روش پارتیشن فضایی برای تجزیه و تحلیل حساسیت مبتنی بر واریانس: طرح پارتیشن بهینه و مطالعه تطبیقی

Variance-based sensitivity analysis has been widely studied and asserted itself among practitioners. Monte Carlo simulation methods are well developed in the calculation of variance-based sensitivity indices but they do not make full use of each model run. Recently, several works mentioned a scatter-plot partitioning method to estimate the variance-based sensitivity indices from given data, where a single bunch of samples is sufficient to estimate all the sensitivity indices. This paper focuses on this space-partition method in the estimation of variance-based sensitivity indices, and its convergence and other performances are investigated. Since the method heavily depends on the partition scheme, the influence of the partition scheme is discussed and the optimal partition scheme is proposed based on the minimized estimator's variance. A decomposition and integration procedure is proposed to improve the estimation quality for high order sensitivity indices. The proposed space-partition method is compared with the more traditional method and test cases show that it outperforms the traditional one.

Sensitivity Analysis (SA) aims at investigating the sensitivity of the model output with respect to its input parameters. It involves a variety of techniques, which are generally classified into local SA techniques and global SA techniques. In the domain of local SA, the sensitivity of the model output about a certain parameter is studied by varying the particular parameter around its nominal value while keeping other parameters at their nominal states [1], [2] and [3]. It is often the case, however, that the parameters in the model are not exactly known, which violates the assumption of the local SA. Typically, it is assumed that the model output yy is linked to the model inputs x=T[x1,x2,…,xn]x=[x1,x2,…,xn]T by a deterministic function g(⋅)g(⋅), i.e. y=g(x)y=g(x), where the model inputs are not known with certainty. Then the uncertainty of model inputs will propagate through g(⋅)g(⋅) and lead to the output non-deterministic. Studying the influence of the inputs’ uncertainty on the output's uncertainty is what global SA concerns. The scope of global SA is to rank the importance of various sources of uncertainty that result in the uncertainty of the output. A number of global SA techniques are available, such as non-parametric techniques [4], [5] and [6], meta-model techniques [7], [8], [9], [10], [11] and [12], variance-based techniques [13], [14], [15] and [16], and moment independent techniques [17], [18] and [19]. Among all these different global SA techniques, the variance-based techniques attract most attention and have asserted themselves among practitioners. To calculate the variance-based sensitivity indices, two popular approaches are Fourier Amplitude Sensitivity Test (FAST) [20], [21], [22] and [23] and Monte Carlo sampling [14], [15], [16] and [24]. In FAST, variance-based sensitivity indices are calculated in terms of the coefficients of the multiple Fourier series expansion of the output function, which is powerful in estimating the “first-order” effect [25] and [26]. Recently, Xu and Gertner [27] show that FAST can also be used to estimate high order effects, which further extends its application. Sampling is always a straightforward way to investigate the response of complex functions and generally effective. However, to estimate the sensitivity indices, a great amount of model runs may be required for the general sampling-based methods. It is easy to handle if the model is simple, but prohibitive for complex models (e.g., a single model run needs several hours or more). To estimate the sensitivity indices with as fewer model runs as possible, many efforts have been devoted to develop efficient algorithms. In this respect, one direction is to fill the input space with proper representative samples; another is to make best use of model runs. To accomplish the former, Latin hypercube sampling (LHS) was proposed in the late 1970s [28] and has been well studied since then. For the latter, different formulas to calculate the variance-based indices have been proposed [14], [29], [30] and [31]. The traditional FAST method by Cukier et al. in 1970s [20] is another nice example in best using model samples with the use of cyclic samples. However, almost all of these methods cannot fully exploit the model runs. In fact, most models runs are used only once in the sensitivity indices calculation [14]. To make best use of the sample runs (and in turn reduce the cost), Plischke et al. [32] proposed a “space-partition” method to estimate the moment-independent importance measure [18]. The space-partition method is universal, which is applicable to any given data and independent of the generation method. Specifically, it can be used to estimate the sensitivity indices of models with independent or dependent inputs. Though they mentioned the extension of the method to estimate other sensitivity indices, such as the variance-based sensitivity indices, they discussed less of the estimator's properties for the variance-based indices (e.g., the convergence of the estimator and the optimal partition scheme). In this paper, we focus on the space-partition approach in the estimation of the variance-based sensitivity indices, including the first order indices and the high order extension. Since the performance of the space-partition method heavily depends on the partition scheme, the partition scheme for the space-partition estimator is discussed and the optimal partition scheme based on the minimized estimator's variance is proposed. The remainder of the paper is organized as follows. Section 2 briefly reviews the variance-based sensitivity indices and their estimation by Monte Carlo simulation. Section 3 introduces the space-partition estimation for the first order sensitivity indices and its extension to high order indices. Section 4 discusses the influence of different partition schemes for the space-partition estimator and proposes an optimal partition scheme based on the minimized estimator's variance. For the high order sensitivity indices, a decomposition and integration procedure is proposed to further improve the quality of the estimation. Section 5 illustrates the performance of the space-partition method and its comparison with the more traditional substituted-column method. The case with dependent inputs is also studied to illustrate the applicability of the proposed method. Conclusions and discussions are given in the end.

Variance-based sensitivity indices are popular and widely used in identifying important factors in various models. This paper focuses on the space-partition method in the estimation of the variance-based sensitivity indices. We investigate the influence of the partition scheme on the estimator and propose the optimal partition scheme MEVPS based on the minimized estimator's variance. For the high order sensitivity indices, a decomposition and integration procedure is proposed to further improve the estimation. By comparing the performance of the space-partition estimator with the more traditional substituted-column method, it is shown that the space-partition method with MEVPS (and DI) always outperforms the traditional one. The case study on the model with correlated inputs shows that the proposed method also performs well for models with dependent inputs. In this paper, we consider the equiprobability partition scheme, whereas other partition schemes may also be applicable. Actually, it is possible to find better partition schemes based on the minimized estimator's variance criterion when unequal-probability partition schemes are used. However, the computation effort will increase when more general partition schemes are considered, especially for high order sensitivity indices. Genetic algorithm is proved to be quite powerful in solving complicated optimization problems [36], [37] and [38] and it may be useful in finding the partition scheme that minimized the estimator's variance.