1 Tokyo Daily Rainfall (mm) Days (mm)

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1 ( ) r-taka@maritime.kobe-u.ac.jp

2 1 Tokyo Daily Rainfall (mm) Days (mm)

3 Tokyo, 1876 Daily Rainfall (mm) Tokyo, 2013 Daily Rainfall (mm) (mm)

4 Tokyo Annual Maximum Daily Rainfall (mm) Year (mm)

5

6

7 [1] Coles, S. G. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer. [2] Katz, R. W., Parlange, M. B. and Naveau, P. (2002). Statistics of extremes in hydrology. Adv. Water Resour 25, [3] (2015). A. B. C

8 GEV GP (PP )

9 2 Annual Maximum Series AMS Partial Duration Series PDS

10 Years f(x) = F (x) AMS

11 Years f(x) = F (x) u PDS

12 X F (x) = P (X x) f(x) AMS n X 1, X 2,..., X n F Z n = max { X 1, X 2,..., X n } P (Z n z) = F n (z) n PDS u X u X > u u P (X u y X > u) = F (u + y) F (u) 1 F (u), y > 0.

13 U(0, 1) Mrn 30

14 Pa(1, 3) Mrn 30

15 Z n a n > 0, b n R (n = 1, 2,...) G(x) Z Z n b n a n d Z : n. P ( Zn b n a n ) x P (Z x) = G(x). G (extreme value distribution) F G F D(G) (a n, b n )

16 G ξ (z) = { exp[ (1 + ξz) 1/ξ ], ξ 0, exp[ exp( z)], ξ = 0. F n ( ) Zn b n P x = P (Z n a n x + b n ) = F n (a n x + b n ) G ξ (x). a n a n x + b n = z P (Z n z) = F n (z) G ξ ( z bn a n ). Z n b n a n G ξ F D(G ξ ).

17 F F D(G ξ ) ξ a n b n F a n = σ b n = µ ( ) { [ ( )] } 1/ξ z µ z µ G ξ = exp 1 + ξ σ σ (µ, σ, ξ)

18 (generalized extreme value) GEV(µ, σ, ξ) < µ < σ > 0 < ξ < G(z) = exp { [ 1 + ξ ( )] } 1/ξ z µ σ = G ξ ( z µ σ G ξ GEV(0, 1, ξ) G ξ (z) = exp [ (1 + ξz) 1/ξ], 1 + ξz > 0, µ σ ξ AMS ),

19 GEV(µ, σ, ξ) G ξ ((z µ)/σ) ξ < 0 Weibull z < µ σ/ξ ξ = 0 Gumbel < z < G 0 ((z µ)/σ) = lim ξ 0 G ξ ((z µ)/σ) = exp{ exp[ (z µ)/σ]} ξ > 0 Fréchet z > µ σ/ξ GEV(0, 1, ξ) G ξ (z) g ξ (z) = { (1 + ξ z) 1/ξ 1 exp [ (1 + ξ z) 1/ξ], 1 + ξz > 0, ξ 0, exp [ z exp( z) ], z R, ξ = 0.

20 GEV( 2.5, 1, 0.4) 0 GEV(0, 1, 0) GEV(2.5, 1, 0.4) 0

21 (Generalized Pareto, GP) H ξ (x) = { 1 (1 + ξx) 1/ξ, ξ 0, 1 e x, ξ = 0. F D(G ξ ) u σ u > 0 P (X u y X > u) H ξ (y/σ u ). ξ G ξ H ξ

22 (generalized Pareto) GP(σ, ξ) σ > 0 < ξ < H(y) = 1 ( 1 + ξ y ) 1/ξ ( y ) = Hξ, 1 + ξy/σ > 0. σ σ H ξ GP(1, ξ) H ξ (y) = 1 (1 + ξy) 1/ξ, 1 + ξy > 0, σ ξ PDS

23 GP(σ, ξ) H ξ (y/σ) ξ < 0 0 < y < σ/ξ ξ = 0 0 < y < H 0 (y/σ) = lim ξ 0 H ξ (y/σ) = 1 e y/σ, ξ > 0 0 < y < GP(1, ξ) H ξ (y) h ξ (y) = { (1 + ξ y) 1/ξ 1, 1 + ξy > 0, ξ 0, exp( y), 0 < y <, ξ = 0.

