### 팀3 (최선아-김시원) - (KAIST) 기계공학과

```CARE Lab.
기계공학에서의 신뢰성공학
Project 1
송지호 교수님
이정환, 이용석 조교
Student
20143677 최선아
20144341 김시원
누적분포함수 추정법
i) 대칭표본누적분포법
ii) 평균 랭크법
iii) 메디안 랭크법
iv) 그 외의 방법
CARE Lab.

− .
=

=
+

− .
=
+ .

− .
=
+ .
확률분포 및 수식
i) 정규 분포

−
= ( +
)

=
ii) 대수정규 분포
iii) 와이불 분포
CARE Lab.
−
−  =

() −
= ( +
)

−
=    −  =

=  − [−

=
−

]
=    −  ()
−
=  − [−(
)]

iv) 이중지수 분포

=
−
−
=

Data set 1 (44개)
567
703
332
621
137
101
521
599
429
30
155
255
123
194
218
203
362
90
197
476
656
700
350
24
43
32
128
686
683
185
103
215
171
105
146
326
83
106
277
338
45
104
667
68
CARE Lab.
대칭표본 누적분포법
3
Equation
3
y = a + b*x
2
Value
y = a + b*
normal
Intercept
-1.1953
0.0868
normal
Slope
0.00419
2.40725E-4
0.95328
Value
Standard Err
2
Standard Erro
lognormal
Intercept
-5.4784
0.18767
lognormal
Slope
1.03595
0.03495
1
Lognormal
1
Normal
Equation
CARE Lab.
0
-1
Normal
-2
0
-1
Log-Normal
-2
Normal
Linear Fit of normal
-3
Lognormal
Linear Fit of lognormal
-3
0
100
200
300
400
500
600
700
3.0
800
3.5
4.0
4.5
5.0
2
Equation
1
2
y = a + b*x
weibull
Intercept
weibull
Slope
Standard Error
-7.51179
0.22755
1.31253
0.04238
1
6.0
6.5
7.0
y = a + b*x
0.73233
Value
Standard Error
biexpo
Intercept
-1.9568
0.16083
biexpo
Slope
0.00486
4.46015E-4
0
Biexponential
0
-1
Weibull
Equation
0.95705
Value
5.5
ln(x)
X
-2
-1
-2
-3
Weibull
-3
-4
Weibull
Linear Fit of weibull
-4
-5
Bi-exponential
Biexponential
Linear Fit of biexpo
-5
3.0
3.5
4.0
4.5
5.0
ln(x)
5.5
6.0
6.5
7.0
0
100
200
300
400
X
500
600
700
800
평균 랭크(Mean rank)
Equation
2
Equation
y = a + b*x
0.88704
Value
Normal
Intercept
Normal
Slope
2
Standard Error
-1.13037
0.07763
0.00396
2.15285E-4
y = a + b*x
0.96093
Value
Standard Error
Lognormal
Intercept
-5.1675
0.16125
Lognormal
Slope
0.97715
0.03003
1
Lognormal
Normal
1
CARE Lab.
0
-1
0
-1
Normal
Log-Normal
Normal
Linear Fit of Normal
-2
0
100
200
300
400
500
600
700
Lognormal
Linear Fit of Lognormal
-2
800
3.0
3.5
4.0
4.5
X
2
Equation
2
0.97274
1
Weibull
Intercept
Weibull
Slope
6.0
6.5
7.0
y = a + b*
0.7631
Standard Error
-7.00031
0.16724
1.22051
0.03115
Value
1
Biexponential Intercept
Biexponential Slope
Standard Error
-1.85034
0.13958
0.00457
3.87098E-4
0
Biexponential
0
Weibull
Equation
Value
5.5
ln(x)
y = a + b*x
5.0
-1
-2
-3
Weibull
Linear Fit of Weibull
3.0
3.5
4.0
4.5
5.0
ln(x)
5.5
6.0
6.5
-2
-3
Weibull
-4
-1
7.0
Bi-exponential
Biexponential
Linear Fit of Biexponential
-4
0
100
200
300
400
X
500
600
700
800
메디안 랭크(Median Rank)
Equation
2
y = a + b*x
Equation
0.88121
Value
Normal
Intercept
Normal
Slope
2
Standard Error
Intercept
0.00409
2.286E-4
Lognormal
Slope
Lognormal
Normal
Standard Error
-5.34101
0.17459
1.00996
0.03252
1
Normal
Linear Fit of Normal
0
100
200
300
400
500
600
700
0
-1
Normal
-2
Log-Normal
Lognormal
Linear Fit of Lognormal
-2
3.0
800
3.5
4.0
4.5
Equation
y = a + b*x
2
0.96538
Value
Weibull
Intercept
Weibull
Slope
5.0
5.5
6.0
6.5
7.0
ln(x)
X
Equation
y = a + b*x
Standard Error
-7.28074
0.197
1.27099
0.03669
0.7474
Value
1
0
Biexponential
Intercept
Biexponential
Slope
Standard Error
-1.90914
0.15065
0.00473
4.178E-4
0
Biexponential
-1
Weibull
Value
Lognormal
-1
1
0.95729
0.08243
0
2
y = a + b*x
-1.16675
1
CARE Lab.
