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Degradation Data Analysis with a Linear Regression Model
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This example validates the results for a degradation analysis with a linear regression model in Weibull++ degradation folios.
Reference Case
The data set is from Example 13.10 on page 338 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.
Data
The test is for the current (i.e., amperage) of a laser device. Failure is defined as a 10% increases in amperage.
| Time
|
Unit Number
|
| 1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
| 250 |
0.47 |
0.71 |
0.71 |
0.36 |
0.27 |
0.36 |
0.36 |
0.46 |
0.51 |
0.41 |
0.44 |
0.4 |
0.3 |
0.44 |
0.51
|
| 500 |
0.93 |
1.22 |
1.17 |
0.62 |
0.61 |
1.39 |
0.92 |
1.07 |
0.93 |
1.79 |
1 |
0.8 |
0.74 |
0.7 |
0.83
|
| 750 |
2.11 |
1.9 |
1.73 |
1.36 |
1.11 |
1.95 |
1.21 |
1.42 |
1.57 |
2.38 |
1.57 |
1.4 |
1.52 |
1.05 |
1.29
|
| 1000 |
2.72 |
2.3 |
1.99 |
1.95 |
1.77 |
2.86 |
1.46 |
1.77 |
1.96 |
3 |
1.96 |
1.7 |
1.85 |
1.35 |
1.52
|
| 1250 |
3.51 |
2.87 |
2.53 |
2.3 |
2.06 |
3.46 |
1.93 |
2.11 |
2.59 |
3.84 |
2.51 |
3 |
2.39 |
1.8 |
1.91
|
| 1500 |
4.34 |
3.75 |
2.97 |
2.95 |
2.58 |
3.81 |
2.39 |
2.4 |
3.29 |
4.5 |
2.84 |
3.6 |
2.95 |
2.55 |
2.27
|
| 1750 |
4.91 |
4.42 |
3.3 |
3.39 |
2.99 |
4.53 |
2.68 |
2.78 |
3.61 |
5.25 |
3.47 |
4 |
3.51 |
2.83 |
2.78
|
| 2000 |
5.48 |
4.99 |
3.94 |
3.79 |
3.38 |
5.35 |
2.94 |
3.02 |
4.11 |
6.26 |
4.01 |
4.4 |
3.92 |
3.39 |
3.42
|
| 2250 |
5.99 |
5.51 |
4.16 |
4.11 |
4.05 |
5.92 |
3.42 |
3.29 |
4.6 |
7.05 |
4.51 |
4.8 |
5.03 |
3.72 |
3.78
|
| 2500 |
6.72 |
6.07 |
4.45 |
4.5 |
4.63 |
6.17 |
4.09 |
3.75 |
4.91 |
7.8 |
4.8 |
5.2 |
5.47 |
4.09 |
4.11
|
| 2750 |
7.13 |
6.64 |
4.89 |
4.72 |
5.24 |
7.7 |
4.58 |
4.16 |
5.34 |
8.32 |
5.2 |
5.5 |
5.84 |
4.83 |
4.38
|
| 3000 |
8 |
7.16 |
5.27 |
4.98 |
5.62 |
8.61 |
4.84 |
4.76 |
5.84 |
8.93 |
5.66 |
6 |
6.5 |
5.41 |
4.63
|
| 3250 |
8.92 |
7.78 |
5.69 |
5.28 |
6.04 |
9.15 |
5.11 |
5.16 |
6.4 |
9.55 |
6.2 |
6.2 |
6.94 |
5.76 |
5.38
|
| 3500 |
9.49 |
8.42 |
6.02 |
5.61 |
6.32 |
9.95 |
5.57 |
5.46 |
6.84 |
10.5 |
6.54 |
7 |
7.39 |
6.14 |
5.84
|
| 3750 |
9.87 |
8.91 |
6.45 |
5.95 |
7.1 |
10.49 |
6.11 |
5.81 |
7.2 |
11.3 |
6.96 |
7.4 |
7.85 |
6.51 |
6.16
|
| 4000 |
10.94 |
9.28 |
6.88 |
6.14 |
7.59 |
11.01 |
7.17 |
6.24 |
7.88 |
12.2 |
7.42 |
7.9 |
8.09 |
6.88 |
6.62
|
Result
The book provides a linear regression model, [math]\displaystyle{ y=b+at\,\! }[/math], but the parameters of the model are not provided. Only the predicted failure times are given. These are: 3702, 4194, 5847, 6172, 5301, 3592, 6051, 6538, 5110, 3306, 5326, 4995, 4721, 5689, and 6102. A Weibull distribution is used to fit the data; however, the model parameters are not provided. Only a probability plot is given.
Results in Weibull++
- The following picture shows the parameters of the linear regression model for each unit.
- The predicted pseudo failure times are shown next.
- The predicted failure times in Weibull++ are slightly different from the values given in the book. Since the book does not provide the model parameters, we couldn’t compare the Weibulll++ results with the results in the book.
- The next picture shows the parameters of the fitted Weibull distribution.
