Fitting and Forecasting Modified Arima Models to Poison Data

Abstract

Certain models, like Autoregressive Moving Average (ARMA) model are commonly known in time series. This method takes into account that past information influences the variables of today. However, a problem occurs when Poisson data have to be modeled. The number of counts in a certain period can only be an integer and that is why the commonly used ARMA-model, which assumes stationarity, seems not very useful anymore, simply because there are some associated problems like outlier, excess zero and over dispersion that can be encountered in the Poisson data, which may actually lead to violation of stationarity assumption of the ARMA model.  For this problem, Integrated Autoregressive Moving Average (ARIMA) model were studied on Poisson data. These models were used to capture different phenomena of Poisson data with different parameter. Data set were simulated in R statistical software with different sample sizes from Poisson process of The four models under study, namely: ARIMA (1,1,1), ARIMA (1,1,2), ARIMA (2,1,1) and ARIMA (2,1,2) were then fitted to the simulated data so as to examine the effect of the proportion of changes in  and sample size on them. Their performances were then compared at different sample sizes and means of Poisson. Thereafter, the forecast performances of the four fitted models were examined at different steps ahead. All cases of simulation were randomized and replicated 1000 times each for the respective selected sample sizes. ARIMA (2,1,2) models are obviously preferred to be the best model that captured the Poisson data with different parameter at lower sample sizes using the selected criteria while ARIMA (1,1,2) is chosen as the best for the data with higher sample sizes