untitled

18 1 2,000,000 2,000,000 2007 2 2 2008 3 31 (1) 6 JCOSSAR 2007pp.57-642007.6.

LCC (1) (2) 2

10mm 1020 14 12 10 8 6 4 40,50,60 2 0

1998 27.5 1995 1960 40 1) 2) 3) LCC LCC LCC 1

1) Vol.42No.5pp.29-322004.5. 2) 2003.4. 3) 2005.11.1. 2

LCC 1 3

7 1 2 3 4 5 6 7 1 2 3 4 5 4

6 7 1) 6 JCOSSAR 2007pp.57-642007.6. 5

1) 3.1 2) RC 3),4) (A) 5)(B) (A)(B) 3.1 3.1 (A) (B),,,, 6

3.1 (B) 3.2 6) 7)14) 15) 7

X(t+1) t X(t) RC 9) H9 H13 25 10)14) a,b,c,d,e RC MCI 16)17) 1) pp.94-962005.11.1. 2) 2002. 8

3) No.767-64pp. 35-462004. 4) No.781-66pp.157-1702005. 5) 2003.3. 6) pp.222006.12. 7) JCOSSAR'95 pp.341-348,1995. 8) RC Vol.47App.277-2842001.3. 9) RC 41 4 pp.3-372002-12. 10) RC Vol.7pp. 1141-11482004. 11) No.744-61pp.29-382003.10. 12) BMS LCC Vol.11pp.111-1222004-12. 13) No.801-73pp.69-822005-10. 14) No.801-73pp.83-962005.10. 15) Hiroshi IIZUKAA STATISTICAL STUDY ON LIFE TIME OF BRIDGESProc. of JSCENo.392-9Vol.5No.1pp.51-60April 1988. 16) Vol. 63No. 1pp.1-152007. 17) 61 2006.9. 9

10

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A B C D E 14

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4.3.1 17

LCC LCC 18

n0,1,2,n=0 n=n N+1 X n 5 Bridge Health Index BHI 5 5 3 2 MCI MCI 5 5.2.1 n n X n 5.2.1 5.2.2 19

5.2.2 RC n=0,1,2, X 0,X 1, 1 2 n-1 X n-1 =i X n-1 X n-1 =i n X n =j p ij p ij n-1 X n-1 =i 20

p ij =P(X n =j X n-1 =i),i,js (5.2.1) i j p ij K n-1 n KK 5.2.1 1 5.2.2 5.2.3 1 5.2.4 5.2.3 1 i 5.2.2 5.2.4 5.2.5 5.2.5 1 i j 5.2.5 5.2.5 21

n X n =i n=0 n=n' n' X n' =i p in' =P[X n' =i] i K n' K p(n)=p 1n,p 2n,,p Kn n n+1 X n+1 =i n X n i n+1 X n+1 j p ij =P(X n+1 =j X n =i),i,js (5.2.3) 22

n+1 X n+1 =j n X n =i p in n X n =i n+1 X n+1 =j p ij p in jn+1 =P(X n =i)p(x n+1 =j X n =i)=p in p ij (5.2.4) p jn+1=p(x n+1 =j)=p in p ij (5.2.5) p 1n+1,p 2n+1,,p Kn+1 = (5.2.6) P(n+1)=p(n)P (5.2.7) n P(n+1)=p(n)P=[p(n-1)P}P ==p(0)[p} n+1 (5.2.8) p(0) P P n X n =i m j p (m) ij p (m) ij i j m p (m) ij (5.2.9) P (m) m m P (m) P m = P (m-1) P= P (m-2) P (2) == PP (m-1) (5.2.10) Chapman-Kolmogorov Chapman-Kolmogorov (5.2.11) 23

