Weibull++ SimuMatic

Reliability analysis using simulation, in which reliability analyses are performed a large number of times on data sets that have been created using Monte Carlo simulation, can be a valuable tool for reliability practitioners. Such simulation analyses can assist the analyst to a) better understand life data analysis concepts, b) experiment with the influences of sample sizes and censoring schemes on analysis methods, c) construct simulation-based confidence intervals, d) better understand the concepts behind confidence intervals and e) design reliability tests. This section explores some of the results that can be obtained from simulation analyses with SimuMatic utility in Weibull++.

Parameter Estimation and Confidence Bounds Techniques

In life data analysis, we use data (usually times-to-failure or times-to-success data) obtained from a sample of units to make predictions for the entire population of units. Depending on the sample size, the data censoring scheme and the parameter estimation method, the amount of error in the results can vary widely. To quantify this sampling error, or uncertainty, confidence bounds are widely used. In addition to the analytical calculation methods that are available, simulation can also be used. SimuMatic generates these confidence bounds and assists the practitioner (or the teacher) to visualize and understand them. In addition, it allows the analyst to determine the adequacy of certain parameter estimation methods (such as rank regression on X, rank regression on Y and maximum likelihood estimation) and to visualize the effects of different data censoring schemes on the confidence bounds.

As an example, we will attempt to determine the best parameter estimation method for a sample of ten units following a Weibull distribution with [math]\displaystyle{ \beta }[/math] = 2 and [math]\displaystyle{ \eta }[/math] = 100 and with complete time-to-failure data for each unit (i.e., no censoring). Using SimuMatic, 10,000 data sets were generated (using Monte Carlo methods based on the Weibull distribution) and we estimated their parameters using RRX, RRY and MLE. The plotted results generated by SimuMatic are shown next.

The results clearly demonstrate that the median RRX estimate provides the least deviation from the truth for this sample size and data type. However, the MLE outputs are grouped more closely together, as evidenced by the confidence bounds. The same figures also show the simulation-based bounds, as well as the expected variation due to sampling error.

This experiment can be repeated in SimuMatic using multiple censoring schemes (including Type I and Type II right censoring as well as random censoring) with the included distributions. We can perform multiple experiments with this utility to evaluate our assumptions about the appropriate parameter estimation method to use for the data set.

Using Simulation to Design Reliability Tests

Good reliability specifications include requirements for reliability and an associated lower one-sided confidence interval. When designing a test, we must determine the sample size to test as well as the expected test duration. The next simple example illustrates the methods available in SimuMatic.

Example 1: Template:LDABOOK SUB

Reliability analysis using simulation, in which reliability analyses are performed a large number of times on data sets that have been created using Monte Carlo simulation, can be a valuable tool for reliability practitioners. Such simulation analyses can assist the analyst to a) better understand life data analysis concepts, b) experiment with the influences of sample sizes and censoring schemes on analysis methods, c) construct simulation-based confidence intervals, d) better understand the concepts behind confidence intervals and e) design reliability tests. This section explores some of the results that can be obtained from simulation analyses with SimuMatic utility in Weibull++.

Parameter Estimation and Confidence Bounds Techniques

In life data analysis, we use data (usually times-to-failure or times-to-success data) obtained from a sample of units to make predictions for the entire population of units. Depending on the sample size, the data censoring scheme and the parameter estimation method, the amount of error in the results can vary widely. To quantify this sampling error, or uncertainty, confidence bounds are widely used. In addition to the analytical calculation methods that are available, simulation can also be used. SimuMatic generates these confidence bounds and assists the practitioner (or the teacher) to visualize and understand them. In addition, it allows the analyst to determine the adequacy of certain parameter estimation methods (such as rank regression on X, rank regression on Y and maximum likelihood estimation) and to visualize the effects of different data censoring schemes on the confidence bounds.

As an example, we will attempt to determine the best parameter estimation method for a sample of ten units following a Weibull distribution with [math]\displaystyle{ \beta }[/math] = 2 and [math]\displaystyle{ \eta }[/math] = 100 and with complete time-to-failure data for each unit (i.e., no censoring). Using SimuMatic, 10,000 data sets were generated (using Monte Carlo methods based on the Weibull distribution) and we estimated their parameters using RRX, RRY and MLE. The plotted results generated by SimuMatic are shown next.

The results clearly demonstrate that the median RRX estimate provides the least deviation from the truth for this sample size and data type. However, the MLE outputs are grouped more closely together, as evidenced by the confidence bounds. The same figures also show the simulation-based bounds, as well as the expected variation due to sampling error.

This experiment can be repeated in SimuMatic using multiple censoring schemes (including Type I and Type II right censoring as well as random censoring) with the included distributions. We can perform multiple experiments with this utility to evaluate our assumptions about the appropriate parameter estimation method to use for the data set.

Using Simulation to Design Reliability Tests

Good reliability specifications include requirements for reliability and an associated lower one-sided confidence interval. When designing a test, we must determine the sample size to test as well as the expected test duration. The next simple example illustrates the methods available in SimuMatic.

Example 1: