Quantile

. New global sensitivity measures based on quantiles of the output are introduced. Such measures can be used for global sensitivity analysis of problems in which  -th quantiles are explicitly the functions of interest and for identification of variables which are the most important in achieving extreme values of the model output. It is proven that there is a link between introduced measures and Sobol’ main effect sensitivity indices. Two different Monte Carlo estimators are considered. It is shown that the double loop reordering approach is much more efficient than the brute force estimator. Several test cases and practical case studies related to structural safety are used to illustrate the developed method. Results of numerical calculations show the efficiency of the presented technique.


Introduction
Global sensitivity analysis (GSA) is the study of how the uncertainty in model output is apportioned to the uncertainty in model inputs. It enables the identification of key parameters whose uncertainty most affects the output. It can be used to rank variables, fix unessential variables and reduce model complexity. Over the years there has been a significant progress in developing global sensitivity measures which quantify the uncertainty of inputs in the uncertainty of outputs [1].
Variance-based method also known as the Sobol' method of global sensitivity indices is one of the most efficient and comprehensive GSA techniques [2]. However, generally variance-based methods require a large number of function evaluations to achieve acceptable convergence. They can become impractical for large engineering problems. Besides, being centered around the first and the second order moments of the output distribution function these methods are not well suited for GSA of problems in which higher moments or other statistical values can play a significant role. These methods also cannot identify variables which are the most important in achieving extreme values of the model output.
A number of alternative GSA techniques have been proposed recently. Derivative based global sensitivity measures (DGSM) have been introduced by Kucherenko and co-authors [3]. Sobol' and Kucherenko [4] proved theoretically that there is a link between DGSM and Sobol' total sensitivity indices. Variance based measures by definition are moment dependent. Borgonovo [5] proposed a moment independent measure.
There are problems in which analysts are interested only in certain regions of output values.
Examples include problems of mechanical engineering (f.e beam under loading), civil engineering (f.e the reliability of buildings under seismic load), environmental science (f.e a dose of contamination of soil, 2 water, air ), chemical engineering (f.e stability behavior of dynamic systems such as chemical reactors). Reliability analysis of system failure in such systems is often based on computing the probability of failure with respect to some performance function g (x), where x is a vector of uncertain variables. In this formulation, g(x) < C denotes the failure state, while g(x) > C denotes the safe state and g(x) = C is known as the limit state. It is possible to reformulate this problem in terms of critical  quantile of the cumulative distribution function (CDF) of the performance function. There are also problems in which  -th quantiles of CDF of the output are used explicitly (reliability analysis, risk analysis in finance, etc ).
For such problems conventional sensitivity measures are not adequate.
In this paper we introduce sensitivity measures based on quantiles of the output. Such measures can be used for global sensitivity analysis of problems in which  -th quantiles are the functions of interest and for problems in which the analysts are interested in ranking of inputs contributing to the extreme values of the output. It is shown that there is a direct link between introduced measures and Sobol' main effect sensitivity indices. We consider two different MC estimators of the introduced sensitivity measures and present the results of numerical tests including two practical case studies related to structural safety.
We note, that accurate and numerically efficient computation of quantiles is a difficult mathematical problem. Importance sampling is a widely used technique for variance reduction of Monte Carlo estimates. The basic idea of importance sampling is to change the sampling distribution so that a greater concentration of samples is generated in a region of the sample space which has a dominant impact on the calculations. Importance sampling has been successfully applied in rare-event simulation.
Glynn suggested to use importance sampling for computing extreme quantiles [6]. He proved a central limit theorem for proposed importance sampling quantile estimators and provided efficiency comparisons in a certain asymptotic setting. A sample-based quantile estimators with adaptive importance sampling was introduced in [7]. Oakley proposed an efficient technique for estimating percentiles of models with uncertain parameters related to water drain design [8]. Design is based on predicting extreme events that are related to the 95 th percentile of the cumulative distribution function of the model output.
This paper is organized as follows: The next Section presents a brief review of Sobol' sensitivity indices. We consider both deterministic and probabilistic approaches. In Section 3 two different sensitivity measures based on quantiles of output are introduced. The brute force MC estimator and the estimator based on the double loop reordering approach are discussed in Section 4. In Section 5 we present Value at Risk which is a measure widely used in financial risk analysis and show its link with importance measures based on quantiles of model output. Linear models with normally distributed variables are considered in Section 6. In this Section we prove a Theorem establishing a direct link between introduced measures and Sobol' main effect sensitivity indices. The results of numerical tests for models given analytically are considered in Section 7. Section 8 presents the results of practical case in which case it is called a decomposition into summands of different dimensions. This decomposition is also known as the ANOVA (ANalysis Of VAriances) decomposition. The ANOVA decomposition is orthogonal, i.e. for any two subsets uv  an inner product and so on.
For square integrable functions, the variances of the terms in the ANOVA decomposition add up to the total variance of the function    Homma and Saltelli [10] introduced the total variance tot y V : One of the most important results obtained by Sobol' is an effective way of computing sensitivity indices using direct formulas in a form of high-dimensional integrals [11]. Sobol' formulas were further improved 5 in a number of papers ( see f.e. [12,13] (2 ) Od .

