تجزیه و تحلیل حساسیت از سهم واریانس با توجه به پارامترهای توزیع با تابع کرنل

Variance based sensitivity indices represent how the input uncertainty influences the output uncertainty. In order to identify how the distribution parameters of inputs influence the variance contributions, this work proposes the sensitivity of the variance contributions, which is defined by the partial derivative of the first-order variance contribution with respect to the distribution parameter. The proposed sensitivity can reflect how small variation of the distribution parameter influences the first-order variance contribution. By simplifying the partial derivative of the first-order variance contribution into the form of expectation via the kernel function, the proposed sensitivity can be seen as a by-product of the variance based sensitivity analysis without any additional output evaluations. For the classical quadratic responses, the proposed sensitivity can be derived analytically based on the integral form, while for the complex responses, the state dependent parameter (SDP) based method, which has been applied in the variance sensitivity analysis, can be employed to compute the proposed sensitivity. Several examples are used to demonstrate the correctness of the analytical solutions and the efficiency of the SDP based method.

Development of probabilistic sensitivity analysis is frequently considered as an essential component of a probabilistic analysis [1]. Traditional sensitivity analysis (SA) can be classified into two groups: local SA and global SA [2]. Local SA usually investigates how small variation of the distribution parameter around the reference point changes the value of the output. One classical local SA is the derivative based SA which is defined as the derivative of response merits with respect to the distribution parameters of inputs. The main drawback of the derivative based SA is that it depends on the choice of the nominal point, while the main advantage of it is the low computational cost that is gained in turn [3]. Global SA studies how the uncertainty in the output of a computational model can be decomposed according to the input sources of uncertainty [4]. Contrary to the local SA, global SA explores the whole range of uncertainty of the model inputs by letting them vary simultaneously [5]. At present, a number of global sensitivity indices have been suggested, e.g. Helton and Saltelli proposed the non-parametric sensitivity indices (input–output correlation) [6] and [7], Sobol, Iman and Saltelli proposed the variance based sensitivity indices [7], [8] and [9], Chun, Liu and Borgonovo proposed the moment independent sensitivity indices [10] and [11], Sobol and Kucherenko proposed the derivative based sensitivity indices and investigated their link with the variance based sensitivity indices [12], [13] and [14]. In this work, we mainly investigate the variance based sensitivity indices which have been applied to design under uncertainty problems and are capable of identifying the contributions of any random variable. However, the variance based sensitivity indices are far more computationally demanding and various methods have been used to compute the variance based sensitivity indices, such as Sobol’s method [15], the FAST method [16], and the meta-model based method [17]. Those methods all have their deficiencies, Sobol’s method and the FAST method which are indeed computationally demanding for the engineering applications, and the meta-model based method cannot estimate the total variance based sensitivity indices if the interactions are higher than the second order in the ANOVA decomposition. It is noticed that in the classical variance based SA, the influences of distribution parameters are not considered. If the variation of a distribution parameter can lead to the considerable changes to the variance contributions, the computational results of the variance based SA will be vulnerable and less reliable. Thus, analysts need to investigate the effects of the distribution parameters on the variance contributions in the variance based SA. In this work, we propose the sensitivity of distribution parameters on variance contributions which is defined by the partial derivative of the first-order variance contribution (FOVC) with respect to the distribution parameter. The proposed sensitivity can reflect how small variations of distribution parameters influence the FOVC. In Ref. [1], Millwater has employed the kernel function which can be derived analytically for various distribution types to simplify the partial derivative based sensitivity of the first two moments and the failure probability, and then the partial derivative based sensitivity can be computed efficiently. Thus, in this work, we also employ the kernel function to simplify the proposed sensitivity first. Then for the simplified sensitivity, solutions for the classical quadratic polynomial without cross-terms response are derived analytically, while for the complex responses, the SDP based method, which is developed recently by Ratto [18] and Li [19], is employed to compute the proposed sensitivity. The remainder of this work is organized as follows: Section 2 gives a brief review of the variance based SA. Section 3 first develops the sensitivity of distribution parameters on the FOVC, then the proposed sensitivity for the classical quadratic polynomial without cross-terms response is derived analytically. Some discussions for the effect of the kernel function and some computational considerations of the proposed sensitivity are given in Section 4. In Section 5, the SDP based method is employed to compute the proposed sensitivity. In Section 6, two numerical examples are first given to validate the correctness of the analytical solutions for the classical quadratic polynomial without cross-terms response and the efficiency of the SDP based method, then a simple cantilever with explicit response and a ten-bar structure with implicit response are employed to validate the reasonability of the proposed sensitivity. Finally, some conclusions are drawn in Section 7.

This work investigates how the distribution parameter of the input influences the variance contributions and proposes the sensitivity of the first-order variance contribution (FOVC). By using some properties of the expectation and the conditional expectation developed via the kernel function, the proposed sensitivity can be simplified into the form of expectation. Comparing the simplified sensitivity of the FOVC with the definition of the FOVC, it can be concluded that the influences of the distribution parameters are transmitted by the kernel function at each reference point of input. Since the proposed sensitivity needs no additional computational cost during the process of the variance based SA, it can be seen as a by-product of the variance based SA. Due to the large computational cost of the variance based sensitivity indices, the analytical solutions of the proposed sensitivity are derived for the classical quadratic polynomial without cross-terms response, and two numerical examples are employed to discuss the effects of the cross-terms in the response. Generally, the cross-terms in the polynomial response may lead to the influences of one input’s distribution parameters on the FOVCs of other inputs. For the complex responses, the SDP based method is employed to compute the proposed sensitivity, the accuracy and high efficiency are validated by several examples. The computational results of several examples show that the influences of the parameters cannot be neglected in the variance based SA, for a small variation of the most influential parameter may lead to different results of the variance based sensitivity indices. Thus, collecting the information and improving the understanding of those most influential parameters are significant and can make the results of the variance based sensitivity indices reliable.