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3 18 1 2,000,000 2,000, (1) 6 JCOSSAR 2007pp

4 LCC (1) (2) 2

5 10mm ,50,60 2 0

9 ) 2) 3) LCC LCC LCC 1

10 1) Vol.42No.5pp ) )

11 LCC 1 3

13 6 7 1) 6 JCOSSAR 2007pp

14 1) 3.1 2) RC 3),4) (A) 5)(B) (A)(B) (A) (B),,,, 6

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

16 X(t+1) t X(t) RC 9) H9 H )14) a,b,c,d,e RC MCI 16)17) 1) pp )

17 3) No pp ) No pp ) ) pp ) JCOSSAR'95 pp , ) RC Vol.47App ) RC 41 4 pp ) RC Vol.7pp ) No pp ) BMS LCC Vol.11pp ) No pp ) No pp ) Hiroshi IIZUKAA STATISTICAL STUDY ON LIFE TIME OF BRIDGESProc. of JSCENo.392-9Vol.5No.1pp.51-60April ) Vol. 63No. 1pp )

18 10

19 11

20 12

21 13

22 A B C D E 14

23 15

24 16

26 LCC LCC 18

27 n0,1,2,n=0 n=n N+1 X n 5 Bridge Health Index BHI MCI MCI n n X n

28 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

29 p ij =P(X n =j X n-1 =i),i,js (5.2.1) i j p ij K n-1 n KK i i j

30 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

31 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

32 24

33 25

34 D X Z T = C RC C RC C C C C RC 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

35 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)

36 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

37 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

38 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

39 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

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

41 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

42 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

43 ( 8 t 3.98 ) St ( ) = exp (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

44 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

45 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

46 [ ] =, [ ] 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

47 (5.4.6) j x ij 39

49 1) , 1998, 20035, 4, 45, 4, 5 4, 4, 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

50 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

51 K 1 π ik = 1 π ij (5.5.10) j = i π ik

52 5.5.4 H0n =0 n H1n0 t- 1 t t-t t->t n A n305% t )

53 ) 4.62%

54 t

56 t- 48

58 CLEN

59 CWID

60 HIRW

61 53

62 54

63 55

64 56

65 57

66 58

67 59

68 60

69 61

70 y = 0.020x R² = y = 0.002x R² = 9E mm mm 62

72 812mm 0.7mm 1mm 2mm

73 good good 65

74 66

75 67

76 68

77 69

78 70

79 71

80 ,

81 73

82 74

83 75

84 76

85 77

86 78

87 good BIC good good BIC good

88 good BIC good good BIC good

89 good BIC good good BIC good BIC BIC BIC

90 ,50, ,50, ,50,

91 ,50, ,50, ,50,

92 yi y% i =, i= 1,..., n Median( y,..., y ) 1 YDAN 0.12 n

93 YDEP LEVEL 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

94 β 0 β 1 exp( ) = 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( ) =

95 m -8m -5m -3m -2.5m -1.7m -1m 0m 1.1m 1.8m 1m 1m m -8m -5m -3m -2.5m -1.7m -1m 0m 1.1m 1.8m 6(24mm^2) 20 (80mm^2 ) 87

96 24mm^ mm^2-1m mm^2 88

97 mm^

98 mm^ mm 90

99 50% Tohman-Bain 1 5, 4, 3, 2, t 2.04 t 2.0 (1) c = s r r (1) 91

100 c s r

101

102 94

103 mm 10.2mm 14 40,50, ,50, ,50,60 40,50, ,50, ,50,

104 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, FVol.62, No.2, pp , D.R. Cox and D. Oakes : Analysis of Survival Data, Monographs on Statistics and Applied Probability 21, Chapman & Hall/CRC, E.T. Lee and J.W. Wang : Statistical Methods for Survival Data Analysis, John Wiley & Sons, ) Cox ) :,, W.N. Venables and B.D. Ripley : Modern Applied Statistics with S-PLUS 3rd edition, Chapter12, Springer-Verlag, 1999 ; : S-PLUS,, C. Cameron and P. Trivedi : Econometric Models Based on Count Data : Comparisions and Applications of Some Estimators, Journal of Applied Econometrics, 1, pp.29-53, T. Yasuno : Activity Analysis on Diary Data, K. Kobayashi et al Eds.: Social Capital and Development Trends in Rural Areas, Chap11, :,, ) 2) 96

105 No pp ) Vol.14pp ) 12 pp

106 98 RC

107 RC 99

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123 LCC LCC LCC LCC 1 115

124 (1) (2) 2 116

125 117

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35