-2
-3
Weibull
Linear Fit of Weibull
-5
3.0
3.5
4.0
4.5
5.0
ln(x)
5.5
6.0
6.5
-2
-3
Weibull
-4
-1
7.0
Bi-exponential
-4
Biexponential
Linear Fit of Biexponential
-5
0
100
200
300
400
X
500
600
700
800
그 외의 방법
3
Equation
3
y = a + b*x
Normal
Intercept
Normal
Slope
y = a + b*x
0.956
Value
Standard Error
-1.17694
0.08392
0.00413
2.32728E-4
2
Lognormal
Intercept
Lognormal
Slope
0
-1
Normal
-2
0.17895
1.0192
0.03333
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
Lognormal
Linear Fit of Lognormal
-3
-3
0
100
200
300
400
500
600
700
800
3.0
3.5
4.0
4.5
X
2
Equation
Weibull
Intercept
Weibull
Slope
5.5
6.0
6.5
7.0
y = a + b*x
-7.36191
0.20705
1.28559
0.03856
0.74235
Value
Standard Error
1
Biexponential
Intercept
Biexponential
Slope
Standard Error
-1.92597
0.1541
0.00478
4.27374E-4
0
Biexponential
0
-1
Weibull
Equation
0.96272
Value
5.0
ln(x)
2
y = a + b*x
1
Standard Error
-5.38991
1
Lognormal
1
Normal
Equation
0.87927
Value
2
CARE Lab.
-2
-3
-1
-2
-3
Weibull
-4
Weibull
Linear Fit of Weibull
-5
Bi-exponential
Biexponential
-4
Linear Fit of Biexponential
-5
3.0
3.5
4.0
4.5
5.0
ln(x)
5.5
6.0
6.5
7.0
0
100
200
300
400
X
500
600
700
800
눈으로 판단한 직선성 결과
CARE Lab.
대칭표본
누적분포법
평균 랭크법
메디안 랭크법
그 외의 방법
Normal

Lognormal

Weibull

Biexponential

육안으로 판단할 경우,
Normal 와 Biexponential 에 비해
Lognormal과 Weibull distribution가 직선성이 우수함을 알 수 있다.
R correlation coefficient
CARE Lab.
메디안
랭크법
그 외의 방법
(Median Rank)
R-square
대칭표본
누적분포법
평균 랭크법
(Mean Rank)
Normal
0.8753
0.8870
0.8812
0.8793
Lognormal
0.9533
0.9609
0.9573
0.9560
Weibull
0.9571
0.9727
0.9654
0.9627
Biexponential
0.7323
0.7631
0.7474
0.7424
Adj. R-square 값을 비교한 결과 눈으로 판단한 직선성과 동일하게
Lognormal과 Weibull distribution 가 적합도가 우수함을 알 수 있다.

유의수준 (
정규 분포 및
대수 정규 분포
이중 지수
및 와이불 분포
)
CARE Lab.
α = 0.05
•  = 0.1326
α = 0.10
•  = 0.1214
α = 0.05
•  = 0.1374
α = 0.10
•  = 0.1262
대칭표본누적분포법 K-S 검정
Symm_Normal
Theoretical curve
a = 5%
a = 10%
1.0
1.0
0.8
0.6
Symm.S.C
Symm.S.C
0.8
0.2
0.0
100
200
300
400
0.6
= .
0.4
= .
= .
0.2
= .
0.0
Symm_Lognormal
Theoretical curve
a = 5%
a = 10%
0.4
0
500
600
700
0
800
100
200
300
= .
Symm.S.C
Symm.S.C
0.8
0.6
0.4
Symm_Weibull
Theoretical curve
a = 5%
a = 10%
0.2
0.0
0
100
200
300
400
X
500
600
700
800
Symm_Biexponential
Theoretical curve
a = 5%
a = 10%
1.0
= .
0.8
400
X
X
1.0
CARE Lab.
500
600
700
800
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
평균 랭크법 K-S 검정
Mean_Normal
Theoretical curve
a = 5%
a = 10%
1.0
1.0
0.8
0.6
Mean rank
Mean rank
0.8
0.4
0.2
0.0
0
100
200
300
400
0.6
0.4
= .
0.2
= .
0.0
500
600
700
= .
= .
Mean_Lognormal
Theoretical curve
a = 5%
a = 10%
0
800
100
200
300
= .
