### Citations

11929 | Maximum likelihood from incomplete data via the EM algorithm
- Dempster, Laird, et al.
- 1977
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Citation Context ...actice. The ingredients for constructing copula-based mixture models are described in Section 2. Section 3 provides the details for maximum likelihood estimation through Expectation-Maximization (EM; =-=Dempster et al., 1977-=-) and proposes relevant procedures for getting starting values from the combination of a partitioning algorithm (like k-medoids) and of component-wise applications of the Inference Functions from Marg... |

1739 | Finite Mixture Models - McLachlan, Peel - 2000 |

983 |
Comparing partitions
- Hubert, Arabie
- 1985
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Citation Context ...s apparent none of the three models performs well in detecting the true shape of the underlying clusters with the misclassification rates ranging between 22.12% to 30.5% and adjusted Rand index (ARI; =-=Hubert and Arabie, 1985-=-) between 0.45 and 0.51. The challenge with the artificial data set in Figure 1 is the tail behaviour that the true groups demonstrate. If we restrict the number of components to four, the demonstrate... |

786 |
Multivariate Models and Dependence Concepts
- Joe
- 1997
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Citation Context ...d which takes into account both the copula and the marginal specification of each component in the mixture model. The procedure is an application of the Inference Functions from Margins (IFM) method (=-=Joe, 1997-=-, Chapter 10) for each component, and relies on an initial partitioning of the observation indices A = {1, . . . , n} into exclusive subsets S1, . . . , Sk, with ∪kj=1Sj = A, of cardinality N1, . . . ... |

469 | Model-Based Gaussian and NonGaussian Clustering - Banfield, Raftery - 1993 |

408 |
Univariate Discrete Distributions
- Johnson, Kemp, et al.
- 2005
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Citation Context ...ultivariate binomial distributions which allow for correlated marginals is not straightforward outside the copula framework (for a discussion on bivariate and multivariate binomial distributions, see =-=Johnson et al., 1997-=-). The one-parameter Frank copula is defined as C(F )(u1, u2, u3;ψ) = − 1 ψ log { 1 + (exp−ψu1 −1)(exp−ψu2 −1)(exp−ψu3 −1) (exp−ψ−1)2 } , (19) where ψ is an association parameter which is common for a... |

322 | Maximum likelihood estimation via the ECM algorithm: A general framework - Meng, Rubin - 1993 |

165 |
Gaussian parsimonious clustering models
- Celeux, Govaert
- 1995
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Citation Context ...I of the clustering with 6 Beta and 1 Gamma marginal, and the ARI of the clustering from the Gaussian mixture model with the lowest BIC (4 components with VEV parameterization with BIC −5026.44; see, =-=Celeux et al. 1995-=- and Fraley et al. 2012 for the parameterizations that mclust uses). Figure 4 shows the results for each metric. Despite the low ARI’s for both fits, all points fall below the 45o line and hence the c... |

61 |
The meta-elliptical distributions with given marginals,”
- Fang, Fang, et al.
- 2002
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Citation Context ...is the 45o line from the origin. 14 contrast closure under general affine transformations is satisfied for all mixture models that are based on elliptical distributions such as Normal and t mixtures (=-=Fang et al., 2002-=-). 4.3.2 Rotated copulas In two dimensions, the survival version of any copula C(u1, u2) is C180(u1, u2) = u1 +u2−1+ C(u1, u2), where “180” denotes that the survival copula is a rotated version of C(u... |

55 |
An introduction to copulas. Springer Series in Statistics.
- Nelsen
- 2006
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Citation Context ...ls 2.1 Mixture models through copulas A copula C(u1, . . . , up) is a distribution function with uniform marginals. The importance of copulas in statistical modelling stems from Sklar’s theorem (see, =-=Nelsen, 2006-=-, §2.3), which shows that every multivariate distribution can be represented via the choice of an appropriate copula and, more importantly, it provides a general mechanism to construct new multivariat... |

