A comprehensive implementation of the log, Box-Cox and log-sinh transformations for skewed and censored precipitation data

Abstract

The log, Box-Cox and log-sinh transformations have been widely used for the purpose of data normalization in hydrological applications. Focusing on the skewness and censoring characteristics of precipitation data, this paper has developed a novel Newton-type algorithm for the comprehensive implementation of the three transformations. Specifically, the likelihood functions are formulated by using the classic z-score and the Jacobian determinants of the three transformations; the gradient vectors are analytically derived to determine the searching directions; and grid-based starts are devised to facilitate the convergence to the global optimal. To test the effectiveness of the algorithm, numerical experiments are devised for monthly precipitation data from the Global Precipitation Climatology Centre. The results show that the log transformation reduces the positive skewness by using a simple offset parameter; the Box-Cox transformation is flexible for data normalization and incurs slightly high computational costs (more than 80 function evaluations); and the log-sinh transformation is computational efficient in normalizing precipitation data (less than 40 function evaluations). For the three transformations, multiple starts are in demand to handle local optimal and 10 grid-based starts for each parameter are shown to be sufficient for the algorithm to reach the global optimal. After transformation, skewness and kurtosis coefficients are reduced to nearly 0 in most regions and the Kolmogorov-Smirnov and Shapiro-Wilk tests suggest that transformed data follow Gaussian distribution at the significance level of 0.05. Overall, the Newton-type algorithm can serve to implement the log, Box-Cox and log-sinh transformations for effective hydrological modelling and water management.