Project I 11.27.2014 Group 8 Saeb Moosavi, Shakil Bin Zaman MAE - - PowerPoint PPT Presentation

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Project I 11.27.2014 Group 8 Saeb Moosavi, Shakil Bin Zaman MAE - - PowerPoint PPT Presentation

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Project I 11.27.2014 Group 8 Saeb Moosavi, Shakil Bin Zaman MAE 430 Introduction To Reliability In Mechanical Engineering Design Professor Ji Ho Song MAE 430 Project I 1 Contents Data set 1 (Saeb) Linearity Test K-S test

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SLIDE 1

MAE 430 Project I

1

Project I

11.27.2014

Group 8 Saeb Moosavi, Shakil Bin Zaman MAE 430 Introduction To Reliability In Mechanical Engineering Design Professor Ji Ho Song

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SLIDE 2

MAE 430 Project I

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Contents

  • Data set 1 (Saeb)
  • Data set 2 (Shakil)
  • Data set 3 (1+2: Shakil + Saeb)
  • Conclusion
  • Linearity Test
  • K-S test
  • Distribution analysis
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SLIDE 3

MAE 430 Project I

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DATA SET 1 – Saeb 26 data

600 38 128 270 376 66 120 47 134 382 187 656 266 661 337 57 681 384 565 53 632 373 141 7 378 253

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SLIDE 4

MAE 430 Project I

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Step1 Cumulative distribution function estimation Step2 Probability estimation using probability paper Step3 Goodness of fit test (Kolomogorov-Smirnov test)

600 38 128 270 376 66 120 47 134 382 187 656 266 661 337 57 681 384 565 53 632 373 141 7 378 253

26 Data

Target: select the best distribution of given data

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SLIDE 5

MAE 430 Project I

5

  • C.D.F. estimation method
  • Simple cumulative distribution(X)

F(xj) = j / n

  • Symmetric simple cumulative distribution

F(xj) = ( j - 0.5) / n

  • Mean rank

F(xj) = j /(n +1)

  • Median rank

F(xj) = ( j - 0.3) /(n + 0.4)

  • (Mode rank) (X)

F(xj) = ( j -1) /(n -1)

  • The rest method

F(xj) = ( j - 0.375) /(n + 0.25)

Simple C.D. & Mode rank can’t be use because F(x) = 0 or 1

C.D.F. estimation method

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SLIDE 6

MAE 430 Project I

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Probability distribution

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SLIDE 7

MAE 430 Project I

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Symmetric simple cumulative distribution

100 200 300 400 500 600 700

  • 2
  • 1

1 2

Norm A

Equation y = a + b*

  • Adj. R-Squar

0.91477 Value Standard Err Norm Intercept

  • 1.2884

0.09701 Norm Slope 0.0043 2.61958E-4

Normal LogNormal Weibull Biexponential

2 3 4 5 6 7

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a + b*

  • Adj. R-Squar

0.9666 Value Standard Erro C Intercept

  • 6.3174

0.21831 C Slope 1.0894 0.04047

2 3 4 5 6 7

  • 2
  • 1

1 2

B A

Equation y = a + b*

  • Adj. R-Squar

0.88426 Value Standard Erro B Intercept

  • 4.3861

0.32344 B Slope 0.83085 0.05996

100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a + b

  • Adj. R-Squ

0.8072 Value Standard Er C Intercept

  • 2.091

0.18333 C Slope 0.0050 4.95067E-4

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SLIDE 8

MAE 430 Project I

8

Normal LogNormal Weibull Biexponential

100 200 300 400 500 600 700

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

C A

Equation y = a + b*

  • Adj. R-Squa

0.92916 Value Standard Err C Intercept

  • 1.1853

0.08076 C Slope 0.0039 2.18083E-4 100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a + b

  • Adj. R-Squ

0.84288 Value Standard Er C Intercept -1.923 0.14794 C Slope 0.0046 3.9949E-4

Mean Rank

2 3 4 5 6 7

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

C A

Equation y = a + b

  • Adj. R-Squa

0.88285 Value Standard Err C Intercept

  • 4.0022

0.29714 C Slope 0.7581 0.05509

2 3 4 5 6 7

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a + b

  • Adj. R-Squa

0.95881 Value Standard Err C Intercept

  • 5.6530

0.21671 C Slope 0.97004 0.04018

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SLIDE 9

MAE 430 Project I

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Normal LogNormal Weibull Biexponential

