Bayesian Time Series Modelling of the Italian Daily Rainfall Data using Mixed Distribution

Abstract

A combination of continuous and discrete elements is referred to as a mixed distribution. For example, daily rainfall data consist of zero and positive values. We aim to develop a Bayesian time series model that captures the evolution of the daily rainfall data in Italy, focussing on directly linking the amount and occurrence of rainfall. Two gamma (G1 and G2) distributions with different parameterisations and lognormal distribution were investigated to identify the ideal distribution representing the amount process. Truncated Fourier series was used to incorporate the seasonal effects which captures the variability in daily rainfall amounts throughout the year. A first-order Markov chain was used to model rainfall occurrence conditional on the presence or absence of rainfall on the previous day. We also built a hierarchical prior structure to represent our subjective beliefs and capture the initial uncertainties of the unknown model parameters for both amount and occurrence processes. The daily rainfall data from Urbino rain gauge station in Italy were then used to demonstrate the applicability of our proposed methods. Residual analysis and posterior predictive checking method were utilised to assess the adequacy of model fit. In conclusion, we clearly found that our proposed method satisfactorily and accurately fits the Italian daily rainfall data. The gamma distribution was found to be the ideal probability density function to represent the amount of daily rainfall.