Independent Mixed-gamma Variables for Modelling Rainfall

摘要

Understanding the rainfall process and characteristics are crucial to the efficient design of flood mitigation and construction of crop growth models. Modelling rainfall is not limited to fit the historical data to a suitable distribution but the model should be able to generate synthetic rainfall data. In this study, we derive sets of formulae of mean and variance for the sum of two and three independent mixed-gamma variables, respectively. Firstly, the positive data is fitted to gamma model marginally and the shape and scale parameters are estimated using the maximum likelihood estimation method. Then, the mixed-gamma model is defined to include zero and positive data. The formulae of mean and variance for the sum of two and three independent mixed-gamma variables are derived and tested using the daily rainfall totals from Pooraka station in South Australia for the period of 1901-1990. The results demonstrate that the values of generated mean and using formula are close to the observed mean. However, the values of the variance are sometimes over-estimated or under-estimated of the observed values. The observed variance is lower possibly due to correlation between the experimental data, that have not been included in the mixed-gamma models. The Kolmogorov-Smirnov and Anderson-Darling goodness of fit tests are used to assess the fit between the observed sum and the generated sum of independent mixed-gamma variables. In both cases, the observed sum is not significantly different from the generated sum of independent mixed-gamma model at 5% significance level. This methodology and formulae derived can be applied to find the sum of more than three independent mixed-gamma variables and the general form of the formulae can be derived.