# Prediction of Machine Inactivation Status Using Statistical Feature Extraction and Machine Learning
## 3 The Proposed Method
### Features Extraction Using Statistical Analysis
#### 3.2.1 Reliability Features  
In this study, the reliability measures that we used for extracting features from raw data included Weibull, lognormal, and exponential distributions. Reliability analysis was a statistical measure for life data [15; 16]. It was determined by deriving the proportion of methodical variation in a scale, which can be performed by determining the association between the scores obtained from divergent administrations of the scale. Therefore, if the association in reliability analysis was high, the scale yielded consistent results and was, therefore, reliable. To compute the reliability of the machines in a manufacturing line, 2 measures, called the time between non-active states (TBNA) and mean time between non-active (MTBNA) states of the machines, were calculated. Figure 2 illustrates the structure of the reliability analysis. Based on the TTF [17; 18] mathematical formulation, we can define the _TBNA_ state, which is the elapsed time between the non-active states of the machines, and _MTBNA_ as follows. \[TBNA=start\ of\ non-active\ time-start\ of\ active\ time \tag{1}\]  
\[MTBNA=\frac{\sum TBNA}{number\ of\ non-active\ states} \tag{2}\]  
Figure 2: Structure of feature extraction in reliability analysis. Once _TBNA_ and _MTBNA_ are calculated, we extract features from raw data using Weibull, lognormal, and exponential distributions. We first obtained the Weibull distribution, which was one of the most widely used lifetime data analyses for reliability engineering [19]. It can flexibly model various types of lifetime distributions [20]. In this paper, we used a 2-parameter Weibull distribution, which had scale and shape parameters. Here, the scale parameter was denoted as \(\eta\), and the shape parameter was denoted as \(\beta\)[21]. When \(\beta\) was less than 1, the distribution showed a decreasing failure rate over time [22]. When \(\beta\) was 1, the distribution had a constant failure rate. When the \(\beta\) parameter was greater than 1, the failure rate increased over time [21]. We estimated the appropriate scale and shape parameters of the Weibull distribution by using the TBNA with maximum likelihood estimation (MLE) method. More specifically, the MLE formula [23] for determining the parameters of the Weibull distribution is written as follows:  
\[ \begin{split}\beta=\left[\frac{\sum\limits_{i=1}^{n}t_{i}\ln(t_{i })}{\sum\limits_{i=1}^{n}t_{i}}-\frac{\sum\limits_{i=1}^{n}\ln(t_{i})}{n} \right]^{-1}\\ \eta=\left(\frac{1}{n}\sum\limits_{i=1}^{n}t_{i}\right)^{\frac{1} {\beta}}\end{split}  \tag{3}\]  
In Equation (3), \(t_{i}\) was TBNA calculated in Equation (1), and \(i\) and \(n\) are the numbers of non-zero data points. Once the shape parameter \(\beta\) and scale parameter \(\eta\) of the Weibull distribution was calculated using Equation (3), we could estimate the non-active state rate of the machine using the probability density function of the Weibull as follows:  
\[f(t)=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}e^{-(t/\eta)^{ \beta}} \tag{4}\]  
Figure 3 illustrates the fitting curve result of the Weibull distribution with 2 parameters set as \(\beta=0.71\), eta \(\eta=144.02\), and the mean life of the non-active state was set as 179.6, which means that the non-active state of the machine will occur when the machine was operating for approximately 179 min. In Figure 3, the \(x\)-axis indicated the elapsed time (in minutes) from which the machine was active, and the \(y\)-axis indicated the probability of the non-active state of the machine. From the figure, we can observe the relationship between active and non-active states of the machine. For example, when the probability of non-activation of a facility was 0.8, it means that about 280 min have passed from the time it was activated. Further, we obtained the lognormal distribution, which was a constant probability distribution of random variables [24]. This distribution was also widely used to model the lives of units whose failure modes were of the fatigue-stress nature [25].