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1 2014 3

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4 Reliability Data Analysis and its Application Based on Linear Bivariate History of Two-Dimensional Time Scale Masahiro Yokoyama Abstract In reliability engineering, the failure mechanism is a key concept to identify the lifetime distribution and its time scale. However a failure phenomenon could be occurred by two or more failure mechanisms. Therefore the analysis of field reliability data may lead us to a conclusion different from the results of lab experiments. This thesis investigates the problem of multiple time scales, especially in bivariate cases, to assess the effects of failure mechanisms on field reliability by estimating the joint distribution function of the product lifetime on these time scales. Pons (1986) proposed a nonparametric estimator of the joint cumulative hazard function of bivariate survival data in the presence of censoring. However this estimator does not consider the fact that each product has a bivariate history up to a failure or a censoring on a two-dimensional space. Intrinsically the lifetime distribution of a product on multiple time scales is univariate. Therefore this thesis proposes a simple estimator of the cumulative hazard function which takes usage histories of each product into consideration for analyzing field failure data of industrial products.

5 This estimator is proposed in Chapter 4 under the assumption that a sample path can be modeled as a straight line. Chapter 5 shows the analysis of an actual field reliability data to demonstrate that it enables estimation of the usage-frequency-dependent failure probability. The difference between this estimator and the estimator proposed by Pons (1986) is discussed in Chapter 6. The estimator in Chapters 4 through 6 is proposed for the cases in which the product bivariate history is linear and only the event point, failure or censoring, is observed. Chapter 7 investigates the cases in which every product history is observed. Variables which affect the failure mechanism are commonly referred to covariates. For example, the temperature and the relative humidity are sometimes included into the analysis of reliability data as covariates. Recently, covariates can be obtained continuously by the use of Information and Communication Technology (ICT). Using a conversion model from a failure time to a new value taking covariate information, a method to estimate the value of a covariate effect on failure mechanism is shown. According to the above studies, it is shown that user s information such as usage-frequency and covariate becomes possible to be utilized for lifetime estimation.

7 i Nelson-Aalen Pons[14]

8 ii Pons[14] Pons[14] Pons[14] Pons[14] ε i j ε i j Pons

9 iii A 72 B MSE kl 76

10 iv 1 :, :, :, :, :, :, :, :, :, :, (5 5) (5 5) E Pons[14] (5 5) E log 10 Ĥ ˆF

11 v 11 η i ,, ε i j 0 0.5, x i j,, ε i j 0 2.5, x i j,, ε i j { < x } ε i j { < x } , x i j,, 200 η i ε i j ε i j { < x } ε i j { < x } z q J = T (β) η = 100 β 1 β1 = ±

12 vi β 1 η = 100 n = β 1 = (5 5) {(x,y) : < x , < y }

13 vii d kl c kl Ĥ Pons[14] Pons[14] 37 9 [ ] ε i j 0.5 n = ε i j 0.5 Pons[14] n = ε i j 0.5 n =

14 viii 13 ε i j 0.5 n = ε i j 0.5 n = [ ] ε i j 0.5 n = Ti β 1 = β 2 = 1.00 µ = 2.0, n = 1000, : Ti (β ) Ti (β) β 1000 β 1 = β 2 = 1.00) T i (β ) β 1000 β 1 = β 2 = 1.00) [ ] ε i j 2.5 n = ε i j 2.5 MSE kl n =

15 ix 23 ε i j 2.5 Pons[14] MSE kl n =

17 (x i,y i,e i ), i = 1,...,8988, i x i i y i i e i = 1 (x i,y i ) e i = 0 (x i,y i ) 1 T C T F(t) C T G(c) i T i C i i x i T i C i x i = min{t,c} T i C i T i > C i

18 1 3 1: :, :, 8988

19 1 4 T 1 T 2 C 1 C 2 (T 1,T 2 ) F(t 1,t 2 ) (C 1,C 2 ) (T 1,T 2 ) G(c 1,c 2 ) i (T 1i,T 2i ) (C 1i,C 2i ) (0,0) T 1i C 1i T 2i C 2i C 1i < T 1i C 2i < T 2i 1.3 (, ) (50,1000) (120,2200) (200,4000) 2 2:

20 1 5 3: :, :, : 3 :, :, :, :,1080

21 x i y i a i = y i /x i i 4 a i a i 4 a i A A A (T 1,T 2 ) F(t 1,t 2 A) A (C 1,C 2 ) (T 1,T 2 ) G(c 1,c 2 A) A

22 1 7 5: 2 :, :, (x i,y i,e i )

23 Information and Communication Technology ICT PC Pons[14] Pons[14] 3

24 Nelson-Aalen Nelson[12], Aalen[1] Pons[14] 4 Pons[14]

25 Pons[14]

26 S(t) = P{T > t}, t 0 F(t) = P{T t} = 1 S(t), t 0 S(t) F(t) f (t) F(t) = t 0 f (u)du λ(t) = f (t) S(t) λ(t)

27 2 12 (2.1) λ(t) Λ(t) Λ(t) = t 0 λ(u)du = log λ(u)du + log λ(u)du t 0 = log t λ(u)du = logs(t) = log(1 F(t)) (2.1) (2.1) Λ(t) F(t) A A A (T 1,T 2 ) S(t 1,t 2 A) F(t 1,t 2 A) S(t 1,t 2 A) = P{T 1 > t 1 T 2 > t 2 A} F(t 1,t 2 A) = P{T 1 t 1 T 2 t 2 A} f (t 1,t 2 A) F(t 1,t 2 A) S(t 1,t 2 A) > 0 F(t 1,t 2 A) f (t 1,t 2 A) λ(t 1,t 2 A) λ(t 1,t 2 A) = f (t 1,t 2 A) S(t 1,t 2 A) Λ(t 1,t 2 A) Λ(t 1,t 2 A) = t1 t2 0 0 λ(u 1,u 2 A)du 1 du 2 (2.2)

28 Λ(t 1,t 2 A) (2.1) F(t 1,t 2 A) f (t 1,t 2 A) F(t 1,t 2 A) f (t 1,t 2 A) A P{A} P{A} A f (t 1,t 2 A) P{A} (t 1,t 2 ) S(t 1,t 2 ) F(t 1,t 2 ) S(t 1,t 2 ) = P{T 1 > t 1 T 2 > t 2 } F(t 1,t 2 ) = P{T 1 t 1 T 2 t 2 } S(t 1,t 2 ) > 0 F(t 1,t 2 ) f (t 1,t 2 ) λ(t 1,t 2 ) λ(t 1,t 2 ) = f (t 1,t 2 ) S(t 1,t 2 ) Λ(t 1,t 2 )

29 2 14 Λ(t 1,t 2 ) = t1 t2 0 0 λ(u 1,u 2 )du 1 du 2 (2.3) Λ(t 1,t 2 ) 3 (2.3) Λ(t 1,t 2 ) F(t 1,t 2 ) [ t1 t2 logs(u 1,u 2 ) du 1 du 2 = logs(u 1,u 2 ) v 1 v ] u1 =t 1 u 2 =t 2 u 1 =0 u 2 =0 ( ) S(0,0) S(t 1,t 2 ) = log S(t 1,0) S(0,t 2 ) (2.4) t1 t2 0 0 t1 t2 logs(u 1,u 2 ) du 1 du 2 v 1 v { 2 S(u 1,u 2 )/ v 1 v 2 S(u 1,u 2 ) = S(u 1,u 2 )/ v 1 S(u 1,u 2 )/ v 2 du 1 du S(u 1,u 2 ) S(u 1,u 2 ) { t1 t2 ( u2 ) ( f (u 1,u 2 ) = 0 0 S(u 1,u 2 ) f (u 1,v 2 )dv u1 ) 2 f (v 1,u 2 )dv } 1 du 1 du 2 S(u 1,u 2 ) S(u 1,u 2 ) { t1 t2 ( u2 ) ( f (u 1,v 2 )dv u1 ) 2 f (v 1,u 2 )dv } 1 = Λ(t 1,t 2 ) du 1 du 2 S(u 1,u 2 ) S(u 1,u 2 ) 0 0 = Λ (t 1,t 2 ) ( ) (2.5) }

30 2 15 (2.4) (2.5) (2.6) ( ) Λ S(0,0) S(t 1,t 2 ) (t 1,t 2 ) = log S(t 1,0) S(0,t 2 ) (2.6) (2.6) S(t 1,t 2 ) (2.7) S(t 1,t 2 ) = exp(λ (t 1,t 2 )) S(t 1,0) S(0,t 2 ) (2.7) F(t 1,t 2 ) (2.8) S(t 1,0) = t1 0 F(t 1,t 2 ) = t1 t2 0 0 f (u 1,u 2 )du 1 du 2 = 1 S(t 1,0) S(0,t 2 ) + S(t 1,t 2 ) (2.8) f (u 1,u 2 )du 1 du 2 S(0,t 2 ) = 0 t2 t 1,t 2 f (u 1,u 2 )du 1 du 2 (2.8) 5 6

31 Nelson-Aalen Nelson-Aalen Nelson[12], Aalen[1] Nelson-Aalen [8] n i i = 1,,n X i T i C i T i C i i X i = min{t i,c i } e i = I {Xi =T i } I {} {} 1 0 N(t) = R(t) = n I {Xi t,e i =1} i=1 n I {Xi t} i=1

32 3 17 [t,t + dt) dn(t) = N((t + dt) ) N(t ) dn(t) [t,t + dt) N(t) N(t ) = lim u t 0 N(u) Nelson[12], Aalen[1] R(t) > 0 Λ(t) ˆΛ(t) = t 0 dn(u) R(u) (3.1) (3.1) Lawless[10] 4 (4.6) (4.6) (3.1) 3.2 Pons[14] Pons[14] n x y i i = 1,,n (T 1i,T 2i ) (C 1i,C 2i ) (T 1i,T 2i ) (C 1i,C 2i ) i P Pons[14] N(t 1,t 2 ) = R P (t 1,t 2 ) = I {T1i <t 1,T 2i <t 2,C 1i >T 1i,C 2i >T 2i } 1 i n I {min{t1i,c 1i }>t 1,min{T 2i,C 2i }>t 2 } 1 i n

33 3 18 [(t 1,t 2 ),(t 1 + dt 1,t 2 + dt 2 )) dn(t 1,t 2 ) = N((t 1 + dt 1,t 2 + dt 2 ) ) N((t 1,t 2 ) ) dn(t 1,t 2 ) [(t 1,t 2 ),(t 1 + dt 1,t 2 + dt 2 )) N(t 1,t 2 ) N((t 1,t 2 ) ) = lim u1 t 1 0 lim u2 t 2 0 N(u 1,u 2 ) Pons[14] Λ(t 1,t 2 ) ˆΛ P (t 1,t 2 ) = t1 t2 0 0 dn(u 1,u 2 ) R P (u 1,u 2 ) (3.2) (3.2) 4.2 (3.2) (4.8) Pons[14] 1 4

34 ( ) a i (x i,y i,e i ), i = 1,...,n, x i,i = 1,...,n, (V k 1,V k ], k = 1,...,K

35 4 20 y i,i = 1,...,n, (W l 1,W l ], l = 1,...,L K L V 0, V K W 0, W L V 0 < min i x i, max i x i V K, W 0 < miny i, maxy i W L, i i 1 K (logv K logv 0 ) = 1 L (logw L logw 0 ) V k,k = 1,...,K 1, W l,l = 1,...,L 1, V k = V 0 10 k{(logv K logv 0 )}/K, k = 1,...,K 1 {(logv K logv 0 )}/K = {(logw L logw 0 )}/L W l = W 0 10 l{(logv K logv 0 )}/L, l = 1,...,L 1 K L E kl = {(x,y) V k 1 < x V k,w l 1 < y W l } K L L K = (logw L logw 0 ) (logv K logv 0 ) logv 0 = log minx i, i logw 0 = log miny i, logv K = log maxx i, logw L = log maxy i i i i

36 4 21 K L 1 K (logv K logv 0 ) = 1 L (logw L logw 0 ) (4.1) K = L = 5, V 0 = W 0 = 10, V K = W L = 100 (4.1) : 0.2 (5 5) 6(b) 6(a) 6(a) 6(b) k l = m A m = { E k l k l = m } (4.2) A m, m = 1 L,...,0,...,K 1, x y y i /x i K + L 1