24

25

D X Z T = 1 0 12 C 0 2 0 25 RC 0 3 1 36 C 1 4 0 5.5. RC 1 5 1 48 C 0 6 0 17 C 0 7 0 30 C 1 8 1 5.5. C 0 9 0 46 RC 1 10 1 53 C 1 N ( i = 1, L, N )i J X i = ( X i1, L, X ij) X ij j T = 0 i K S Y i( S) = ( Yi 1( S), L, YiK( S)) Yik ( S) k 26

M Z = ( Z 1, L, Z ) Z im m i i im X = ( X 1, L, X ) Z = ( Z 1, L, Z ) i i ij i i im ii ( = 1, K, N) ρ i α ( ) ρ = exp X α + Z β, i = 1, K, N (5.3.1) i i i = ( α1, L, α J ) J = ( 1, L, M ) M β β β N i T T T T (Right Censoring) 1970 1980 1990 2000 27

0 10 20 30 N T = ( T %, L, 1 T % N) i c i c= ( c1, L, c N ) i T i c i T % i i = 1, L, N if, T% if, T% i i > c i c i (5.3.2) T = 0 c i T = 0 T% i( ci ) Ti = min{ T% i, ci} (5.3.3) i D i D i = 1{ T i = T% i } (5.3.4) i D i 1 if, Ti = T% i( lifetime) = 0 if, Ti = ci( Censoring) (5.3.5) i t λ() t 28

Pr( t Ti < t+ t t Ti) λ() t = lim t 0 t (5.3.6) dlog S( t) / dt i T i t St () St () = Pr( t< T i ) (5.3.7) Ft () = 1 St () f () t f () t = ds() t / dt dlog S( t) ds( t) / dt f( t) λ() t = = = (5.3.8) dt S() t S() t f () t = λ() t S() t Λ() t t () t λ( u) du Λ = (5.3.9) 0 St ( ) = exp( Λ ( t)) (5.3.10) v( < t) (Truncation) v Sv () St ()/ Sv ( ), f () t / S( v) f () t / S( v) f() t λ() t = = St ()/ Sv ( ) St () (5.3.11) i T % i f ( T% i) = λ( T% i)exp{ Λ( T% i)} (5.3.12) c i D i = 1Ti = T% i T i f ( T) = λ( T)exp{ Λ ( T)} (5.3.13) i i i D i = 0 T i = c i c i ST ( ) = exp{ Λ ( T)} (5.3.14) i D i = 1 D i = 0 i i = 1, L, N i D l = λ( T) i exp{ Λ( T)} (5.3.15) i i i i 29

N N Di L= λ( Ti) exp{ Λ( Ti)} (5.3.16) i= 1 λ() t St () u τ MRL( u) Mean Residual Life MRL( u) = E( τ u u < τ ) (5.3.17) τ u τ τ u u ( t u) f( t) dt u E( τ u u< τ) = (5.3.18) Su ( ) f () t = ds() t / dt [ t ust] ( St ( )) ( t u) f( t) dt = ( t u) [ ds( t) / dt] dt (5.3.19) u u = ( ) ( ) u dt ( t ust ) () ( t ust ) () Stdt () u t t= u u = + (5.3.20) = Stdt () (5.3.21) u t St () u () u MRL( u) = Stdt (5.3.22) Su ( ) t t λ() t t St () 30

ρ > 0, t 0 t λ() t ρ t St () exp ( ρt) t λ() t θ θ, ρ > 0,t ρ t t St () ρ θ t θ κ, ρ > 0, t 0 t λ() t ρ exp( κ t) t St () ρ exp 1 exp t κ { ( κ )} 31

κ, ρ > 0, t 0 t λ() t κρt 1+ ρt κ 1 κ t St () 1 1+ ρt κ 32

κ, ρ > 0, t 0 t λ() t t St () κρ 1 t κ exp ( ρt κ ) T Weibull( ρ, κ) ρ > κ > 0 t 0 λ() t ρκ 1 = t κ (5.3.23) κ > 1 κ = 1 κ < 1 33

