ARMA Models for Zero Inflated Count Time Series
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Zero inflation is a common nuisance while monitoring disease progression over time. This article proposes a new observation driven model for zero inflated and over-dispersed count time series and applies it to Indian dengue counts. The disease counts given the past history of the process and available information on covariates is assumed to be distributed as a mixture of a Poisson distribution and a distribution degenerate at zero, with a time dependent mixing probability, π_t. Since, count data usually suffers from overdispersion, a Gamma distribution is used to model the excess variation, resulting in a zero inflated Negative Binomial (NB) regression model with mean parameter λ_t. Linear predictors with auto regressive and moving average (ARMA) type terms, covariates, seasonality and trend are fitted to λ_t and π_t through canonical link generalized linear models. Estimation is done using maximum likelihood aided by iterative algorithms, such as Newton Raphson (NR) and Expectation and Maximization (EM). Stationarity of the model is discussed under certain conditions. In-depth simulation studies provide an understanding about the properties of the estimators. Since, dengue occurrence has been linked to various climatic factors, the disease model includes, relative humidity, amount of rainfall, temperature, as covariates. A detailed comparative study of the proposed zero inflated NB-ARMA dengue model is also conducted.
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