## 3 Problem Formulation and Proposed Model
### The Proposed Model: Prognostics Generation Model  
In the prognostic literature, different data-driven techniques are available to predict future failure probabilities. However, the most used methods in this context are relevance vector machine (RVM), Gaussian process regression (GPR), and neural networks (NNs). The flexibility offered by neural networks in the feed-forward configuration is crucial for the entire project's practical feasibility [51]. In this paper, we developed the prognostic model through feed-forward neural networks. However, the general logic and the framework as a whole is still valid in the case of RVM and GPR. Therefore, in the prognostic stage, we used a feed-forward neural network in a time series prediction configuration. This type of neural network application is known as a multi-step prediction in which the network employs the previously predicted value to forecast the future values iteratively. In this way, it is possible to predict a series of values depending on the required number of time steps. The first value predicted from the network (\(y_{t}\)) is a function of the last \(n\) recorded values at preceding time steps (\(x_{t-1}\)):  
\[y_{t}=f(x_{t-n},x_{t-n+1},x_{t-n+2},\ldots,x_{t-1}) \tag{3}\]  
In Equation (3), each recorded value represents an input associated with a different node of the NN. In predicting the next value, \(y_{t+1}\), the last \(n-1\) recorded values and the last predicted value are used:  
\[y_{t+1}=f(x_{t-n+1},x_{t-n+2},\ldots,x_{t-1},y_{t}) \tag{4}\]  
Then, it is possible to predict all the future desired \(h\) values iteratively:  
\[y_{t+h}=f(x_{t-n+h},x_{t-n+h+1},\ldots,x_{t-1},y_{t},y_{t+1},\ldots,y_{t+h-1}) \tag{5}\]  
In this context, it is essential to define and control each aspect of its development and implementation. As a consequence, each network, which is linked to certain equipment or machine with unique features, presents a different neural structure (specifically, the number of nodes and the activation function of each different node). However, thanks to the uniformity of the value that logistic regression provides, it is possible to standardise the input data activity for each neural network. In our approach, the architecture of the neural network consists of an output layer composed of only one output (the predicted value) and an input layer consisting of the first n recorded values. According to [PERSON] et al. [52], it is advisable to introduce nonlinearity into the network by using some nonlinear activation functions (i.e., the _tanh_ or _logit_ function) as a combination of functions from input to output layer. Hence, it is important to find the dimension of the network (i.e., the number of nodes), the weight of nodes, and define each activation function. In particular, the number of active hidden neurons in the network is a critical factor. If the network has a small number of hidden nodes, the performance tends to be ineffective. Meanwhile, if the network has too many hidden nodes, the risk of data over-fitting arises and impedes the desired generalisation. In the literature, it is usual to define the dimension and the activation functions of a neural network with a trial and error strategy. When data acquisition takesplace, a set of data is extracted and used for estimating the dimension of the neural network. Starting with the most straightforward network, composed of only one hidden and one output node, an iterative training of these networks (varying the number of nodes and the activation functions) is made with the previously collected data. For each trial, root mean square deviation (RMSD), which represents a good measure of accuracy to compare forecasting errors of different models, is estimated until the minimum of this value is found.