Reliability Project 1
Team 9
Philippe Delelis
Florian Brouet
SungHyeok Lee
Data 1
N=21
2
Probabiblity Distribution
Data 1 : Symmetric Simple
cumulative Distribution
Normal
Weibull
Log normal
Bi-exponential
Data 1 : Mean Rank
Normal
Weibull
Log normal
Bi-exponential
Data 1 : Median rank
Normal
Weibull
Log normal
Bi-exponential
Data 1 : The rest method
Normal
Log normal
Bi-exponential
Weibull
Data 1 : Linearity Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Log-Normal
X
X
X
X
Weibull
0
0
0
0
Bi-exponential
0
0
0
0
Data 1 : R (Correlation Coefficient Comparaison)
Symmetric .S.C
Mean Rank
R
SD
R
SD
Normal
0.95883
0.20172
0.96493
0.16754
Log-Normal
0.84415
0.3925
0.83983
0.35807
Weibull
0.93251
0.32349
0.92194
0.30615
Bi-exponential
0.85644
0.4718
0.88683
0.36864
Median Rank
The Rest Method
R
SD
R
SD
Normal
0.96225
0.18434
0.95883
0.20172
Log-Normal
0.84255
0.37647
0.9612
0.19002
Weibull
0.92866
0.31387
0.8432
0.38198
Bi-exponential
0.87197
0.42048
0.93023
0.31675
n = 21
Data 1 : Value of Dnα
Normal
α = 0.05
Dnα =0,1882
Weibull, Biexponential
α = 0.05
Dnα =0.1932
Normal
α = 0.15
Dnα =0.1636
Weibull, Biexponential
α = 0.15
Dnα =0.1668
K-S test : Symmetric Simple
Cumulative Distribution
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 288,431
• σ = 196,078
Weibull
• m = 1,166
• ξ = 326,693
Bi-exponential
• ξ = 163,934
• x0= 378,885
Mean Rank
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 288,696
• σ = 217,391
Weibull
• m = 1.021
• ξ = 338.885
Bi-exponential
• ξ = 188,185
• x0= 386,741
Median Rank
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 286,939
• σ = 204,082
Weibull
• m = 1,099
• ξ = 332,047
Bi-exponential
• ξ = 171,527
• x0= 378,851
The Rest Method
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 285,8
• σ = 200
Weibull
• m = 1,112
• ξ = 331,007
Bi-exponential
• ξ = 169,492
• x0= 380,339
Data 1 : K-S Test Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Weibull
x
x
0
0
Bi-exponential
x
x
x
0
Data 2
N=26
16
Data 2 : Symmetric S. C. Distribution
Normal
Weibull
Log-Normal
Bi-Exponential
17
Data 2 : Mean Rank
Normal
Log-Normal
Weibull
Bi-exponential
18
Data 2 : Median Rank
Normal
Weibull
Log-Normal
Bi-exponential
19
Data 2 : The Rest Method
Normal
Weibull
Log-Normal
Bi-exponential
20
Data 2 : Linearity Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Log-Normal
X
X
X
X
Weibull
0
0
0
0
Bi-exponential
0
0
0
0
21
Data 2 : R (Correlation Coefficient Comparaison)
Symmetric .S.C
Mean Rank
R
SD
R
SD
Normal
0.98122
0.1364
0.98162
0.1304
Log-Normal
0.89569
0.32143
0.89569
0.32143
Weibull
0.9646
0.2353
0.9646
0.2353
Bi-exponential
0.9005
0.39445
0.92236
0.35894
Median Rank
The Rest Method
R
SD
R
SD
Normal
0.9619
0.12848
0.9818
0.13082
Log-Normal
0.89421
0.31095
0.89482
0.3145
Weibull
0.9636
0.22677
0.96421
0.22889
Bi-exponential
0.9120
0.35259
0.90832
0.36637
22
n = 26
Data 2 : Value of Dnα
Normal
α = 0.05
Dnα =0.1702
Weibull, Biexponential
α = 0.05
Dnα =0.175
Normal
α = 0.15
Dnα =0.1474
Weibull, Biexponential
α = 0.15
Dnα =0.1514