37 A k l E kl E kl A k l E kl { Ek l k l = k l, k k, l l } E kl R A kl A A k l d kl c kl d kl = i:(x i,y i ) E kl e i, c kl = i:(x i,y i ) E kl (1 e i ) (4.3) R A kl min{k k,l l} R A kl = ( ) dk+ j,l+ j + c k+ j,l+ j j=0 (4.4) E kl (V k V k 1 ) (W l W l 1 ) δ kl h A kl ˆ h A kl = d kl R A kl δ kl (4.5) A k l E kl H A kl ˆ H A kl = 0 ( ) ha ˆ k+ j,l+ j δ k+ j,l+ j = j= min{k 1,l 1} 0 j= min{k 1,l 1} d k+ j,l+ j R A k+ j,l+ j (4.6)

38 4 23 7: (5 5) E 23 {logv 0 = 1.0,logV 1 = 2.0,...,logV 5 = 6.0},{logW 0 = 1.0,logW 1 = 2.0,...,logW 5 = 6.0} 25(= 5 5) E 23 7 E 23 R A 23 7 E 23 Ĥ23 A Pons[14] Pons[14] E kl R P kl RP kl } R P K L kl = {d j1 j2 + c j1 j2 (4.7) j 1 =k j 2 =l

39 4 24 E kl δ kl ĥ P kl d kl ĥ P kl = R P kl δ kl Ĥ P kl ĤP kl Ĥ P kl = = k j 1 =1 k j 1 =1 l j 2 =1 l j 2 =1 {ĥ Pj1 j2 δ j1 j2 } { d j1 j 2 R P j 1 j 2 } (4.8) 8 7 (5 5) E 23 E 23 R P 23 8 E 23 Ĥ23 P 8 8: Pons[14] (5 5) E 23

40 Pons[14] (1) x y 9(a) 9(b) x V 0 = , V K = y W 0 = , W L = (c) 0.1 (18 28)

41 5 26 9:

42 5 27 (2) d kl 1 c kl 2

43 5 28 1: 0.1 d kl 2: 0.1 c kl

44 5 29 (3) { < x , < y } 1 2 R A kl :

45 { < x , < y } log 10 Ĥ 4 4: Ĥ 4 x y log 10 Ĥ 3 3

46 log 10 Ĥ ˆF ˆF = 1 exp( Ĥ) 10: 3 log 10 Ĥ ˆF 10

47 Ĥ ˆF = 1 exp( Ĥ) 5 (4.6) 5:

48 { < x } { < y } A { < x } { < y } B A B 5 A B { < x } A 2.9% B 22.2% { < y } A 15.5% B 10.5%

49 : 5 6 B B 5 6

50 Pons[14] Pons[14] 5.1 (1) (2) (3) 7 Pons[14] { < x , < y } R P kl 3 7: Pons[14]

51 Pons[14] 8 Pons[14]

52 5 37 8: Pons[14] 9: [ ]

53 Pons[14] 6.1 i j, ( j = 1...,J i 1), ( j = J i ) x i j j 1 j y i j j 1 j x i,y i x i = J i j=1 x i j y i = J i j=1 y i j (6.1) η i ε i j

54 6 39 y i j = η i x i j + ε i j (6.1) η i 11 η i , : η i ,, 200

55 ε i j ε i j ε i j x i j = 5 12: ε i j 0 0.5, x i j,, 200

56 ε i j : ε i j 0 2.5, x i j,, 200

57 , Pons[14] F(t 1,t 2 ) = P{T 1 t 1 T 2 t 2 } F = n I n {T1i t 1 T 2i t 2 } I {T1i t = i=1 I {T1i 1 T 2i t 2 } 0 T 2i 0} i=1 n (6.2)

58 ε i j ε i j % % 75% { < x } 50 Pons[14]

59 : ε i j 0.5 n = : ε i j 0.5 Pons[14] n = : ε i j 0.5 n =

60 : ε i j { < x }

61 ε i j ε i j ε i j MSE ε i j 2.5 MSE(mean squared error) k l ˆF kl F kl 50 k l MSE MSE kl MSE kl = 50 ( ˆF kl F kl ) 2 r=1 50 MSE kl MSE kl Pons[14] MSE kl MSE kl B

