شبیه سازی مونت کارلو برای تجزیه و تحلیل حساسیت مستقل لحظه ای

The moment-independent sensitivity analysis (SA) is one of the most popular SA techniques. It aims at measuring the contribution of input variable(s) to the probability density function (PDF) of model output. However, compared with the variance-based one, robust and efficient methods are less available for computing the moment-independent SA indices (also called delta indices). In this paper, the Monte Carlo simulation (MCS) methods for moment-independent SA are investigated. A double-loop MCS method, which has the advantages of high accuracy and easy programming, is firstly developed. Then, to reduce the computational cost, a single-loop MCS method is proposed. The later method has several advantages. First, only a set of samples is needed for computing all the indices, thus it can overcome the problem of “curse of dimensionality”. Second, it is suitable for problems with dependent inputs. Third, it is purely based on model output evaluation and density estimation, thus can be used for model with high order (>2) interactions. At last, several numerical examples are introduced to demonstrate the advantages of the proposed methods.

Sensitivity analysis (SA) is a study of how “uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input” [1]. It is a very useful tool for model simplification, importance ranking, risk reduction and other purposes. During the past few decades, a variety of SA techniques have been proposed by researchers for different purposes [2], [3], [4], [5], [6], [7], [8] and [9]. Among these techniques, the variance-based one developed by Sobol [3] and Homma and Saltelli [4] and the moment-independent one developed by Borgonovo [9] are the most popular. The variance-based SA aims at distributing the model output variance to different sets of model inputs by looking at the entire distribution ranges of those inputs. It is global, quantitative and model free, thus has been studied widely by practitioners in the past years. Nowadays, there have been a lot of smart methods available for computing the variance-based SA indices [10], [11], [12] and [13]. The moment-independent SA focuses on finding those inputs that, if fixed at their distribution ranges, will lead to the greatest shift in the probability density function (PDF) of model output on average. It is also global, quantitative and model free, and additionally, it is moment-independent, thus attracts more and more attentions of practitioners recently. During the past few years, some efforts have been devoted to developing efficient and robust methods for computing the moment-independent SA indices (also called delta indices). In the original paper, Borgonovo proposed the PDF-based method for computing the delta indices [9]. This is a computationally expensive double-loop simulation method, and also the precision of estimates suffers from calculating the intersection points of the unconditional and conditional PDF of the model output. To improve the accuracy of the estimates, Liu developed the CDF-based method, however, as pointed by Liu “for a computationally intensive model, when the total computational time is mainly due to the time of running the model, the improvement of the computational efficiency by the CDF-based method can be negligible” [14]. In another paper, Borgonovo proposed the emulation method for expressing the input–output relationship by metamodel, and then computing the delta indices based on the metamodel [15]. Both the State Dependent Parameter (SDP) [12], [16] and [17] and kriging emulator are investigated. This method can drastically reduce the computational cost, thus receives more and more attentions in the area of SA. The estimate precision of these methods mainly depends on the metamodel. If the input–output mapping contains high order interactions, the metamodel may often fail to capture the structure feature of the model. In Ref. [18], Castaings used the quadrature method to deal with the one dimensional integral given in the definition of delta indices for individual input. Compared with the PDF-based and CDF-based methods, this method is computationally more efficient. However, it is still a double-loop method, and the computational cost increases with the number input variables. Despite these works by different researchers, compared with the variance-based SA indices, robust and efficient methods are less available for computing the delta indices. In this paper, we firstly propose a double-loop Monte Carlo simulation (MCS) method for computing the delta indices. This method is purely based on model evaluation and univariate density estimation, and it has the advantages of high-accuracy and usability, but its computational cost is unacceptable for engineering models. Then, to substantially improve the computational efficiency, a single-loop MCS method is developed. This single-loop MCS method has several advantages. First, only a set of samples are needed for calculating all the delta indices, thus it is computationally efficient and overcomes the problem of “curse of dimensionality”. Second, it is suitable for models with dependent inputs. Thirdly, compared with the emulation method, it is based purely on model evaluation and density estimation, thus can be employed to deal with the problem with high order (>2) interactions terms. The rest of this paper is organized as follows. Section 2 reviews the original definition of the delta indices. Section 3 proposes the double-loop MCS method and Section 4 develops the single-loop MCS method for computing the delta indices. Section 5 introduces three test examples to demonstrate the advantages of the proposed methods. Section 6 gives conclusions.

In this paper, the computation of the delta indices is concerned. The double-loop MCS method has been developed firstly. This method is purely based on the model evaluation and univariate density estimation. Compared with the PDF-based and CDF-based methods, the double-loop MCS method doesnot need to compute the intersection points between f Y(y ) and fY|Xi(y)fY|Xi(y), thus is more accurate. Compared with the meta-model method, the double-loop method can be accurately applied to model with high order (>2) interactions, in which meta-model methods may introduce additional error. However, this method is computationally expensive, especially for the time-consuming models. To substantially improve the efficiency for computing the delta indices, the single-loop MCS method has been developed in this contribution. This method has three main advantages. First, only a set of samples is needed for computing all the delta indices, thus it is free of “curse of dimensionality”. Second, it can be applied to models with dependent inputs. Third, it is purely based on model evaluation and the density estimation, thus it can be used for problems with even high order (>2) interactions. Numerical examples have verified these advantages of the proposed methods.