24 GP(1, ξ) ξ = 0.4, 0, 0.4

25 3 GEV) (GP) n

26 GEV) {z 1, z 2,..., z n } GEV(µ, σ, ξ) = +

27 GEV(µ, σ, ξ) l(µ, σ, ξ) = n log σ (1 + 1/ξ) n i=1 [ 1 + ξ n log i=1 ( zi µ σ [ 1 + ξ )] 1/ξ 1 + ξ(z i µ)/σ > 0, i = 1,..., n. ( )] zi µ ( µ, σ, ξ) R σ

28 GEV(µ, σ, ξ) I(θ) = I(µ, σ, ξ) (Prescott and Walden, 1980) 1 σ 2 ξ 2 ξ 2 p ξ { Γ(2 + ξ) p } ( ) p σξ ξ q [ Γ(2 + ξ) Γ(2 + ξ) + p σ + q p ] ξ 1 + γ σ 2 [ π ξ ( 1 γ + 1 ξ ) ] 2 2q ξ + p ξ 2 θ = (µ, σ, ξ) Γ( ) ψ(r) = d log Γ(r)/dr p = (1 + ξ) 2 Γ(1 + 2ξ) q = Γ(2 + ξ){ψ(1 + ξ) + (1 + ξ)/ξ} γ = Euler

29 {GEV(µ, σ, ξ), µ R, σ > 0, ξ R} ξ > 0.5 (Smith, 1985) ξ 0.5 Hosking et al. (1985) 0.5 < ξ < 0.5 (2010) 0.4 < ξ < 0.6 θ = ( µ, σ, ξ) ξ > 0.5 θ N(θ, I(θ) 1 /n)

30 /T * * G g 1/T z_t * z T T G

31 GEV(µ, σ, ξ) 1 1/T z T ( ) zt µ G(z T ) = G ξ = 1 1/T σ z T = { µ + σ {[ log(1 1/T ) ] ξ 1 }/ ξ, ξ 0, µ + σ { log [ log(1 1/T ) ]}, ξ = 0. z T (return period) T (return level) T = 100 z n T

32 z T (µ, σ, ξ) ẑ T = µ + σ {[ log(1 1/T ) ] ξ 1 }/ ξ, ξ 0, µ + σ { log [ log(1 1/T ) ]}, ξ = 0. ξ 95% { ξ : 2 { l( µ, σ, ξ) max µ, σ l(µ, σ, ξ)} χ 2 1(0.05) }

33 i = 1, 2,..., n µ(t i ) = α 0 + α 1 t i + α 2 t 2 i, σ(t i ) = exp(β 0 + β 1 t i ), ξ(t i ) = γ 0 + γ 1 t i. t i z i (α 0, α 1, α 2, β 0, β 1, γ 0, γ 1 ) M ijk (i = 0, 1, 2, j = 0, 1, k = 0, 1) M ijk µ(t) log σ(t) ξ(t) i j k M 110 µ(t i ) = α 0 + α 1 t i, σ(t i ) = exp(β 0 + β 1 t i ), ξ(t i ) = ξ = γ 0 σ(t) = exp(β 0 + β 1 t) σ(t) > = 12 AIC

34 (GP) {y 1, y 2,..., y n } GP(σ, ξ) GP(σ, ξ) l(σ, ξ) = n log σ (1 + 1/ξ) n log(1 + ξ y i /σ), i=1 1 + ξ y i /σ > 0, i = 1, 2,..., n. ( σ, ξ)

35 GP(σ, ξ) 1 (1 + ξ)(1 + 2ξ) ( (1 + ξ)/σ 2 1/σ 1/σ 2 ). ξ > 1/2 n (σ, ξ) 1 n ( ) 2σ 2 (1 + ξ) σ(1 + ξ) σ(1 + ξ) (1 + ξ) 2 Smith, 1985

36 (threshold) Y GP(σ, ξ) ξ < 1 E(Y ) = ω 0 (1 H ξ (y/σ))dy = ω 0 ( 1 + ξ y ) 1/ξ σ dy = σ 1 ξ. ω = sup{y H ξ (y/σ) < 1}

