Effect of Normal Transformation Methods on Performance of Multivariate Normal Distribution

Abstract

Multivariate normal distribution is used widely to characterize the uncertainties and correlations for correlated geotechnical data. The success of constructing a multivariate normal distribution depends on the reliable estimation of the marginal probability density functions (PDFs) and the correlation matrix. This paper focused on the normal transformation which is related to the fitted marginal PDFs and investigated its effect on the performance of the constructed multivariate normal distributions, i.e., the normality of the multivariate normal distribution, the fitness of the simulated data with the original data, the rationality of the derived point estimate equations, and validation of the equations based on validation data sets. Three normal transformation methods with different types of fitted marginal PDF, namely Johnson transformation, three-parameter lognormal transformation, and Box–Cox transformation, were compared based on their application to a real soil database. It was found that all the three normal transformation methods are applicable in the framework of multivariate normal distribution, although the transformed variables do not follow the multivariate normal distribution. The consistence of normality of the transformed variables with the performance of the constructed multivariate normal distribution in estimating the unknown parameters using Bayesian updating technique was verified. The Johnson transformation method is the recommended method for constructing the multivariate normal distribution for the real databases due to its robustness and superiority in normal transformation.