Fig. 2 Hierarchical structure of ship’s engine failure Since the naval ship data is nonlinear time series data, polynomial and spline regression are considered. In polynomial regression, to achieve flexibility, a degree should be increased; however, the risk of overfitting becomes higher with its degree. To prevent this and to endow the model a form of locality, the B-Spline model is suggested: overall life span, or age period, is first divided into several sections. Then low-dimensional polynomial is fitted for each section to form a piecewise polynomial spline. The third layer of ROK naval ship hierarchy, representing the ship engine, is modeled with B-Spline. As can be seen from Equation 1, parameters corresponding to each layer are modeled in a way that could enable the pooling possible; though hyperparameter sharing structures.
\begin{equation} Y_{s}\ \sim\ \text{Normal}(\mu_{s},\ \sigma_{y})\nonumber \\ \end{equation}\begin{equation} \mu_{s}\ =\ \alpha_{s}+\ \Sigma_{k=1}^{K}\ w_{k,\ \ s}\ B_{k}\nonumber \\ \end{equation}\begin{equation} \alpha_{s}\ \sim\ \text{Normal}(\ \overset{\overline{}}{\alpha_{e}},\ \sigma_{\alpha})\nonumber \\ \end{equation}\begin{equation} w_{s}\ \sim\ \text{Normal}(\ \overset{\overline{}}{w_{e}},\ \sigma_{w})\nonumber \\ \end{equation}\begin{equation} \ \overset{\overline{}}{\alpha_{e}}\ \sim\ \text{Normal}(\ \overset{\overline{}}{\alpha_{0}},\ \sigma_{\overset{\overline{}}{\alpha}})\nonumber \\ \end{equation}\begin{equation} \overset{\overline{}}{w_{e}}\ \sim\ \text{Normal}(\ \overset{\overline{}}{w_{0}},\ \sigma_{\overset{\overline{}}{w}})\nonumber \\ \end{equation}\begin{equation} \sigma_{\alpha}\ \sim\ \text{Gamma}(10,\ 10)\nonumber \\ \end{equation}\begin{equation} \sigma_{w}\ \sim\ \text{Gamma}(10,\ 10)\nonumber \\ \end{equation}\begin{equation} \sigma_{\overset{\overline{}}{\alpha}}\ \sim\ \text{Exponential}(1)\nonumber \\ \end{equation}\begin{equation} \sigma_{\overset{\overline{}}{w}}\ \sim\ \text{Exponential}(1)\nonumber \\ \end{equation}
\(\sigma_{y}\ \sim\ \text{Exponential}(1)\) Equation 1
To be more specific, based on the overall average failure rate of 98 ships, B-spline is pre-fitted to obtain the hyperparameters.\(\overset{\overline{}}{\alpha_{0}}\), \(\overset{\overline{}}{w_{0}}\)are set as the posterior sample mean with their priors given as Equation 2. We used the intercept from linear regression fit as the prior mean of\(\overset{\overline{}}{\alpha_{0}}\). Next, with these layer1 parameters, engine-specific parameters,\(\overset{\overline{}}{\alpha_{e}}\ \)and\(\overset{\overline{}}{w_{e}}\ \) are learned. ­ Note that weight, w, is a vector whose length is determined by the number of knots.
\begin{equation} \overset{\overline{}}{\alpha_{0}}\ \sim\ \text{Normal}(\ I,\ 1)\nonumber \\ \end{equation}
\(\overset{\overline{}}{w_{0}}\ \ \sim\ \text{Normal}(\ 0,\ 1)\)Equation 2
We used Stan as a probalistic programming language which has the advantage of faster estimation of Bayesian models in ecology in MCMC. Stan HMC is effective as model complexity is large and correlated variables are large. Stan was used in consideration of future developments (complexity and variables to be added) of the proposed model. Stan code of the proposed model is included in Appendix. The working process is organized as shown in Fig. 3. From failure rate data, a rough trend of the failure rate over a lifetime is deduced. This trend is the basis for the number of B-spline knot and hyperprior estimation in the construct hierarchical model. Apply the estimated b-spline and hyperprior to the model, and parameter fit by learning the data of each layer through MCMC sampling.