Comparison of sum of two correlated gamma variables for Alouini's model and McKay distribution

Abstract

A statistical model for rainfall is useful to describe the relationship between rainfall at a given location and other weather-related variables. It is also able to provide a principled way to quantify the uncertainty that associates rainfall processes, which is crucial to the efficient design of environmental projects and to improve crop production. Various statistical models have been used for rainfall such as a right-skewed distribution including the exponential, gamma or mixed-exponential to model the rainfall intensities and Markov chain model to model rainfall occurrences. Two models of the sum of two correlated gamma variables, namely Alouini's model and McKay distribution are studied. Alouini's model is an extension of Moschopoulos results for the sum of n correlated gamma variables. There are two forms of McKay distribution, the Type I is defined for sum of two correlated gamma variables whereas Type II is defined for difference of two correlated gamma variables. In this study, the Alouini's model and Type I of McKay distribution are compared using monthly rainfall totals to form water catchment from two meteorological stations, Hume and Beechworth within the Murray-Darling Basin. Rainfall totals during the summer season at both stations are selected based on the significant correlations. The procedure for the analysis is as follows: (1) select positive pairs set of data from two stations (2) calculate the pairwise Spearman's correlation and check if the correlation is significant (3) fit the data marginally with the gamma distribution and use maximum likelihood estimation method to estimate the parameters for the gamma distribution (4) calculate the average value of shape parameters and re-estimate the values of scale parameters (5) apply the model to generate synthetic sum of monthly rainfall totals (6) compare the sum of monthly rainfall totals between the observed and generated data using the Kolmogorov-Smirnov goodness of fit test. The results of the analysis show that the values of mean and variance of the observed and estimated from the McKay distribution is closer than the values of mean and variance estimated from the Alouini's model. Based on the Kolmogorov-Smirnov goodness of fit test, the P-value of observed versus McKay distribution is much higher than the P-value of observed versus Alouini's model. It shows that the McKay distribution fits the data better even though both models pass the test. However, when the Alouini's model and McKay distribution are compared, the rainfall totals generated by the two models fail the test of being from the same population. Zakaria (2011) shows that the Alouini's model is suitable in modelling the sum of four correlated gamma variables and can easily be extended to more than two variables. On the other hand, the McKay model is not easily extendable to more than two variables because of extensive algebraic manipulation. Thus, in this particular example of two stations and three months, it is instructive to note that the McKay formulation can well represent the sum of individual months at the two locations, but if we wanted to represent the sum over the three months of the season, we would have to use Alouini's method. Both models are able to be used to generate synthetic sum of rainfall totals and Alouini's model can be used to model the sum of more than two correlated gamma variables. It may well be that catering for this extra flexibility is the reason that the Alouini's model does not perform so well for the two variable case. In future work, we will be comparing these two approaches with other formulations as well, such as the use of Maximum Entropy methods.