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.