Introduction to Life Data Analysis
If the lifetime (or any other damage-equivalent quantity such as distance or load cycles) of a unit is considered to be a continuous random variable T, then the probability that a unit has failed at t is defined by its CDF (cumulative distribution function) F(t).
\[ P(T\leq t) = F(t) \]
In order to obtain an estimate of the CDF for each observation \(t_1, t_2, ..., t_n\) two approaches are possible. Using a parametric lifetime distribution requires that the underlying assumptions for the sample data are valid. If the distribution-specific assumptions are correct, the model parameters can be estimated and the CDF is computable. But if assumptions are not held, interpretations and derived conclusions are not reliable.
A more general approach for the calculation of cumulative failure probabilities is to use non-parametric statistical estimators \(\hat{F}(t_1), \hat{F}(t_2), ..., \hat{F}(t_n)\). In comparison to a parametric distribution no general assumptions must be held. For non-parametric estimators, an ordered sample of size \(n\) is needed. Starting at \(1\), the ranks \(i \in \{1, 2, ..., n \}\) are assigned to the ascending sorted sample values. Since there is a known relationship between ranks and corresponding ranking probabilities a CDF can be determined.
But rank distributions are systematically skewed distributions and thus the median value instead of the expected value \(E\left[F\left(t_i\right)\right] = \frac{i}{n + 1}\) is used for the estimation 1. This skewness is visualized in Figure 1.
library(dplyr) # data manipulation
library(ggplot2) # visualization
x <- seq(0, 1, length.out = 100) # CDF
n <- 10 # sample size
i <- c(1, 3, 5, 7, 9) # ranks
r <- n - i + 1 # inverse ranking
df_dens <- expand.grid(cdf = x, i = i) %>%
mutate(n = n, r = n - i + 1, pdf = dbeta(x = x, shape1 = i, shape2 = r))
densplot <- ggplot(data = df_dens, aes(x = cdf, y = pdf, colour = as.factor(i))) +
geom_line() +
scale_colour_discrete(guide = guide_legend(title = "i")) +
theme_bw() +
labs(x = "Failure Probability", y = "Density")
densplot
Figure 1: Densities for different ranks i in samples of size n = 10.
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:
\[j_i = j_{i-1} + x_i \cdot I_i, \;\; with \;\; j_0 = 0\]
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 \(\mu\) and \(\sigma\) 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: \(y = \Phi^{-1}_{SEV}\left[F(t)\right]\), which is the quantile function of the SEV (smallest extreme value) distribution and can be written out with \(\log\left\{-\log\left[1-F(t)\right]\right\}\).
Another version of the Weibull CDF with parameters \(\eta\) and \(\beta\) 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:
\[ \log\left\{-\log\left[1-F(t)\right]\right\} = \beta \cdot \log(t) - \beta \cdot \log(\eta)\]
Transformations regarding the axes are:
- Abscissa: \(x = \log(t)\)
- Ordinate: \(y = \log\left\{-\log\left[1-F(t)\right]\right\}\).
It can be easily seen that the parameters can be converted into each other. The corresponding equations are:
\[\beta = \frac{1}{\sigma}\]
and
\[\eta = \exp\left(\mu\right).\]
