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.

Flood frequency analysis of Manas River Basin in China under non‐stationary condition

“…Therefore, scholars have used indirect methods, mainly relying on the rainfall-runoff relation of the basin as well as the decomposition and composition of time series or the hydrological model, which are modified to eliminate the influence of changes and construct stationary time series. Meanwhile, to restore the original hydrological time series, researchers have proposed a series of direct calculation methods, including the mixed distribution (MD) method (Ibrahim, 2019;Singh & Sinclair, 1972), time-variant moment method (Sheu et al, 2017;Zeng et al, 2016), and conditional probability distribution (CPD) method (Singh et al, 2005;Tanaka et al, 2016). Singh and Sinclair (1972) and Waylen and Woo (1982) found that individuals in the extreme value series selected by frequency analysis sometimes did not come from the same population, that is, they did not obey the same distribution.…”