100 200 300 400 500 600 700

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

C A

Equation y = a + b*

  • Adj. R-Squa

0.92212 Value Standard Err C Intercept

  • 1.2423

0.08907 C Slope 0.0041 2.40535E-4

Median Rank

100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a + b*x

  • Adj. R-Squar

0.84288 Value Standard Error C Intercept

  • 1.9237

0.14794 C Slope 0.00464 3.9949E-4

2 3 4 5 6 7

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

C A

Equation y = a + b

  • Adj. R-Squ

0.88386 Value Standard Er C Intercept

  • 4.212

0.31123 C Slope 0.7979 0.0577 2 3 4 5 6 7

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a +

  • Adj. R-Squ

0.9635 Value Standard Er C Intercep

  • 6.008

0.21666 C Slope 1.0339 0.04017

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SLIDE 10

MAE 430 Project I

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Normal LogNormal Weibull Biexponential The Rest Method

100 200 300 400 500 600 700

  • 2
  • 1

1 2

C A

Equation y = a + b*

  • Adj. R-Squar

0.91972 Value Standard Err C Intercept

  • 1.2586

0.09174 C Slope 0.0042 2.4773E-4

100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a + b*

  • Adj. R-Squar

0.81911 Value Standard Err C Intercept

  • 2.041

0.17181 C Slope 0.0049 4.63975E-4 2 3 4 5 6 7

  • 4
  • 3
  • 2
  • 1

1 2

C A

Equation y = a +

  • Adj. R-Squ

0.96477 Value Standard Er C Intercep

  • 6.115

0.21693 C Slope 1.0530 0.04022 2 3 4 5 6 7

  • 2
  • 1

1 2

C A

Equation y = a +

  • Adj. R-Squ 0.88405

Value Standard Er C Intercep

  • 4.273

0.31543 C Slope 0.8094 0.05848

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SLIDE 11

MAE 430 Project I

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Symm.S.C Mean Rank Median Rank Rest R2 R R2 R R2 R R2 R Normal 0.9181 0.95822 0.9319 0.9654 0.9252 0.96189 0.9229 0.96069 Log-Normal 0.8888 0.94281 0.7667 0.87565 0.8884 0.9426 0.974 0.98694 Weibull 0.9679 0.98384 0.756 0.8695 0.8954 0.9463 0.787 0.88717 Bi-exponential 0.8149 0.90272 0.8491 0.9215 0.8491 0.9215 0.8263 0.90903

R-correlation coefficient

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SLIDE 12

MAE 430 Project I

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α = 0.01 α = 0.05

n = 26

Dn

α = 0.198

Normal & LogNormal

Dn

α = 0.204

Weibull & Extreme Value

Dn

α = 0.170

Normal & LogNormal

Dn

α = 0.175

Weibull & Extreme Value

Kolmogorov-Smirnov Test (K-S test)

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MAE 430 Project I

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100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Biexponential a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Weibull a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

LogNormal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Normal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

σ = 232.56 μ = 299.53 σ = 1.203 μ = 5.279 m = 1.0894 ξ = 329.96 ξ = 200 X0 = 418.2 Symmetric simple cumulative distribution

K-S Test

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SLIDE 14

MAE 430 Project I

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K-S Test

Mean Rank σ = 256.41 μ = 303.92 σ = 1.319 μ = 5.279 m = 0.97 ξ = 339.54 ξ = 217.39 X0 = 418.04

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Normal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

LogNormal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Weibull a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Biexponential a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

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SLIDE 15

MAE 430 Project I

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K-S Test

Median Rank σ = 243.9 μ = 303 σ = 1.253 μ = 5.279 m = 1.0339 ξ = 333.95 ξ = 215.52 X0 = 414.59

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Normal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

LogNormal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Weibull a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Biexponential a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

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SLIDE 16

MAE 430 Project I

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K-S Test

The Rest Method σ = 238.09 μ = 299.66 σ = 1.235 μ = 5.279 m = 1.053 ξ = 332.69 ξ = 204.08 X0 = 416.53

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Normal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

LogNormal a = 0.01, D = 0.198 a = 0.05, D = 0.170

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Weibull a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Biexponential a = 0.01, D = 0.204 a = 0.05, D = 0.175

B A

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SLIDE 17

MAE 430 Project I

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Discussion

According to K-S test results the best distribution function for this data set is Weibull distribution because data are more concentrated in the center of limit line in this distribution.

