## 2 Proposing a New Failure Threshold
### PCA Algorithm  
The features introduced in the previous subsection were employed as the inputs of the PCA algorithm. This algorithm uses the matrix characteristics to create a new feature that describes the maximum data variance. This feature is known as the first principal component (PC\({}_{1}\)). The PC\({}_{1}\) can be used as a degradation indicator in prognostic problems. Some researchers have assumed a linear relationship between PC\({}_{1}\) and the size of the growing defect [23].  
The PCA algorithm is an orthogonal linear transform used to reduce the dimensions of data. This algorithm creates new features based on their importance (eigenvalues) so that the primary features have more data variance and better illustrate the nature of data. Assume \(\rm{X}\;_{n\;\times\;L}\;=\;[x_{1}\,x_{2}\,x_{3..\;N_{L}}]\) represents the input data, where L is the number of observations in an n-dimensional space of which n is the number of rows (features). The PCA algorithm converts X data into \(\rm{\hat{X}}\;_{m\;\times\;L}\;=\;[\hat{x}_{1}\,\hat{x}_{2}\,\hat{x}_{3..\;N_{L }}]\) via Equation (1):  
\[\rm{\hat{X}}\;_{m\;\times\;L}\;=\;W_{m\;\times\;n}^{T}X_{n\;\times\;L} \tag{1}\]  
where m is smaller than n and \(\rm{W}_{m\;\times\;n}^{T}\) is the transformation matrix. The algorithm then uses eigenvalues and eigenvectors, as in Equation (2), to calculate matrix W, where \(\lambda\) is the eigenvalue, \(\rm{V_{i}}\) is the eigenvector and \(\rm{C_{x}}\) is the covariance matrix of the X matrix that is calculated through Equation (3). As a result, the covariance matrix of the new components (new features) is diagonal and these components lack any correlation.  
\[\rm{\lambda}\;V_{i}\;=\;C_{x}\;V_{i} \tag{2}\]  
\[\rm{C_{x}}\;=\;\left(\begin{array}{cccc}\sigma_{1\;\times\;1}&\cdots&\sigma_ {1\;\times\;n}\\ \vdots&\ddots&\vdots\\ \sigma_{n\;\times\;1}&\cdots&\sigma_{n\;\times\;n}\end{array}\right) \tag{3}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature** & **Definition** & **Feature** & **Definition** \\ \hline Peak amplitude & \(\rm{\frac{1}{2}(max(x(n))\;-\;min(x(n))\;)}\) & Skewness & \(\rm{\frac{k\sum_{m=1}^{N}\left(x(n)-\tilde{x}\right)^{3}}{RMS(x(n))^{6}}}\) \\ Peak-to-peak & \(\rm{max(x(n))-min(x(n))}\) & Margin factor & \(\rm{\frac{Feak(n)}{(\tilde{k}\sum_{m=1}^{N}\sqrt{[x(n)]})}}\) \\ Root mean square & \(\sqrt{\rm{\frac{1}{N}\sum\limits_{n=1}^{N}\left(x(n)\right)^{2}}}\) & Shape factor & \(\rm{\frac{RMS(x(n))}{\tilde{k}\sum_{m=1}^{N}\left(x(n)\right)}}\) \\ Kurtosis & \(\rm{\frac{\frac{1}{k\sum_{m=1}^{N}\left(x(n)-\tilde{x}\right)^{2}}}{RMS(x(n))^{ 4}}}\) & Clearance factor & \(\rm{\frac{max(x(n))}{\left(\tilde{k}\sum_{m=1}^{N}\sqrt{[x(n)]}\right)^{2}}}\) \\ Crest factor & \(\rm{\frac{Feak(x(n))}{RMS(x(n))}}\) & Mean frequency & \(\rm{\frac{1}{k\sum\limits_{m=1}^{N}X(k)}}\) \\ Mean amplitude & \(\rm{\frac{1}{N}\sum\limits_{n=1}^{N}x(n)}\) & Frequency center & \(\rm{\frac{k\sum_{m=1}^{N}\left(\tilde{k}\right)\;x(k)}{\sum_{m=1}^{N}X(k)}}\) \\ Standard deviation & \(\rm{\sqrt{\frac{1}{N}\sum\limits_{n=1}^{N}\left(Finally, the PCA algorithm outputs the eigenvalues in descending order (Equation (4)). The first component of the new features (PC\({}_{1}\)) that corresponds to the largest eigenvalue explains the most data variance and is as assumed to be the best indicator of data. In the rest of this paper, this component is called the degradation indicator.  
\[\lambda_{\;1}>\;\lambda_{\;2}>\;\lambda_{\;3}\ldots\;>\;\lambda_{\;m} \tag{4}\]