Reliability HotWire

Issue 112, June 2010

Degradation Model Selection in Weibull++

Degradation modeling is an effective reliability analysis tool for products with failures caused by degradation. Weibull++ provides several commonly used degradation models including linear, exponential, power, logarithmic, Gompertz and Lloyd-Lipow. If the mechanism of the degradation is known, a mechanism-based model should be used. An example of this would be the light intensity of an LED that degrades exponentially with time and therefore an exponential model is appropriate for the analysis. If the physics of degradation are not clear, a model that provides the best statistical fit to the data can be used. This model usually is referred to as a statistics-based model. In this article, we will explain how to use Weibull++ to select the best degradation model for your data.

Before we go further, we will first discuss the definition of failures that are caused by degradation.[1] In general, there are two types of degradation failures: soft failures and hard failures.

Soft Failures: For some products, there is a gradual loss of performance (e.g. decreasing light output from a fluorescent light bulb). The failure would be defined at a specified level of degradation (e.g. 60% of initial output). This type of failure is called a soft failure.

Hard Failures: For some products, failure is defined as the event when the product stops working due to the degradation (e.g. when the resistance of a resistor deviates too much from its nominal value, causing the oscillator in an electronic circuit to stop oscillating). This type of failure is called a hard failure.

In this article, we only focus on the modeling of soft failures.

Example

The tread depths of several tires are measured every 5,000 miles. Failure is defined as the time when the depth is less than 2 millimeters. The following table gives the tire tread degradation measurements.

Table 1: Tire Tread Depth

In this case, it is not clear what model should be used based on the physics of the degradation. We need choose a model based on the available data. Weibull++ has a tool called the Degradation Model Wizard that can help you choose the right model. Figure 1 shows the data entered into the Weibull++ Degradation Analysis Folio.

Figure 1: Degradation Data in Weibull++

When you click the Analyze button in the Degradation Model Wizard for this data set, you will see the following results.

Figure 2: Degradation Model Wizard: Overall Rank Summary

Among all the selected models, Exponential is ranked as number 1. Ranks are calculated based on the Sum of Squares of Error, which is an index of how well a model can fit the data.*

Next, click the Analysis Detail tab. Figure 3 shows the overall rank of each model and also the individual rank for each tire. Due to the randomness of the materials and the initial depth of each tire, a model that is good for one test tire may not be the best for other tires. For this case study, we can see that the logarithmic model is the best model for analyzing the data collected for Unit_g while the exponential model is the number 1 model for the rest of the tires.

Figure 3: Degradation Model Wizard Showing the Ranking

To see how these ranks are obtained, click the second sheet.

Figure 4: Degradation Model Wizard

The second column displays the sum of all the errors for each individual tire whose sum of squares errors are shown in the Unit columns. The exponential model has the smallest sum of squares error value, so it has the highest rank.

Using the Degradation Model Wizard, we found out that the exponential model is the best model. The exponential model uses the formula:

Y = b*exp(a*X)

where:

• Y is the degradation measurement.
• X is the mileage.
• a and b are the model parameters.

We implement the exponential model and get the following results.

Figure 5: Results for Model Parameters

Using the above model parameters, we can calculate the sum of squares errors that are given in Figure 4. For instance, the sum of squares errors for Unit_a is calculated in this way:

Table 2: Calculation of Sum of Squares Error for Unit_a

The error is the square of the difference of the observed Y and the predicted Y values. For the measurement at 5,000 miles, the observed tread depth is 6.1. The predicted Y value is b*exp(a*X) = 7.4166 * exp(-0.0404*5) = 6.06. So the square of error is (6.1-6.06)^2 = 0.001596. SSE is the sum of all the error squares at each observation. We can see in Table 2 that the calculated sum of squares error for Unit_a is 0.027. It is very close to the sum of squares error value in Figure 4. The difference is caused by the rounding up error that occurs during the calculation.

Once the degradation model has been obtained, we can use it to predict when the tread depth will reach 2 millimeters (i.e. failure) for each unit. Figure 6 gives the predicted failure times for each of the units in the data set.

Figure 6: Extrapolated Failure Time for Each Tire

Figure 7 shows the raw data points and the fitted degradation lines.

Figure 7: Degradation vs. Time (Mileage) Plot

Conclusion

In this article, we illustrate how to choose a degradation model when the physics model of the degradation is not clear. It should be noticed that the statistics-based model is purely based on the available data and the statistical techniques used in the analysis. When the sample size is small, or when other factors that affect degradation are not considered in the model, non-realistic prediction results may occur. So when conducting degradation analysis, it is recommended to combine engineering knowledge in the analysis. If no engineering knowledge is available, large sample sizes can increase the confidence of the analysis results.

*Note: Mean Squares of Error (MSE) is the Sum of Squares Error (SSE) divided by a constant value in the degradation model. Both MSE and SSE calculations provide the same ranking. Weibull++ uses SSE for its rankings, but uses the term "MSE" to refer to this calculation.

Reference

[1] W. Meeker and L. Escobar, Statistical Methods for Reliability Data. John Wiley & Sons, New York, 1998.

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