23
K-S test : Symmetric Simple Cumulative
Distribution
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 330.367
• σ = 166.667
Weibull
• m = 1.9055
• ξ = 340.52
Bi-exponential
• ξ = 125
• x0= 368.75
24
Data 2 : Mean Rank
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 301.54
• σ = 160.527
Weibull
• m = 1.8251
• ξ = 337.25
Bi-exponential
• ξ = 145.85
• x0= 384.26
25
Data 2 : Median Rank
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 316.54
• σ = 166.06
Weibull
• m = 1.84
• ξ = 343.7
Bi-exponential
• ξ = 142.85
• x0= 405.28
26
Data 2 : The Rest Method
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 321.53
• σ = 166.6
Weibull
• m = 1.81
• ξ = 342.87
Bi-exponential
• ξ = 142.85
• x0= 409.97
27
Data 2 : K-S Test Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
X
0
0
0
Weibull
0
0
0
0
Bi-exponential
0
0
X
X
28
Data 3
N=29
29
Data 3 : Symmetric Simple
cumulative Distribution
Normal
Log normal
Weibull
Bi-exponential
Data 3 : Mean Rank
Normal
Weibull
Log normal
Bi-exponential
Data 3 : Median rank
Log normal
Normal
Weibull
Biexponential
Data 3 : The rest method
Log normal
Normal
Weibull
Bi-exponential
Data 3 : Linearity Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Log-Normal
X
X
X
X
Weibull
0
0
0
0
Bi-exponential
0
0
0
0
Data 3 : R (Correlation Coefficient Comparaison)
Symmetric .S.C
Mean Rank
R
SD
R
SD
Normal
0.98703
0.14944
0.98605
0.15517
Log-Normal
0.86032
0.47444
0.86453
0.4685
Weibull
0.92709
0.42972
0.93019
0.42177
Bi-exponential
0.94846
0.36331
0.94377
0.37985
Median Rank
The Rest Method
R
SD
R
SD
Normal
0.98419
0.17302
0.98352
0.17899
Log-Normal
0.86901
0.48339
0.87037
0.48743
Weibull
0.94004
0.41502
0.94303
0.41164
Bi-exponential
0.94377
0.37985
0.93343
0.44388
n = 29
Data 3 : Value of Dnα
Normal
α = 0.05
Dnα =0.1612
Weibull, Biexponential
α = 0.05
Dnα =0.1660
Normal
α = 0.15
Dnα =0.1486
Weibull, Biexponential
α = 0.15
Dnα =0.1436
K-S test : Symmetric Simple
Cumulative Distribution
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 262.55
• σ = 178.23
Weibull
• m = 0.80
• ξ = 306.43
Bi-exponential
• ξ = 156.01
• x0= 332.30
Mean Rank
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 262.78
• σ = 180.17
Weibull
• m = 0.80
• ξ = 334.45
Bi-exponential
• ξ = 167.50
• x0= 351.75
Median Rank
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 263.04
• σ = 182.68
Weibull
• m = 0.86
• ξ = 316.02
Bi-exponential
• ξ = 159.24
• x0= 350.33
The Rest Method
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 260.89
• σ = 175.52
Weibull
• m = 0.87
• ξ = 331.82
Bi-exponential
• ξ = 157.23
• x0= 350.62
Data 3 : K-S Test Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Weibull
X
X
X
X
Bi-exponential
0
0
0
X
Data 1+2
N=47
Data 1+2: Symmetric Simple
Cumulative Distribution
Normal
Log normal
Bi-exponential
Weibull
Data 1+2 : Mean Rank
Normal
Log normal
Bi-exponential
Weibull
Data 1+2 : Median Rank
Normal
Log normal
Bi-exponential
Weibull
Data 1+2 : The Rest Method
Normal
Log normal
Bi-exponential
Weibull
