## 3 Methodology
### Assumption of Curve Morphology in Peak Region of Acceleration Spectrum  
The Gaussian distribution curve can represent the data by the mean and covariance of the data. However, because the data are symmetric according to the mean and have a fixed shape [26], Gaussian distribution is often unsuitable for fitting the real vibration signal data. In order to overcome the limitations of Gaussian distribution curves, the asymmetric generalized Gaussian distribution (AGGD) is proposed to represent asymmetric Gaussian data [27]. It also has a good effect in simulating the asymmetric pattern and properties of the curve in the peak region of the acceleration spectrum. The expression of the asymmetric generalized Gaussian curve is as follows:  
\[f(X|\mu,\sigma_{l},\sigma_{r},\lambda)=\frac{\lambda\left[\frac{\Gamma(3/ \lambda)}{\Gamma(1/\lambda)}\right]^{1/2}}{(\sigma_{l}+\sigma_{r})\Gamma(1/ \lambda)}\left\{ \begin{array}{ll}\exp\left[-A(\lambda)\left(\frac{\mu-X}{ \sigma_{l}}\right)^{\lambda}\right]&if\ X<\mu\\ \exp\left[-A(\lambda)\left(\frac{X-\mu}{\sigma_{r}}\right)^{\lambda}\right]&if \ X\geq\mu\end{array} \right. \tag{7}\]  
where \(\mu\) is the mean, \(\sigma_{l}\) is the left standard deviation, \(\sigma_{r}\) is the right standard deviation, \(\lambda\) is the shape parameter, \(A(\lambda)\) is the function of \(\lambda\) and the expression is as follows:  
\[A(\lambda_{ij})=\left[\frac{\Gamma(3/\lambda)}{\Gamma(1/\lambda)}\right]^{ \lambda/2} \tag{8}\]  
where \(\Gamma(. )\) is the gamma function, and the expression is as follows:  
\[\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}dt\quad x>0 \tag{9}\]  
The asymmetric generalized Gaussian distribution curve introduces three parameters: shape parameter \(\lambda\), left standard deviation \(\sigma_{l}\) and right standard deviation \(\sigma_{r}\) to enhance the flexibility of fitting the shape of the curve, which can better fit the asymmetric data. The shape of the curve is altered by changing the value of the shape parameter to make it flatten or sharpen. Figure 6 shows the influence of different parameters of asymmetric generalized Gaussian curve distribution on the shape of the curve. In the actual acceleration signal data processing, the acceleration spectrum curve cannot be simply represented by a single asymmetric generalized Gaussian distribution curve. The asymmetric generalized Gaussian distribution lacks flexibility in fitting the real acceleration signal data. The asymmetric Gaussian mixture model [28] not only effectively solves the problem of the limitation of the curve shape of Gaussian distribution, but is also  
Figure 6: Asymmetric generalized Gaussian curves with different parameter Settings. more flexible in fitting the actual acceleration signal data and can fit the asymmetric data better and faster. The expression of the asymmetric Gaussian mixture model is as follows. \[f(X|\xi_{j})=\prod_{i=1}^{n}\prod_{j=1}^{m}\frac{\lambda_{ij}\Big{[}\frac{\Gamma(3 /\lambda_{ij})}{\Gamma(1/\lambda_{ij})}\Big{]}^{1/2}}{\left(\sigma_{l_{ij}}+ \sigma_{r_{ij}}\right)\Gamma(1/\lambda_{ij})}\left\{ \begin{array}{ll}\exp \left[-A(\lambda_{ij})\Big{(}\frac{\mu-X_{i}}{\sigma_{l_{ij}}}\Big{)}^{ \lambda_{ij}}\right]&\text{if }X_{i}<\mu\\ \exp\left[-A(\lambda_{ij})\Big{(}\frac{X_{i}-\mu}{\sigma_{r_{ij}}}\Big{)}^{ \lambda_{ij}}\right]&\text{if }X_{i}\geq\mu\end{array} \right.