Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers

Journal of Applied Statistics

2018

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Insurance and economic data are often positive, and we need to take into account this peculiarity in choosing a statistical model for their distribution. An example is the inverse Gaussian (IG), which is one of the most famous and considered distributions with positive support. With the aim of increasing the use of the IG distribution on insurance and economic data, we propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various branches of statistics, and we give some examples. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and does not allocate probability mass to unrealistic negative values. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers. Finally, for modelbased clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain maximum likelihood estimates of the parameters of the mixture and contaminated models. We use insurance data about bodily injury claims, and economic data about incomes of Italian households, to illustrate the models.

Section: Discussionmentioning

confidence: 99%

A new look at the inverse Gaussian distribution with applications to insurance and economic data

Journal of Applied Statistics

2018

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“…In this literature, HMRMs play a special role . The standard approach that we call HMRMFCs focuses on modeling the conditional state‐specific distribution

f ({bold-italicY}_{i t} = {bold-italicy}_{i t} | {bold-italicX}_{i t} = {bold-italicx}_{i t}, S_{i t} = k)

by assuming a functional form for the expectation

E ((), Y_{i t} = y_{i t} false| X_{i t} = x_{i t}, S_{i t} = k)

. An implicit assumption of HMRMFCs, in a clustering perspective, is the so‐called assignment independence: the assignment of the data points

((), x_{i t}, y_{i t})

to the hidden states is independent from the covariates distribution.…”

Section: Methodsmentioning

confidence: 99%

Multivariate generalized hidden Markov regression models with random covariates: Physical exercise in an elderly population

Ingrassia

Maruotti

2018

Statistics in Medicine

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A time-varying latent variable model is proposed to jointly analyze multivariate mixed-support longitudinal data. The proposal can be viewed as an extension of hidden Markov regression models with fixed covariates (HMRMFCs), which is the state of the art for modelling longitudinal data, with a special focus on the underlying clustering structure. HMRMFCs are inadequate for applications in which a clustering structure can be identified in the distribution of the covariates, as the clustering is independent from the covariates distribution. Here, hidden Markov regression models with random covariates are introduced by explicitly specifying state-specific distributions for the covariates, with the aim of improving the recovering of the clusters in the data with respect to a fixed covariates paradigm. The hidden Markov regression models with random covariates class is defined focusing on the exponential family, in a generalized linear model framework. Model identifiability conditions are sketched, an expectation-maximization algorithm is outlined for parameter estimation, and various implementation and operational issues are discussed. Properties of the estimators of the regression coefficients, as well as of the hidden path parameters, are evaluated through simulation experiments and compared with those of HMRMFCs. The method is applied to physical activity data.

“…These methods robustify, by down-weighting, the estimation of the component means and covariance matrices with respect to mixtures of MN distributions, but they do not automatically detect bad points, although an a posteriori procedure (i.e., a procedure taking place once the model is fitted) to detect bad points with M t Ms is illustrated by McLachlan and Peel (2000). To overcome this problem, Punzo and McNicholas (2016) introduced MCN mixtures (MCNMs); for further recent uses of the MCN distribution in model-based clustering, see (Punzo et al (2020), Punzo and McNicholas (2017), Punzo and Maruotti (2016), Maruotti and Punzo (2017) and Farcomeni and Punzo (2019).…”

Section: Mixtures Of Mscn Distributionsmentioning

confidence: 99%

Multiple scaled contaminated normal distribution and its application in clustering

Tortora

2019

Statistical Modelling

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The multivariate contaminated normal (MCN) distribution represents a simple heavytailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers, referred to as "bad" points. The MCN can also automatically detect bad points. The price of these advantages is two additional parameters, both with specific and useful interpretations: proportion of good observations and degree of contamination. However, points may be bad in some dimensions but good in others. The use of an overall proportion of good observations and of an overall degree of contamination is limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution with a proportion of good observations and a degree of contamination for each dimension. Once the model is fitted, each observation has a posterior probability of being good with respect to each dimension. Thanks to this probability, we have a method for simultaneous directional robust estimation of the parameters of the MN distribution based on down-weighting and for the automatic directional detection of bad points by means of maximum a posteriori probabilities. The term "directional" is added to specify that the method works separately for each dimension. Mixtures of MSCN distributions are also proposed as an application of the proposed model for robust clustering. An extension of the EM algorithm is used for parameter estimation based on the maximum likelihood approach. Real and simulated data are used to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails.