associated with the distrIbutionof X underBo. Likewise, let '"'"' n L 1(8) = I1f(Xs;8) denote the likelihood based only on the training sample Xl, ...,Xn. 8=1 Wang, et a1.(1997) and Sain, Gray, Woodward, and Fisk (1999) used the modified likel1l1ood-ratio test statistic (2) 3. The usuallikelihood ratio involvesa secondfactor in the denominator,h(:cu), where h(z) is the densityfunction of the outlier populationand Xu denotesa singleobservation availablefrom that popuJation.However, ~~1Jg h(x) is very difficult with only one observationavailable and whena priori informationis not availableconcerning the outlier distnoution. Thus, it makes senseto estimate h(x) nonparametrically. Moreover, given any of the potentialnonparametric density estimators of h(z) in the caseof only one data point, the factor h(xu) will not vary with (Jin the maximizationprocess nor will h(:cu) vary as Xu variesfrom sampleto sample. Consider,for example,a histogramestimator of h (z ). With a singledata value,such an estimatorwould be a constantregardless of the value 0 f :1:. Thus, for simplicity we useW in (2). It is easilyseen m (2) that if Xu doesnot belongto ll, then W will tend to be small. Eencethe rejectionregion is of the form W ~ WQ for someW Q pickedto provide a level a test. Sincethe null distn"butionofW hasno known closedform, Wang,et at. (1997) useda bootstrapprocedure (see Efron ~ibshirani, 1993)to derivethe critical valueWa. Wheneversome of the training data are un1abel~ the parameters~,JJ.i., and Ei of the mixture modelare estimatedyja the Expectation-M~~tiQn (EM) algorithm (see Dempster, Laird, and Rubin, 1977, Mclachlan and Krishnan, 1997, and Redner and Walker, 1984). Based on simulations, Wang, et al (1997) showed that in this setting, the modified likelihood ratio test can be used successfully for outlier detection. Sain, et at (1999) extend the results of Wang, et at (1997) to the case in which no data are labeled and in which the number of components in the mixture is unknown.; They demonstrated their resuhs using simulations similar to those of Wang, et a1.(1997) and showed little or no loss of power when no training data are labeled. S~ et aI. (1999) obtained excellent results using their procedme on actual seismic data from the VogtJand region near the Czech-German border and from the WMQ station m western China. Using the China data, the authors de~nstrated that a mixture model may be preferable to the use of a single multivariate normal model due to apparent non-normality of the data 4 even when there are not any identifiable groups of observation types represented in the training data. In this paper we consider the case in which some of the variables may be missing for some observations in the training sample. For example, in the seismic setting log(Pg/Lg) ratios at higher frequency bands are often missing becauseof attenuation effects on high frequencies. We consider the case in which d variables are observed on the new observation,and we denotethis observationby Xu = (Xu!, Xu2,..., Xud)', where Xu; denotesvariable j observedon the new observation. We further ass~ that there existsa training sample Xl = (XII, X12,X13, ..., Xld)' X2 = (X21' X22, X23: ~1' "« Xn = (Xnl' Xn2, Xn8, ftom ll. When SOIMof the trainingdata includes missing data, the training samplehas the generalappearance Xl = (XUt , X13' ... Xl;, , X l.J+20 . XIII)' X2 = ( - , X22, X23, ., X2.j, X2.Jt-l. '.. Xlii)' X,,= :Xnl, -, ~, ... Xnj, Xn.j+l. XnJ+2' ... Xnd)' where" " denotesthat the particularvariable is missingfor that observation.Thus, to apply standardor recentlydeveloped outlier detectionmethodology, one must reducethe datato a subsetof the original trajningdata that includesonly those1 (where < n) data 5 vectorsfor which all of the variableswere observed. It is clearthat sucha procedurecan result in a lossof information and shouldlead to a reductionin detectionpower. To denx>nstratethe extent oftbis problem, Woodward,Sain, Gray,Zbao, and Fisk (2002) showthat, dependingon the missingdata probabilities and the numberof variables,the number of complete vectors available for analysis may be dramatically smaller than the numberof original cases. For example,in the caseof d 4 variablesand a 25% chance that an observationwill be missing,we expectfewer than one-thirdof the data vectorsto be retamedfor analysisusjng the strategyof analyzingonly the completedata vectors. This is in spite of the fuct that about75% of the origjnal data set shouldbe availablefor use. Another problem arisesif there are no casesor only a very few casesin which an d of the variablesare observed. If the strategyofusing only the completevectors is used, then someof the variablesmay need to be deleted. This may alsoresult m loss of some jmportantinformation. It shouldaJso be pointed'Outthat we are not consideringthe case in which there is missingdata in the outlier. In an applicationof the techniquesdeveloped here,the variablesobserved m the outlier determinethe variablesto be usedin the outlier test. The pm-poseof this paperis to examinethe extent to which detectionpower can be improvedby
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]