Multivariate generalized Poisson geometric process model with scale mixtures of normal distributions

This paper proposes a new model named as the multivariate generalized Poisson log-t geometric process (MGPLTGP) model to study multivariate time-series of counts with overdispersion or underdispersion, non-monotone trends within each time-series and positive or negative correlation between pairs of time-series. This model assumes that the multivariate counts follow independent generalized Poisson distributions with an additional parameter to adjust for different degrees of dispersion including overdispersion and underdispersion. Their means after discounting the trend effect geometrically by ratio functions form latent stochastic processes and follow a multivariate log-t distribution with a flexible correlation structure to capture both positive correlation and negative correlation. By expressing the multivariate Student's t-distribution in scale mixtures of normals, the model can be implemented through Markov chain Monte Carlo algorithms via the user-friendly WinBUGS software. The applicability of the MGPLTGP model is illustrated through an analysis of the possession and/or use of two illicit drugs, amphetamines and narcotics in New South Wales, Australia.