Š ñ { d ñ ` ` U Û M 32 10 Vol.32, No.10 2012 10 Sstems Engineering Theor & Practice Oct., 2012 : 1000-6788(2012)10-2345-06 : TV122.5 : A "!#$%&'(*)+*,-*.*/02143#*5 687:9, ;:<>= (@ACBEDFCBGEHIJHKLFMNCOQP, RS 310058) T U VXWZYZ[]\X^]_X`XacbZdXe]fZgZhZiX, k]lzmznzozpzqzrzs, t]u L- v, wzx hz zz{z Z}Z~ pzqzz `] X, ƒz ZnX ]Z `]ˆX ]ŠZ. k]œz ZŽZ Z Z Z 24h {Z Z rzs, š œ Ÿšœ š p 213mm `œš { 200mm. ª «š ^œ_š`ša b dšeœf g h iš œ ± ² ³š ψ eœf g hš`œä Å Æ Ç VšÈ nš œ, µ š œ¹ g h `œº» ¼ ½ ¾ ~šàœ `. MC,. É Ê, Ë ^œ_š`šaìb dšeœf g h iš]íš ]Î lzï ÐZÑ p qšòœ. Ó Ô ÕÖ^œ_ aìb dšeœf g hš ; l m n o p q ³š ; nš œ ; L- v Application of an improved adaptive importance sampling method in extreme hdrological analsis TONG Yang-bin, XU Yue-ping (Institute of Hdrolog and Water Resources Engineering, Zheiang Universit, Hangzhou 310058, China) Abstract This paper proposes an improved adaptive importance sampling method that can be used for variables following a GEV distribution. The L-moment method is emploed to build the relationship between sample statistics and distribution parameters. In the case stud, the 24-hour design storm of Yungang Basin in Zheiang Province is considered, and the probabilit of extreme precipitation is calculated for two rainfall gauge stations, Jinzhuling and Xianrentan. The results show that the proposed method performs well in simulating extreme rainfall, and the iterative number decreases as the number of samples increases. Compared with traditional MC simulation, the improved adaptive importance sampling method has better efficienc. Besides, this method has huge potential to be used for other distributions. Kewords improved adaptive importance sampling; GEV distribution; extreme hdrological events; L- moment method 1 ØÚÙ Û Ü Ý ß à á â ã ä å æ Ý Û, è é ê ëíìïî ðíìïñ òíìïó ôíìïõ öíìï ø ù ú û ü ý, þ ÿ Ý Ý, 98 ñ ì 2008 ì ó ô ù Û Ü Ý á â ã ä ß., ñ ò! "#$ % & ' ( ) * +, 4 5 6 Û. -., / 0 1 2 3, 7 8 9 ó : ; Ý á â < ä = >@@A @B C Û Ü Ý á â ã ä [1]. ñ D ð@e F@G@H I@H J@K@L M@N OZ F P Q. R S ð T U. V J K W X Y Z [ \ ] ^@_ ` á âzã ä å az Ý d. b / c O e f, Y g h i & k [2 5] ìml n o p M N [6] ì q r [7] ù s ú t, T u U v w x l n z ã ä. / ñ D { ü ý, R S u U / GLUE J } ~ é V M N [8 9], O t! R S Û Ü Ý @d / U, ƒ ß v 2 ˆ @M Š@ Œ@ Ž@ [10]. ñ D F Û ÝZ w x ú. T T U. Ž Û ; M N, O M Š } Pearson- Cì h Pearson- Cì e _ Û, ~ é -. U ú 4 h Kappa M Šíì 5 h Wakeb M Š ù ; ` š œ L 2 c 9 Ÿ : 2010-06-24 : ª «(50809058); ± ² ³ ± µ (2010DFA24320); ¹ ² º» ¼ ½ ¾ (200803351029) À Â Ò : Ã Ä Å (1984 ), Æ, Ç È É Ê Ë, Ì», Í Î Ï Ð : Ñ Ñ Ó Ô ; Õ Ö µ : Ø Ù (1975 ), Ú, Ç È Û Ü Ë, Ý, Í Î Ï Ð : Ñ Ñ Ó Ô, E-mail: uepingxu@zu.edu.cn.
«é _ / d ß ß ß 2346 ßáàáâäãáåáæ Û Ü M Š W ê U R S J K ñ D Û M Š ò, ó ô î R S / ñ D 2 õ öùøùúùûùüäýáþáÿ U 2.1 Û Ü Ý Ý Û Ü Òááè 32 w x. ë D ì í î é ï Ž ð ñ d R S, L ~ e d M N U. á â ã ä Ý ( ) w x, F O Monte Carlo (MC), ð E I H R S Ž Û Ü Ý á â ã ä æ J K. v, U MC J K W S Z ë X l, Û î F G. ) û!, Y g R S Ü " íì R S Eì M # R S ù ù. R S ò U p 2 $ (change of measure) Û Ü Ý á â ã ä, ò U % M Š Ž R S, & ' ( ( ) z F. Œ I H 2 x ] ) ã ä * ß p + h f(x),, w(x) = f(x)/g(x), w(x) - + h, ÿ. /102+ h, g(x) ÿ R S (importance sampling, 3 4 IS) * p + 5. 6 7 8 9 : ; < % = > * + 5 81@2A1B2C D E F (G H I J K L M E F ) N O P Q, R S T U V W X E F 8 Y Z [, \ ] ^ _ ` T D > [10]. ` T 8 c d e f g h : i s(x i ) r k l m H I J K, n = > p = P {s(x i ) r} o p. q W r s t 5 I {s(xi)}, < u v w I {s(xi)} = g(x) z { f(x), } ~ M N ƒ 8 T U, n l 8 ˆ k { 1, x s(x i ) r 0, x s(x i ) < r p = I(x)f(x) dx = I(x)w(x)g(x) dx (2) ˆp IS = 1 N N I{s(X i ) r}w(x i ) (3) d Š 8 Œ ; g Ž 8 = > t 5 g(x), ^ _ ` T D >. 2.2 2 š œ 8 IS e f1@ i, IS ƒ Ÿ m 5 g(x θ), 51B2 θ < e p. e f k ª, ˆ. Bucher [11] Š Y P ƒ 1@2^ ± ² m ³ µ ` T (adaptive importance sampling, AIS) e f. Stadler Ro [12] ¹ ³ºµ ` T» k 5 e f ¼ 5 ef ½¾, À 5 e f ƒ k ½ : K (Ãe ) f KÄ5 f, ÅÆÇÈ8 ÉÊÌËͽ f ˆ8 5, G K KÄ [13] 5, Î Z Ï Ð ` T t 58 5. Ñ Ò Ó Ô Õ K Ä 5 ˆ f Ö ØÇ, Ù X f Ú ~ M mûnyüýtu, ÛTUÚ ˆßà8_áâ t58 5, Ö ãzäå, u ÉÊÌËÍ e f p = > J K ; o æ D 8. 2.2.1 è é ê ë b í î K  f 8 ì g h : 1) Ù l ß à  µ init Ú { ï ð ` T (ñ ò ó ô ƒ ) 8  µ 0, Å Æ ~ M m Û (n ) T U x; ± V W H I E F 8 T U xe, u 5 ne Ê s ; õ ns = 1 Ê s ö 1 Å Æ, ˆ  µ is (1)øùe σ is (1) = > P is (1). ú û ü ý [12] 8 þ ÿ, K  øùe = > 8 g h : ˆµ = ˆσ 2,f = 1 p w(n i ) 1 N f 1 N f (1) p n i w(n i ) (4) (x f,i ˆµf )2 (5) ˆp f = 1 N N g(x i ) (6)
i p h 8 N N N + a b p 8 10, :! 2347 2) õ " # ns = ns + 1,  e $ k % 8 ` T t 5 8 5 Ú Ö ö ns Å Æ, Å Æ & K  a µ is (ns)øùe σ is (ns)øù = > P is (ns). " # í î b + 3) ª, ' ( ) *. 2.2.2 è, - ë K Ä 5 f ; T U 8 Ä 5 ˆ z { ² Â, Ä 5  e 8 ˆ $ k m Å Æ. ` T 8 5, /. Ö h m Å Æ, 0 1 2 3 à K  f 4. K Ä 5 f 8 ð 5 ; : ó ô ƒ 1@76 æ 8 m 8, 9 : Z < t 5 ;1@2 H I J K æ Y Z L M 8 E F, \ ] < Ï Ð Â. 3 =>@BADCEGFIHIJGKILIM N O P Q i 8 ; m R [ 8 ³ºµ ` T ef, u N w S T Š } U V \ óô ƒ, ] d c1@7w X U a [13 15] [16 17] ; N V \ ó ô ƒ 8, Y u ; Z [ 9 \ ] 8 ^ ü U, g _ ` ø N ˆ b c Ô Ô. Š H I d c J K 8 e > ƒ 1@, f g H ƒ (generalized extreme value, GEV) ; 7 8 m, Æ ) D É h N [18 20]. f g H ƒ 8 = > t 5 à ƒ t 5 g h : u1@, = f(x) = α 1 e (1 k) e (7) F (x) = e e (8) { k 1 log{1 k(x ξ)/α}, k 0 (x ξ)/α, k = 0 GEV ƒ 8 3 5 ƒ k k l 5 køm 5 α n o 5 ξ. k < 0, ; GEV- qºâ ƒ, r s k Frechet ƒ p ; k > 0, ; GEV- t2â ƒ, r s k Weibull ƒ p ; k = 0, ; GEV- u â, h s k Gumbel ƒ. O v Õ f g H ƒ, ^ ± ² w Ö 8 ³ºµ ` Tf. U k ó ô ƒ, ˆ T U 8  e í z ˆ., x & ² % t 5 8 5, /. Ö h m ; { Š H ƒ 1@, ˆ T U 8  e., N Z & t 5 8 5, } ~ m % 8 ˆ e f, U ü s S k K 5 f., H ƒ 8 Ä 5 N Ï Ð ƒ 8  ( m à ó ô ƒ N ), } h N ƒ K Ä 5 f. b í î K 5 f 8 ì g h : 1) ` T. Ù ß à 8 f g H ƒ Å Æ ~ M m Û (n ) T U x; ± V W H I E F 8 (ne ) T U + xe; /. 6 2 H ƒ 5 à T U ˆ 8 \ ƒ 5 kø αø ξ. N O ˆ + K  f K Ä 5 f Î ; Šf ƒ 5 +, T ƒ 5 q W ² L- Š f [21]. \ L- Š f 8 U = Œ ø g 4 Ž L- Š ˆ 8 ˆ Y ü ý [21 22]. & 4 Ž L- Š ˆ., xy : TU8 U 5 (L-CV)ø ô 5 (L-skewness) ` 5 (L-kurtois), u ˆ g h : τ = l 2 /l 1 τ 3 = l 3 /l 2 τ 4 = l 4 /l 3 ] L- Š ø L-CVø L-Skewness L-Kurtois à H ƒ 5 8 \ g h [21] : ˆl 1 = ξ + α[1 Γ (1 + k)]/k ˆl 2 = α(1 2 k )Γ (1 + k)/k ˆl 3 = 2(1 3 k )/(1 2 k ) 3 ˆl 4 = [5(1 4 k ) 10(1 3 k ) + 6(1 2 k )]/(1 2 k ) + 6 2 (11)H ƒ 8 5, $ k Å Æ 8 ß à 5. 2) õ ns = 1 Ê s ö 1 Å Æ, í î 1) @28 ß à 5 $ k ` T = > ƒ 8 5, M n T U, ± u1@28 H I T U, (10) ø (11)ˆ H ƒ 8 5, ƒ k k is (1)ø α is (1) ξ is (1); ] ö 1 Å Æ. H I J K 8 = > p is (1) (6)ˆ. 3) ns = ns+1, í î 2) @ ˆ & ± 8 5 $ k % 8 ` T t 5 8 5 Ú Ö ö ns Å Æ. k d Å Æ É 8 [, ƒ m Å Æ 8 É, À ö ns Å Æ ˆ 8 5 = > k k i (ns)ø α i (ns)ø (9) (10) (11)
F a " b ³ 8 2348 Ĩ IšG IœI IIŸI ξ i (ns) p i (ns). ] ö ns Å Æ. ` T t 5 8 5 s k k is a (ns)ø α is (ns)ø ξ is (ns) p is (ns). í î b + 4) ª 3), ' ( ) *, Ï Ð O û Y ó 2 8 z K  f. 32 [13], 5 = > Ê 4 4.1 ª «- ± ² ³ F T µ ¹ º» ¼, k % ¹ ½, ³ O ¾ ³ F 251.8km 2, ì  ³ F Ÿ Ã Ä Å Æ Ç â È É, c Ê Ë, d ^ ;1@, ; ¹ º b c1@2a S m. U ü \ ] 8 } U k 24 p [22] d c x, ] x > a ; m H I d c J K, a k m r d c. W Ì Í Ô ± ² ³ L- Š f ˆ F 8 ˆ b c, & ² N d Î 8 ˆ b c, Ï Ð u É Ê1Ë 24 p ˆ b c V \ f g H ƒ. U ü Ñ ] i ³ 24 p ˆ b c 8 5 Ò ƒ t Ó k f g H ƒ, Ú ƒ w Ö ³ºµ ` T f h G K Ó ± ² ³ f Å Æ H I J K 8 Y [. Ê 1 9 s k F Ô Õ Ö Ø Ù ½ c Ú 8 ß à ƒ Ó ø a 8 Û x > a m J K Š 5 Ò 8 = >. ` T T U Ó n ƒ Û 200ø 500ø 1000ø 2000; Å Æ 8 Ü W Ý Û 0.001. 1 ß àáâ ã ä å æ è é ê ë ì í î ï ð ñ ò ó ô õ ò ó ö ò ó a ú û ü ý þ ÿ œ ø ù 0.127 30.368 96.400 213.0 5.124E-3 0.131 91.464 28.551 200.0 5.224E-3 62 km, À  d 1.45%. 4.2 Õ ½ Ú ø N ` T Ó Î Ö ² ½ Å Æ. Ê 2 Ê 3 ; K Ó f Å Æ 8 ˆ É, ± ² \ ] 8 Ó, f Ï Ð H ƒ 8 Ó, : H I J K 8 = > Ô. 2 ß àáâ ã ä å æ è é ó ó ð ñ ò ó ô õ ò ó ö ò ó ø ù (%) 200 84 0.263 7.359 223.413 5.100E-3 0.463 200 77 0.294 6.077 221.192 5.122E-3 0.034 500 65 0.173 6.737 222.652 5.126E-3 0.055 500 59 0.237 6.339 221.855 5.099E-3 0.475 1000 40 0.232 6.803 222.178 5.130E-3 0.135 1000 35 0.197 6.706 222.335 5.121E-3 0.058 2000 35 0.300 6.867 221.889 5.133E-3 0.182 2000 27 0.225 6.405 221.753 5.133E-3 0.187 3 ß àáâ ã ä å æ è é ó ó ð ñ ò ó ô õ ò ó ö ò ó ø ù (%) 200 102 0.273 5.477 207.690 5.205E-3 0.365 200 93 0.201 6.727 209.211 5.241E-3 0.319 500 59 0.314 5.169 206.799 5.213E-3 0.215 500 60 0.296 5.572 207.573 5.208E-3 0.305 1000 48 0.217 6.178 208.505 5.235E-3 0.210 1000 44 0.198 6.039 208.560 5.227E-3 0.055 2000 36 0.210 6.195 208.540 5.227E-3 0.044 2000 23 0.175 6.291 208.903 5.221E-3 0.070 Ê1@2ö 2 ; f 9 ~ 8 z Ó, Y :1Ë ± : ` T Ó, 9 ~ 8 z Ó. Õ H ƒ 8 Ó, k l Ó Š Å Æ 8 U"!, ] m Ó n o Ó 4 Õ"#. $ 8 Ý ; r Å Æ 8 = > Ã Ê 1 @75 Ò = > 8 4 Õ Ý p, f %, 4 Õ Ý Î p 0.5%, Ê1Ë2Å Æ = > ¼ 7 Ü Q 5 Ò = >, h G w Ö ³ºµ ` T f Z o & Å Æ H I J K 8 L M. 6 2('2 h Y ),
«í X c b " ƒ X ª V X & 10, :! 2349 ` T Ó, Å Æ N, g n k 500 8 ½ É 4 o p ; ` T Ó k 2000, Å Æ É. 4.3 * +1 2, - k ² Ö m È. w Ö 8 ³ºµ ` T e f 8 D É ˆ, 8 Monte Carlo (MC) e f, G } ` T e f Ú Å Æ X H I J K, Ï Õ ½ e f 8 Å Æ É Ö ('2. Å Æ T U Ó \ 1000 / à, /. ƒ Û 1 0 ø 10 0 ø 50 0 ø 100 0 ø 500 0. Å Æ É g Ê 4 9 s. \ Ê 4 @2Y : ±, Å Æ = > 8 4 Õ b 5 Ý T U Ó8 R S] 1p, { 1 p 2 3 2 u 3 4. g É Ý Š 1% : h 6 ` T 7 É, N O < MC f 8 ~ 100 0 : 6 9 :, ; ; _ Ó f 9 ~ 6 9 : Ó.= F : ü(>6 24 p 2 d c @ A B"6 U, MC f ` Û C 0 9 : D E"6 Ï"G H, I É @ ª < U V W X (>6 U, J K L M N 6 O P. Q R S T N N 1 Y Z [ 9 \, ] ^ _ ` a [ 9 7 b. 4 defghi MC klmnopq rstu vwx z{ } ~ } ~ (%) 1000 0.004 28.088 0.004 30.605 1000 0.0053 3.330 0.0043 21.493 100000 0.00478 7.187 0.00508 2.838 100000 0.00505 1.456 0.00513 1.796 1000000 0.00508 0.976 0.00516 1.303 5000000 0.00513 0.099 0.00524 0.306 (%) 5 ƒ X U V W X W œ Z ˆ Š Œ Ž [ T š, S Ÿ ª «b, ] ± ² ³ µ \ ¹ º» ¼, ½ ¾ Ž S À Z a,  à K Š Œ Ž [ Ä Å X MC À(ÆÇ. È É Ê(Ë, Ì ˆ Š Œ Ž [ _ M Í Î Ÿ Ï Ð Ð Ø, ] ^ N N Ñ Ò Z [ Ó b, Ž Ô Õ Ÿ (Ö S. Ù"Ú U"V"W"X «" "U"V ", T" "Û"Ü"Ý" ""ß"à"á Å"š "â"ã, ä"m"å" "Ž" ", æ""_ ª ««ï V «ö S è Ö, é ê S è ë ì í î. ð à ñ ò(öé K ó ô K ë S õ ë ì Ö. ø ù ú û [1] üý, þÿ, Gemmer M. [J]., 2008, 19(5): 650 655. Jiang T, Su B D, Gemmer M. Trends in precipitation extremes over the Yangtze River Basin[J]. Advances in Water Science, 2008, 19(5): 650 655. [2] Lu D, Yao K. Improved importance sampling technique for efficient simulation of digital communication sstems[j]. IEEE Journal on Selected Areas in Communications, 1988, 6(1): 67 75. [3] Sadowsk J S, Bucklew J A. On large deviations theor and asmptoticall efficient Monte Carlo estimation[j]. IEEE Transactions on Information Theor, 1990, 36(3): 579 588. [4] Chen J C, Lu D Q, Sadowsk J S, et al. On importance sampling in digital communications Part I: Fundamentals[J]. IEEE Journal on Selected Areas in Communications, 1993, 11(3): 289 299. [5] Sadowsk J S. On the optimalit and stabilit of exponential twisting in Monte Carlo estimation[j]. IEEE Transactions on Information Theor, 1993, 39(1): 119 128. [6] De A, Mahadevan S. Ductile structural sstem reliabilit analsis using adaptive importance sampling[j]. Structural Safet, 1998, 20(2): 137 154. [7],,. r! " # $ % & ' [J]. ( ) $ % *, 2007, 19(18): 4107 4110. Zhou H, Qiu Y, Wu X J. Rare event simulation method based on importance sampling technolog[j]. Journal of Sstem Simulation, 2007, 19(18): 4107 4110. [8] Kuczera G, Parent E. Monte Carlo assessment of parameter uncertaint in conceptual catchment models: The Metropolis algorithm[j]. Journal of Hdrolog, 1998, 211(1/4): 69 85.
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