Mean rank
Mean rank
0.8
0.6
0.4
Mean_Weibull
Theoretical curve
a = 5%
a = 10%
0.2
0.0
0
100
200
300
400
X
500
600
700
800
Mean_Biexponential
Theoretical curve
a = 5%
a = 10%
1.0
= .
0.8
400
X
X
1.0
CARE Lab.
500
600
700
800
0.6
0.4
0.2
= .
0.0
= .
0
100
200
300
400
X
500
600
700
800
메디안 랭크법 K-S 검정
Median_Normal
Theoretical curve
a = 5%
a = 10%
1.0
1.0
0.8
Median rank
Median rank
0.8
0.6
0.2
0.0
100
200
300
400
0.6
= .
0.4
= .
= .
0.2
= .
0.0
Median_Lognormal
Theoretical curve
a = 5%
a = 10%
0.4
0
500
600
700
0
800
100
200
300
= .
Median rank
Median rank
0.8
0.6
0.4
Median_Weibull
Theoretical curve
a = 5%
a = 10%
0.2
0.0
0
100
200
300
400
X
500
600
700
800
Median_Biexponential
Theoretical curve
a = 5%
a = 10%
1.0
= .
0.8
400
X
X
1.0
CARE Lab.
500
600
700
800
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
그 외의 방법 K-S 검정
0.8
The rest method
0.8
The rest method
1.0
Rest_Normal
Theoretical curve
a = 5%
a = 10%
1.0
0.6
0.4
0.2
0.0
0
100
200
300
400
0.6
0.4
= .
0.2
= .
0.0
500
600
700
= .
= .
Rest_Lognormal
Theoretical curve
a = 5%
a = 10%
0
800
100
200
300
= .
0.4
Rest_Weibull
Theoretical curve
a = 5%
a = 10%
0.2
0.0
100
200
300
400
X
500
600
700
800
The rest method
The rest method
0.8
0.6
0
500
600
700
800
Rest_Biexponential
Theoretical curve
a = 5%
a = 10%
1.0
= .
0.8
400
X
X
1.0
CARE Lab.
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
대칭표본누적분포법/평균랭크법 α = 0.25
1.0
Log-Normal
1.0
0.6
0.4
Symm_Lognormal
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
0.0
0
100
200
300
Log-Normal
400
500
600
700
0.6
0.4
0.0
800
0
X
1.0
0.8
0.8
0.6
0.6
0.4
Symm_Lognormal
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
0.0
0
100
200
300
400
X
500
600
700
800
Symm_Weibull
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
Mean rank
Mean rank
1.0
Weibull
0.8
Symm.S.C
Symm.S.C
0.8
CARE Lab.
100
200
300
400
500
600
700
800
X
Weibull
0.4
Symm_Weibull
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
0.0
0
100
200
300
400
X
500
600
700
800
메디안 랭크법/그 외 방법들 α = 0.25
1.0
Log-Normal
0.8
0.8
0.6
0.6
0.4
Symm_Lognormal
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
0.0
0
100
200
300
400
500
600
700
Median rank
Median rank
1.0
Weibull
0.4
Symm_Weibull
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
0.0
800
0
100
200
300
X
1.0
400
500
600
700
800
X
Log-Normal
1.0
Weibull
0.8
0.6
0.4
Symm_Lognormal
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
0.0
0
100
200
300
400
X
500
600
700
800
The rest method
0.8
The rest method
CARE Lab.
0.6
0.4
Symm_Lognormal
Theoretical curve
a = 5%
a = 10%
a = 25%
0.2
0.0
0
100
200
300
400
X
500
600
700
800
적합도가 가장 높은 분포
CARE Lab.
Lognormal과 Weibull distribution은
유의수준 10%에서 기각되는 data가 없었기 때문에
둘 중 더 적합도가 높은 분포를 찾기 위해서
α = 0.25 일 때 Log-Normal  =0.1052, Weibull  =0.1082 로 하여
비교해 보았다.
하지만 유의수준 25%에서도 역시 기각되는 data는
하나도 없었고 판단할 수 있는 기준이 모호했다.
그래서 Theoretical curve와
더 가깝고 유사한 분포를 보이는
Lognormal distribution이
가장 적합도가 높다고
결론지었다.
DATA-1 분석결과
CARE Lab.
 눈으로 판단한 직선성 결과
Weibull > Log-normal > Normal > Biexponential
 Adj. R-square 값으로 판단한 적합성 결과
Weibull > Log-normal > Normal > Biexponential
Weibull distribution이 모든 누적분포함수 추정법에서 가장 우수한 적합성을 보였다.
① Weibull distribution – Mean rank (0.9727)
② Weibull distribution – Median rank (0.9654)
③ Weibull distribution – The rest method (0.9627)

K-S 검정에 따른 적합성 결과
1)
Log-Normal과 Weibull distribution이 유의수준 25%에서도 기각되지 않았다.
