1. Stracener_EMIS 7305/5305_Spr08_02.28.08 1 Reliability Data Analysis and Model Selection Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7305/5305 Systems Reliability, Supportability and Availability Analysis Systems Engineering Program Department of Engineering Management, Information and Systems

2. Stracener_EMIS 7305/5305_Spr08_02.28.08 2 Reliability Model Selection Estimation of Reliability Model Parameters Probability Plotting

3. Stracener_EMIS 7305/5305_Spr08_02.28.08 3 Estimation of Reliability Model Parameters Estimation of Binomial Distribution Parameters Estimation of Normal Distribution Parameters Estimation of Lognormal Distribution Parameters Estimation of Exponential Distribution Parameters Estimation of Weibull Distribution Parameters

4. Stracener_EMIS 7305/5305_Spr08_02.28.08 4 Estimation - Binomial Distribution Estimation of a Proportion, p X 1, X 2, …, X n is a random sample of size n from B(n, p), where Point estimate of p: where f s = # of successes ^ _

5. Stracener_EMIS 7305/5305_Spr08_02.28.08 5 Estimation - Normal Distribution

6. Stracener_EMIS 7305/5305_Spr08_02.28.08 6 Estimation of the Mean - Normal Distribution X 1, X 2, …, X n is a random sample of size n from N( ,  ), where both µ & σ are unknown. Point Estimate of  Point Estimate of s ^

7. Stracener_EMIS 7305/5305_Spr08_02.28.08 7 Estimation - Lognormal Distribution

8. Stracener_EMIS 7305/5305_Spr08_02.28.08 8 Estimation of Lognormal Distribution Random sample of size n, X 1, X 2,..., X n from LN ( ,  ) Let Y i = ln X i for i = 1, 2,..., n Treat Y 1, Y 2,..., Y n as a random sample from N( ,  ) Estimate  and  using the Normal Distribution Methods

9. Stracener_EMIS 7305/5305_Spr08_02.28.08 9 Estimation of the Mean of a Lognormal Distribution Mean or Expected value or MTBF Point Estimate of MTBF where and are point estimates of and respectively. Median time to failure Point estimate of median time To Failure ^

10. Stracener_EMIS 7305/5305_Spr08_02.28.08 10 Estimation - Exponential Distribution

11. Stracener_EMIS 7305/5305_Spr08_02.28.08 11 Estimation of Exponential Distribution Random sample of size n, X 1, X 2, …, X n, from E(  ), where  is unknown.

12. Stracener_EMIS 7305/5305_Spr08_02.28.08 12 Estimation - Weibull Distribution

13. Stracener_EMIS 7305/5305_Spr08_02.28.08 13 Estimation of Weibull Distribution Random sample of size n, T 1, T 2, …, T n, from W( ,  ), where both  &  are unknown. Point estimates is the solution of g(  ) = 0 where ^ ^ ^

14. Stracener_EMIS 7305/5305_Spr08_02.28.08 14 Estimation of the Mean of a Weibull Distribution Mean or Expected value or MTBF Point Estimate of MTBF where and are point estimates of and respectively. ^

15. Stracener_EMIS 7305/5305_Spr08_02.28.08 15 Example The following data represents a random sample from the normal distribution N( ,  ) : 94 74 105 126 124 135 56 95 122 78 86 66 63 80 85 58 89 92 103 93 Estimate the population parameters. Then estimate the 90% percentile, and plot estimates of the probability density and distribution functions.

16. Stracener_EMIS 7305/5305_Spr08_02.28.08 16 Example solution T ~ N( ,  ), then the estimates of  are  is:

17. Stracener_EMIS 7305/5305_Spr08_02.28.08 17 Example solution Normal Model N(, ): ^ Standard Deviation

18. Stracener_EMIS 7305/5305_Spr08_02.28.08 18 Example solution

19. Stracener_EMIS 7305/5305_Spr08_02.28.08 19 - Probability Plotting

20. Stracener_EMIS 7305/5305_Spr08_02.28.08 20 Probability Plotting Data are plotted on special graph paper designed for a particular distribution - Normal - Weibull - Lognormal- Exponential If the assumed model is adequate, the plotted points will tend to fall in a straight line If the model is inadequate, the plot will not be linear and the type & extent of departures can be seen Once a model appears to fit the data reasonably well, parameters and percentiles can be estimated.

21. Stracener_EMIS 7305/5305_Spr08_02.28.08 21 Probability Plotting Procedure Step 1: Obtain special graph paper, known as probability paper, designed for each of the following distributions: Weibull, Exponential, Lognormal and Normal. http://www.http://www.weibull.com/GPaper/index.htm Step 2: Rank the sample values from smallest to largest in magnitude i.e., X 1  X 2 ..., X n.

22. Stracener_EMIS 7305/5305_Spr08_02.28.08 22 Probability Plotting General Procedure Step 3: Plot the X i ’s on the probability paper versus depending on whether the marked axis on the paper refers to the % or the proportion of observations. The axis of the graph paper on which the X i ’s are plotted will be referred to as the observational scale, and the axis for as the cumulative probability scale. or

23. Stracener_EMIS 7305/5305_Spr08_02.28.08 23 Probability Plotting General Procedure The formula is an approximation that can be used to estimate median ranks, called Benard’s approximation. where n is the sample size and i is the sample order number. Tables of median ranks can be found in may statistics and reliability texts.

