تجزیه و تحلیل حساسیت با استفاده از کمک به نمونه طرح واریانس: برنامه کاربردی برای مدل چکش آب

This paper presents “contribution to sample variance plot”, a natural extension of the “contribution to the sample mean plot”, which is a graphical tool for global sensitivity analysis originally proposed by Sinclair. These graphical tools have a great potential to display graphically sensitivity information given a generic input sample and its related model realizations. The contribution to the sample variance can be obtained at no extra computational cost, i.e. from the same points used for deriving the contribution to the sample mean and/or scatter-plots. The proposed approach effectively instructs the analyst on how to achieve a targeted reduction of the variance, by operating on the extremes of the input parameters' ranges. The approach is tested against a known benchmark for sensitivity studies, the Ishigami test function, and a numerical model simulating the behaviour of a water hammer effect in a piping system.

Let us consider a simulation model represented by an input/output function Y=G(X), where Y is a scalar model output and X=(X1,X2,…,Xn) defines the generic vector of n input parameters. The values of the model parameters are not perfectly known; in other words, they are affected by some uncertainty. Therefore, although it is not the case, each input can be considered as a random variable characterized statistically by its probability density function pi(Xi). The model output Y can also be thought as a random variable; the estimation of its pdf is the objective of uncertainty analysis and is carried out by evaluating the function G(X) on a sample of points generated from the pdf pi(Xi). The analysis described here is in the context of deterministic models. Probabilities have only been introduced to represent imprecision about model parameter values. Graphical sensitivity tools provide valuable information on the relationship between uncertain model inputs and model outputs. In 1989, Sacks et al. proposed the use of scatter plots, i.e. projections on a two-dimensional plane of the hyper-surface describing the input/output mapping [1]. A number of other graphical techniques are discussed in [2], [3], [4] and [5]. In 1993, Sinclair [6] introduced the contribution to the sample mean plot (CSM) which was further developed by Bolado-Lavin et al. [7]. The idea behind CSM is to use a given random sample of the input parameters—that is generally used for uncertainty analysis, to draw conclusions about the sensitivity of the model output. This paper presents an extension of CSM plots, called “contribution to sample variance plots” (CSV). When investigating input–output relationship, CSV contains a considerable higher amount of information than that provided by standard global sensitivity indices [8]. Once the most important input has been detected, global sensitivity indices do not inform the analyst about how to act operatively in order to reduce the range of uncertainty of the important input for a given target reduction of the output variance. Contrarily, CSV does give us the amount of the variance reduction that would be achieved for any arbitrarily chosen restriction of the input uncertainty range. The CSV measure is defined and its properties are described in Section 2. The interpretation of CSV is given in terms of change of variance that can be achieved by trimming the range of the input parameters in Section 3. The CSV plot can use the same sample points utilized for the CSM, for the scatter-plots, and for uncertainty analysis in general. Therefore, new information can be obtained without additional model evaluations. As the CSM is a powerful tool to identify local regions of the input space that contribute substantially to the mean value of the model output, the CSV is powerful to localize areas where the contribution to the variance of the model output is considerable. The theoretical results are shown on the Ishigami test function in Section 4. The CSV technique is also applied to a numerical model simulating the behaviour of a water hammer in a pipeline. The model is described in Section 5 and the results of CSV and the comparison with the results from CSM are shown in Section 6. Table 1 presents notation used throughout the article. Table 1. Notation. n Number of parameters N Sample size Xi Model parameter i Y Model output value V(·) Variance E(·) Mean value p(·) Probability density function F(·) Cumulative distribution function G(·) Model function CSMXi(·) Contribution to sample mean for parameter Xi CSVXi(·) Contribution to sample variance for parameter Xi q quantile

The CSV technique was developed as an extension of the CSM plot. The definition of the CSV proposed in this article is able to provide information how much the variance as defined in (5) with respect to the full range mean can be reduced and what is the most effective way to do it. The theoretical study of Ishigami function showed some new findings which are not visible by applying mean based methods like CSM or first order FAST indices, e.g. importance of the parameter X3 is neglected as having zero effect on the mean of Y, but it has a significant effect on the variance of Y, which can be quantified by using the CSV. The CSM and CSV techniques were applied to water hammer model case study. The CSV plots indicate a number of new findings compared to the CSM plots and provide the most effective ways to reduce variance of the model output. The results were also compared to a sensitivity study by using EFAST method. In a number of cases the results of this complex model study indicate that the CSM and CSV tools complement each other and both should be studied in a single framework.