Linearity Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Log-Normal
X
X
X
X
Weibull
0
0
0
0
Bi-exponential
0
0
0
0
Data 1+2 : R (Correlation Coefficient Comparaison)
Symmetric .S.C
Mean Rank
R
SD
R
SD
Normal
0.97609
0.15421
0.97995
0.13307
Log-Normal
0.85885
0.37466
0.85304
0.36025
Weibull
0.96044
0.25098
0.94964
0.26211
Bi-exponential
0.87964
0.43778
0.90253
0.36467
Median Rank
The Rest Method
R
SD
R
SD
Normal
0.97828
0.14319
0.9776
0.14675
Log-Normal
0.85644
0.36809
0.85734
0.37037
Weibull
0.95623
0.25496
0.95787
0.25323
Bi-exponential
0.89121
0.40196
0.88741
0.41395
n = 47
Data 1+2 : Value of Dnα
Normal
α = 0.05
Weibull, Biexponential
α = 0.05
Normal
α = 0.15
Weibull, Biexponential
α = 0.15
Dnα = 0,1282
Dnα = 0,1332
Dnα = 0,111
Dnα = 0,1175
Data 1+2 : K-S Test (Symmetric Simple
Cumulative Distribution)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 292,04
• σ = 168,06
Weibull
• m = 1,45
• ξ = 348,25
Bi-exponential
• ξ = 139,86
• x0= 372,16
Data 1+2 : K-S Test (Mean Rank)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 292,44
• σ = 178,25
Weibull
• m = 1.35
• ξ = 343,13
Bi-exponential
• ξ = 149,25
• x0= 374,04
Data 1+2 : K-S Test (Median Rank)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 292,18
• σ = 172,41
Weibull
• m = 1.42
• ξ = 340,09
Bi-exponential
• ξ = 143,88
• x0= 372,86
Data 1+2 : K-S Test (The Rest Method)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 292,30
• σ = 170,94
Weibull
• m = 1,44
• ξ = 339,24
Bi-exponential
• ξ = 142,45
• x0= 372,63
Data 1+2 : K-S Test Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Weibull
X
X
X
X
Bi-exponential
X
X
X
X
Data 2+3
N=55
55
Data 2+3 : Symmetric Simple Cumulative
Distribution
Normal
Log-Normal
Weibull
Bi-Exponential
56
Data 2+3 : Mean Rank
Normal
Log-Normal
Weibull
Bi-exponential
57
Data 2+3 : Median Rank
Normal
Weibull
Log-Normal
Bi-exponential
58
Data 2+3 : The Rest Method
Normal
Weibull
Log-Normal
Bi-exponential
59
Linearity Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Log-Normal
X
X
X
X
Weibull
0
0
0
0
Bi-exponential
0
0
0
0
60
Data 2+3 : R (Correlation Coefficient Comparaison)
Symmetric .S.C
Mean Rank
R
SD
R
SD
Normal
0.97801
0.14793
0.98381
0.1204
Log-Normal
0.74931
0.4995
0.73564
0.48649
Weibull
0.89612
0.40745
0.86468
0.43368
Bi-exponential
0.88209
0.43411
0.90544
0.36254
Median Rank
The Rest Method
R
SD
R
SD
Normal
0.98109
0.13404
0.9801
0.13863
Log-Normal
0.74288
0.49423
0.74509
0.49619
Weibull
0.88134
0.42199
0.88643
0.41743
Bi-exponential
0.89376
0.39929
0.8899
0.4110
61
n = 55
Data 2+3 : Value of Dnα
Normal
α = 0.05
Dnα = 0.119
Weibull, Biexponential
α = 0.05
Dnα = 0.124
Normal
α = 0.15
Weibull, Biexponential
α = 0.15
Dnα = 0.1035
Dnα = 0.107
62
Data 2+3 : K-S Test (Symmetric Simple
Cumulative Distribution)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 279.55
• σ = 166.667
Weibull
• m = 1.1293
• ξ = 341.3
Bi-exponential
• ξ = 138.8
• x0= 360.27
63
Data 2+3 : K-S Test (Mean Rank)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 319.2
• σ = 200
Weibull