Λ () t = ρt κ (5.3.24) St ( ) = exp{ Λ ( t)} = exp{ ρt κ } (5.3.25) T κ q q/ κ q ET [ ] = ρ Γ (1 + q/ κ) ET = ρ Γ + / κ (5.3.26) 1/ κ [ ] (1 1 ) Var[ T ] = ρ Γ (1 + 2 / κ) Γ (1 + 1 / κ) (5.3.27) 2/ κ 2 c > 0 T Weibull( ρ, κ ) ct Weibull( ρc κ, κ) T c Weibull( ρ, κ/ c) M Weibull( ρ, κ ) min{ T, L, T M } Weibull( M ρ, κ ) (minimum 1 stable) N 1 D κ i κ L= ρκti ρt i i= 1 exp{ } (5.3.28) N i= 1 log L Dilog( ) Di( 1) logti T κ = ρκ + κ ρ i (5.3.29) κ ρ 34

( 8 t 3.98 ) St ( ) = exp 3.036 10 (25%-quantile) (75%-quantile) Su ( ) = exp{ ρu κ } κ exp{ ρt } dt u MRL( u) = (5.3.30) κ exp{ ρu } κ IMRL ( u, ρκ, ) = exp{ ρt } dtκ > 0 y = ρt κ 1 κ dt = t dy/ ( ρκ ) ρ > 0, κ > 0 t y( = ρt κ ) t 1 κ 1 1 / κ 1 / κ 1 u 1 κ (,, ) exp{ } t IMRL u ρκ = y dy (5.3.31) κ ρu ρκ = ρ y 11 / κ (,, ) ρ exp{ } 1/ κ 1 IMRL u ρκ = y y dy κ ρκ (5.3.32) ρu Γ ( ν, z) 1 ( z) exp{ w} w ν ν dw ( ν 0 z 0) z Γ, =, >, > (5.3.33) (,, ) ρ 1 IMRL u ρκ = Γ, u (5.3.34) 1/ κ κ ρ κ κ 35

MRL( u) = ρ 1 / 1 κ κ Γ, ρu κ κ κ exp( ρu ) (5.3.35) u = 0 1/ κ ρ 1 0 Γ, / 1 κ κ κ ρ 1 MRL(0) = = Γ, 0 (5.3.36) exp(0) κ κ MRL(0) u 36

Y i n Y i µ i µ i > 0 x i k xi = ( xi 1,..., x ik) x i Y i y exp( µ ) i i µ i Pr( Yi = yi µ i) = µ i > 0, yi = 0,1, 2,... (5.4.1) y! i i µ i k µ = exp( x β ) (5.4.2) i β k i 37

[ ] =, [ ] E y x µ i i i Var y x = µ (5.4.3) i i i n n y exp( µ ) i i µ i L = (5.4.4) y! i= 1 n µ i y i n n y exp( µ ) i i µ i log L = log i= 1 yi! n i [ µ i yilog( µ i) log( yi!) ] [ exp( x iβ) yix iβ log( yi!) ] (5.4.5) = + = + i= 1 i= 1 n β ˆβ [ ] ˆ E y x = exp( x β ), i = 1,..., n (5.4.6) i i i [ x ] E y i x ij i = ˆ β exp( x ˆ β), i= 1,..., n; j = 1,..., k j i (5.4.7) 38

(5.4.6) j x ij 39

52117 52 5.5.1 40

1) 1 2 3 1993, 1998, 20035, 4, 45, 4, 5 4, 4, 5 2 10 1 OK 5, 4, 3, 2, 1 2) 3) i(i=1,,k) S={1,,K} t (t)=i t+1 RC (t+1)=j Pr ob [ ( t + 1) = j( t) = i] = (5.5.1) ij π11 π1k Π = M O M (5.5.2) 0 π KK K j = 1 π = 1 ij RC ς f ς ) F ς ) i i RC i y i+1 λ i ( y i ) y i i ( i i F ~ ( ) i y i i ( i i 41

