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 LloydLipow. If the mechanism of the degradation is known, a mechanismbased 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 statisticsbased 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
Mileage (K) 
Unit_a 
Unit_b 
Unit_c 
Unit_d 
Unit_e 
Unit_f 
Unit_g 
5 
6.1 
5.9 
5.9 
6.1 
6.3 
5.7 
6 
10 
5 
5.1 
5 
5 
5.3 
5 
4.8 
15 
4 
4.3 
4.05 
4.1 
4.2 
3.9 
3.7 
20 
3.2 
3.3 
3.5 
3.4 
3.5 
3.2 
3.2 
25 
2.8 
2.9 
2.9 
2.8 
2.5 
2.6 
2.9 
30 
2.2 
2.4 
2.4 
2.3 
2.1 
2 
2.2 
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
Mileage (K)

Tread Depth

Predicted Y

Error

5 
6.1 
6.06 
0.001596 
10 
5 
4.95 
0.00234 
15 
4 
4.05 
0.002112 
20 
3.2 
3.31 
0.011221 
25 
2.8 
2.70 
0.00975 
30 
2.2 
2.21 
5.16E05 

SSE 
0.027 
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 (6.16.06)^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 statisticsbased 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, nonrealistic 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.
