Stochastics and StatisticsGeneralized moment-independent importance measures based on Minkowski distance
Introduction
Importance measures are very useful in ranking components in complex systems, especially when further improvement decisions have to be made. It is closely related to sensitivity analysis (Castillo et al., 2008, Helton et al., 2006, Kleijnen, 2005, Kleijnen and Helton, 1999, Saltelli et al., 2000, Xie, 1987, Xie and Bergman, 1992). With the information of the relative importance of the parameters, one can quickly identify the improvement for the system performance instead of directly searching for a global optimal solution (Borgonovo, 2010, Kuo and Zhu, 2012a). The importance measures in reliability analysis can be classified into structure importance measure, reliability importance measure and lifetime importance measure (Kuo & Zhu, 2012b). A variety of importance measures have been proposed for different systems and different purposes, such as the traditional Birnbaum measure (Birnbaum, 1968) and Barlow–Proschan measure (Barlow and Proschan, 1975, Eryilmaz, 2013), the joint importance of components (Gao et al., 2007, Hong et al., 2002), the importance measures for multi-state systems (Levitin et al., 2003, Si et al., 2010, Si et al., 2012) as well as the more recently importance measures for Markovian systems (Do Van et al., 2008, Do Van et al., 2009, Do Van et al., 2010) or for semi-Markov systems (Distefano et al., 2012, Hellmich and Berg, 2013). On the other hand, the importance measures can be considered as “local” or “global” from the viewpoint of sensitivity analysis. The global sensitivity analysis techniques, including non-parametric techniques (Storlie and Helton, 2008, Storlie et al., 2013, Storlie et al., 2009), variance-based techniques (Iman, 1987, Li et al., 2011, Sobol, 2001, Sobol, 2003, Zhou et al., 2013) and moment-independent techniques (Borgonovo, 2007), provide an attractive perspective in identifying the influence of the inputs on the output.
The moment-independent importance measures, which consider the impact of a certain parameter on the distribution of the output and bear the merits of model-free and moment-independent, have been proposed recently and received considerable attention. Borgonovo (2007) introduced a probability density function (PDF)-based importance measure, whereas Liu and Homma (2010) proposed a similar importance measure but in terms of the cumulative distribution function (CDF). In order to carry out the sensitivity analysis of the parameter inside the distribution function, Cui, Lu, and Wang (2012) suggested a CDF-based importance measure utilizing the distance between the original CDF and the conditional CDF instead of distance.
In fact, all these importance measures weigh the importance of the parameter in terms of the distance between the conditional distribution function and the original distribution function of the output, where the distance can be distance or distance. So it is natural to extend the measure into a more general case by Lp distance, i.e. Minkowski distance. On the other hand, since the distribution of the output can be characterized by its PDF, CDF or quantile function, it is reasonable to define importance measures on any one of these functions. With these considerations, three classes of generalized Minkowski distance based (MD) importance measures, i.e. PDF-based MD importance measure, CDF-based MD importance measure and quantile-based MD importance measure are proposed. The generalized MD importance measures have considerable flexibility and the existing moment-independent importance measures can be seen as their special cases. The properties of the generalized MD importance measures are studied and some new and promising moment-independent importance measures are derived from the generalized MD importance measures.
The remainder of this paper is organized as follows. In Section 2, three unified MD importance measures are proposed. In Section 3, properties of the proposed MD importance measures are investigated. In Section 4, some new importance measures derived from the generalized MD importance measures are discussed and illustrated with case studies. Conclusions are given in the end.
Section snippets
Minkowski distance (MD)
Minkowski distance is derived from the well-known Minkowski inequality. In general, the Minkowski distance of order p could be defined as:where g1(x) and g2(x) are functions of x and is the range of the integration. For instance, if is an index set, , and g1(x) and g2(x) are real, then Eq. (1) can be rewritten aswhich is the distance between two points and in Rn. Alternatively, if and g1
Some properties of the proposed MD importance measures
In this section, we focus on the properties of the proposed MD importance measures and some analytical results are shown. They are useful for the applications of the proposed MD importance measures. Proposition 1 is monotonically increasing with p ⩾ 1, i.e. holds for any 1 ⩽ p1 < p2. Proof Let . The derivative with respect to p is
Some new measures derived from MD importance measures
The most commonly used cases of the Minkowski distance are these with the order set to 1, 2 and ∞, which are rectilinear distance, Euclid distance and Chebyshev distance, respectively. Therefore, it is intuitive to use the MD importance measures with order 1, 2 or ∞ for practical applications. The MD importance measures with p = 1, i.e. and , which measure the area bounded by the original distribution function and the conditional distribution function, have clear
Conclusions
This paper presented a general formulation of moment-independent importance measures used in system reliability analysis. It is noted that almost all the existing moment-independent importance measures are based on Minkowski distance (MD). We have thus generalized the definitions of the existing importance measures by Minkowski distance. Three classes of MD importance measures are proposed: PDF-based, CDF-based and quantile-based MD importance measure. Their properties are investigated and some
Acknowledgements
The authors are particularly grateful to the editor and three anonymous reviewers who gave useful suggestions and considerably helped in improving the manuscript. This research was supported by Grant 11001005 from the National Nature Science Foundation of China, the Fundamental Research Funds for the Central Universities under Grant No. YWF-14-KKX-008, the Innovation Foundation of BUAA for PhD Graduates under Grant No. YWF-14-YJSY-035 and a research grant from City University of Hong Kong
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