Probabilistic approach
Consider a model function Here is x a real-valued random variable with a continuous probability distribution function (PDF) Here i X is a realization of the random variable The first order i S sensitivity index for one input has a form: 6 Here i S is defined in terms of variances of conditional expectation and tot i S is defined in terms of expectation of conditional variance are the same sensitivity indices as defined by (4). We note, that the importance measure (6) (although not normalized by V ) was firstly suggested by Iman and Hora [14].

Sensitivity measures based on quantiles of output
Quantiles of the output CDF are used in reliability analysis, risk analysis in finance and some other areas. For such problems conversional sensitivity measures are not adequate.
where () Y y  and () Y Fy are PDF and CDF of the output Y, respectively. Equation (8) can be presented in a different form which is more useful for practical use: The moment independent measure based on quantiles was proposed by Chun et al in [15]: Here CHT does not depend on  and it can be formally written as We define new quantile-based sensitivity measures (1) We note that measures (1) ()  , correspondingly. 7 We also introduce normalized versions of quantile-based sensitivity measures We notice that by construction. These measures can be evaluated explicitly We note a formal structural similarity in definitions of (1) (2) () i q  and a moment independent measure proposed by Borgonovo [5]: Once CDF's are computed, MC/QMC estimates of quantiles can be computed as The Monte Carlo (MC) or Quasi-Monte Carlo (QMC ) estimators (1) () i q  given by (15) and (16) have the form: In practice for the j -th trial we generate two independent points   1 ,..., To achieve a good convergence N should be large, which means that T N depends on N quadratically, hence such a simple brute force algorithm requires high computational efforts.
Oakley proposed an efficient technique for estimating percentiles of models with uncertain parameters in which the output as a function of its inputs is modelled as a Gaussian process [8]. A few initial runs of the model are used for building a metamodel, to choose further suitable design points and to make inferences about the percentile of interest.
Finally, the MC/QMC estimators of integrals (15), (16) have the form: The subdivision in bins is done in the same way for all inputs using the same set of sampled points.

Value at Risk
Finance is one of the potential areas in which the proposed new sensitivity measures can be used. In this Section we briefly consider a risk measure widely used in risk analysis in finance, namely Value at Risk (VaR). VaR is the amount of potential loss with given probability over the specific time period. VaR is a technique used to measure and quantify the level of financial risk within a firm or investment portfolio.
As before we will use two types of distances [] d :

Linear model with normally distributed variables
Consider the following model is the inverse error function. Hence Applying formula (12) we obtain  (2) i Q (14).

Test models given analytically
In this Section we present several test cases given analytically to illustrate the developed method.

Independent inputs
For the first two cases considered in this Section we found analytical values of sensitivity indices, so that they can be used as benchmarks for verification of numerical estimates.  , , , x x x x (in descending order). We note that  The convergence comparison of the brute force (23) and DLR (26) MC estimators show that the brute force estimator is much less efficient than DLR (Fig. 2). Further we present only the results 13 obtained with the DLR method.  Fig. 3 (a).

Test 2. Consider the following model
It is possible to find analytical values for Sobol' sensitivity indices The values of quantile measures versus  are shown in Fig. 3(b). We notice that dependence of quantile measures versus  is non-linear and non-monotonic. The ranking of variables using

Correlated inputs
All presented formulas can be used in the case of correlated input variables. Estimation of Sobol' global i S 17 sensitivity indices i S , tot i S for models with dependent variables was proposed in [20].
Test 5. Consider the following model For numerical test we used the following parameters: The numerical values of Sobol' sensitivity indices and Borgonovo's measure are given in Table 2.

Practical case studies
In this section we consider applications of proposed measures to practical test cases related to structural safety.
Case study 1. Roof Truss structure. Consider a roof truss structure shown in Fig. 7 (a)  Uniformly distributed load q can be transformed into the nodal load P=qL/4 , where L is the length of the steel bar ( Fig.7 (b)). The perpendicular deflection C  of node C can be is obtained through basic structural mechanics. It the following function of the input random variables , , , representing respectively sectional areas and elastic moduli of the concrete and steel bars: Considering as the safety criteria the deflection C  not exceeding an admissible maximal deflection 3 cm, the performance response model can be constructed by using the limit state function () gX = 0.03-C  . We assume that all the input variables are normally distributed with distribution parameters given in Table 4.  PDF of the model random response () gX is skewed ( Fig. 8 (a)) which causes non monotonic nonlinear behavior of i Q versus ( Fig. 8 (b)). The ranking of variables determined by , and is the same with q being the most important and L being the least significant input, respectively. The ranking of variables using (1) i Q and (2) i Q measures is different comparing with that of (or ) for only for inputs 3 ( S A ) and 4 ( S E ), although the values of sensitivity indices of these two inputs differ in less than 3%.  Table 5.

 
, ranking is given in brackets. Creep-fatigue failure model PDF of the output () gX is symmetric ( Fig. 9 (a)). It results in practically linear behavior of i Q versus ( Fig. 9 (b)). The ranking of variables determined by all sensitivity measures is the same with being the most important and being the least significant input, respectively.