2)
Normal과 Biexponential distribution은 유의수준 5%에서도 기각되었다.
 최종결과
Theoretical curve와 더 가깝고 유사한 분포를 보이는
Lognormal distribution 중에서도 R값이 가
장 높은 Mean rank가 가장 적합도가 높다고
Data set 2 (51개)
245
281
542
273
411
35
33
136
291
648
685
598
251
704
198
516
528
292
112
483
203
275
12
662
500
207
53
553
61
523
171
587
216
460
238
629
337
517
408
415
448
222
454
297
55
501
511
69
363
532
451
CARE Lab.
대칭표본누적분포법
3
2
Equation
y = a + b*x
Weight
No Weightin
Pearson's r
0.98159
0.96278
3
2
Value
Intercept
-1.7760
0.05624
normal
Slope
0.00498
1.384E-4
1
0
-1
Normal
-2
-3
100
200
No Weigh
Pearson's
0.91037
0.82527
300
400
500
600
700
Intercept
lognormal
Slope
Value Standard
-5.615
0.36926
1.0012
0.06502
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
0
y = a + b*
Weight
Standard Err
normal
LogNormal
Normal
1
Equation
CARE Lab.
LogNormal
Linear Fit of LogNormal
-3
800
2
3
4
5
X
1
Equation
y = a + b*x
Weight
No Weighti
2
Pearson's r
0.97452
1
0.94866
Value
0
Weibull
weibull
Intercept
Slope
Standard Er
-8.1834
0.25348
1.3572
0.04463
-2
-3
Weibull
-4
4
5
lnx
6
Bi-exponential
-1
-2
Equation
y = a + b*x
Weight
No Weighti
-3
-4
Pearson's r
0.95486
0.90996
Value
Weibull
Linear Fit of Weibull
-5
3
Biexponential
Linear Fit of Biexponential
0
-1
2
7
lnx
Biexponential
2
6
-5
7
Biexpo
0
100
200
300
400
X
Standard Er
Intercept
-2.759
Slope
0.0061 2.72589E-4
500
600
0.11078
700
800
평균 랭크(Mean rank)
2
y = a + b*x
Equation
y = a + b*x
Weight
No Weighti
Weight
No Weighti
2
Pearson's r
0.98625
Pearson's r
0.90859
0.97213
0.82197
Value
Normal
Intercept
Normal
Slope
Standard Err
-1.6873
Value
1
0.04602
Lognormal
0.0047 1.13243E-4
LogNormal
1
Normal
Equation
0
-1
Intercept
Slope
CARE Lab.
Standard Er
-5.2991
0.35245
0.9449
0.06206
0
-1
Normal
-2
Log-Normal
-2
Normal
Linear Fit of Normal
0
100
200
300
400
500
600
700
800
LogNormal
Linear Fit of LogNormal
2
3
4
5
X
2
1
Equation
y = a + b*x
Weight
No Weightin
Pearson's r
0.96984
0.93938
Standard Erro
Intercept
-7.5887
0.25596
Slope
1.25527
0.04507
0
-1
-2
Weibull
-3
3
4
5
lnX
6
-1
Bi-exponential
-2
-3
Weibull
Linear Fit of Weibull
-4
2
Biexponential
Linear Fit of Biexponential
1
Biexponential
Weibull
Weibull
7
lnX
2
Value
0
6
y = a + b*x
Weight
No Weighting
Pearson's r
-4
7
Equation
Intercept
Slope
Biexpo
0
100
200
300
0.96744
0.93462
400
X
500
Value
Standard Error
-2.60888
0.08772
0.00577
2.15848E-4
600
700
800
메디안 랭크(Median rank)
3
2
Equation
y = a + b*x
Weight
No Weightin
Pearson's r
0.98395
0.96751
3
2
Value
Normal
-1.7370
0.05127
Slope
0.00487
1.26149E-4
-1
Normal
-2
-3
200
0.90967
0.82398
Value
Standard Err
-5.4744
0.36162
0.97616
0.06367
300
400
500
600
700
LogNormal
Intercept
Slope
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
100
No Weightin
Pearson's r
1
0
0
y = a + b*x
Weight
Standard Err
Intercept
LogNormal
Normal
1
Equation
LogNormal
Linear Fit of LogNormal
-3
2
800
3
4
5
Equation
y = a + b*x
Weight
No Weightin
2
1
7
Biexponential
Linear Fit of Biexponential
1
0.9727
Pearson's r
0.94504
Value
Weibull
0
Standard Err
Intercept
-7.9118
0.25374
Slope
1.3106
0.04468
Biexponential
0
-1
Weibull
6
lnX
X
2
CARE Lab.
