Stracener_EMIS 7305/5305_Spr08_ 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

Stracener_EMIS 7305/5305_Spr08_ Reliability Model Selection Estimation of Reliability Model Parameters Probability Plotting

Stracener_EMIS 7305/5305_Spr08_ 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

Stracener_EMIS 7305/5305_Spr08_ 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 ^ _

Stracener_EMIS 7305/5305_Spr08_ Estimation - Normal Distribution

Stracener_EMIS 7305/5305_Spr08_ 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 ^

Stracener_EMIS 7305/5305_Spr08_ Estimation - Lognormal Distribution

Stracener_EMIS 7305/5305_Spr08_ 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

Stracener_EMIS 7305/5305_Spr08_ 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 ^

Stracener_EMIS 7305/5305_Spr08_ Estimation - Exponential Distribution

Stracener_EMIS 7305/5305_Spr08_ Estimation of Exponential Distribution Random sample of size n, X 1, X 2, …, X n, from E(  ), where  is unknown.

Stracener_EMIS 7305/5305_Spr08_ Estimation - Weibull Distribution

Stracener_EMIS 7305/5305_Spr08_ 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 ^ ^ ^

Stracener_EMIS 7305/5305_Spr08_ 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. ^

Stracener_EMIS 7305/5305_Spr08_ Example The following data represents a random sample from the normal distribution N( ,  ) : Estimate the population parameters. Then estimate the 90% percentile, and plot estimates of the probability density and distribution functions.

Stracener_EMIS 7305/5305_Spr08_ Example solution T ~ N( ,  ), then the estimates of  are  is:

Stracener_EMIS 7305/5305_Spr08_ Example solution Normal Model N(, ): ^ Standard Deviation

Stracener_EMIS 7305/5305_Spr08_ Example solution

Stracener_EMIS 7305/5305_Spr08_ Probability Plotting

Stracener_EMIS 7305/5305_Spr08_ 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.

Stracener_EMIS 7305/5305_Spr08_ 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. Step 2: Rank the sample values from smallest to largest in magnitude i.e., X 1  X 2 ..., X n.

Stracener_EMIS 7305/5305_Spr08_ 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

Stracener_EMIS 7305/5305_Spr08_ 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.

Stracener_EMIS 7305/5305_Spr08_ 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.

Stracener_EMIS 7305/5305_Spr08_ 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.

Stracener_EMIS 7305/5305_Spr08_ Weibull Probability Plotting Paper If, the cumulative probability distribution function is We now need to linearize this function into the form y = ax +b:

Stracener_EMIS 7305/5305_Spr08_ Weibull Probability Plotting Paper Then which is the equation of a straight line of the form y = ax +b,

Stracener_EMIS 7305/5305_Spr08_ Weibull Probability Plotting Paper where and

Stracener_EMIS 7305/5305_Spr08_ 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

Stracener_EMIS 7305/5305_Spr08_ Weibull Probability Plotting Paper cumulative probability (in %) x

Stracener_EMIS 7305/5305_Spr08_ Probability Plotting - example To illustrate the process let 10, 20, 30, 40, 50, and 80 be a random sample of size n = 6.

Stracener_EMIS 7305/5305_Spr08_ 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, ^ ^

Stracener_EMIS 7305/5305_Spr08_ Weibull Probability Plotting Paper cumulative probability (in %) x

Stracener_EMIS 7305/5305_Spr08_ 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 ^

Stracener_EMIS 7305/5305_Spr08_ Probability Plotting- example The line represents the estimated relationship between x and F(x): F(x) (in %) x ^

Stracener_EMIS 7305/5305_Spr08_ 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.

Stracener_EMIS 7305/5305_Spr08_ Probability Plotting - example

Stracener_EMIS 7305/5305_Spr08_ Probability Plotting - example

Stracener_EMIS 7305/5305_Spr08_ Probability Paper - Normal

Stracener_EMIS 7305/5305_Spr08_ Probability Paper - Lognormal

Stracener_EMIS 7305/5305_Spr08_ Probability Paper - Exponential

Stracener_EMIS 7305/5305_Spr08_ Example - Probability Plotting Given the following random sample of size n=8, which probability distribution provides the best fit?

Stracener_EMIS 7305/5305_Spr08_ 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

Stracener_EMIS 7305/5305_Spr08_ Time Series plot: By visual inspection of the scatter plot, there seems to be no trend. 40 Specimens

Stracener_EMIS 7305/5305_Spr08_ Specimens Using the descriptive statistics function in Excel, the following were calculated:

Stracener_EMIS 7305/5305_Spr08_ 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:

Stracener_EMIS 7305/5305_Spr08_ Specimens Box Plot The lower quartile The median is 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

Stracener_EMIS 7305/5305_Spr08_ Specimens

Stracener_EMIS 7305/5305_Spr08_ Specimens

Stracener_EMIS 7305/5305_Spr08_ Specimens

Stracener_EMIS 7305/5305_Spr08_ Specimens The point estimates for μ and σ are:

Stracener_EMIS 7305/5305_Spr08_ The tensile strength distribution can be estimated by 40 Specimens ^ ^

Stracener_EMIS 7305/5305_Spr08_ Specimens Estimate of Probability that P(x<50) is: or