A new computational method of a moment-independent uncertainty importance measure
Introduction
Uncertainty is an integral part of the concept of risk [1]. The importance of identifying and understanding uncertainty has been well stressed in risk assessment problems [2], [3], [4], [5]. The sources of uncertainty include model structure, model parameters and data uncertainty. Given that the uncertainty of input parameters exists, it will be propagated through the model to the model output and will lead to the output uncertainty. “The study of how the uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input” is the work of sensitivity analysis [6]. In other words, sensitivity analysis addresses “the contributions of individual uncertain analysis inputs to the uncertainty in analysis results” [7], [8]. Given the limited research resources, sensitivity analysis is helpful for a decision maker to identify the most important input parameters that control the uncertainty in the model output. In addition to risk assessment problems [9], [10], [11], [12], [13], [14], sensitivity analysis has also been applied in other fields, including physiochemical systems [15], [16], ecosystems [17], atmospheric environment systems [18], geographic information systems [19], economics [20], etc. A detailed description of the existed sensitivity analysis techniques is given in Ref. [21].
Back to 17 years ago, in their paper on “a robust measure of uncertainty importance for use in fault tree system analysis”, Iman and Hora [22] advocated that an ideal uncertainty importance measure should “provide an unconditional measure of the contribution of an event to the top-event uncertainty”, and should be “easy to interpret in the natural scale of the analysis” and “easy to compute”. In addition, it should also provide “stable results from sample to sample or simulation to simulation”. In short, an ideal measure should be “unconditional, easy to interpret, easy to compute, and stable”. Here, “unconditional” means that a measure is not conditional on any assumed value of a variable. Three measures were discussed in Ref. [22]. One measure is , in which Var(Y) is the variance of the model output Y obtained when all model input parameters (X1,X2, …, Xj, …, Xn) are sampled over their variation range, and E(Var(Y∣Xj)) is the expected variance of Y obtained when the input parameter of interest Xj is fixed. The second measure is , which is log-based. The third measure is a bivariate measure (Rα,R1−α), where and . Yα and Y1−α represent the α and 1−α quantiles of the unconditional distribution of the model output Y, and represent the α and 1−α quantiles of the conditional distribution of Y given that an input parameter Xi is fixed at its nominal value. Iman and Hora [22] commented that none of the three measures satisfies the above requirements and suggested them to be used together to evaluate the uncertainty importance of input parameters.
Similar to Iman and Hora's proposal, Saltelli [6] pointed out that a sensitivity analysis technique should be “global, quantitative, and model free”. Here, “global” has two meanings. One is that, for one input parameter whose uncertainty importance is evaluated, the effect of the entire distribution of this parameter should be considered. The other one is that the importance of this input parameter should be evaluated with all other input parameters varying as well. “Model free” means that the calculation results should be stable and be independent from presumptions about the model, such as linearity, additivity, and so on. Saltelli [6] described two variance-based sensitivity indicators which satisfy the above requirements. One is the first order sensitivity index , which represents the expected fractional reduction of output variance that will be achieved when the input of interest Xi is known [23]. The other one is the total effect sensitivity index , which represents the expected fractions of variance that will be left when all input parameters except Xi are known [23]. The direct contribution to the theoretical and computational development of and can be referred to Refs. [24], [25], [26], [27].
The use of variance as a measure of uncertainty, as Saltelli [6] mentioned, “implicitly assume that this moment is sufficient to describe output variability”. Borgonovo [28] illustrated that variance is not always sufficient to describe uncertainty and pointed out that a sensitivity measure should refer to the entire output distribution instead of a particular moment. In his recent paper, Borgonovo [29] extended Saltelli's three requirements by adding the fourth feature, moment-independence, and proposed a new global sensitivity indicator, δi.
In this paper we propose a new computational method of δi. The remainder of this paper is organized as follows. Firstly, for the readers’ convenience, we give a short introduction about the definition of δi in Section 2. Then we propose a new method for computing δi in Section 3. In Section 4 we give a comparison of this new method with the previous one [29]. In Section 5 we study the feasibility of this new method by applying it to two models and compare the results of both methods. Finally, conclusions of this work are highlighted in Section 6.
Section snippets
Definition of δi [29]
Given a model Y=g(X1,X2, …, Xm), where Y is the model output, and X1,X2, …, Xk are input parameters with uncertainty. The unconditional probability density function (PDF) fY(y) and the unconditional cumulative distribution function (CDF) FY(y) of the model output Y are obtained with all input parameters varying over their variation range. Suppose now that the input of interest Xi is fixed at one value , we can obtain the conditional PDF and CDF of Y, respectively.
The difference
A new computational method of δi
It is known from statistics textbooks that if F(u) and f(u) are the CDF and the PDF of a random variable U, then F(u) can be integrated from f(u):
Here F(u) is non-negative and monotonically increasing, and has the property of 0⩽F(u)⩽1.
Based on Eq. (5), we can convert the area s(Xi) closed by fY(y) and (see Fig. 1) to a distance metric between FY(y) and . In this section we use two cases to show what this distance metric is. We then propose a new method for
A comparison of the two computational methods of δi
From the introduction in 2 Definition of, 3.3 Summary, it is known that the difference between these two methods is on the calculation of s(Xi). In the CDF-based method, to calculate s(Xi), we know from Eq. (16) (or Eq. (17)) that firstly it is necessary to get FY(y) and . FY(y) and can be obtained by Monte Carlo simulations. If the number of samples is n, FY(y) (or ) can be approximated by the empirical distribution Sn(y), which can be easily obtained based on Eqs. (18)
Examples
In this section we apply the CDF-based computational method of δi to two examples to show its feasibility. Section 5.1 is a non-linear, non-monotonic mathematical model and Section 5.2 is a fault tree model. Because we do not know the analytical δi value of each input parameter of both models, the PDF-based method was also implemented for direct comparison of the results of both methods.
In the CDF-based method, the first derivative of was approximated by using the simple
Conclusions
In this study, we analyzed a moment-independent sensitivity measure δi [29] and proposed a new method for computing it. Compared with the previous computational method [29], this new method is simpler and easier to implement with no additional computational cost in terms of number of model evaluations. The applicability of this new method is confirmed with two test models. For simple models, when a large number of samples are needed for performing the analysis, the computational efficiency of
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