روش تجزیه و تحلیل حساسیت جهانی مرتبه اول عمومی

Fourier amplitude sensitivity test (FAST) is one of the most popular global sensitivity analysis techniques. The main mechanism of FAST is to assign each parameter with a characteristic frequency through a search function. Then, for a specific parameter, the variance contribution can be singled out of the model output by the characteristic frequency. Although FAST has been widely applied, there are two limitations: (1) the aliasing effect among parameters by using integer characteristic frequencies and (2) the suitability for only models with independent parameters. In this paper, we synthesize the improvement to overcome the aliasing effect limitation [Tarantola S, Gatelli D, Mara TA. Random balance designs for the estimation of first order global sensitivity indices. Reliab Eng Syst Safety 2006; 91(6):717–27] and the improvement to overcome the independence limitation [Xu C, Gertner G. Extending a global sensitivity analysis technique to models with correlated parameters. Comput Stat Data Anal 2007, accepted for publication]. In this way, FAST can be a general first-order global sensitivity analysis method for linear/nonlinear models with as many correlated/uncorrelated parameters as the user specifies. We apply the general FAST to four test cases with correlated parameters. The results show that the sensitivity indices derived by the general FAST are in good agreement with the sensitivity indices derived by the correlation ratio method, which is a non-parametric method for models with correlated parameters.

Identification and representation of uncertainty are recognized as essential components in model application [1], [2], [3], [4], [5] and [6]. In the study of uncertainty, we need to know how much uncertainty there is in the model output (uncertainty analysis) and where the uncertainty comes from (sensitivity analysis). There are two groups of sensitivity analyses: local sensitivity analysis and global sensitivity analysis [7]. The local sensitivity analysis examines the local response of the output(s) by varying input parameters one at a time while holding other parameters at central values. The global sensitivity analysis examines the global response (averaged over the variation of all the parameters) of model output(s) by exploring a finite (or even an infinite) region. The local sensitivity analysis is easy to implement. However, local sensitivity analysis can only inspect one point at a time and the sensitivity index of a specific parameter is dependent on the central values of the other parameters. Thus, more studies currently are using global sensitivity analysis methods instead of local sensitivity analysis. Many global sensitivity analysis techniques are now available [7] and [8], such as Fourier Amplitude Sensitivity Test (FAST) [8], [9], [10], [11], [12], [13], [14] and [15]; the design of experiments method [16], [17], [18], [19] and [20]; regression-based methods [5], [21], [22], [23] and [24]; Sobol's method [25]; McKay's one-way ANOVA method [26]; and moment independent approaches [27], [28] and [29]. One of the most popular global sensitivity analysis techniques is FAST. The theory of FAST was first proposed by Cukier et al. in 1970s [10], [11], [12] and [15]. The main mechanism of FAST is to assign each parameter with a characteristic frequency through a search function. Then, for a specific parameter, the variance contribution can be singled out of the model output by the characteristic integer frequency. Koda et al. [9] and McRae et al. [13] provided the computational codes for FAST. FAST is computationally efficient and can be used for nonlinear and non-monotonic models. Thus, it has been widely applied in sensitivity analysis of different models, such as chemical reaction models [10], [11], [12], [14], [15] and [30]; atmospheric models [31], [32] and [33]; nuclear waste disposal models [34]; soil erosion models [35]; and hydrological models [36]. Although FAST has been widely applied, there are two limitations. First, there will be an aliasing effect among parameters when using integer characteristic frequencies. In order to avoid the aliasing effect, large sample sizes are needed when there are many candidate parameters. Second, FAST can only be applied for models with independent parameters [32]. However, in many cases, the parameters are correlated with one another. For example, in meteorology, the central pressure of the storm is correlated with the radius of the maximum wind [3]. Recently, there have been two improvements aimed at overcoming the aliasing and independency limitation. The aliasing effect limitation can be circumvented by using a random balance design [1]. For the independence limitation, it can be overcome by reordering [2]. In this paper, we propose to synthesize the two improvements using both a random balance design and reordering. In this way, FAST can be a general first-order global sensitivity analysis method for linear/nonlinear models with as many correlated/uncorrelated parameters as the user specifies. Section 2.1 is a brief review of the traditional FAST method. We introduce the improvement to overcome the independence limitation in Section 2.2 and the improvement to overcome the aliasing effect limitation in Section 2.3. In Section 2.4, we propose to synthesize the two improvements described in 2.2 and 2.3. We introduce a general rule to select the maximum harmonic order for the synthesized method in Section 2.5 and provide a detailed procedure for the proposed method in Section 2.6. Then, we provide four test cases in Section 3. In Section 4, we discuss the random error effect which is introduced by the random balance design and compare our proposed method with other sensitivity analysis methods for models with correlated parameters. Finally, we summarize the main contribution of this paper in Section 5.

For real problems, it would be common for the parameters of the model to be mutually correlated. This paper synthesizes two recent improvements in FAST to overcome the aliasing effect and the independence limitation; and proposes a general first-order global sensitivity approach for linear/nonlinear models with as many correlated/uncorrelated parameters as the user specifies. The main mechanism of the synthesized FAST is to reorder the FAST parameter sample (based on one common characteristic frequency) to honor a specified correlation structure so that it can be used for models with correlated parameters. By using a common characteristic frequency for all parameters, the synthesized FAST method is not limited by the aliasing effect. Thus, it is computationally efficient and would be a good choice for uncertainty and sensitivity analysis for models with correlated parameters.