Clustering Multivariate Longitudinal Observations: The Contaminated Gaussian Hidden Markov Model

Abstract

The Gaussian hidden Markov model (HMM) is widely considered for the analysis of heterogeneous continuous multivariate longitudinal data. To robustify this approach with respect to possible elliptical heavy-tailed departures from normality, due to the presence of outliers, spurious points, or noise (collectively referred to as bad points herein), the contaminated Gaussian HMM is here introduced. The contaminated Gaussian distribution represents an elliptical generalization of the Gaussian distribution and allows for automatic detection of bad points in the same natural way as observations are typically assigned to the latent states in the HMM context. Once the model is fitted, each observation has a posterior probability of belonging to a particular state and, inside each state, of being a bad point or not. In addition to the parameters of the classical Gaussian HMM, for each state we have two more parameters, both with a specific and useful interpretation: one controls the proportion of bad points and one specifies their degree of atypicality. A sufficient condition for the identifiability of the model is given, an expectation-conditional maximization algorithm is outlined for parameter estimation and various operational issues are discussed. Using a large scale simulation study, but also an illustrative artificial dataset, we demonstrate the effectiveness of the proposed model in comparison with HMMs of different elliptical distributions, and we also evaluate the performance of some well-known information criteria in selecting the true number of latent states. The model is finally used to fit data on criminal activities in Italian provinces.

The multivariate leptokurtic‐normal distribution and its application in model‐based clustering

“…However the kurtosis which is is only defined for and can assume any value in the interval , with the extreme values being assumed for and , respectively (see, e.g., Zografos, ). For MCNMs is defined as one of the two parameters governing the tail weight; it is the inflation parameter denoting the degree of outlierness with higher values related to heavier tails (see Maruotti & Punzo, 1998; Punzo & Maruotti, ; Punzo & McNicholas, for details). Also in this case the kurtosis which is given in Equation can assume any value in the interval ; details are given in Appendix G. For MPEMs, is a shape parameter; heavy tails are obtained for , light tails are obtained for , whereas normal tails are obtained for .…”

Model‐based clustering for noisy longitudinal circular data, with application to animal movement

“…However, even when a longitudinal circular data set is made up mainly of clusters of the projected normal type, there may be (noisy) observations that do not fit the prevailing pattern of clusters, and, as currently specified, the model does not allow to account for those (noisy) observations. However, for a wide class of HMMs, maximum likelihood estimates are not robust (Punzo & Maruotti, ). This implies that deviations from the nominal model, such as a small proportion of noisy points in the data, can lead to poor estimates and clustering.…”

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

“…The M‐step, on the same iteration, requires the maximization of Q ( ϑ ) with respect to ϑ . As the 4 terms on the right‐hand side of have 0 cross‐derivatives, they can be maximized separately . In particular, the maximization of and —expected counterparts of in and in —with respect to π and Π , respectively, subject to the constraints on these parameters, yields The maximization of —expected counterpart of in —with respect to β u k and γ u k , u =1,…, d Y and k =1,…, K , is equival...…”

“…As the 4 terms on the right-hand side of (6) have 0 cross-derivatives, they can be maximized separately. 30 In particular, the maximization of Q 1 ( ) and Q 2 (Π)-expected counterparts of c 1 ( ) in (7) and c 2 (Π) in (8)-with respect to and , respectively, subject to the constraints on these parameters, yields…”