Optimal sequential Bayesian analysis for degradation tests

Abstract

Degradation tests are especially difficult to conduct for items with high reliability. Test costs, caused mainly by prolonged item duration and item destruction costs, establish the necessity of sequential degradation test designs. We propose a methodology that sequentially selects the optimal observation times to measure the degradation, using a convenient rule that maximizes the inference precision and minimizes test costs. In particular our objective is to estimate a quantile of the time to failure distribution, where the degradation process is modelled as a linear model using Bayesian inference. The proposed sequential analysis is based on an index that measures the expected discrepancy between the estimated quantile and its corresponding prediction, using Monte Carlo methods. The procedure was successfully implemented for simulated and real data.

Appendices

Appendix 1: Sensitivity analysis

Here we perform a sensitivity analysis for our sequential method in two perspectives, (a) by modifying the hyperparameters of the prior distribution and (b) by increasing the maximum experiment duration time $t_c$. In both cases we used synthetic data as in Sect. 4.1, always obtaining reasonable robust results, both in the resulting sequential observation times and the final estimates.

Beyond the examples outlined here, we also carried out several other similar tests, both modifying the priors and the maximum observation time $t_c$, and with other data sets, always obtaining reasonable robust outputs (results not shown).

1.1 Modifying the hyperparameters

To perform our sensitivity analysis with respect to the prior distribution chosen for the parameters in our model, our sequential analysis was conducted using the same true parameter values and other settings as in Sect. 4.1 (synthetic data), but considering altered hyperparameters values. We present here the same hyperparameter values as used in Sect. 4.1 and an altered set of hyperparameters; see rows 1 and 2 of Table 5, respectively. These altered hyperparameter values follow the guidelines and restrictions established in Sect. 2.1 . We increased $b_{0}$ and $b_{1}$ 10 % from row 1 and increased considerably the variances for parameters $\beta _{0}$, $\beta _{1},$ $\theta _{0}^{j},$ and $\theta _{1}^{j},$ ($v_0$ increased 32 % and $v_1$ increased 133 %). Moreover, note that now there is no correlation between $\beta _{0}$ and $\beta _{1}$ nor among $\theta _{0}^{j}$ and $\theta _{1}^{j}$ since $\nu _{01}$ and $u_{01}$ are zero.

Table 5 Values of the hyperparameters for the sensitivity analysis in the synthetic degradation example; changing the prior hyperparameters from their original values in row 1 (see Table 1) to the altered values in row 2 (in bold)

The results obtained are summarized in Table 6. The optimum new observation time values are close to the former values. In fact, the third observation time remains exactly the same while the fourth and fifth observation times increased only slightly. We see that these results suggest robustness of our model to prior selection.

Table 6 Initial, maximum and optimal observation times for the sensitivity analysis of the synthetic degradation example; changing the prior hyperparameters from their original values (results in row 1) to the altered values (results in row 2)

1.2 Modifying $t_{c}$

In this case we also took the same setting as in Sect. 4.1, with the only modification that $v_{01}=u_{01}=0$, and changing $t_{c}=4.5$ and $t_{c}=5.5$, the maximum observation time. Results are shown in Table 7 and vary slightly within such table and also with respect to Table 6. This suggests that our method also shows robustness to the selection of the maximum experimental time $t_c$. Note, however, that $t_c$ cannot be arbitrarily large beyond unit failure unreasonably increasing experimental costs.

Table 7 Initial, maximum and optimal observation times for the sensitivity analysis of the synthetic degradation example, changing the maximum time to $t_{c}=4.5$ and $t_c =5.5$ (same hyperparameter values as in Table 5, row 1, only changing $v_{01}=u_{01}=0$)

1.3 A simulation study for the precision

To address the issue of the precision of the results obtained by our optimal designs we proceed as follows. We simulated two sets of 50 degradation paths under the following conditions

$$\begin{aligned} \left( \begin{array}{c} \beta _{0} \\ \beta _{1} \end{array} \right) =\left( \begin{array}{c} -1.01611 \\ 0.45 \end{array} \right) ,\quad \quad \left( \begin{array}{c} \theta _{0}^{j} \\ \theta _{1}^{j} \end{array} \right) \sim \mathrm {i.i.d.N}\left( \left( \begin{array}{c} 0 \\ 0 \end{array} \right) ,\left( \begin{array}{cc} 0.0071 &{} -0.0043 \\ -0.0043 &{} 0.0028 \end{array} \right) \right) , \end{aligned}$$

and $\varepsilon _{ij}\sim \mathrm {i.i.d.N}\left( 0,0.04\right) .$

Using these data and under the same hyperparameter settings, we conducted the optimal sequential designs. The results are presented in Table 8 and Fig. 7. The resulting designs are basically the same and the sequential analysis is very similar apart from negligible variations. We performed other repetitions, even down to 20 degradation paths, always obtaining very consistent designs (results not shown). Therefore, precise results should be expected by our sequential analysis.

Fig. 7

Sample 1 a sequential index $\widehat{I}\left( t\right) ,$ with $k=2,3,4$, and b sequential and total distribution function with $k=2,3,4,5,35$ for $m=50$ simulated degradation data. Sample 2 c sequential index $\widehat{I}\left( t\right) ,$ with $k=2,3,4$, and d sequential and total distribution function with $k=2,3,4,5,35$ for $m=50$ simulated degradation data

Table 8 Optimal observation times for the repeated synthetic degradation data. Very similar optimal designs are obtained in both cases

Appendix 2: Goodness of fit

There are some Bayesian goodness of fit tests developed recently in the literature. We choose to use the one proposed by Gelman et al. (1996). All Bayesian goodness of fit approaches are based on the idea that predicted statistics should be coherent with their observations. We use the 0.2 and 0.8 quantiles denoted by $t_{0.2}\left( \mathbf {y}\right)$ and $t_{0.8}\left( \mathbf {y}\right)$, respectively, as the test statistics. We generated a large number of J data replicates $\mathbf {y}^{rep}$ using the posterior predictive distribution obtaining the quantiles $t_{0.2}\left( \mathbf {y}^{rep}\right)$ and $t_{0.8}\left( \mathbf {y}^{rep}\right)$ in each replicated sample. We then compare with the observed quantiles to obtain the posterior predicted p value $p_{q}=\frac{\#\left( t_{q}\left( \mathbf {y}^{rep}\right) >t_{q}\left( \mathbf {y}\right) \right) }{J}$ ($q=0.2,0.8$), see Gelman et al. (1996). As a rule of thumb a p value of 0.5 is ideal since this indicates that the observed statistic is in the middle of the posterior predicted distribution.

We evaluated the goodness of fit of the linear model obtained for the lacquer data shown in Fig. 6. We simulated $J=15,000$ data replicates obtaining the distribution for the posterior predicted quantiles $t_{0.2}\left( \mathbf {y}^{rep}\right)$ and $t_{0.8}\left( \mathbf {y} ^{rep}\right)$ shown in Fig. 8. The observed quantiles are 186.6376 and 243.234 respectively. These are shown in Fig. 8 as black vertical lines both nicely placed near the middle of the posterior predicted distributions. The obtained p values are $p_{0.2}=0.375$ and $p_{0.8}=0.45$ which are not extreme values showing evidence of an adequate goodness of fit.

Fig. 8

Simulated posterior predicted distributions for quantiles 0.2 and 0.8 and the observed quantiles shown as black vertical lines. The observed quantiles are correctly located in the middle of the distributions with a p value of 0.375 and 0.45 respectively, providing adequate evidence of a correct goodness of fit for the lacquer data