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SLIDE 18

MAE 430 Project I

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DATA SET 2 – Shakil 21 data

262 500 328 91 608 119 15 164 211 89 668 278 319 116 330 449 74 128 622 223 98

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SLIDE 19

MAE 430 Project I

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Symmetrical Simple Cumulative Distribution

100 200 300 400 500 600 700

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

C Linear Fit of C C A

Equation y = a + b*x

  • Adj. R-Square

0.85896 Value Standard Error C Intercept

  • 1.15535

0.14318 C Slope 0.00452 4.41996E-4

Normal

0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

D Linear Fit of D D A

Equation y = a + b*x
  • Adj. R-Square
0.82697 Value Standard Error D Intercept
  • 3.10377
0.35302 D Slope 3.90427 0.4305

LogNormal 0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 3
  • 2
  • 1

1

E Linear Fit of E E A

Equation y = a + b*x
  • Adj. R-Square
0.89182 Value Standard Error E Intercept
  • 4.44447
0.34076 E Slope 4.9369 0.41554

Weibull

100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

K Linear Fit of K K A

Equation y = a + b*x
  • Adj. R-Square
0.71158 Value Standard Error K Intercept
  • 1.88455
0.25577 K Slope 0.00517 7.89547E-4

Biexponential

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SLIDE 20

MAE 430 Project I

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Mean Rank Distribution

100 200 300 400 500 600 700

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

E Linear Fit of E E A

Equation y = a + b*x

  • Adj. R-Squar

0.86499 Value Standard Erro E Intercept

  • 1.0308

0.12459 E Slope 0.00403 3.8459E-4

Normal

0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

G Linear Fit of G G A

Equation y = a + b*x
  • Adj. R-Square
0.82697 Value Standard Error G Intercept
  • 3.10377
0.35302 G Slope 3.90427 0.4305

LogNormal

0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 3
  • 2
  • 1

1

H Linear Fit of H H A

Equation y = a + b*x
  • Adj. R-Square
0.89182 Value Standard Error H Intercept
  • 4.44447
0.34076 H Slope 4.9369 0.41554

Weibull

100 200 300 400 500 600 700

  • 3
  • 2
  • 1

1

H Linear Fit of H H A

Equation y = a + b*x
  • Adj. R-Square
0.74457 Value Standard Error H Intercept
  • 1.6937
0.2092 H Slope 0.00459 6.45777E-4

Biexponential

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SLIDE 21

MAE 430 Project I

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Median Rank Distribution

100 200 300 400 500 600 700

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

L Linear Fit of L L A

Equation y = a + b*x
  • Adj. R-Square
0.8624 Value Standard Error L Intercept
  • 1.09873
0.13425 L Slope 0.0043 4.14407E-4

Normal

0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

J Linear Fit of J J A

Equation y = a + b*x
  • Adj. R-Square
0.8338 Value Standard Error J Intercept
  • 3.32544
0.36928 J Slope 4.18312 0.45032

LogNormal

0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 4
  • 3
  • 2
  • 1

1 2

K Linear Fit of K K A

Equation y = a + b*x
  • Adj. R-Square
0.90626 Value Standard Error K Intercept
  • 4.81111
0.34247 K Slope 5.37043 0.41763

Weibull

100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

O Linear Fit of O O A

Equation y = a + b*x
  • Adj. R-Square
0.7281 Value Standard Error O Intercept
  • 1.79623
0.23303 O Slope 0.00491 7.19353E-4

Biexponential

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MAE 430 Project I

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Rest Distribution

100 200 300 400 500 600 700

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

S Linear Fit of S S A

Equation y = a + b*
  • Adj. R-Squar
0.86134 Value Standard Erro S Intercept
  • 1.1185
0.13727 S Slope 0.00438 4.23738E-4

Normal

0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

M Linear Fit of M M A

Equation y = a + b
  • Adj. R-Squa
0.8359 Value Standard Err M Intercept
  • 3.3913
0.37378 M Slope 4.2660 0.45581