-2
-3
-1
Bi-exponential
-2
Equation
y = a + b*x
Weight
No Weighting
-3
Weibull
-4
2
3
4
5
lnX
6
0.92235
-4
Weibull
Linear Fit of Weibull
-5
0.9612
Pearson's r
Value
Intercept
Biexpo
Slope
Standard Erro
-2.692
0.09952
0.00597
2.44896E-4
600
700
-5
7
0
100
200
300
400
X
500
800
그 외의 방법
3
2
Equation
y = a + b*x
Weight
No Weightin
Pearson's r
0.98318
0.96596
3
2
Value
Normal
-1.7509
0.05294
Slope
0.00491
1.30273E-4
-1
Normal
-2
-3
200
No Weightin
0.90994
0.82448
300
400
500
600
700
Lognormal
Intercept
Slope
Value
Standard Err
-5.5242
0.36428
0.98504
0.06414
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
100
y = a + b*x
Weight
Pearson's r
1
0
0
Equation
Standard Err
Intercept
LogNormal
Normal
1
CARE Lab.
LogNormal
Linear Fit of LogNormal
-3
800
2
3
4
5
X
2
Equation
y = a + b*x
Weight
No Weightin
2
1
7
Biexponential
Linear Fit of Biexponential
1
0.9734
0.94644
Value
Weibull
0
Standard Err
Intercept
-8.006
0.2534
Slope
1.3269
0.04462
Biexponential
Pearson's r
0
-1
Weibull
6
lnX
-2
-3
-1
Bi-exponential
-2
Equation
y = a + b*x
Weight
No Weighting
-3
Weibull
-4
2
3
4
5
lnX
6
0.95911
0.91826
-4
Weibull
Linear Fit of Weibull
-5
Pearson's r
Value
Biexpo
Intercept
Biexpo
Slope
Standard Erro
-2.71579
0.1033
0.00603
2.54196E-4
-5
7
0
100
200
300
400
X
500
600
700
800
눈으로 판단한 직선성 결과
CARE Lab.
대칭표본
누적분포법
평균 랭크법
메디안 랭크법
그 외의 방법
Normal
Distribution

LogNormal
Distribution

Weibull
Distribution

Biexponential
Distribution

육안으로 판단할 경우,
대수정규분포 및 이중지수분포에 비해
정규분포 및 와이불 분포가 직선성이 우수함을 알 수 있다.
R correlation coefficient
CARE Lab.
대칭표본
누적분포법
평균 랭크법
메디안 랭크법
그 외의 방법
Normal
Distribution
0.9628
0.9721
0.9675
0.9660
LogNormal
Distribution
0.8253
0.8220
0.8240
0.8245
Weibull
Distribution
0.9487
0.9394
0.9450
0.9464
Biexponential
Distribution
0.9100
0.9346
0.9224
0.9183
눈으로 판단한 직선성과 동일하게 정규분포 및 와이불 분포의
적합도가 우수함을 알 수 있다.

유의수준 (
정규 분포 및
대수 정규 분포
이중 지수
및 와이불 분포
)
CARE Lab.
α = 0.05
•  = 0.1230
α = 0.10
•  = 0.1131
α = 0.05
•  = 0.1280
α = 0.10
•  = 0.1171
대칭표본누적분포법 K-S 검정
SSCD_LogNormal
SSCD_Normal
이론곡선
1.0
이론곡선
1.0
a = 5%
a = 10%
a = 5%
a = 10%
0.8
0.6
0.4
= .
0.2
SSCD_LogNormal
0.8
SSCD_Normal
CARE Lab.
0.6
0.4
= .
0.2
= .
= .
0.0
0.0
0
100
200
300
400
500
600
700
800
0
100
200
300
X
SSCD_Weibull
600
700
800
SSCD_Biexponential
이론곡선
1.0
a = 5%
a = 10%
a = 5%
a = 10%
0.8
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
SSCD_Biexponential
0.8
SSCD_Weibull
500
X
이론곡선
1.0
400
0.6
0.4
0.2
= .
0.0
= .
0
100
200
300
400
X
500
600
700
800
평균 랭크법 K-S 검정
Mean_Normal
Mean_LogNormal
이론곡선
1.0
이론곡선
1.0
a = 5%
a = 10%
a = 5%
a = 10%
0.8
0.6
0.4
= .
0.2
Mean_LogNormal
Mean_Normal
0.8
0
100
200
300
400
500
600
700
0.6
0.4
= .
0.2
= .
0.0
= .
0.0
800
0
100
200
300
X
400
500
600
700
800
X
Mean_Biexponential
Mean_Weibull
이론곡선
1.0
이론곡선
1.0
a=5
a = 10%
a=5
a = 10%
0.8
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
Mean_Biexponential
0.8
Mean_Weibull
CARE Lab.
0.6
0.4
0.2
= .
0.0
= .
0
100
200
300
400
X
500
600
700
800
메디안 랭크법 K-S 검정
Median_Normal
Median_LogNormal
이론곡선
1.0
이론곡선
1.0
a=5
a = 10%
a=5
a = 10%
0.8
0.6
0.4
= .
