Life Data Analysis Part I - Estimation of Failure Probabilities

Failure Probability Estimation

In practice, a simplification for the calculation of the median value, also called median rank, is made. The formula of Benard’s approximation is given by

$$\hat{F}(t_i) \approx \frac{i - 0,3}{n + 0,4} $$

and is described in The Plotting of Observations on Probability Paper ².

However, this equation only provides valid estimates for failure probabilities if all units in the sample are defectives (estimate_cdf(methods = "mr", ...)).

In field data analysis, however, the sample mainly consists of intact units and only a small fraction of units failed. Units that have no damage at the point of analysis and also have not reached the operating time or mileage of units that have already failed, are potential candidates for future failures. As these, for example, still are likely to fail during a specific time span, like the guarantee period, the CDF must be adjusted upwards by these potential candidates.

A commonly used method for correcting probabilities of (multiple) right-censored data is Johnson’s method (estimate_cdf(methods = "johnson", ...)). By this method, all units that are included in the period looked at are sorted in an ascending order of their operating time (or any other damage-equivalent quantity). If there are units that have not failed before the i-th failure, an adjusted rank for the i-th failure is formed. This correction takes the potential candidates into account and increases the rank number. In consequence, a higher rank leads to a higher failure probability. This can be seen in Figure 1.

The rank adjustment is determined with:

j_i = j_i − 1 + x_i ⋅ I_i,  with  j₀ = 0

Here, j_i − 1 is the adjusted rank of the previous failure, x_i is the number of defectives at t_i and I_i is the increment that corrects the considered rank by the potential candidates.

$$I_i=\frac{(n+1)-j_{i-1}}{1+(n-n_i)}$$

The sample size is n and n_i is the number of units that have a lower t than the i-th unit. Once the adjusted ranks are calculated, the CDF can be estimated according to Benard’s approximation.

Other methods in {weibulltools} that can also handle (multiple) right-censored data are the Kaplan-Meier estimator (estimate_cdf(methods = "kaplan", ...)) and the Nelson-Aalen estimator (estimate_cdf(methods = "nelson", ...)).

Probability Plotting

After computing failure probabilities a method called probability plotting is applicable. It is a graphical goodness of fit technique that is used in assessing whether an assumed distribution is appropriate to model the sample data.

The axes of a probability plot are transformed in such a way that the CDF of a specified model is represented through a straight line. If the plotted points (plot_prob()) lie on an approximately straight line it can be said that the chosen distribution is adequate.

The two-parameter Weibull distribution can be parameterized with parameters μ and σ such that the CDF is characterized by the following equation:

$$F(t)=\Phi_{SEV}\left(\frac{\log(t) - \mu}{\sigma}\right)$$

The advantage of this representation is that the Weibull is part of the (log-)location-scale family. A linearized representation of this CDF is:

$$\Phi^{-1}_{SEV}\left[F(t)\right]=\frac{1}{\sigma} \cdot \log(t) - \frac{\mu}{\sigma}$$

This leads to the following transformations regarding the axes:

• Abscissa: x = log (t)
• Ordinate: y = Φ_SEV⁻¹[F(t)], which is the quantile function of the SEV (smallest extreme value) distribution and can be written out with log {−log [1 − F(t)]}.

Another version of the Weibull CDF with parameters η and β results in a CDF that is defined by the following equation:

$$F(t)=1-\exp\left[ -\left(\frac{t}{\eta}\right)^{\beta}\right]$$

Then a linearized version of the CDF is:

log {−log [1 − F(t)]} = β ⋅ log (t) − β ⋅ log (η)

Transformations regarding the axes are:

• Abscissa: x = log (t)
• Ordinate: y = log {−log [1 − F(t)]}.

It can be easily seen that the parameters can be converted into each other. The corresponding equations are:

$$\beta = \frac{1}{\sigma}$$

and

η = exp (μ).

Weibull Probability Plot

# Weibull grid for estimated probabilities: 
weibull_grid <- plot_prob(
  cdf_tbl,
  distribution = "weibull", 
  title_main = "Weibull Probability Plot", 
  title_x = "Mileage in km", 
  title_y = "Probability of Failure in %",
  title_trace = "Method",
  plot_method = "ggplot2"
)

weibull_grid

Figure 3: Plotting positions in Weibull grid.

Figure 3 shows that the consideration of survivors (orange points, Method: johnson) decreases the failure probability in comparison to the sole evaluation of failed items (green points, Method: mr).

Log-normal Probability Plot

Finally, we want to use a log-normal probability plot to visualize the estimated failure probabilities.

# Log-normal grid for estimated probabilities: 
lognorm_grid <- plot_prob(
  cdf_tbl,
  distribution = "lognormal",
  title_main = "Log-normal Probability Plot",
  title_x = "Mileage in km",
  title_y = "Probability of Failure in %",
  title_trace = "Method",
  plot_method = "ggplot2"
)

lognorm_grid

Figure 4: Plotting positions in log-normal grid.

On the basis of Figure 3 and Figure 4 we can subjectively assess the goodness of fit of Weibull and log-normal. It can be seen that in both grids, the plotted points roughly fall on a straight line. Hence one can say that the Weibull as well as the log-normal are good model candidates for the shock data.