54 |
mvtnorm: Multivariate normal and t distribution. R package version
- Genz, Bretz, et al.
- 2011
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Citation Context ...an be calculated through accurate deterministic approximation methods like those of Joe (1995). Such methods are implemented in the mprobit R package by Joe, Choy and Zhang and the mvtnorm R package (=-=Genz et al., 2013-=-). The following example concerns the use of copulas to construct mixtures of trivariate Binomial distributions that allow for dependence. Example 6.1: This example relates to cognitive diagnosis mode... |

47 | Parsimonious Gaussian mixture models’, - McNicholas, Murphy - 2008 |

41 | A primer on copulas for count data. - Genest, Neslehova - 2007 |

34 | mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation.
- Fraley
- 2012
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Citation Context ...rrelation structure. A (?) denotes the best BIC for each copula specification and a (??) the best BIC overall. of the copula-based mixture model to that of a Gaussian mixture fitted using the mclust (=-=Fraley et al., 2012-=-) R package as follows: each metric is broken into I intervals, whose endpoints are calculated using the empirical quantiles at I + 1 equidistant probabilities ranging from 0 to 1. For each metric and... |

28 | Methods for merging Gaussian mixture components. - Hennig - 2010 |

27 | Combining mixture components for clustering. - Baudry, Raftery, et al. - 2010 |

21 | Bayesian Inference for Finite Mixtures of Univariate and - Frühwirth-Schnatter, Pyne - 2010 |

20 |
On model-based clustering, classification, and discriminant analysis
- McNicholas
- 2011
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Citation Context ...tions their joint treatment has been largely overlooked, mainly because of shortage in appropriate and easy to handle models. Typically, models based on latent variables are considered for such data (=-=Browne and McNicholas, 2012-=-) which may have limitations for practical purposes because of assumptions like conditional independence. Current work focuses on extending the methods presented here to mixed-mode data. Furthermore, ... |

20 | Approximations to multivariate normal rectangle probabilities based on conditional expectations - Joe - 1995 |

18 | 2002. Vines – a new graphical model for dependent random variables - Bedford, Cooke |

15 |
Analytic calculations for the EM algorithm for multivariate skew-t mixture models
- Vrbik, McNicholas
- 2012
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Citation Context ...stributions (see, Andrews and McNicholas, 2011), multivariate skew-Normal and skew-t distribution (see, for example, Cabral et al., 2012; Frühwirth-Schnatter and Pyne, 2010; Lee and McLachlan, 2014; =-=Vrbik and McNicholas, 2012-=-), multivatiate skew student-t-Normal distributions (Lin et al., 2013), multivariate Normal inverse Gaussian distributions (Karlis and Santourian, 2009). Other attempts can be found in Forbes and Wrai... |

14 |
Model-based clustering with non-elliptically contoured distributions
- Karlis, Santourian
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Citation Context ...chnatter and Pyne, 2010; Lee and McLachlan, 2014; Vrbik and McNicholas, 2012), multivatiate skew student-t-Normal distributions (Lin et al., 2013), multivariate Normal inverse Gaussian distributions (=-=Karlis and Santourian, 2009-=-). Other attempts can be found in Forbes and Wraith (2013) for finite mixtures of multivariate scaled Normal distributions and (Morris and McNicholas, 2013) for mixtures of shifted asymmetric Laplace ... |

12 | Likelihood inference for archimedean copulas in high dimensions under known margins - Hofert, Mächler, et al. - 2012 |

10 |
Finite mixtures of multivariate skew t-distributions: some recent and new results
- Lee, McLachlan
- 2014
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Citation Context ...include multivariate t distributions (see, Andrews and McNicholas, 2011), multivariate skew-Normal and skew-t distribution (see, for example, Cabral et al., 2012; Frühwirth-Schnatter and Pyne, 2010; =-=Lee and McLachlan, 2014-=-; Vrbik and McNicholas, 2012), multivatiate skew student-t-Normal distributions (Lin et al., 2013), multivariate Normal inverse Gaussian distributions (Karlis and Santourian, 2009). Other attempts can... |

