Forecasting Non-Gaussian Time Series with TB Data

AbstractConventional forecasting models require time series that are stationary over time in terms of mean andvariance. However, we often encounter data that rarely meet this condition. The data may have Non-Gaussian (N-G) distribution or contain heavy tails or extreme values. In order to improve and strengthenthe predictive performance, various (N-G) models have been used, each of which has a different propertyfrom the other models. The combined formulas of discrete distributions such as Poisson or Negative –Binomial (NB) distribution with Autoregressive Integrated Moving Average (ARIMA) models provide aninterpretable methodology when modeling time series data by following the characteristics of count databecause it relies on the distributional properties represented by the general linear model based on countdata and the time dependence represented by the ARIMA model of the residuals. Predicting time-dependent patterns of count data involves complexities resulting from the discrete and positive nature ofthe data, which is not compatible with the classical ARIMA methodology. To address this shortcoming,models combining the two were used as an alternative solution. These models are Gamma-ARIMA,Poisson-ARIMA, and NB- ARIMA. To fit discrete data to a continuous gamma distribution, a newframework, the transformed Gamma-ARIMA model, was proposed. By applying a mathematicaltransformation to discrete data, the series formation becomes more consistent, and the Gamma-ARIMAtechnique is successful on non-Gaussian discrete data sets.. Four different mathematical formulationswere used, and the Enhanced Grey Wolf Optimizer (EGWO) algorithm was used to compare them. Theresults show that the square root transformation is the best using the No-U-Turn Sampler (NUTS)algorithm, and that the Bayesian estimation performance is robust and suitable for reliable inference andfuture predictions. Using an annual time series of the number of pulmonary Tuberculosis (TB) cases inIraq, the results showed that the Poisson-ARIMA model outperformed the other models using MeanSquare Error (MSE)and Mean Absolute Percentage Error (MAPE).Keywords: Non-Gaussian; Gamma-ARIMA; EGWO algorithm; Bayesian inference; Tuberculosis; Iraq.