• m = 1.0349
• ξ = 349.87
Bi-exponential
• ξ = 147.05
• x0= 360.29
64
Data 2+3 : K-S Test (Median Rank)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 328.4
• σ = 200
Weibull
• m = 1.0855
• ξ = 344.94
Bi-exponential
• ξ = 142.86
• x0= 362
65
Data 2+3 : K-S Test (The Rest Method)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 330
• σ = 200
Weibull
• m = 1.1
• ξ = 342.53
Bi-exponential
• ξ = 140.84
• x0= 359.97
66
Data 2+3 : K-S Test Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
X
X
X
Weibull
X
X
X
X
Bi-exponential
0
0
0
0
67
Data 1+2+3
N=76
Data 1+2+3 : Symmetric Simple Cumulative
Distribution
Normal
Weibull
Log-Normal
Bi-Exponential
69
Data 1+2+3 : Mean Rank
Normal
Log-Normal
Weibull
Bi-exponential
70
Data 1+2+3 : Median Rank
Log-Normal
Normal
Weibull
Bi-exponential
71
Data 1+2+3 : The Rest Method
Normal
Weibull
Log-Normal
Bi-exponential
72
Linearity Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Log-Normal
X
X
X
X
Weibull
0
0
0
0
Bi-exponential
0
0
0
0
73
Data 1+2+3 : R (Correlation Coefficient Comparaison)
Symmetric .S.C
Mean Rank
R
SD
R
SD
Normal
0.99056
0.13213
0.99012
0.13519
Log-Normal
0.88292
0.45247
0.88564
0.44766
Weibull
0.95282
0.36607
0.95435
0.36040
Bi-exponential
0.94838
0.38246
0.94495
0.39482
Median Rank
The Rest Method
R
SD
R
SD
Normal
0.98881
0.14718
0.98836
0.15108
Log-Normal
0.88867
0.45249
0.88958
0.4537
Weibull
0.96072
0.34544
0.96261
0.34023
Bi-exponential
0.94495
0.39482
0.93729
0.43776
74
n = 76
Data 1+2+3 : Value of Dnα
Normal
α = 0.05
Dnα =0.1018
Weibull, Biexponential
α = 0.05
Dnα =0.1058
Normal
α = 0.15
Weibull, Biexponential
α = 0.15
Dnα = 0.0884
Dnα =0.0914
75
Data 1+2+3 : K-S Test (Symmetric Simple
Cumulative Distribution)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 280.89
• σ = 170.05
Weibull
• m = 1.09
• ξ = 342.02
Bi-exponential
• ξ = 145.14
• x0= 357.04
76
Data 1+2+3 : K-S Test (Mean Rank)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 281.28
• σ = 172.29
Weibull
• m = 1.09
• ξ = 345.17
Bi-exponential
• ξ = 150.15
• x0= 364.86
77
Data 1+2+3 : K-S Test (Median Rank)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 281.65
• σ = 174.42
Weibull
• m = 1.14
• ξ = 329.72
Bi-exponential
• ξ = 146.41
• x0= 363.10
78
Data 1+2+3 : K-S Test (The Rest Method)
Dash dot : α = 0.15
Line : α = 0.05
Normal
• µ = 280.46
• σ = 171.83
Weibull
• m = 1.15
• ξ = 333.18
Bi-exponential
• ξ = 145.35
• x0= 363.38
79
Data 1+2+3 : K-S Test Results
Symmetric .S.C
Mean Rank
Median Rank
The Rest
Method
Normal
0
0
0
0
Weibull
X
X
X
X
Bi-exponential
X
X
X
X
80
Conclusion
• R value comparison
- Normal > Weibull > Bi-Exponential > Lognormal
but R value and C.D.F doesn’t guarantee optimal distribution
• The best distribution
Data
The fittest distribution
C. D. F
Data 1
Normal distribution
Mean rank
Data 2
Weibull distribution
Symmetric .S.C
Data 3
Normal distribution
Mean rank
Data 1+2
Normal distribution
Mean rank
Data 2+3
Bi-Exponential distribution
Mean rank
Data 1+2+3
Normal distribution
Symmetric .S.C