24. Stracener_EMIS 7305/5305_Spr08_02.28.08 24 Probability Plotting General Procedure Median ranks represent the 50% confidence level (“best guess”) estimate for the true value of F(t), based on the total sample size and the order number (first, second, etc.) of the data.

25. Stracener_EMIS 7305/5305_Spr08_02.28.08 25 Probability Plotting General Procedure Step 4: If a straight line appears to fit the data, draw a line on the graph, ‘by eye’. Step 5: Estimate the model parameters from the graph.

26. Stracener_EMIS 7305/5305_Spr08_02.28.08 26 Weibull Probability Plotting Paper If, the cumulative probability distribution function is We now need to linearize this function into the form y = ax +b:

27. Stracener_EMIS 7305/5305_Spr08_02.28.08 27 Weibull Probability Plotting Paper Then which is the equation of a straight line of the form y = ax +b,

28. Stracener_EMIS 7305/5305_Spr08_02.28.08 28 Weibull Probability Plotting Paper where and

29. Stracener_EMIS 7305/5305_Spr08_02.28.08 29 Weibull Probability Plotting Paper which is a linear equation with a slope of b and an intercept of Now the x- and y-axes of the Weibull probability plotting paper can be constructed. The x-axis is simply logarithmic, since x = ln(T) and

30. Stracener_EMIS 7305/5305_Spr08_02.28.08 30 Weibull Probability Plotting Paper cumulative probability (in %) x

31. Stracener_EMIS 7305/5305_Spr08_02.28.08 31 Probability Plotting - example To illustrate the process let 10, 20, 30, 40, 50, and 80 be a random sample of size n = 6.

32. Stracener_EMIS 7305/5305_Spr08_02.28.08 32 Probability Plotting – Example Solution Based on Benard’s approximation, we can now calculate F(t) for each observed value of X. These are shown in the following table: For example, for x 2 =20, ^ ^

33. Stracener_EMIS 7305/5305_Spr08_02.28.08 33 Weibull Probability Plotting Paper cumulative probability (in %) x

34. Stracener_EMIS 7305/5305_Spr08_02.28.08 34 Probability Plotting- example Now that we have y-coordinate values to go with the x- coordinate sample values so we can plot the points on Weibull probability paper. F(x) (in %) x ^

35. Stracener_EMIS 7305/5305_Spr08_02.28.08 35 Probability Plotting- example The line represents the estimated relationship between x and F(x): F(x) (in %) x ^

36. Stracener_EMIS 7305/5305_Spr08_02.28.08 36 Probability Plotting - example In this example, the points on Weibull probability paper fall in a fairly linear fashion, indicating that the Weibull distribution provides a good fit to the data. If the points did not seem to follow a straight line, we might want to consider using another probability distribution to analyze the data.

37. Stracener_EMIS 7305/5305_Spr08_02.28.08 37 Probability Plotting - example

38. Stracener_EMIS 7305/5305_Spr08_02.28.08 38 Probability Plotting - example

39. Stracener_EMIS 7305/5305_Spr08_02.28.08 39 Probability Paper - Normal

40. Stracener_EMIS 7305/5305_Spr08_02.28.08 40 Probability Paper - Lognormal

41. Stracener_EMIS 7305/5305_Spr08_02.28.08 41 Probability Paper - Exponential

42. Stracener_EMIS 7305/5305_Spr08_02.28.08 42 Example - Probability Plotting Given the following random sample of size n=8, which probability distribution provides the best fit?

43. Stracener_EMIS 7305/5305_Spr08_02.28.08 43 40 specimens are cut from a plate for tensile tests. The tensile tests were made, resulting in Tensile Strength, x, as follows: Perform a statistical analysis of the tensile strength data and estimate the probability that tensile strength on a new design will be less than 50, i.e, reliability at 50. Example: 40 Specimens 1/4/2016

44. Stracener_EMIS 7305/5305_Spr08_02.28.08 44 Time Series plot: By visual inspection of the scatter plot, there seems to be no trend. 40 Specimens

45. Stracener_EMIS 7305/5305_Spr08_02.28.08 45 40 Specimens Using the descriptive statistics function in Excel, the following were calculated:

46. Stracener_EMIS 7305/5305_Spr08_02.28.08 46 40 Specimens From looking at the Histogram and the Normal Probability Plot, we see that the tensile strength can be estimated by a normal distribution. Using the histogram feature of excel the following data was calculated: and the graph:

47. Stracener_EMIS 7305/5305_Spr08_02.28.08 47 40 Specimens Box Plot 40 4550556065 The lower quartile 49.45 The median is 53.03 The mean 52.6 The upper quartile 55.3 The interquartile range is 5.86 lower extreme upper extreme lower quartile upper quartile median mean

48. Stracener_EMIS 7305/5305_Spr08_02.28.08 48 40 Specimens

49. Stracener_EMIS 7305/5305_Spr08_02.28.08 49 40 Specimens

50. Stracener_EMIS 7305/5305_Spr08_02.28.08 50 40 Specimens

51. Stracener_EMIS 7305/5305_Spr08_02.28.08 51 40 Specimens The point estimates for μ and σ are:

52. Stracener_EMIS 7305/5305_Spr08_02.28.08 52 The tensile strength distribution can be estimated by 40 Specimens ^ ^

53. Stracener_EMIS 7305/5305_Spr08_02.28.08 53 40 Specimens Estimate of Probability that P(x<50) is: or