LogNormal

0.0 0.2 0.4 0.6 0.8 1.0 1.2

  • 4
  • 3
  • 2
  • 1

1 2

N Linear Fit of N N A

Equation y = a + b*x
  • Adj. R-Square
0.91071 Value Standard Error N Intercept
  • 4.92406
0.34179 N Slope 5.50395 0.4168

Weibull

100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

V Linear Fit of V V A

Equation y = a + b*x
  • Adj. R-Square
0.72264 Value Standard Error V Intercept
  • 1.82686
0.24066 V Slope 0.005 7.42901E-4

Biexponential

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MAE 430 Project I

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R-correlation coefficient

  • Symm. S.C

R SD Bie 0.85355 0.6664 Log 0.91496 0.36748 Nor 0.93126 0.37306 Wei 0.94773 0.35471

Median

R SD Bie 0.86261 0.60716 Log 0.91847 0.3844 Nor 0.9330 0.34977 Wei 0.95487 0.35649

The rest

R SD Bie 0.85963 0.62703 Log 0.91954 0.38908 Nor 0.93247 0.35765 Wei 0.95706 0.35578

Mean

R SD Bie 0.87155 0.54506 Log 0.91496 0.36748 Nor 0.93431 0.32461 Wei 0.94773 0.35471

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MAE 430 Project I

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Linearity test

Symm. S.C. Mean Median The rest Normal O O O O LogNormal O O O O Weibull O O O O Bi-exponential X X X X

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Kolmogorov-Smirnov test (K-S test)

  • Weibull, Bi-exponential
  • Normal, LogNormal

α=0.05

  • Weibull, Bi-exponential
  • Normal, LogNormal

α=0.15

n

D

= 0.207

n

D

= 0.202

n

D

= 0.178

n

D

= 0.175

n=21

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SLIDE 26

MAE 430 Project I

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K-S test (Simple Symmetrical Coefficient)

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Normal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Weibull

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Biexponential

Red : alpha = 0.05 Blue : alpha = 0.15

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B A LogNormal

µ= 5.29 σ=1.886 µ= 255.61 σ=312.88 m= 0.8369 ξ= 202.48 x0= 364.5 ξ= 193.42

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SLIDE 27

MAE 430 Project I

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K-S test Mean rank

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Normal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Weibull

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Biexponential

Red : alpha = 0.05 Blue : alpha = 0.15

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B A LogNormal

µ= 5.29 σ=1.886 x0= 364.3 ξ= 198.42 µ= 255.78 σ=322.88 m= 0.8469 ξ= 212.48

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MAE 430 Project I

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K-S test Median rank

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

D D A Normal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

D D A Weibull

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Biexponential

Red : alpha = 0.05 Blue : alpha = 0.15

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B A LogNormal

µ= 5.29 σ=1.886 x0= 362.5 ξ= 191.52 µ= 252.61 σ=314.22 m= 0.8769 ξ= 232.48

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SLIDE 29

MAE 430 Project I

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K-S test Rest method

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

F F A Normal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

F F A Weibull

Red : alpha = 0.05 Blue : alpha = 0.15

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B B A Biexponential

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

B A LogNormal

µ= 5.29 σ=1.886 x0= 364.5 ξ= 193.42 µ= 254.21 σ=311.86 m= 0.8369 ξ= 202.48

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SLIDE 30

MAE 430 Project I

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Data set 2 Results (21 items)

  • Linearity test observation

– Weibull distribution makes the best distribution of the given data followed by Normal distribution (according to R-value and Standard deviation).(O) – It is imperative that the Bi-exponential distribution is skewed from the straight line. (X)

  • K-S Test Observation

– Normal distribution gives a better distribution (all values inside the significance level) with no outliers.(O) – LogNormal distribution is rejected undoubtedly. – Weibull and Bi-exponential distributions show a scattered distribution as far as the significance value is concerned. (X) – Both distributions contain some outliers outside the 5% significance level.

