Finite mixtures of unimodal beta and gamma densities and the $$k$$ -bumps algorithm

Bagnato, Luca; Punzo, Antonio

doi:10.1007/s00180-012-0367-4

Cited by 68 publications

(44 citation statements)

References 46 publications

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“…The choice of the starting values for EM‐based algorithms constitutes an important issue (see, e.g. Biernacki et al., ; Karlis and Xekalaki, ; Bagnato and Punzo, ). For the ECM algorithm described before, two natural strategies are: …”

Section: Further Aspectsmentioning

confidence: 99%

Parsimonious mixtures of multivariate contaminated normal distributions

Punzo

McNicholas

2016

Biometrical J

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A mixture of multivariate contaminated normal distributions is developed for model-based clustering. In addition to the parameters of the classical normal mixture, our contaminated mixture has, for each cluster, a parameter controlling the proportion of mild outliers and one specifying the degree of contamination. Crucially, these parameters do not have to be specified a priori, adding a flexibility to our approach. Parsimony is introduced via eigen-decomposition of the component covariance matrices, and sufficient conditions for the identifiability of all the members of the resulting family are provided. An expectation-conditional maximization algorithm is outlined for parameter estimation and various implementation issues are discussed. Using a large-scale simulation study, the behavior of the proposed approach is investigated and comparison with well-established finite mixtures is provided. The performance of this novel family of models is also illustrated on artificial and real data.

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Section: Further Aspectsmentioning

confidence: 99%

Parsimonious mixtures of multivariate contaminated normal distributions

Punzo

McNicholas

2016

Biometrical J

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“…corresponds to the univariate discrete beta probability mass function defined in Punzo and Zini (2012) and parameterized, as in Punzo (2010) and Bagnato and Punzo (2013), according to the mode m Z ∈ Z and according to another parameter h Z > 0 that is closely related to the distribution variability. In particular, for h Z → 0 + , K h Z (z; m Z ) tends to a Dirac delta function in z = m Z , while for h Z → ∞, K h Z (z; m Z ) tends to a discrete uniform distribution on Z. Discrete beta kernels possess two peculiar characteristics.…”

Section: The Modelmentioning

confidence: 99%

Bivariate discrete beta Kernel graduation of mortality data

Mazza

Punzo

2014

Lifetime Data Anal

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Various parametric/nonparametric techniques have been proposed in literature to graduate mortality data as a function of age. Nonparametric approaches, as for example kernel smoothing regression, are often preferred because they do not assume any particular mortality law. Among the existing kernel smoothing approaches, the recently proposed (univariate) discrete beta kernel smoother has been shown to provide some benefits. Bivariate graduation, over age and calendar years or durations, is common practice in demography and actuarial sciences. In this paper, we generalize the discrete beta kernel smoother to the bivariate case, and we introduce an adaptive bandwidth variant that may provide additional benefits when data on exposures to the risk of death are available; furthermore, we outline a cross-validation procedure for bandwidths selection. Using simulations studies, we compare the bivariate approach proposed here with its corresponding univariate formulation and with two popular nonparametric bivariate graduation techniques, based on Epanechnikov kernels and on P-splines. To make simulations realistic, a bivariate dataset, based on probabilities of dying recorded for the US males, is used. Simulations have confirmed the gain in performance of the new bivariate approach with respect to both the univariate and the bivariate competitors.

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“…According to Bagnato and Punzo (2013), the two parameterizations in Equations (1), (2) are equivalent for α, β > 1. This subclass excludes beta densities that are bimodal, unlimited (reverse) J-shaped, or uniform.…”

Section: Properties Of the Beta Densitymentioning

confidence: 99%

“…In particular, we adopt an alternative method of specifying the beta distribution that allows only unimodal univariate densities, first explored by Bagnato and Punzo (2013) focusing on restricted unimodal reparameterizations of the gamma and beta densities. This subclass of beta densities is given by:…”

Section: Properties Of the Beta Densitymentioning

confidence: 99%

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Clustering student skill set profiles in a unit hypercube using mixtures of multivariate betas

Dean

Nugent

2013

Adv Data Anal Classif

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July 2013Abstract This paper presents a finite mixture of multivariate betas as a new model-based clustering method tailored to applications where the feature space is constrained to the unit hypercube. The mixture component densities are taken to be conditionally independent, univariate unimodal beta densities (from the subclass of reparameterized beta densities given by Bagnato and Punzo, 2013). The EM algorithm used to fit this mixture is discussed in detail, and results from both this beta mixture model and the more standard Gaussian model-based clustering are presented for simulated skill mastery data from a common cognitive diagnosis model and for real data from the Assistment System online mathematics tutor (Feng et al, 2009). The multivariate beta mixture appears to outperform the standard Gaussian modelbased clustering approach, as would be expected on the constrained space. Fewer components are selected (by BIC-ICL) in the beta mixture than in the Gaussian mixture, and the resulting clusters seem more reasonable and interpretable.

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Finite mixtures of unimodal beta and gamma densities and the $$k$$ -bumps algorithm

Cited by 68 publications

References 46 publications

Parsimonious mixtures of multivariate contaminated normal distributions

Parsimonious mixtures of multivariate contaminated normal distributions

Bivariate discrete beta Kernel graduation of mortality data

Clustering student skill set profiles in a unit hypercube using mixtures of multivariate betas

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