37 u > 0 Y u Y > u P (Y u > y Y > u) = 1 H ξ((y + u)/σ) 1 H ξ (u/σ) ( y = 1 + ξ σ + ξu ) 1/ξ = ( 1 + ξ(y + u)/σ ) 1/ξ ( 1 + ξu/σ ) 1/ξ ξ GP(σ + ξu, ξ) Y u Y > u GP(σ u, ξ u ) σ u = σ + ξ u u, ξ u = ξ σ = σ u ξ u u, ξ = ξ u

38 e(u) Y (mean excess) e(u) = E(Y u Y > u) Y u Y > u GP(σ + ξu, ξ) e(u) = σ + ξ u 1 ξ = σ 1 ξ + ξ 1 ξ u u ξ = 0 e(u) ê n (u) ê n (u) = 1 N u n (X i u) +, N u u i=1 X 1, X 2,..., X n (a) + = max(a, 0)

39 1) u u GP(σ u, ξ u ) ( σ u, ξ u ) σ = σ u ξ u u ξ u u 2) u u

40 m F F F (y p ) = F (VaR p ) = 1 p y p = VaR p (Value at Risk) p = 1/m m m y 1/m n u y p

41 /m * * F(u) F f u H : GP * y 1/m y 1/m m F

42 F (x) = P (X x) x > u P (X x) = P (X u) + P (u < X x) = P (X u) + P (u < X x) P (X > u) P (X > u) = P (X u) + P (X u x u X > u) P (X > u) u P (X u x u X > u) GP H ξ ( ) x u F (x) = F (u) + H ξ [1 F (u)] σ ζ u = 1 F (u) F (y p ) = 1 p { (ζu ) ξ 1} y p = u + σ ξ p

43 u N u GP(σ, ξ) ( σ, ξ) ζ u N u /n y p ( ŷ p = u + σ ξ ζu p ) ξ 1 ξ 95% {ξ : max σ l(σ, ξ) l( σ, ξ) 1.921}.

44 Point Process u PP

45 4 F X 1, X 2,... F D(G ξ ) a n > 0 b n R lim n[1 F (a nz + b n )] = log G ξ (z) = (1 + ξz) 1/ξ n 1 F (a n z + b n ) (X i b n )/a n z n[1 F (a n z + b n )] n (X 1 b n )/a n,..., (X n b n )/a n z n z

46 F X 1, X 2,... Z n = max 1 i n X i a n > 0 b n R P { (Z n b n )/a n z } G ξ (z) = exp[ (1 + ξz) 1/ξ ], n α ω F {( i n + 1, X ) i b n a n } : i = 1,..., n n z > α [0, 1] (z, ω) A = [t 1, t 2 ] (z, ω) ([t 1, t 2 ] [0, 1]) Λ(A) = (t 2 t 1 )(1 + ξz) 1/ξ

47 (a n, b n ) (σ, µ) P n = {( ) i n + 1, X i } : X i > u, i = 1,..., n ( Xi b n P (X i > u) = P a n > u b n a n ) = P ( Xi b n > u µ ) a n σ x = (u b n )/a n u = a n x + b n ω F = sup{x F (x) < 1}, n u

48 P n u A = [t 1, t 2 ] (u, ω) Λ(A) = (t 2 t 1 ) [ 1 + ξ ( u µ σ )] 1/ξ P PP(µ, σ, ξ) u u n y A = [0, 1] (u, ω) { (t1, x 1 ),..., (t N(A), x N(A) ) } A P n P

49 n y Λ(A) = n y [1 + ξ ( u µ σ )] 1/ξ (µ, σ, ξ) (GEV) L A (µ, σ, ξ; x 1,..., x N(A) ) = exp { Λ(A) } N(A) i=1 λ(t i, x i ) exp { n y [ 1 + ξ ( u µ σ )] 1/ξ } N(A) i=1 1 σ [ 1 + ξ ( xi µ σ )] 1/ξ 1 λ(t, x) = [1 + ξ(x µ)/σ] 1/ξ 1 /σ ( µ, σ, ξ)