f ( ~ ) i yi yi λ i( yi ) yi = (5.5.3) Fi ( yi) λ i ( y i ) y i i [ y i, y i + y i ] i+1 RC y i θi > 0( i = 1,, K) λ i( yi ) = θi (5.5.4) RC i y F ~ ( ) i i y i ~ F ( y ) = exp( y ) (5.5.5) i i θ i i τ A i y A i y A ~ z ( 0) i F ( y + z ζ y ) ~ F ( y + z ζ y ) i A i i i A i A i i A = Pr ob{ ζ y + z ζ y } (5.5.6) i A i i F ~ ( ) A i y i ~ Fi ( ya + zi) exp{ θ i( ya + zi)} ~ = = exp( θ izi) (5.5.7) Fi ( ya) exp{ θ i ya} y A i y i ω Pr ob[ ( y ) i ( y ) = i] = exp( Z) B ω A θ = (5.5.8) i = y Z B A + Z Pr ob[ ω( y ) = iω( y ) i] B A = π ii π ii θ i Z y A, yb 2) y A y B i j π π = Pr ob[ h( y ) = j h( y ) i] ij B A = ii j m = K = i m= i θ θ θ k 1 j 1 m θm exp( θ kz) θ θ k m= k m+ 1 k (5.5.9) 42

K 1 π ik = 1 π ij (5.5.10) j = i π ik 5.5.2 5.5.1 5.5.3 43

5.5.4 H0n =0 n H1n0 t- 1 t t-t t->t 0.025 n A n305% t 1.96 2.0 4) 5.5.5 50 5.5.2 44

5.5.2 1 7 7 8 5.5.5 17 3 4 2) 4.62% 1 7 7 5.5.2 18 8 5.5.6 45

5.5.3 5.5.7 5.5.8 t- 5.5.9 50 5.5.4 46

5.5.4 5.5.10 47

5.5.5 5.5.11 5.5.12 t- 48

5.5.13 50 5.5.6 5.5.6 49

CLEN 0.175 0.150 0.125 0.100 0.075 0.050 0.025-5 0 5 10 15 20 25 30 35 50

CWID 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 2 4 6 8 10 12 14 16 18 51

HIRW 0.025 0.020 0.015 0.010 0.005-20 0 20 40 60 80 100 120 140 160 180 52

53

54

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8 7 6 5 4 3 2 y = 0.020x + 2.597 R² = 0.005 8 7 6 5 4 3 2 y = 0.002x + 2.055 R² = 9E-05 1 0 0 10 20 30 40 10.2mm 1 0 0 10 20 30 40 10.2mm 62

0.150 0.125 0.100 0.075 0.050 0.025 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 63

812mm 0.7mm 1mm 2mm 5.5.31 64

good good 65

66

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70

71

10 102020 15 0.5 0.4,0.6 510 50100 72

73

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79 0.688 0.486 0.765 0.680 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 good 23.20 27.15 20.87 23.37 0 5 10 15 20 25 30 BIC good 0.848 0.695 0.868 0.854 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 good 17.42 22.97 16.27 17.09 0 5 10 15 20 25 BIC good

80 0.698 0.504 0.757 0.695 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 good 22.91 26.87 21.13 22.98 0 5 10 15 20 25 30 BIC good 0.893 0.727 0.910 0.901 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 good 14.57 22.09 13.14 13.93 0 5 10 15 20 25 BIC good

81 0.771 0.561 0.827 0.770 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 good 20.72 25.88 18.42 20.72 0 5 10 15 20 25 30 BIC good 0.905 0.758 0.914 0.913 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 good 13.62 21.12 12.79 12.95 0 5 10 15 20 25 BIC good BIC BIC BIC

14 12 10 8 6 4 2 0 40,50,60 14 12 10 8 6 4 2 0 40,50,60 14 12 10 8 6 4 2 40,50,60 0 82