10 |
Vine copulas with asymmetric tail dependence
- Nikoloulopoulos, Joe, et al.
- 2012
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Citation Context ... can be prohibitive. A class of copulas that does not satisfy the property of closure under marginalisation is the class of vine copulas (see, for example, Bedford and Cooke 2002 for introduction and =-=Nikoloulopoulos et al. 2012-=- for applications). Example 5.1: In Example 4.2, the mixture components were defined using the Gaussian copula. Hence, the bivariate marginal density of Xt and Xs (s, t = 1, . . . , 7; s 6= t) corresp... |

9 |
Multivariate mixture modelling using skew-normal independent distributions
- Cabral, Lachos, et al.
- 2012
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Citation Context ...ed. Prominent examples of alternative component densities include multivariate t distributions (see, Andrews and McNicholas, 2011), multivariate skew-Normal and skew-t distribution (see, for example, =-=Cabral et al., 2012-=-; Frühwirth-Schnatter and Pyne, 2010; Lee and McLachlan, 2014; Vrbik and McNicholas, 2012), multivatiate skew student-t-Normal distributions (Lin et al., 2013), multivariate Normal inverse Gaussian d... |

7 |
Mixtures of modified t-factor analyzers for model-based clustering, classification, and discriminant analysis
- Andrews, McNicholas
- 2011
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Citation Context ...s have resulted in an expanding literature where other special component distributions are considered. Prominent examples of alternative component densities include multivariate t distributions (see, =-=Andrews and McNicholas, 2011-=-), multivariate skew-Normal and skew-t distribution (see, for example, Cabral et al., 2012; Frühwirth-Schnatter and Pyne, 2010; Lee and McLachlan, 2014; Vrbik and McNicholas, 2012), multivatiate skew... |

7 | A new family of multivariate heavy-tailed distributions with variable marginal amounts of tailweight: Application to robust clustering - Forbes, Wraith - 2013 |

7 | Rotations for n-dimensional graphics - Hanson - 1995 |

7 |
Dimension reduction for model-based clustering via mixtures of shifted asymmetric Laplace distributions. Statistics and Probability Letters
- Morris, McNicholas
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Citation Context ...riate Normal inverse Gaussian distributions (Karlis and Santourian, 2009). Other attempts can be found in Forbes and Wraith (2013) for finite mixtures of multivariate scaled Normal distributions and (=-=Morris and McNicholas, 2013-=-) for mixtures of shifted asymmetric Laplace distributions. The results of such studies indicate that the introduction of heavy tails and/or skewness allows the construction of more parsimonious model... |

6 | Pair Copula Constructions for Multivariate Discrete Data,” - Panagiotelis, Czado, et al. - 2012 |

4 | Densities of nested Archimedean copulas
- Hofert, Pham
- 2013
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Citation Context ...thermore, closure under marginalization holds for every Archimedean and nested Archimedean copula, because its generator function necessarily takes value 0 at 1 (see, for example Hofert et al., 2012; =-=Hofert and Pham, 2013-=-, for definitions and results for multivariate Archimedean and nested Archimedean copulas). The results here extend to the case of discrete data by replacing the integration of density functions with ... |

3 |
Copula functions in model based clustering
- Jajuga, Papla
- 2006
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Citation Context ...ed by the mixture components. A few application-specific attempts have already been made in the direction of facilitating the flexibility that copulas offer in model-based clustering (see,for example =-=Jajuga and Papla, 2006-=-; Lascio and Giannerini, 2012; Vrac et al., 2012). The current paper sets a thorough framework for constructing mixture models using copulas, highlighting the benefits but also the challenges of their... |

