Abstract
The most suitable statistical method for explaining serial dependency in time series count data is that based on Hidden Markov Models (HMMs). These models assume that the observations are generated from a finite mixture of distributions governed by the principle of Markov chain (MC). Poisson-Hidden Markov Model (P-HMM) may be the most widely used method for modelling the above said situations. However, in real life scenario, this model cannot be considered as the best choice. Taking this fact into account, we, in this paper, go for Generalised Poisson Distribution (GPD) for modelling count data. This method can rectify the overdispersion and underdispersion in the Poisson model. Here, we develop Generalised Poisson Hidden Markov model (GP-HMM) by combining GPD with HMM for modelling such data. The results of the study on simulated data and an application of real data, monthly cases of Leptospirosis in the state of Kerala in South India, show good convergence properties, proving that the GP-HMM is a better method compared to P-HMM.
Highlights
Poisson model is the most commonly used method for modelling time series count data
The four-state model is fitted to the leptospirosis series by using Hidden Markov Models (HMMs) with Poisson distribution, while the two-state model is fitted to the data using Generalised Poisson Distribution (GPD) as state dependent distribution
We propose to deal with overdispersion and underdispersion in time series of count data by introducing GPD in HMM
Summary
Poisson model is the most commonly used method for modelling time series count data. Though equidispersion is the unique feature of Poisson distribution, in practical cases, either the mean will be greater than variance or vice-versa, making the Poisson assumption wrong. In many populations of Poisson nature, the probability of the occurrence of an event does not remain constant and is affected by previous occurrences, resulting in unequal mean and variance in the data (Kendall & Stuart 1963). To deal with such situations, modification and generalization of the Poisson distribution were considered by Greenwood & Yule (1920) and by Neyman (1931). The variance of GPD model is greater than, equal to, or less than the mean when the second parameter λ2 is positive, zero or, negative respectively. The following part of this paper has been categorized into four sections, detailing the methods and estimation of parameters of GP-HMM, a simulation study and a real data application of GPHMM
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