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26
Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
, 2010
"... A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAPEM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is describe ..."
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Cited by 55 (8 self)
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A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAPEM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAPEM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost. 1 I.
mclust Version 4 for R: Normal Mixture Modeling for ModelBased Clustering, Classification, and Density Estimation
, 2012
"... mclust is a contributed R package for modelbased clustering, classification, and density estimation based on finite normal mixture modeling. It provides functions for parameter estimation via the EM algorithm for normal mixture models with a variety of covariance structures, and functions for simul ..."
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Cited by 32 (1 self)
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mclust is a contributed R package for modelbased clustering, classification, and density estimation based on finite normal mixture modeling. It provides functions for parameter estimation via the EM algorithm for normal mixture models with a variety of covariance structures, and functions for simulation from these models. Also included are functions that combine modelbased hierarchical clustering, EM for mixture estimation and the Bayesian Information Criterion (BIC) in comprehensive strategies for clustering, density estimation and discriminant analysis. There is additional functionality for displaying and visualizing the models along with clustering, classification, and density estimation results. Several features of the software have been changed in this version, in particular the functionality for discriminant analysis and density estimation has
Mixtures of shifted asymmetric Laplace distributions
, 2012
"... A mixture of shifted asymmetric Laplace distributions is introduced and used for clustering and classification. A variant of the EM algorithm is developed for parameter estimation by exploiting the relationship with the general inverse Gaussian distribution. This approach is mathematically elegant a ..."
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Cited by 15 (13 self)
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A mixture of shifted asymmetric Laplace distributions is introduced and used for clustering and classification. A variant of the EM algorithm is developed for parameter estimation by exploiting the relationship with the general inverse Gaussian distribution. This approach is mathematically elegant and relatively computationally straightforward. Our novel mixture modelling approach is demonstrated on both simulated and real data to illustrate clustering and classification applications. In these analyses, our mixture of shifted asymmetric Laplace distributions performs favourably when compared to the popular Gaussian approach. This work, which marks an important step in the nonGaussian modelbased clustering and classification direction, concludes with discussion as well as suggestions for future work.
Parsimonious Skew Mixture Models for ModelBased Clustering and Classification
"... In recent work, robust mixture modelling approaches using skewed distributions have been explored to accommodate asymmetric data. We introduce parsimony by developing skewt and skewnormal analogues of the popular GPCM family that employ an eigenvalue decomposition of a positivesemidefinite matri ..."
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Cited by 8 (7 self)
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In recent work, robust mixture modelling approaches using skewed distributions have been explored to accommodate asymmetric data. We introduce parsimony by developing skewt and skewnormal analogues of the popular GPCM family that employ an eigenvalue decomposition of a positivesemidefinite matrix. The methods developed in this paper are compared to existing models in both an unsupervised and semisupervised classification framework. Parameter estimation is carried out using the expectationmaximization algorithm and models are selected using the Bayesian information criterion. The efficacy of these extensions is illustrated on simulated and benchmark clustering data sets. 1
Finite state space non parametric Hidden Markov Models are in general identifiable. arXiv :1306.4657v1
, 2013
"... In this paper, we prove that finite state space non parametric hidden Markov models are identifiable as soon as the transition matrix of the latent Markov chain has full rank and the emission probability distributions are linearly independent. We then propose several non parametric likelihood based ..."
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Cited by 8 (2 self)
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In this paper, we prove that finite state space non parametric hidden Markov models are identifiable as soon as the transition matrix of the latent Markov chain has full rank and the emission probability distributions are linearly independent. We then propose several non parametric likelihood based estimation methods, which we apply to models used in applications. We finally show on examples that the use of non parametric modeling and estimation may improve the classification performances. 1
Clusters and water flows: a novel approach to modal clustering through Morse theory
, 2014
"... The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the investigation of some theoretical aspects of clustering has been re ..."
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Cited by 5 (2 self)
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The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, unlike the situation with other statistical problems as regression or classification, for some of the cluster methodologies it is quite difficult to specify a population goal to which the databased clustering algorithms should try to get close. This paper aims to provide some insight into the theoretical foundations of the usual nonparametric approach to clustering, which understands clusters as regions of high density, by presenting an explicit formulation for the ideal population clustering.
Parsimonious shifted asymmetric Laplace mixtures
, 2013
"... A family of parsimonious shifted asymmetric Laplace mixture models is introduced. We extend the mixture of factor analyzers model to the shifted asymmetric Laplace distribution. Imposing constraints on the constitute parts of the resulting decomposed component scale matrices leads to a family of par ..."
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Cited by 3 (3 self)
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A family of parsimonious shifted asymmetric Laplace mixture models is introduced. We extend the mixture of factor analyzers model to the shifted asymmetric Laplace distribution. Imposing constraints on the constitute parts of the resulting decomposed component scale matrices leads to a family of parsimonious models. An explicit twostage parameter estimation procedure is described, and the Bayesian information criterion and the integrated completed likelihood are compared for model selection. This novel family of models is applied to real data, where it is compared to its Gaussian analogue within clustering and classification paradigms.
Integrative analysis of histone ChIPseq and transcription data using Bayesian mixture models
 Bioinformatics
, 2014
"... Motivation: Histone modifications are a key epigenetic mechanism to activate or repress the transcription of genes. Datasets of matched transcription data and histone modification data obtained by ChIPseq exist, but methods for integrative analysis of both data types are still rare. Here, we presen ..."
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Cited by 1 (0 self)
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Motivation: Histone modifications are a key epigenetic mechanism to activate or repress the transcription of genes. Datasets of matched transcription data and histone modification data obtained by ChIPseq exist, but methods for integrative analysis of both data types are still rare. Here, we present a novel bioinformatics approach to detect genes that show different transcript abundances between two conditions putatively caused by alterations in histone modification. Results:We introduce a correlation measure for integrative analysis of ChIPseq and gene transcription data measured by RNA sequencing or microarrays and demonstrate that a proper normalization of ChIPseq data is crucial. We suggest applying Bayesian mixture models of different types of distributions to further study the distribution of the correlation measure. The implicit classification of the mixture models is used to detect genes with differences between two conditions in both gene transcription and histone modification. The method is applied to different datasets, and its superiority to a naive separate analysis of both data types is demonstrated. Availability and implementation: R/Bioconductor package epigenomix. Contact:
A Separability Index for Distancebased Clustering and Classification Algorithms
"... We propose a separability index that quantifies the degree of difficulty in a hard clustering problem under assumptions of a multivariate Gaussian distribution for each cluster. A preliminary index is first defined and several of its properties are explored both theoretically and numerically. Adjust ..."
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We propose a separability index that quantifies the degree of difficulty in a hard clustering problem under assumptions of a multivariate Gaussian distribution for each cluster. A preliminary index is first defined and several of its properties are explored both theoretically and numerically. Adjustments are then made to this index so that the final refinement is also interpretable in terms of the Adjusted Rand Index between a true grouping and its hypothetical idealized clustering, taken as a surrogate of clustering complexity. Our derived index is used to develop a datasimulation algorithm that generates samples according to the prescribed value of the index. This algorithm is particularly useful for systematically generating datasets with varying degrees of clustering difficulty which can be used to evaluate performance of different clustering algorithms. The index is also shown to be useful in providing a summary of the distinctiveness of classes in grouped datasets.