Table 8.

Formulas of statistical features extracted from the vibration signal.

Statistical Feature	Formula	Statistical Feature	Formula
Peak value	$X_p=\underset{i}{\max }\left|x_i\right|$	5th normalized moment	$HOMn5=\frac{\frac{1}{n}{\displaystyle \sum _{i=1}^N{\left(x_i-\mu \right)}^5}}{{\left(\sqrt{\frac{1}{N-1}{\displaystyle \sum _{i=1}^N{\left(x_i-\mu \right)}^2}}\right)}^5}$
Root-mean square	$X_{RMS}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{{\displaystyle x}}_i^2}}$	6th normalized moment	$HOMn6=\frac{\frac{1}{n}{\displaystyle \sum _{i=1}^N{\left(x_i-\mu \right)}^6}}{{\left(\sqrt{\frac{1}{N-1}{\displaystyle \sum _{i=1}^N{\left(x_i-\mu \right)}^2}}\right)}^6}$
Kurtosis	$X_{kurtosis}=\frac{1}{N}\left(\frac{{\displaystyle \sum _{i=1}^N{\left(x_i-\mu \right)}^4}}{{\sigma }^4}\right)$	Skewness	$X_{kurtosis}=\frac{1}{N}\left(\frac{{\displaystyle \sum _{i=1}^N{\left(x_i-\mu \right)}^3}}{{\sigma }^3}\right)$
Crest factor	$C_f=\frac{X_p}{X_{RMS}}$	Shape factor RMS	$S{F}_{RMS}=\frac{X_{RMS}}{\mu }$
Clearance factor	$L=\frac{X_p}{{\left(\left(1/N\right){\displaystyle \sum _{i=1}^N\sqrt{\left|x_i\right|}}\right)}^2}$	Peak-to-peak value	$x_{ptp}=\max \left|x\right|-\min \left|x\right|$
Impulse factor	$L=\frac{\max \left\{\left|x_i\right|\right\}}{\left(\left(1/N\right){\displaystyle \sum _{i=1}^N\left|x_i\right|}\right)}$	Energy of signal	$e={\displaystyle \sum _{i=1}^N{x}_i^2}$
Shape factor SMR	$S{F}_{SMR}=\frac{X_{SMR}}{\mu }$	Frequency center	$FC=\frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}fs\left(f\right)df}}{{\displaystyle \underset{0}{\overset{\infty }{\int }}s\left(f\right)}df}$
Entropy	$H(x)=-{\displaystyle \sum _{i=1}^{N}P\left(x_i\right)}\cdot {\log }_{2}P\left(x_i\right)$	RMS frequency	$RMSF=\sqrt{\frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}{f}_i^{2}s\left(f_i\right)df}}{{\displaystyle \underset{0}{\overset{\infty }{\int }}s\left(f_i\right)}df}}$
Mean	$\mu =\frac{1}{N}{\displaystyle \sum _{i-1}^N{x}_i}$	Root variance frequency	$RVF=\sqrt{\frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}{(f_i-FC)}^{2}s\left(f_i\right)df}}{{\displaystyle \underset{0}{\overset{\infty }{\int }}s\left(f_i\right)}df}}$
Square mean root	$X_{SMR}={\left(\frac{{\displaystyle \sum _{i=1}^N\sqrt{x_i}}}{N}\right)}^2$