62 : ε i j { < x }

63 η i ε i j j x i j = 5 16:, x i j,, 200 η i ε i j

64 , MSE kl Pons[14] MSE kl { < x } 17 14

65 : ε i j 0.5 n = : ε i j 0.5 n = : [ ]ε i j 0.5 n =

66 : ε i j { < x }

67 6 52 Pons[14] 18 Pons[14] MSE kl Pons[14] MSE kl Pons[14] 6.3 Pons Pons[14] Pons[14] Pons[14] 18

68 : ε i j { < x }

69 T T 7.2

70 Hong and Meeker[5][6] Nelson[13] (7.1) t ( ) z(t) (7.1) T T (β) T (β) = T 0 exp[β z(s)]ds (7.1) z(t) = z 1 (t) z 2 (t). z Q (t) : t Q β = [β 1 β 2 β Q ] : (7.1) Meeker and Escobar[11] proportional quantities(pq) scale accelerated failure-time(saft)

71 7 56 β β Hong and Meeker[5][6] T (β ) β T (β ) β T (β ) T (β ) Hong and Meeker[5][6] β T (β) T (β ) Hong and Meeker[5][6] ˆβ T ( ˆβ) T (β ) T (β) β

72 µ σ T (β ) (7.2) L LN (β,µ,σ) = { } n exp[β z(ti, o 1 B i )] i=1 2πσT i (β) exp[ (lnt i (β) µ)2 2σ 2 ] (7.2) (7.2) β T (β ) 0 β g ln (β) = n i=1 (z(t o i,b i )) n i=1 Ti Ti (β) (β) 1ˆσ n i=1 {( log(t i (β)) ˆµ ˆσ ) } T i (β) Ti (β) (7.3) T i (β ) (log-location-scale) (7.4) i=1 n logl wei (β) = (β z(ti,b o i )) log( ˆσ) logti (β) i=1 i=1 i=1 { } n log(ti + (β)) ˆµ exp( log(t i (β)) ˆµ ) ˆσ ˆσ n n (7.4) g wei (β) (7.5) {( ) n g wei (β) = (z(ti,b o n Ti i )) n (β) (β) 1ˆσ exp( log(t i (β)) ˆµ ) 1 ˆσ i=1 i=1 T i i=1 T i } (β) i (β) T (7.5) T (β ) 0 β

73 7 58 exp( log(t i (β)) ˆµ ˆσ ) = exp(x) x = 0 Taylor exp(x) = exp(0) + exp(0) x = 1 + x (7.6) g wei (β) n i=1 (z(t o i,b i )) n i=1 Ti Ti (β) (β) 1ˆσ n i=1 {( ) } ( log(t i (β)) ˆµ ) T i (β) ˆσ Ti (β) = g ln (β) (7.6) T (β ) β

74 t o j, ( j = 1,...,J), t o j 19: z q J = 4 T : t o j J q : ( j = 1,...,J) : : (q = 1,...,Q) z q (t o j ) : to j z q T = t o J

75 T T (β) 20: T (β) i = 1,...,n, T i z(t o i j ) to i j, ( j = 1,...,J i) Ti (β) T i (β ) β z(t o i j ) T i

76 7 61 T i (β ) T i (β ) z(t o i j ) 19 z(t o i j ) z 1,z 2 i ti o j z 1(ti o j ), z 2(ti o j ) β 1,β T i T i (β ) z(t o i j ) T i ti1 o = (to i2 to i1 ) = = (to i, J i 1 to i, J i 2 ) = 1 (: ) (ti, o J i ti, o J i 1 ) < 1 (: 1 ) Ti (β J i 1 ) = j=1 exp[β 1 z 1 (t o i j) + β 2 z 2 (t o i j)] + exp[β 1 z 1 (t o i, J i ) + β 2 z 2 (t o i, J i )] (t o i, J i t o i, J i 1 ) t o i, J i = T i T i

77 7 62 Kordonsky and Gertsbakh[9] T c.v.(t i (β)) = T i i (β) (β) (7.7)

78 T i (β ) T i (β ) T i (β ) 16 T i (β ) β 1 = β 2 = 1.00 µ σ β σ β 16: Ti β 1 = β 2 : 500 µ σ ˆβ1 ˆβ2 ˆβ1 ˆβ = 1.00 µ = 2.0, n = 1000,

79 7 64 T i (β ) Ti (β) β β 1 = β 2 = 1.00 T i (β ) β m η n η = 100 β β m β m β β 1 η = 100 n = m

