Issue 12, February 2002

Multimodal Analysis

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

In life data analysis, the goal is often to fit a good model to the data set. Once a good model has been fitted to the data, other analyses and predictions can be made. Life data can be represented visually on a probability plot. Ideally, the data points on the probability plot should fall in a straight line, indicating a good fit to the specific distribution. If the data points do not fall in a straight line, this is an indication that the chosen distribution is not a good choice to model the data, and another distribution should be chosen.

However, instances may occur when none of the standard life distributions fit a given data set. This may occur, for example, if a data set has been "contaminated" by errors in the test or collection process or if not enough data has been collected to form a good model. Unfortunately, there is not much that can be done about such situations other than to retest under better conditions or collect sufficient data so that a better model can be developed.

Another reason that a straight line may not fit the data is that more than one mode is present in the data set. In life data analysis, this may be indicative of multiple failure modes for the product. Multimodal data sets can often be recognized by the pattern evident on the probability plot. If the points on the plot appear to have a distinct "dogleg," or seem as if they could be modeled by two straight lines of different slopes, this may be indicative of a multimodal data set. The following Weibull probability plot gives an example of this.

As can be seen from the plot, the data points from 0.01 to 100 seem to have one slope, while the data points from 100 on have a much higher slope. This is an indicator that more than one failure mode may be present. In the Weibull++ software, there are two ways to deal with the analysis of such data sets: competing failure modes analysis or mixed Weibull analysis.

Competing Failure Modes Analysis 
Sometimes, it is not necessary to rely on a probability plot to deduce the presence of more than one failure mode. Simple observation during the test or subsequent failure analysis can indicate with little doubt the presence of more than one failure mode. In these instances, competing failure modes analysis can be performed, provided that the products being tested are nonrepairable, and one can assume that the modes are statistically independent.

Competing failure modes analysis assumes that for a given product with multiple potential failure modes, these modes compete to cause the failure in the product. The overall reliability is calculated in the same way that the overall reliability for a series system is calculated:

R_S(t) = R₁(t)*R₂(t)*...*R_n(t)

Where R_S is the reliability of the system, n is the number of failure modes and R_i(t) is the reliability function for the ith failure mode. The reliability functions for each of the failure modes are calculated by analyzing the complete data set for the failures for each failure mode individually, while considering all other failure mode points to be suspensions. For example, for a data set with three failure modes, R₁(t) would be calculated by analyzing the data set for mode 1 failures only, with the failures for modes 2 and 3 considered to be suspensions.

The competing failure modes method is useful when one can make distinctions between failure modes without resorting to the appearance of the probability plot to make the distinction. Furthermore, with failure modes that occur in the same time range, one may not be able to infer the presence of multiple modes as there will be no distinct inflection point on the probability plot. For more information on competing failure modes analysis, see the article in issue 3 of the Reliability Edge newsletter.

Mixed Weibull Analysis 
Another method of dealing with multimodal data sets is the use of the mixed Weibull distribution. Essentially, this is a distribution is composed of a number of distinct Weibull distributions that have been "patched together" to form one continuous function. This analysis is useful when dealing with data sets with distinct "doglegs" or inflection points, or when the assumption of statistical independence is not valid. This method is particularly applicable to data sets that represent different phases in a product's life, e.g. infantile failures, useful life and wearout phases.

The mixed Weibull distribution is developed by combining several Weibull distributions with different parameters, along with a factor that indicates the proportion of the data in each mode subpopulation. The equation for the mixed Weibull pdf is given by:

where:

• S is the number of modes present
• N is the total number of data points
• N_i is the number of data points for the ith failure mode
• β_i is the Weibull slope for the ith failure mode
• η_i is the Weibull scale parameter for the ith failure mode 

The reliability equation for the mixed Weibull distribution is given by:

This equation indicates that the overall reliability function for the mixed Weibull distribution is the sum of the proportional reliability contributions for each subpopulation.

Mixed Weibull Example
Consider a set of complete data from a life test of 20 mechanical units. It is desired to make a preliminary reliability analysis, pending the results from the failure analysis lab. The failure times for the units are as follows:

At first blush, this seems to be an ordinary life data set; there is no obvious pattern to the raw data. When plotted on a Weibull probability plot, the data points do not appear to fall on a straight line; the points at times greater than 100 appear to have a steeper slope than those with values less than 100. This is illustrated in the following graphic. 

It appears as if there may be two failure modes at work with the products on this test. We will use the Weibull++ software to analyze the data using a two-population mixed Weibull distribution. In order to perform this type of analysis in Weibull++, select the 2-Subpop-Mixed Weibull distribution on the control panel, as shown next.

Weibull++ allows you to select up to four subpopulations. Note the minimum number of required distinct points for each subpopulation selection. We will select the two population selection for the current analysis.

The results of the analysis indicate that the first subpopulation comprises approximately 62% of the data is in a subpopulation with β=0.94 and η=92.24 and approximately 38% of the data is in a subpopulation with β=7.87 and η=195.42. A plot of the pdf of the mixed Weibull distribution helps to illustrate this breakdown.

The pdf illustrates how the distributions of the two subpopulations are combines. From time zero to approximately 100, the pdf exhibits the "decreasing with time" trend characteristic of a Weibull distribution with a beta value less than or equal to one -- this is the contribution of subpopulation 1. From time 100 onward, the pdf exhibits the bell shape typical of Weibull distributions with a beta value in the vicinity of five.

It will also be useful to take a look at the failure rate plot of this distribution. Note that the classic "bathtub curve" could be considered to be a three-subpopulation mixed Weibull. With our example, we appear to have a similar curve with the first subpopulation representing the useful life or random failures and the second subpopulation representing wearout failures.

The failure rate decreases slightly from time zero to 100 and then begins to increase rapidly with time from 100 onward. This indicates that the early failures in the data represent random or useful life failures, while the latter failures may be wearout related. This shape of failure rate function is not unusual for mechanical products.

Conclusion
As we have seen, multimodal analysis is useful for dealing with sets of data that do not appear to fall on a straight line on a probability plot, particularly if there is a distinctive "dogleg" in the data. Competing failure modes analysis can be employed when specific failure modes can be assigned to the individual failures, as through failure analysis activities. Also, the objects under analysis must be nonrepairable, and the failure modes must be considered to be statistically independent. If one cannot identify separate failure modes in the data, or if the failure modes cannot be assumed to be statistically independent, mixed Weibull analysis can be used to perform the analysis.

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