50 (mm) 33 0 (mm) GEV GP PP 138 = = % 33 /138 =

51 Tokyo Daily Rainfall (mm) Days (mm)

52 Tokyo Annual Maximum Daily Rainfall (mm) Year (mm)

53 GEV GEV(µ, σ, ξ) µ = (3.35), σ = (2.58), ξ = (0.075). ξ Fréchet

54 Probability Plot Quantile Plot Model Empirical Empirical Model Return Level Plot Density Plot Return Level f(z) e 01 1e+00 1e+01 1e+02 1e+03 Return Period z

55 G n Ĝ Tokyo Fn(x) x G n (z) = i n + 1, z (i) z < z (i+1).

56 GEV z (1) z (2) z (n) (Probability Plot) {( ) i n + 1, Ĝ(z (i) ) } { [ ( )] : i = 1, 2,..., n, Ĝ(z (i) ) = exp 1 + ξ z(i) µ 1/ ξ}. σ (Quantile Plot) {( Ĝ 1 ( ) ) i, z (i) n + 1 ( Ĝ 1 } : i = 1, 2,..., n, i n + 1 ) [ { ( i = µ + σ log n + 1 )} ξ 1] / ξ. (Empirical) (Model) z (i)

57 Probability Plot Quantile Plot Model Empirical Empirical Model Return Level Plot Density Plot Return Level f(z) e 01 1e+00 1e+01 1e+02 1e+03 Return Period z

58 (Return Level Plot) {( 1 / log ( ) ) i, z (i) n + 1 : i = 1, 2,..., n } T {( 1/ log(1 1/T ), µ+ σ [{ } ξ 1 1/ log(1 1/T ) ]/ ξ 95% ) } : 0.1 < T < 1000 x

59 Gumbel Weibull Fréchet {( [ ( )] ) i log log, z (i) n + 1 } : i = 1, 2,..., n

60 12 AIC AIC M 010 µ(y) = , σ(y) = exp( y ), ξ(y) = 0.097, y = (y 1945)/69, y = 1876,..., 2013 Fréchet

61 Residual Probability Plot Residual Quantile Plot (Gumbel Scale) Empirical Model Model Empirical M 010

62 Tokyo p = Annual Maximum Daily Rainfall (mm) p = 0.01 p = 0.02 median scale Year p

63 Tokyo Daily Rainfall (mm) Days (mm)

64 (GP) GP

65 Mean Excess u

66 Mean Excess u

67 Modified Scale Threshold Shape Threshold

68 u = 46 GP(σ, ξ) σ = (1.14), ξ = (0.044). ξ GEV Pareto

69 Probability Plot Quantile Plot Empirical Empirical Model Return Level Plot Model Density Plot Return level f(x) Return period (years) GP x

70 GEV AIC (σ, ξ) AIC M , σ(t) = exp( t), ξ(t) = 0.226, 0 t [0, 1]

71 Residual Probability Plot Residual Quantile Plot (Exptl. Scale) Empirical Empirical Model Model GP M 10

72 (PP) (PP) PP(µ, σ, ξ)

73 Location Threshold Scale Threshold Shape Threshold

74 PP GP u = 46 µ σ ξ PP(µ, σ, ξ) µ = (2.23), σ = (1.90), ξ = (0.044). ξ GP µ = (3.35), σ = (2.58), ξ = (0.075). PP

75 Probability plot Quantile Plot Empirical Empirical Model Model PP

76 GEV GP 12 AIC M µ(t) = t, σ(t) = exp( t), ξ(t) = 0.226, 0 t [0, 1]

77 Residual Probability Plot Residual quantile Plot (Exptl. Scale) Empirical Empirical Model Model PP M 110

78 6

79 p σ(t) log σ(t) p m n y x 1,..., x n x 1,..., x N(A) p λ(t, x) = [1 + ξ(x µ)/σ] 1/ξ 1 /σ

(m/s)

(m/s) ( ) r-taka@maritime.kobe-u.ac.jp IBIS2009 15 20 25 30 1900 1920 1940 1960 1980 2000 (m/s) 1900 1999 -2-1 0 1 715900 716000 716100 716200 Daily returns of the S&P 500 index. 1960 Gilli & Këllezi (2006).

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