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SLIDE 31

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DATA SET 3 – 1+2 47 data

262 500 328 91 608 119 15 164 211 89 668 278 319 116 330 449 74 128 622 223 98

600 38 128 270 376 66 120 47 134 382 187 656 266 661 337 57 681 384 565 53 632 373 141 7 378 253

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SLIDE 32

MAE 430 Project I

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Symmetrical Simple Cumulative Distribution

100 200 300 400 500 600 700

  • 3
  • 2
  • 1

1 2 3

ND Linear Fit of ND ND A

Equation y = a + b*x

  • Adj. R-Square

0.91505 Value Standard Error ND Intercept

  • 1.3099

0.07248 ND Slope 0.00457 2.04908E-4

Normal

2 3 4 5 6 7

  • 3
  • 2
  • 1

1 2 3

LN Linear Fit of C LN A

Equation y = a + b*x
  • Adj. R-Square
0.91185 Value Standard Error C Intercept
  • 4.90233
0.22862 C Slope 0.92787 0.04249

LogNormal

2 3 4 5 6 7

  • 5
  • 4
  • 3
  • 2
  • 1

1 2

W Linear Fit of D W A

Equation y = a + b*x

  • Adj. R-Squar

0.98428 Value Standard Erro D Intercept

  • 7.0104

0.12215 D Slope 1.2187 0.0227

Weibull

100 200 300 400 500 600 700

  • 5
  • 4
  • 3
  • 2
  • 1

1 2

BI Linear Fit of BI BI A

Equation y = a + b*x
  • Adj. R-Square
0.78826 Value Standard Error BI Intercept
  • 2.1124
0.1448 BI Slope 0.00537 4.0935E-4

Bi-exponential

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MAE 430 Project I

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Mean Rank Distribution

100 200 300 400 500 600 700

  • 2
  • 1

1 2

E Linear Fit of E E A

Equation y = a + b*x
  • Adj. R-Square
0.92616 Value Standard Error E Intercept
  • 1.24164
0.06368 E Slope 0.00433 1.80021E-4

Normal

2 3 4 5 6 7

  • 2
  • 1

1 2

F Linear Fit of F F A

Equation y = a + b*x
  • Adj. R-Square
0.91059 Value Standard Error F Intercept
  • 4.61646
0.21696 F Slope 0.87377 0.04033

LogNormal

2 3 4 5 6 7

  • 4
  • 3
  • 2
  • 1

1 2

G Linear Fit of G G A

Equation y = a + b*x
  • Adj. R-Square
0.9802 Value Standard Error G Intercept
  • 6.49557
0.12689 G Slope 1.12585 0.02358

Weibull

100 200 300 400 500 600 700

  • 4
  • 3
  • 2
  • 1

1 2

F Linear Fit of F F A

Equation y = a + b*x
  • Adj. R-Square
0.81779 Value Standard Error F Intercept
  • 1.9997
0.12433 F Slope 0.00506 3.51498E-4

Bi-exponential

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MAE 430 Project I

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Median Rank Distribution

100 200 300 400 500 600 700

  • 3
  • 2
  • 1

1 2 3

D Linear Fit of D D A

Equation y = a + b*x
  • Adj. R-Square
0.92064 Value Standard Error D Intercept
  • 1.27988
0.06825 D Slope 0.00446 1.92944E-4

Normal

2 3 4 5 6 7

  • 3
  • 2
  • 1

1 2 3

I Linear Fit of I I A

Equation y = a + b*x
  • Adj. R-Square
0.91152 Value Standard Error I Intercept
  • 4.77495
0.22313 I Slope 0.90376 0.04147

LogNormal

2 3 4 5 6 7

  • 5
  • 4
  • 3
  • 2
  • 1

1 2

C Linear Fit of C C A

Equation y = a + b*x

  • Adj. R-Square

0.98304 Value Standard Error C Intercept

  • 6.77503

0.12254 C Slope 1.1763 0.02278

Weibull

100 200 300 400 500 600 700

  • 5
  • 4
  • 3
  • 2
  • 1

1 2

J Linear Fit of J J A

Equation y = a + b*x
  • Adj. R-Square
0.80277 Value Standard Error J Intercept
  • 2.06197
0.13496 J Slope 0.00523 3.81545E-4

Bi-exponential

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Rest Distribution

100 200 300 400 500 600 700

  • 3
  • 2
  • 1

1 2 3

M Linear Fit of M M A

Equation y = a + b*x
  • Adj. R-Square
0.91879 Value Standard Error M Intercept
  • 1.29059
0.06968 M Slope 0.0045 1.96998E-4

