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1、?31?2?Vol.31, No.2 2011?2?Systems Engineering Theory ?;?;?;? Computational strategy for uncertainty importance measure ranking based on norm XU Xin, LU Zhen-zhou, LUO Xiao-peng (School of Aeronautics, Northwestern Polytechnical University, Xian 710072, China) Abstract The probability density functio
2、n integral of Borgonovo input uncertainty importance measure is hard to calculate. Hereby based on the defi nition of the input uncertainty importance measure, the concept of norm is introduced into the uncertainty importance ranking analysis for the fi rst time, and a new importance ranking computa
3、tional strategy is developed, for which some equivalent norms are selected. This strategy replaces the integral by its equivalent norm, and introduces a kind of regularization computing method for uncertainty importance measure at the same time, which has more applicability. Theoretically, this stra
4、tegy can provide many kinds equivalent norms for estimating the uncertainty importance ranking. Comparisons of present work with Borgonovo method and Liu method subsequently show that the new method is the easiest one. Finally, two examples are illustrated the feasibility of the present work. Keywor
5、dsuncertainty importance measure; sensitivity; cumulative distribution function; probability density function; norm 1? ?,? ?1.? ?.?Apostolakis?1:? ?. Saltelli2?,? ?,?2?37,?Saltelli?Sobol?Rabitz ?34,816?. Saltelli?2?:? ?.?Borgonovo17?,? ?: 2009-09-15 ?:?(50875213) ?:?(1983),?,?,?,?:?,?. 334?31? ?18,?
6、 ?i. Liu?19?i?(CDF) ?,?Borgonovo?(PDF)?18,? ?. ?,? ?.?,?.?2 ?Borgonovo?,?3? ?,?.?4? ?.?5?,?.? ?. 2? ?Y?n?X = (X1,X2,Xn)? Y = g(X1,X2,Xn). x = (x1,x2,xn)?X?. FY(y)?fY(y)?Y? ?(CDF)?(PDF). FY |Xi(y)?fY |Xi(y)?Xi?Y? ?CDF?PDF. fY(y)?fY |Xi(y)?s(Xi)?(?1?),? s(Xi) = ? + |fY(y) fY |Xi(y)|dy(1) ?1 fY(y)?fY |
7、Xi(y)? ?s(Xi)?Xi?,?Xi?PDFfXi(xi)?,? EXis(Xi)?: EXis(Xi) = ? + fXi(xi)s(xi) dxi(2) EXis(Xi)?Xi?PDFfXi(xi)?, Xi?.? ?Xi?0, 1,?Xi?Xi? ?i? 18 i= 1 2EXis(Xi) (3) i?18. i?,? ?,?i,? ?. i= i n ? i=1 i (4) ?2?,?:?335 Borgonovo?18?(3)?,? ?fY(y)?fY |Xi(y),?i,? ?PDF?Borgonovo?. Liu?19?PDF?CDF?,? fY(y)?fY |Xi(y)?
8、s(Xi)?FY(y)?FY |Xi(y)?.? ?,?fY(y)?fY |Xi(y)?m?,?y = a1,y = a2,y = am. ?FY(a1) FY |Xi(a1) 0,?s(Xi)? s(Xi) = 2 ?F Y(a1) FY |Xi(a1) ? ?F Y(a2) FY |Xi(a2) ? + + (1)(m1) ?F Y(am) FY |Xi(am) ? (5) ?FY(a1) FY |Xi(a1) yk)/n(15) T(y yk) = ? 1,y yk 0,y yk (16) ?, k?. Liu?fY(y)?fY |Xi(y)?,?FY(y) FY |Xi(y)(?2?)
9、? ?,?(17)? d(FY(y) FY |Xi(y) dy = 0(17) ?. Liu?FY(y) FY |Xi(y)?,?Y? ?.?,?. ?,?fY(y)?fY |Xi(y)?.?Y? m?,?Y?j?,?j?fY(y)?fY |Xi(y)?.? ?s(Xi)?s?(Xi).?(9)?,?|f(j) Y (y)f(j) Y |Xi(y)| ?.?s(Xi)?. ?2?,?:?337 ?,? ?FY(y)(?fY(y)? ?FY |Xi(y)(?fY |Xi(y),?Xi ?,? ?. ?Borgonovo?,? ?fY(y)? fY |Xi(y)?fY(y)? fY |Xi(y)?
10、 ? ?+ |fY(y) fY |Xi(y)|dy. ?Liu? ?.?,? ?,? ?,?s?(Xi)? ?Borgonovo?Liu? ?. ?2 FY(y) FY |Xi(y)? ?,?,?s?(Xi)?.?,? ?,? ?+ |fY(y) fY |Xi(y)|dy,? ?. 5? ?.? ?,?Ishigami?,?.? Borgonovo?Liu?. 5.1? ?.?: Y = 3 + 0.1X1+ 5X2(18) ?,?X1?X2?. ?N = 100,?Y?M = 60.?,? ?,?50?,?.?1?i? ?,?. ?1?1?i? ?X1X2 ? ?0.00440.9956 1-?0.00560.9944 2-?0.01110.9889 3-?0.00580.9942 -?0.01090.9891 ? ?0.000160.00074 1-?0.000970.00103 2-?0.000370.00042 3-?0.000580.00059 -?0.000240.00031 ? ?
