## 2 Methodology
### Combined DBN and Particle Filter-Based Approaches for RUL Prediction
The state transition model and its respective variance in Equation (10) \(\omega_{t-1}\) can be modeled by taking \(B\) bootstraps, which randomly samples (with replacement) each row of the training data \(TD\), \(|TD|\) times. The DBN is then trained \(B\) times for each bootstrap and the predicted values will be defined as:  
\[ST_{b}(\mathbf{z})=F(\mathbf{z};ST_{b}^{*}) \tag{18}\]  
where, \(\mathbf{z}\) represents the generic testing input vector of the past \(d\) measurements [Equation (16)] and \(F(\mathbf{z};ST_{b}^{*})\) represents the DBN-FNN output [Equation (17)] with respect to the \(b\)th bootstrap. The mean and variance are simply then calculated as:  
\[ST_{avg}(\mathbf{z})=\frac{1}{B}\sum_{b=1}^{B}ST_{b}(\mathbf{z}) \tag{19}\]  
\[\partial_{ST}^{2}(\mathbf{z})=var(ST_{b}(\mathbf{z})) \tag{20}\]  
and then samples \(x_{t}^{i}\) can be generated from the following Gaussian distribution:  
\[\mathbb{N}(ST_{avg}(\mathbf{z}),\partial_{ST}^{2}(\mathbf{z})) \tag{21}\]  
The measurement distribution can be modeled in a somewhat similar approach to the state transition distribution. The technique [13] requires a dataset of tuples of \(\left(x_{t},\mathbf{z}_{t}\right)_{t=1}^{N_{training}}\) where \(B\) bootstraps are sampled from this training data set, and an interpolator \(\varphi(x)\) is created. \(\mathbf{z}_{t}\) is typically a multi-dimensional vector, which can be reduced to a single mono-dimensional vector for simplicity and ease of calculation. This can be performed by the DBN in its unsupervised stage of learning, by setting the last layer of the DBN's hidden layer to a size of one. A simple FFN can also be used as an interpolator for computing the following:  
\[\varphi_{avg}(x)=\ \frac{1}{B}\sum_{b=1}^{B}\varphi(x;T_{b}^{*}) \tag{22}\]  
where, \(T_{b}^{*}\) is the \(b\)th bootstrap sampled from the dataset. The measurement model in Equation (11) can be hypothesized as:  
\[z(x)=f(x)+v(x) \tag{23}\]  
From Equation (23), we can subtract \(\varphi_{avg}(x)\) from both sides such that:  
\[\left[\mathbf{x}_{t},\mathbf{x}_{t+1},\ldots,\mathbf{x}_{N}\right] \tag{17}\]0  
The error term \(z(x)-\varphi_{avg}(x)\) is then a function of \(\left[f(x)-z(x)\right]\) and \(v(x)\), which we will call the model error and intrinsic noise respectively. Their variances are then denoted as \(\sigma_{m}^{2}(x)\) and \(\alpha^{2}(x)\) respectively. The model error variance can be modeled by taking \(M\) groups of interpolators from \(\varphi\left(x;T_{b}^{*}\right)\) of length \(K\) networks and computing the average as:  
\[\left[\mathbf{x}_{t},\mathbf{x}_{t+1},\ldots,\mathbf{x}_{N}\right] \tag{17}\]1The set, \(\Psi=\left\{\varphi_{com}^{m}(x)\right\}_{m=1}^{M}\) is then resampled with replacement using \(P\) bootstraps, denoted as \(\left\{\Psi_{p}\right\}_{p=1}^{P}\), where \(\Psi_{p}\) is the \(p\)th bootstrap of \(\Psi\). The estimate of the model error variance can then be computed as:  
\[\left[\mathbf{x}_{t},\mathbf{x}_{t+1},\ldots,\mathbf{x}_{N}\right] \tag{17}\]2  
The noise variance can be modeled by building up a training set \((x_{t},\hat{\alpha}_{t}^{2})_{t=1}^{N_{training}}\), where \(\hat{\alpha}_{t}^{2}\) is defined as:  
\[\left[\mathbf{x}_{t},\mathbf{x}_{t+1},\ldots,\mathbf{x}_{N}\right] \tag{17}\]3  
A single FNN can then be trained on the aforementioned training set in order to estimate \(\hat{\alpha}_{t}^{2}(x)\) for an arbitrary \(x\) vector. Finally \(p(z_{t}|x_{t})\) can be approximated by:  
\[\left[\mathbf{x}_{t},\mathbf{x}_{t+1},\ldots,\mathbf{x}_{N}\right] \tag{17}\]4  
The confidence intervals for a generic prediction \(\hat{x}\) can then be obtained with 1-\(\alpha\)% confidence by the following formula [36]:  
\[\left[\mathbf{x}_{t},\mathbf{x}_{t+1},\ldots,\mathbf{x}_{N}\right] \tag{17}\]5  
The training/testing procedure for the integrated DBN and particle filter is then as follows: