Random function model for dependent random variables

Abstract

To characterize the dependency between basic random variables is of paramount importance in engineering practice. However, it is usually difficult to deal with joint probability density function of dependent variates directly. In the present paper, a random function model is proposed for the probabilistic description of 2-dimensional dependent random variables. This random function converts a 2-dimensional dependent random vector into an independent random vector. The undetermined functions in the random function model are found to be the conditional mean and conditional standard deviation function, which could be further specified by observed data. Specifically, in the present paper it is suggested that the shape of the undetermined function, i.e., the conditional mean and standard deviation function in this case, be extracted based on the insight into the embedded physical mechanism, and then the undetermined parameters be identified from observed data. The procedure is illustrated in detail by adopting the relationship between the modulus of elasticity and compressive strength of concrete as an example. The one-dimensional damage evolution mechanism is firstly introduced, yielding the lower limit of the data. Then the viscoelasticity mechanism is advocated to determine the shapes of conditional mean and conditional standard deviation function. The parameters of the model are consequently identified from experimental data. Thereby, a pragmatic model, which could character the dependency between the modulus of elasticity and compressive strength, is proposed for engineering purposes. The comparison to the Copula model shows that the proposed model could capture the probabilistic characteristics of observed data. It is noted that in the proposed model the random function is weakly nonlinear and thus will not worsen the well-posedness of the problem. Besides, the direct dealing with joint probability density function is avoided. The proposed approach could be extended to the probabilistic description of more non-independent random variables.