0.2
Median_LogNormal
Median_Normal
0.8
= .
0.0
0
100
200
300
400
500
600
700
0.6
0.4
= .
0.2
= .
0.0
800
0
100
200
300
X
400
500
600
700
800
X
Median_Biexponential
Median_Weibull
이론곡선
1.0
이론곡선
1.0
a=5
a = 10%
a=5
a = 10%
0.8
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
Median_Biexponential
0.8
Median_Weibull
CARE Lab.
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
그 외의 방법 K-S 검정
Etc_Normal
Etc_LogNormal
이론곡선
1.0
이론곡선
1.0
a=5
a = 10%
a=5
a = 10%
0.8
0.6
0.4
= .
0.2
Etc_LogNormal
Etc_Normal
0.8
0
100
200
300
400
500
600
700
0.6
0.4
= .
0.2
= .
0.0
= .
0.0
800
0
100
200
300
X
500
600
700
800
Etc_Biexponential
이론곡선
1.0
이론곡선
1.0
400
X
Etc_Weibull
a=5
a = 10%
a=5
a = 10%
0.8
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
Etc_Biexponential
0.8
Etc_Weibull
CARE Lab.
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
X
500
600
700
800
DATA-2 분석결과
CARE Lab.
 눈으로 판단한 직선성 결과
Normal > Weibull > Bi-exponential > Lognormal
 Adj. R-square 값으로 판단한 적합성 결과
Normal > Weibull > Bi-exponential > Lognormal
Normal distribution이 모든 누적분포함수 추정법에서 가장 우수한 적합성을 보였
다.
① Normal distribution – Mean rank (0.9721)
② Normal distribution – Median rank (0.9675)
③ Normal distribution – The rest method (0.9660)

K-S 검정에 따른 적합성 결과
1)
Normal distribution만 5% 유의수준에서 기각되지 않았다.
 최종결과
K-S 검정 시 5% 유의수준에서 기각되지 않은,
Normal distribution 중에서도 R값이 가장 높은 Mean rank가 가장 적합도가 높다고 결론지었
Data set 3 (통합, 95개)
CARE Lab.
245
53
281
553
542
61
273
523
411
171
128
35
587
101
686
33
216
521
683
136
460
185
291
238
599
648
629
685
337
567
350
703
24
332
43
621
32
137
429
103
30
215
598
517
155
171
251
408
255
105
704
415
123
146
198
448
326
516
222
194
528
454
218
83
292
297
203
106
112
55
362
277
483
501
90
338
203
511
197
45
275
69
104
12
363
476
662
532
656
667
500
451
700
68
207
대칭표본 누적분포법
3
Equation
3
y = a + b*
Normal
Intercept
-1.4840
0.04611
Normal
Slope
0.00459
1.19514E-4
2
Standard Error
Lognormal
Intercept
-5.58656
0.18378
Lognormal
Slope
1.02317
0.03318
1
Lognormal
Normal
0.90993
Value
Standard Error
1
0
-1
Normal
-2
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
-3
Lognormal
Linear Fit of Lognormal
-3
0
100
Equation
2
200
300
400
500
600
700
800
2
x
y = a + b*x
2
0.97751
Value
1
Weibull
Intercept
Weibull
Slope
Standard Error
-7.93839
0.11684
1.34876
0.0211
3
Equation
4
1
0.83311
Biexpo
Intercept
Biexpo
Slope
5
6
7
ln(x)
y = a + b*x
Value
Standard Error
-2.35309
0.09781
0.0055
2.53481E-4
0
Biexponential
0
-1
Weibull
y = a + b*
Value
2
Equation
0.93994
CARE Lab.
-2
-3
Weibull
-4
-2
-3
Bi-exponential
-4
Biexponential
Linear Fit of Biexpo
-5
Weibull
Linear Fit of Weibull
-5
-1
-6
-6
2
3
4
5
ln(x)
6
7
0
100
200
300
400
x
500
600
700
800
평균 랭크(Mean rank)
3
Equation
3
y = a + b*x
0.9126
Value
Standard Err
Normal
Intercept
-1.4400
0.04079
Normal
Slope
0.00445
1.05712E-4
2
Lognormal
Intercept
Lognormal
Slope
Standard Error
-5.40141
0.17478
0.98926
0.03156
1
Lognormal
1
Normal
y = a + b*x
Value
2
Equation
0.94959
0
-1
Normal
-2
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
Lognormal
Linear Fit of Lognormal
-3
-3
0
100
200
300
400
500
600
700
800
2
3
4
5
x
Equation
2
1
2
0.98258
Weibull
Intercept
Weibull
Slope
6
7
ln(x)
y = a + b*x
Value
Equation
y = a + b*x
0.85662
Standard Error
-7.59145
0.09794
1.28792
0.01769
Value
1
Biexpo
Intercept
Biexpo
Slope
Standard Error
-2.27718
0.08635
0.00531
2.2378E-4
0
Biexponential
0
Weibull
CARE Lab.
