e 4β e 4β + e β ˆβ =0.765

Size: px

Start display at page:

Download "e 4β e 4β + e β ˆβ =0.765"

Jessie Brooks
5 years ago
Views:

1 SIMPLE EXAMPLE COX-REGRESSION i Y i x i δ i Model: z(t x) =z 0 (t) exp{βx} Partial likelihood: L(β) = e 10β e 10β + e 3β + e 5β + e 3β + e 4β + e β e 3β e 3β + e 4β + e β e 4β e 4β + e β ˆβ =

2 BRESLOW-ESTIMATOR so 1 Ẑ 0 (t) = T j t i R j e ˆβ x i Ẑ 0 (10) = 1 =4.57 e 10ˆβ + e 3ˆβ + e 5ˆβ + e 3ˆβ + e 4ˆβ eˆβ Ẑ 0 (120) = e 3ˆβ + e 4ˆβ + = eˆβ Ẑ 0 (400) = e 4ˆβ + eˆβ =

3 ESTIMATED SURVIVAL FUNCTION P (T >t)=r(t; x) so ˆR(t; x) = exp{ Ẑ 0 (t)e ˆβ x } ˆR(10; x) = exp{ e 0.765x } ˆR(120; x) = exp{ e 0.765x } ˆR(400; x) = exp{ e 0.765x } x ˆR(10; x) ˆR(120; x) ˆR(400; x) KM Baseline survival function KM-estimator does not use the value of x 159

4 COX-SNELL RESIDUALS ˆV i = Ẑ 0 (Y i )e ˆβ x i which should behave like expon(1) i Y i x i δ i ˆV i Ẑ 0 (0)e = Ẑ 0 (10)e = Ẑ 0 (10)e = Ẑ 0 (10)e = Ẑ 0 (120)e = Ẑ 0 (400)e = Ẑ 0 (400)e =

5 REPAIRABLE SYSTEMS/RECURRENT EVENTS/COUNTING PROCESSES Definition of repairable system (Ascher and Feingold 1984): A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method, other than replacement of the entire system. 0 S 1 S 2 S 3 t T 1 T 2 T 3 161

6 HAPPY AND SAD SYSTEMS Ascher and Feingold presented the following example of a happy and sad system: Their claim: Reliability engineers do not recognize the difference between these cases since they always treat times between failures as i.i.d. and fit probability models like Weibull. Their conclusion: Use point process models to analyze repairable systems data! 162

7 Today: Recurrent events extensively studied 0 S 1 S 2 S N τ Applications: engineering and reliability studies, public health, clinical trials, politics, finance, insurance, sociology, etc. Reliability applications: breakdown or failure of a mechanical or electronic system discovery of a bug in an operating system software the occurrence of a crack in concrete structures the breakdown of a fiber in fibrous composites Warranty claims of manufactured products 163

8 Nelson (1995): Valve seat data Times of valve-seat replacements in a fleet of 41 diesel engines Event Plot for Valve Seat Replacements Diesel Engine Time

9 Bhattacharjee et al. (2003): Nuclear plant failure data Failure data for closing valves in safety systems at two nuclear reactor plants in Finland. Failures type: External Leakage, follow-up 9 years for 104 valves. 88 valves had no failures Event Plot for Closing Valves Valve Time

10 Aalen and Husebye (1991): Migratory motor complex (MMC) periods in 19 patients, 1-9 events per individual. Individual Observed periods (minutes) (54) (30) (4) (87) 5 67 (131) (23) (111) (110) (44) (122) (85) (72).. 166

11 CONDITIONAL ROCOF BY MINIMAL REPAIR (NHPP) AND PERFECT REPAIR (RENEWAL PROCESS) 167

12 SIMPLE EXAMPLE WITH THREE SYSTEMS Sys. 1: 0 S 11 =5 S 21 =12 S 31 =17 τ 1 =20 Sys. 2: 0 S 12 =9 S 22 =23 τ 2 =30 Sys. 3: 0 S 13 =4 τ 3 =10 Proj: t Y(t): Y (t) =3 Y (t) =2 Y (t) =1 168

13 COMPUTATIONS FOR THE NELSON-AALEN ESTIMATOR t 1/Y (t) 1/Y (t) 2 Ŵ (t) VarŴ (t) SDŴ (t) 4 1/3 1/9 1/3 1/ /3 1/9 2/3 2/ /3 1/9 1 1/ /2 1/4 3/2 7/ /2 1/4 2 5/ /

14 ESTIMATED W (t) with 95% confidence limits (Nelson-Aalen) W(t) t 170

15 Simple Example With 3 Systems 171

16 Simple Example With 3 Systems 172

17 Simple Example With 3 Systems Results for: SimpleNHPP.MTW Nonparametric Growth Curve: Time System: ID Nonparametric Estimates Table of Mean Cumulative Function Mean Cumulative Standard 95% Normal CI Time Function Error Lower Upper System 4 0, , , , , , , , , , , , , , , , , , , , , , , ,

18 Simple Example With 3 Systems 174

19 Nelson-Aalen estimator for Cumulative ROCOF W (t) 1. Order all failure times as t 1 <t 2 <...t n. 2. Let d j (t i ) = # events in system j at t i. 3. Let d(t i )= m j=1 d j(t i ) = # events in all systems at t i. 4. Let Y j (t) = { 1 if system j is under observation at time t 0 otherwise 5. Let Y (t) = m j=1 Y j(t) = # systems under observation at time t. Then Under general assumptions: Ŵ (t) = Assuming NHPP: Var Ŵ (t) = t i t t i t d(t i ) {Y (t i )} 2 d(t i ) Y (t i ). Under general assumptions (MINITAB): Var Ŵ (t) = { m j=1 t i t Y j (t i ) Y (t i ) [ d j (t i ) d(t ] } 2 i) Y (t i ) 175

20 Illustration of last formula for Simple NHPP Example (Compare with MINITAB Output): Var Ŵ (4) = { 1 3 [ { 1 + 3]} 3 [ { 1 + 3]} 3 [ ]} 2 = 6 81 = Var Ŵ (5) = + + { 1 3 { 1 3 { 1 3 [ ] [ ] [ ] [ 1 1 ]} 2 3 [ 0 1 ]} 2 3 [ 0 1 ]} 2 3 = 6 81 =

21 Var Ŵ (9) = { 1 3 { { = 0 [ ] [ ] [ ] [ ] [ ] [ ] [ 0 1 ]} 2 3 [ 1 1 ]} 2 3 [ 0 1 ]} 2 3 Var Ŵ (12) = + + { 1 3 { 1 3 { 1 3 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]} [ 1 1 ]} 2 2 [ 0 1 ]} 2 2 = 1 8 =

22 Simple Example With 3 Systems Power Law NHPP Model: W (t; α, θ) =(t/θ) α 178

23 Simple Example With 3 Systems 179

24 RESIDUAL PROCESS: SIMPLE EXAMPLE. Data points (and endpoints on axes) are transformed with the estimated cumulative ROCOF, Ŵ (t) = t 1.20 Sys. 1: Sys. 2: Sys. 3: Times between events, plus censored times at the end of each axis, are on the next slide anlysed by MINITAB as a set of censored exponential variables. 180

25 181

26 182

27 VALVESEAT DATA 183

28 184

29 VALVESEAT DATA 185

30 VALVESEAT DATA 186

31 VALVESEAT DATA 187

32 188

33 189

34 Grampus- data: Plot of (T i,t i+1 ) to investigate whether times between failures can be assumed independent. The figure does not indicate a correlation between successive times. 190

35 191

36 192

37 PROFILE LIKELIHOOD FOR BETA ( SIMPLE EXAMPLE ) ˆβ =1.20, ˆλ = profile beta 193

38 CONNECTION BETWEEN LAMBDA OG BETA ( SIMPLE EXAMPLE ) ˆβ =1.20, ˆλ = lambda beta 194

39 195

40 196

41 197

42 198

43 Valveseat Data 199

44 Valveseat Data 200

45 TTT-analysis Simple Example Row STTT ID Scaled , , , , , , ,00000 Parameter Estimates Standard 95% Normal CI Parameter Estimate Error Lower Upper Shape 1, ,511 0, ,25186 Scale 0, ,160-0, , Trend Tests MIL-Hdbk-189 Laplace s Anderson-Darling Test Statistic 9,59 0,12 0,24 P-Value 0,697 0,906 0,977 DF

46 Total Time on Test Plot for Simple Example System Column in ID 1,0 P arameter, M LE Shape Scale 1, , Scaled Total Time on Test 0,8 0,6 0,4 0,2 0,0 0,0 0,2 0,4 0,6 Scaled Failure Number 0,8 1,0 202

47 TTT-analysis of Valve Seat Data Parametric Growth Curve: C1 Model: Power-Law Process Estimation Method: Maximum Likelihood Parameter Estimates Standard 95% Normal CI Parameter Estimate Error Lower Upper Shape 1, ,202 1, ,79229 Scale 0, ,026 0, , Trend Tests MIL-Hdbk-189 Laplace s Anderson-Darling Test Statistic 68,72 2,03 3,17 P-Value 0,032 0,043 0,022 DF

48 Total Time on Test Plot for Valve Seat Data 1,0 Parameter, MLE Shape Scale 1, , Scaled Total Time on Test 0,8 0,6 0,4 0,2 0,0 0,0 0,2 0,4 0,6 Scaled Failure Number 0,8 1,0 204

Introduction to repairable systems STK4400 Spring 2011

Introduction to repairable systems STK4400 Spring 2011 Bo Lindqvist http://www.math.ntnu.no/ bo/ bo@math.ntnu.no Bo Lindqvist Introduction to repairable systems Definition of repairable system Ascher and