## 3 Problem Formulation and Proposed Model
### The Proposed Model: Degradation Modelling of the Monitored Machine/Equipment  
As discussed in the Literature Review section, the degradation assessment of industrial machines and equipment for maintenance purposes is often accomplished by means of logistic regression. For instance, [PERSON] et al. [48] applied logistic regression for the degradation assessment of the motor motion of an elevator door with the use of an ARMA-based model for the estimation of the remaining useful life. [PERSON] et al. [49] instead proposed the use of logistic regression to estimate the failure degradation of bearings. While based on a run-to-failure data set, their research considered the result as target vectors of failure probabilities, exploiting an RVM (relevance vector machine) algorithm as an intelligent system for predicting the future failure probabilities. As a consequence of the previous studies and the flexibility of logistic regression, we adopted this regression model for the degradation assessment in the proposed algorithm. Logistic regression considers a binary \"outcome\"; that is, a binary dependent variable whose values are usually coded as zero for the negative outcome (no failure event) and as one for the positive outcome (failure event).  
Equation (1) gives the probability of this event:  
\[Prob(event)=P(\vec{x})=\frac{e^{(\alpha+\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots +\beta_{n}x_{n})}}{1+e^{(\alpha+\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots+\beta_{n} x_{n})}} \tag{1}\]  
where \(\vec{x}=(x_{1}+x_{2}+\ldots+x_{3})\) is the input vector of independent variables reproducing the system's state, while \(\alpha\), \(\beta_{1}\), \(\beta_{2}\),..., \(\beta_{n}\) represent the regression coefficients. The estimation of these coefficients is one of the critical activities of model fine-tuning (as the regression coefficients synthesise the physical knowledge and behaviour of the monitored machine/equipment). If an extensive database of historical data is available, these coefficients can be derived with the maximum likelihood estimation method by finding the estimator values that maximise the probability of making the input data given as parameters [48; 50].  
If historical data are not available, it is possible to acquire only the independent vector variables, without the knowledge of conditions linked to the regular behaviour of the monitored machine/equipment. This knowledge is not sufficient for performing a regression analysis. However, in this case, it is possible to take advantage of expert maintenance technicians' expertise by running different tests on the monitored machine/equipment and setting in advance the corresponding performance level of the monitored condition. Another, and more intuitive, formulation of the logistic regression is shown in Equation (2). It relates to the effect of the dependent variable, not to the probability of the event, but the \"odd\" of the event (the relationship between the probability of an event and the probability of a non-event):  
\[ln\left[\frac{Prob(\vec{x})}{1-Prob(\vec{x})}\right]=\alpha+\beta_{1}x_{1}+ \beta_{2}x_{2}+\ldots+\beta_{n}x_{n}. \tag{2}\]In Equation (2), the first term is the logarithm of the odds ratio, known as \"logit\", i.e., the term \"Logistic Regression\". In this form, there is an apparent linear association between the \"logit\" and the dependent variables, in contrast to the previous formulation (Equation (1)). In this formulation, logistic regression returns the probability of a specific event as a value in the range 0-1. In the proposed model, it is necessary to build a degradation model, which summarises in a single dependent variable the failure probability of the machine. It is worth pinpointing that, at this stage, we are still in the diagnostic phase. The next step is the prediction of dynamic failure probabilities for the future periods, starting from the performance degradation model.