-1
-2
-3
Weibull
-1
-2
-3
Bi-exponential
-4
-4
Weibull
Linear Fit of Weibull
-5
2
3
4
5
ln(x)
6
Biexponential
Linear Fit of Biexpo
-5
7
0
100
200
300
400
x
500
600
700
800
메디안 랭크(Median Rank)
3
2
3
Equation
y = a + b*x
Weight
No Weighting
Pearson's r
0.97222
0.94461
2
Value
Intercept
Normal
Slope
Standard Error
-1.46502
0.04361
0.00453
1.1302E-4
0
-1
Normal
-2
-3
100
200
300
No Weightin
Pearson's r
0.95514
0.91135
400
500
600
700
Standard Err
LogNormal
Intercept
-5.5056
0.17955
LogNormal
Slope
1.00835
0.03242
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
0
y = a + b*x
Weight
Value
1
LogNormal
Normal
1
Normal
Equation
LogNormal
Linear Fit of LogNormal
-3
2
800
3
4
5
1
Equation
y = a + b*x
Weight
No Weightin
Pearson's r
0.99026
0.98041
1
Standard Err
Intercept
-7.7830
0.10669
Slope
1.32152
0.01926
0
Biexponential
Weibull
Weibull
-1
-2
-3
Weibull
-4
-1
Bi-exponential
-2
-3
Equation
y = a + b*x
Weight
No Weighting
Pearson's r
0.91985
-4
0.84447
-5
Biexpo
Value
Weibull
Linear Fit of Weibull
-5
2
3
7
Biexponential
Linear Fit of Biexponential
2
Value
0
6
lnX
X
2
CARE Lab.
4
5
lnX
6
7
0
100
200
300
400
X
Standard Erro
Intercept
-2.31969
0.09238
Slope
0.00541
2.39408E-4
500
600
700
800
그 외의 방법
3
2
No Weightin
Normal
Normal
0.97141
0.94304
Intercept
Slope
2
Value
Standard Err
-1.4718
0.04447
0.00455 1.15248E-4
0
-1
Normal
-2
-3
100
200
300
y = a + b*x
Weight
No Weighti
Pearson's r
0.9549
0.91089
400
500
600
700
Lognormal
Intercept
Lognormal
Slope
Standard Er
-5.5345
0.181
1.0136
0.03268
0
-1
Log-Normal
-2
Normal
Linear Fit of Normal
0
Equation
Value
1
LogNormal
Normal
y = a + b*x
Weight
Pearson's r
1
3
Equation
CARE Lab.
LogNormal
Linear Fit of LogNormal
-3
2
800
3
4
5
2
1
Equation
y = a + b*x
Weight
No Weightin
Pearson's r
0.98982
0.97953
1
Standard Erro
Intercept
-7.83788
0.10992
Weibull
Slope
1.33113
0.01985
0
Biexponential
Weibull
Weibull
-1
-2
-3
Weibull
-4
-5
-6
3
4
5
lnX
6
-1
Bi-exponential
-2
Equation
y = a + b*x
-3
Weight
No Weightin
-4
Pearson's r
-5
Weibull
Linear Fit of Weibull
2
7
Biexponential
Linear Fit of Biexponential
2
Value
0
6
lnX
X
0.91779
0.84064
Intercept
Slope
Biexpo
Biexpo
Value
Standard Err
-2.3315
0.09423
0.00544 2.44213E-4
-6
7
0
100
200
300
400
X
500
600
700
800
눈으로 판단한 직선성 결과
CARE Lab.
대칭표본
누적분포법
평균 랭크법
메디안 랭크법
그 외의 방법
Normal

Lognormal

Weibull

Biexponential

육안으로 판단할 경우,
Normal , Lognormal 및 Biexponential 에 비해
Weibull distribution가 직선성이 우수함을 알 수 있다.
R correlation coefficient
CARE Lab.
메디안
랭크법
그 외의 방법
(Median Rank)
R-square
대칭표본
누적분포법
평균 랭크법
(Mean Rank)
Normal
0.9399
0.9496
0.9446
0.9430
Lognormal
0.9099
0.9126
0.9114
0.9109
Weibull
0.9775
0.9826
0.9804
0.9795
Biexponential
0.8331
0.8566
0.8445
0.8406
Adj. R-square 값을 비교한 결과 눈으로 판단한 직선성과 동일하게
Weibull distribution 이 적합도가 우수함을 알 수 있다.
결론
정규 분포 및
대수 정규 분포
이중 지수
및 와이불 분포
CARE Lab.