3 |
Using multinomial mixture models to cluster internet traffic
- Jorgensen
- 2004
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Citation Context ...ucture. Some successful, but limited in application examples, are finite mixtures of multivariate Poisson distributions (Karlis and Meligkotsidou, 2007), finite mixtures of multinomial distributions (=-=Jorgensen, 2004-=-) and models based on conditionally independent Poisson distributions (see, for example Alfo et al., 2011). Mixture models with latent structures have been considered in Browne and McNicholas (2012), ... |

3 |
A copula-based algorithm for discovering patterns of dependent observations
- Lascio, Giannerini
- 2012
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Citation Context ...ents. A few application-specific attempts have already been made in the direction of facilitating the flexibility that copulas offer in model-based clustering (see,for example Jajuga and Papla, 2006; =-=Lascio and Giannerini, 2012-=-; Vrac et al., 2012). The current paper sets a thorough framework for constructing mixture models using copulas, highlighting the benefits but also the challenges of their use in practice. The ingredi... |

2 |
A mixture model for multivariate counts under endogenous selectivity
- Alfo, Maruotti, et al.
- 2011
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Citation Context ...on distributions (Karlis and Meligkotsidou, 2007), finite mixtures of multinomial distributions (Jorgensen, 2004) and models based on conditionally independent Poisson distributions (see, for example =-=Alfo et al., 2011-=-). Mixture models with latent structures have been considered in Browne and McNicholas (2012), but these can have limitations because of assumptions like conditional independence. 1.3 Setting and fitt... |

2 |
Finite multivariate Poisson mixtures with applications
- Karlis, Meligkotsidou
- 2007
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Citation Context ...easy to work with models that allow practical flexibility on the dependence structure. Some successful, but limited in application examples, are finite mixtures of multivariate Poisson distributions (=-=Karlis and Meligkotsidou, 2007-=-), finite mixtures of multinomial distributions (Jorgensen, 2004) and models based on conditionally independent Poisson distributions (see, for example Alfo et al., 2011). Mixture models with latent s... |

2 |
Flexible mixture modelling using the multivariate skew-t-normal distribution
- Lin, Ho, et al.
- 2013
(Show Context)
Citation Context ...kew-t distribution (see, for example, Cabral et al., 2012; Frühwirth-Schnatter and Pyne, 2010; Lee and McLachlan, 2014; Vrbik and McNicholas, 2012), multivatiate skew student-t-Normal distributions (=-=Lin et al., 2013-=-), multivariate Normal inverse Gaussian distributions (Karlis and Santourian, 2009). Other attempts can be found in Forbes and Wraith (2013) for finite mixtures of multivariate scaled Normal distribut... |

1 | 2013, 2). Modeling dependence with c- and d-vine copulas: The r package cdvine - Brechmann, Schepsmeier |

1 | Comparing different clustering models on the unit hypercube - Dean, Nugent - 2011 |

1 |
Clustering student skill set profiles in a unit hypercube using mixtures of multivariate betas
- Dean, Nugent
- 2013
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Citation Context ...s is because the three attributes share items, and this association needs to be taken into account when clustering the students. Such data have also been analysed in the past via mixture models (see, =-=Dean and Nugent, 2013-=-) but only after transforming the scores into percentages and treating those as realizations of continuous random variables. Such transformations are not necessary when using copulabased mixture model... |

1 |
CDM: Cognitive diagnosis modeling. R package version
- Robitzsch, Kiefer, et al.
- 2014
(Show Context)
Citation Context ... Hence, attribute scores for each student can be obtained by counting the number of successful items out of the total items that belong to each attribute. The data are available in the CDM R package (=-=Robitzsch et al., 2014-=-) and its documentation describes which items belong to which attribute. The aim of this example is to use some of the attribute scores of the students for the construction of performance clusters of ... |

1 | Copula analysis of mixture models - Vrac, Billard, et al. - 2012 |