Normal

2 3 4 5 6 7

  • 3
  • 2
  • 1

1 2 3

J Linear Fit of J J A

Equation y = a + b*x
  • Adj. R-Square
0.91169 Value Standard Error J Intercept
  • 4.82005
0.225 J Slope 0.9123 0.04182

Log Normal

2 3 4 5 6 7

  • 5
  • 4
  • 3
  • 2
  • 1

1 2

K Linear Fit of K K A

Equation y = a + b*x
  • Adj. R-Square
0.98362 Value Standard Error K Intercept
  • 6.85714
0.12192 K Slope 1.19112 0.02266

Weibull

100 200 300 400 500 600 700

  • 5
  • 4
  • 3
  • 2
  • 1

1 2

N Linear Fit of N N A

Equation y = a + b*x
  • Adj. R-Square
0.79792 Value Standard Error N Intercept
  • 2.07978
0.13829 N Slope 0.00528 3.90961E-4

Bi-exponential

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MAE 430 Project I

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R-correlation coefficient

  • Symm. S.C

R SD Bie 0.8904 0.58066 Log 0.9551 0.2961 Nor 0.9151 0.2961 Wei 0.9928 0.1582

Median

R SD Bie 0.8984 0.2737 Log 0.9557 0.2890 Nor 0.9604 0.2737 Wei 0.9917 0.1587

The rest

R SD Bie 0.8957 0.5546 Log 0.9558 0.2914 Nor 0.9595 0.2794 Wei 0.9920 0.1579

Mean

R SD Bie 0.9065 0.4986 Log 0.9553 0.2810 Nor 0.9632 0.2554 Wei 0.9903 0.1643

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Linearity test

  • Symm. S.

C. Mean Median The rest Normal O O O O LogNormal X X O O Weibull O O O O Biexponential X X X X

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Kolmogorov-Smirnov test (K-S test)

  • Weibull, Bi-exponential
  • Normal, LogNormal

α=0.01

  • Weibull, Bi-exponential
  • Normal, LogNormal

α=0.05

n

D

= 0.154

n

D

= 0.143

n

D

= 0.149

n

D

= 0.128

n=47

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100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

SSC SSC A Bi-exponential

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

SSC SSC A Weibull

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

SSC SSC A LogNormal

K-S test (Simple Symmetrical Coefficient)

Red : alpha = 0.05 Blue : alpha = 0.01

µ= 5.29 σ=1.886 µ= 218.8 σ=286.7 m= 0.9369 ξ= 302.48 x0= 364.5 ξ= 193.42

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

SSC SSC A Normal

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100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Mean Mean A Bi-exponential

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Mean Mean A Weibull

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Mean Mean A LogNormal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Mean Mean A Normal

K-S test Mean rank

Red : alpha = 0.05 Blue : alpha = 0.01

µ= 5.31 σ=1.686 x0= 354.5 ξ= 192.62 µ= 276.7 σ=316.9 m= 0.8969 ξ= 312.48

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100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Median Median A Weibull

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Median Median A LogNormal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Median Median A Normal

K-S test Median rank

Red : alpha = 0.05 Blue : alpha = 0.01

µ= 5.32 σ=1.686 x0= 364.5 ξ= 193.42 µ= 276.7 σ=216.9 m= 0.8969 ξ= 312.48

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Median Median A Bi-exponential

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K-S test Rest method

Red : alpha = 0.05 Blue : alpha = 0.01

µ= 5.31 σ=1.686 x0= 364.5 ξ= 193.42 µ= 276.7 σ=216.9 m= 0.8969 ξ= 312.48

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Rest Rest A Normal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Median Median A LogNormal

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Rest Rest A Weibull

100 200 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Rest Rest A Bi-exponential

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Data set 3 Results (47 items)

  • Linearity test observation

– Bi-exponential and Lognormal distributions are not linear in comparison to that of Weibull and Normal. – Weibull shows a better distribution of the linearity test with smallest R-value and SD.

  • K-S Test Observation

– Weibull distribution has the best fit distribution with all points inside the 5% significance range. – LogNormal and Bi-exponential distributions have values outside the 5% significance level and is therefore rejected. (X) – Normal distribution shows the poorest distribution of K-S test and is therefore rejected.(X) – Normal distribution has points outside the significance level of 5% which is unacceptable.

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