On Taylor Series Expansion for Statistical Moments of Functions of Correlated Random Variables

Lukáš Novák,Drahomír Novák

doi:10.3390/sym12081379

Abstract

The paper is focused on Taylor series expansion for statistical analysis of functions of random variables with special attention to correlated input random variables. It is shown that the standard approach leads to significant deviations in estimated variance of non-linear functions. Moreover, input random variables are often correlated in industrial applications; thus, it is crucial to obtain accurate estimations of partial derivatives by a numerical differencing scheme. Therefore, a novel methodology for construction of Taylor series expansion of increasing complexity of differencing schemes is proposed and applied on several analytical examples. The methodology is adapted for engineering applications by proposed asymmetric difference quotients in combination with a specific step-size parameter. It is shown that proposed differencing schemes are suitable for functions of correlated random variables. Finally, the accuracy, efficiency, and limitations of the proposed methodology are discussed.

Highlights

Mathematical modeling in civil engineering is often represented by the finite element method (FEM)
Schlune et al proposed the ECoV method based on linear Taylor series expansion (TSE) with a simple asymmetric differencing, there are no studies on its limitations and possible generalizations, TSE is a highly modifiable technique via differencing schemes and a truncation order of an approximation
This paper presents a novel methodology to estimate the coefficient of variation for functions of correlated input random variables

Summary

Introduction

Mathematical modeling in civil engineering is often represented by the finite element method (FEM). From a practical point of view, it is necessary to decrease the number of FEM calculations as much as possible while satisfying the given safety requirements of the analyzed structure. A solution can be represented by a semi-probabilistic approach widely accepted in the engineering field [1] and implemented into the national codes such as Eurocode [2]. Such approach is able to greatly reduce the number of necessary calculations for the design and an assessment of structures. The basic reliability concept is given as Z = R − E, where Z is a safety margin, which is defined as the difference between the structural resistance R and the load effect E. According to the semi-probabilistic approach, the resistance of a structure R is separated, and the design value

Methods

Discussion

Conclusion