14 12 10 8 6 4 2 40,50,60 0 14 12 10 8 6 4 2 40,50,60 0 14 12 10 8 6 4 2 40,50,60 0 83

yi y% i =, i= 1,..., n Median( y,..., y ) 1 YDAN 0.12 n 0.10 0.08 0.06 0.04 0.02 0 10 20 30 40 50 60 70 80 84

YDEP 0.25 0.20 0.15 0.10 0.05-2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 0.225 LEVEL 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025-12 -11-10 -9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 µ Dan i Dan µ = exp( β + x β ), i= 1,..., n i L 0 i 1 x L i β 0 β 1 85

β 0 β 1 exp(2.10525) = 8.20 µ Pit i µ = exp( β + x β + x β ), i= 1,..., n Pit L Dan i 0 i 1 i 2 x Dan i β 2 x L i β 0 β 1 β 0 β 1 β 2 exp(1.31622) = 3.72 86

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0-10m -8m -5m -3m -2.5m -1.7m -1m 0m 1.1m 1.8m 1m 1m 16 14 12 10 8 6 4 2 0-10m -8m -5m -3m -2.5m -1.7m -1m 0m 1.1m 1.8m 6(24mm^2) 20 (80mm^2 ) 87

24mm^2 6 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 6 24mm^2-1m 10 15 20 80mm^2 88

20 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 20 80mm^2 10 15 20 89

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 0.5 1 4mm^2 0.5 0.20.50.1mm 90

50% 600800 Tohman-Bain 1 5, 4, 3, 2, 1 5.4.8 t 2.04 t 2.0 (1) c = s r r (1) 91

c s r 5.5.1 92

5.5.3 5.5.5 93

94

10 1020 20 10 1020 20 10.2mm 10.2mm 14 40,50,60 12 10 8 6 4 2 0 14 14 12 10 8 6 4 2 0 40,50,60 14 40,50,60 40,50,60 12 12 10 10 8 8 6 6 4 4 2 2 0 0 14 40,50,60 12 10 8 6 4 2 0 14 40,50,60 12 10 8 6 4 2 0 10 5 95

2 2 1H.C. Shin and S. Madanat : Development of A Stochastic Model of Pavement Distress Initiation, JSCE, No.744/IV-61, pp.61-67, 2003. 2 FVol.62, No.2, pp.240-257, 2006. 3D.R. Cox and D. Oakes : Analysis of Survival Data, Monographs on Statistics and Applied Probability 21, Chapman & Hall/CRC, 1998. 4E.T. Lee and J.W. Wang : Statistical Methods for Survival Data Analysis, John Wiley & Sons, 2003. 5) Cox 2001. 6) 2004. 7 :,, 1990. 8W.N. Venables and B.D. Ripley : Modern Applied Statistics with S-PLUS 3rd edition, Chapter12, Springer-Verlag, 1999 ; : S-PLUS,, 2001. 1C. Cameron and P. Trivedi : Econometric Models Based on Count Data : Comparisions and Applications of Some Estimators, Journal of Applied Econometrics, 1, pp.29-53, 1986. 2T. Yasuno : Activity Analysis on Diary Data, K. Kobayashi et al Eds.: Social Capital and Development Trends in Rural Areas, Chap11, 2005. 3 :,, 1990. 1) http://psa2.kuciv.kyoto-u.ac.jp/bms/index.html 2) 96

No.801-73pp.69-822005-10. 3) Vol.14pp.199-2102005. 4) 12 pp.781995. 97

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246.1 0.31 468.5 0.58 86.4 0.11 801.1 1.00 250 200 150 100 50 0 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 110

111 0 50 100 150 200 250 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

112 0 20 40 60 80 100 120 140 160 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 0 5 10 15 20 25 30 35 40 45 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

113 0 10 20 30 40 50 60 70 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 0 20 40 60 80 100 120 140 160 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

114

LCC LCC LCC LCC 1 115

(1) (2) 2 116

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