80 : Ti (β ) Ti (β) β 1000 β1 = β 2 = 1.00)

81 7 66 (a) m = 1 (b) m = 2 (c) m = 3 (d) m = 4 (e) m = 5 (f) m = 6 21: 17 η = 100 β 1 β 1 = ±

82 : 17 β 1 η = 100 n = β 1 = 1.00

83 m 7.6 T i (β ) β 1,β 2 T (β ) m

84 : T i (β ) β 1000 β 1 = β 2 = 1.00)

87 A 72 A 23(a) {logx 0 = 1.0,logX 1 = 1.2,,logX 5 = 2.0},{logY 0 = 1.0, logy 1 = 1.2,,logY 5 = 2.0} (5 5) {(x,y) : < x , < y } 23(b) {(x,y) : < x , < y } 23(a) 23(b) 23(a) 24(a)(b)

88 A 73 23: 0.2 (5 5) {(x,y) : < x , < y } 24:

89 A : 24

90 A 75 20: [ ]

91 B MSE KL 76 B MSE kl 21 ε i j ε i j MSE kl MSE kl MSE kl

92 B MSE KL 77 21: ε i j 2.5 n = (a) {1.6 < x 18} (b) {18 < x 200}

93 B MSE KL 78 22: ε i j 2.5 MSE kl n = (a) {1.6 < x 18} (b) {18 < x 200}

94 B MSE KL 79 23: ε i j 2.5 Pons[14] MSE kl n = (a) {1.6 < x 18} (b) {18 < x 200}

95 80 [1] Aalen, O. (1978) : Nonparametric inference for a family of counting processes, The Annals of Statistics, Vol.6, pp [2] Burke, M. D. (1984) : Estimation of a bivariate distribution function under random censorship, Biometrika, Vol.75, pp [3] Dabrowska, D. M. (1988) : Kaplan-Meier estimate on the plane, The Annals of Statistics, Vol.16, pp [4] Fleming, T. R. and Harrington, D. P. (1991) : Counting Processes and Survival Analysis, Wiley. [5] Hong, Y. and Meeker, W. Q. (2010): A model for field failure prediction using dynamic environmental data, Mathematical and Statistical Models and Methods in Reliability Statistics for Industry and Technology, Springer, pp [6] Hong, Y. and Meeker, W. Q. (2013): Field-failure predictions based on failure-time data with dynamic covariate information, Technometrics, Vol. 55, pp [7] Hougaard, P. (2000) : Analysis of Multivariate Survival Data (Statistics for Biology and Health), Springer. [8], (1983) :,.

96 81 [9] Kordonsky, K. B. and Gertsbakh, I. B. (1993): Choice of the best time scale for system reliability analysis, European Journal of Operational Research, Vol. 65, pp [10] Lawless, J. F. (2003) : Statistical Models and Methods for Lifetime Data Second Edition, Wiley. [11] Meeker, W. Q. and Escobar, L. A. (1998) : Statistical Methods for Reliability Data, Wiley. [12] Nelson, W. (1972) : Theory and applications of hazard plotting for censored failure data, Technometrics, Vol.14, pp [13] Nelson, W. (2001): Prediction of fieldreliability of units, each underdifferingdynamicstresses, from accelerated test data, Handbook of Statistics, Elsevier, Vol. 20, pp [14] Pons, O. (1986) : A test of independence between two censored survival times, Scandinavian Journal of Statistics, Vol.13, pp

97 82

98 ,, Vol.36, pp : Masahiro Yokoyama and Kazuyuki Suzuki Integrated Reliability and Safety Information System for Personalized Risk Communication 22 10, The 8th ANQ Congress, Delhi, JP25. : Masahiro Yokoyama, Toshie Yamashita, Watalu Yamamoto and Kazuyuki Suzuki Personalized Prediction of Optimal Replacement Point Using Data Assimilation 23 9, The 9th ANQ Congress, Ho Chi Minh City, JP11.

JFE.dvi

JFE.dvi ,, Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112 8551, JAPAN E-mail : atsu1005@kc.chuo-u.ac.jp E-mail : kawa@civil.chuo-u.ac.jp SATO KOGYO CO., LTD. 12-20, Nihonbashi-Honcho