α = 0.01
•  = 0.1060
α = 0.05
•  = 0.0915
α = 0.01
•  = 0.1110
α = 0.05
•  = 0.0955
대칭표본누적분포법 K-S 검정
Symm_normal
Theoretical curve
a = 5%
a = 1%
1.0
0.8
0.8
0.6
SSCD
SSCD
Symm_Lognormal
Theoretical curve
a = 5%
a = 1%
1.0
0.6
CARE Lab.
0.4
0.4
0.2
= .
0.2
0.0
= .
0.0
0
100
200
300
400
500
600
700
= .
= .
0
800
100
200
300
x
Symm_Weibull
Theoretical curve
a = 5%
a = 1%
1.0
0.8
400
500
600
700
800
x
Symm_Biexponential
Theoretical curve
a = 5%
a = 1%
1.0
0.8
0.6
SSCD
SSCD
0.6
0.4
= .
0.2
0
100
200
300
400
x
500
600
700
= .
0.2
= .
0.0
0.4
800
= .
0.0
0
100
200
300
400
x
500
600
700
800
평균 랭크법 K-S 검정
Symm_Normal
Theoretical curve
a = 5%
a = 1%
Mean rank 95
0.8
0.8
0.6
0.4
= .
0.2
100
200
300
400
500
600
700
0.6
0.4
= .
0.2
= .
0.0
0
Symm_Lognormal
Theoretical curve
a = 5%
a = 1%
1.0
Mean rank 95
1.0
= .
0.0
800
0
100
200
300
x
0.8
0.6
0.4
0.2
= .
= .
0.0
500
600
700
800
Symm_Biexponential
Theoretical curve
a = 5%
a = 1%
1.0
Mean rank 95
Mean rank 95
0.8
400
x
Symm_Weibull
Theoretical curve
a = 5%
a = 1%
1.0
CARE Lab.
0.6
0.4
= .
0.2
= .
0.0
0
100
200
300
400
x
500
600
700
800
0
100
200
300
400
500
600
700
800
메디안 랭크법 K-S 검정
Median_Normal
Median_LogNormal
이론곡선
1.0
1.0
이론곡선
a = 1%
a = 5%
a = 1%
a = 5%
0.8
0.6
0.4
= .
0.2
Median_LogNormal
Median_Normal
0.8
= .
0.0
0
100
200
300
400
500
600
700
0.6
0.4
= .
0.2
= .
0.0
0
800
100
200
300
X
400
500
600
700
800
X
Median_Weibull
Median_Biexponential
이론곡선
1.0
이론곡선
1.0
a = 1%
a = 5%
a = 1%
a = 5%
0.8
0.6
0.4
0.2
= .
= .
0.0
0
100
200
300
400
X
500
600
700
800
Median_Biexponential
0.8
Median_Weibull
CARE Lab.
0.6
0.4
0.2
= .
= .
0.0
0
100
200
300
400
X
500
600
700
800
그 외의 방법 K-S 검정
Etc_Normal
Etc_LogNormal
이론곡선
1.0
1.0
이론곡선
a = 1%
a = 5%
a = 1%
a = 5%
0.8
0.6
0.4
= .
0.2
Etc_LogNormal
Etc_Normal
0.8
0
100
200
300
400
500
600
700
0.6
0.4
= .
0.2
= .
0.0
= .
0.0
800
0
100
200
300
X
400
500
600
700
800
X
Etc_Weibull
Etc_Biexponential
이론곡선
1.0
이론곡선
1.0
a = 1%
a = 5%
a = 1%
a = 5%
0.8
Etc_Biexponential
0.8
Etc_Weibull
CARE Lab.
0.6
0.4
0.2
= .
= .
0.0
0
100
200
300
400
X
500
600
700
800
0.6
0.4
0.2
= .
= .
0.0
0
100
200
300
400
X
500
600
700
800
DATA-3(1+2) 분석결과
CARE Lab.
 눈으로 판단한 직선성 결과
Weibull > Log-normal > Normal > Biexponential
 Adj. R-square 값으로 판단한 적합성 결과
Weibull > Normal > Log-normal > Biexponential
Weibull distribution이 모든 누적분포함수 추정법에서 가장 우수한 적합성을 보였
다.
① Weibull distribution – Mean rank (0.9826)
② Weibull distribution – Median rank (0.9804)
③ Weibull distribution – The rest method (0.9795)

K-S 검정에 따른 적합성 결과
1)
유의수준 5%에서는 모든 분포가 기각되었다.
2)
유의수준 1%에서는 Weibull distribution만 채택되었다.
 최종결과
K-S 검정 유의수준 1%에 채택되고 있는
Weibull distribution 중 R값이 가장 높은 Mean rank가 가장 적합도가 높다고 결론지었다.
최종 결론
CARE Lab.
 Data 1 : Mean rank를 사용한 Lognormal distribution
 Data 2: Mean rank를 사용한 Normal distribution
 Data 3: Mean rank를 사용한 Weibull distribution
CARE Lab.
감사합니다
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