Optimization of Probability Density Functions Applicable for Hourly Rainfall

Abstract

In order to improve the calculation accuracy of the rainfall probability distribution in related applications, this study aimed to select a theoretical function from applicable functions for three classes of the class-conditional probability density function (CCPD) of hourly rainfall series. The three applicable functions are generalized gamma distribution (GΓD), generalized normal distribution (GND), and Weibull distribution. For the reason that it is hard to distinguish the advantages and disadvantages of the three functions by the probability plot and error analysis of fitted values, optimization criteria are proposed, which are the Bayesian information criterion (BIC) and the estimated accuracy of both the annual average rainfall (AAR) and the annual average continuous rainfall (AACR). The results show that by using three applicable functions in 15 regions, the relative fitting deviations for CCPD1 were less than 2.3% and less than 3.3% for ln(CCPD1). The goodness-of-fit values were all above 0.98 for CCPD1 and greater than 0.94 for ln(CCPD1). The fitting effect of the Weibull distribution was relatively poor from the perspective of the probability plot and error analysis of the fitted values, while the three applicable functions could be used to fit CCPD. GΓD had the highest fitting accuracy for the three classes of CCPDs, but there is concern about overfitting due to its broad spectrum. GND, with fewer parameters, had comparable performance to GΓD, and when fitting CCPD1 using GND, the mean relative fitting deviation was 0.6%, the coefficient of determination was 0.999, and for ln(CCPD1), they were 1.45% and 0.989. At the same time, GND performed well in estimating the AARs, with an 8.6% relative error and a 0.92 correlation coefficient in the fifteen regions, indicating that GND can well reflect the spatial variation characteristics of the AAR. Moreover, the function form of GND is simple. GND follows the parsimonious principle, and it is suitable for the whole domain. Therefore, GND is recommended as the theoretical density function based on the optimization criteria. The genetic algorithm was adopted to obtain the approximate solution of the parameters through optimization, which can simplify the derivation and calculation steps. The multiplicative and additive fitting errors were both used in the objective functions, which gave comprehensive consideration to both ends of the fitting curve.