## 2 Method
### Calculation of MD  
#### 2.2.1 Normalization of Vibration Data for Creation of Unit Space  
With the proposed method, normal data are used to generate the unit space; this unit space is then used as the criterion to evaluate the characteristic difference between normal and anomalous vibration. Table 1 summarizes the normal data before and after normalization. The physical properties of the lifter gradually deteriorate with the passage of operating time; the vibration characteristics gradually change as a consequence. The operation properties are expressed in terms of the multidimensional MD, and judgment as to whether the operation is proceeding normally or abnormally is validated by using the criterion.As shown in Figure 2a,b, the vibration waveform reflecting the movement of the carriage is acquired by an accelerometer placed in an appropriate position on the body of the lifter. The details of the positioning of the sensors are indicated in Section 4.1. For simplicity, to obtain basic knowledge regarding the effect on the value of MD of damage to each of the various parts, we limited the targeted lifter movement to a single condition: one-way movement from the bottom to the top of the lifter. (We were aware from actual measurement that the influence of the lifter's movement on its vibration characteristics differed slightly depending on whether the movement is rising or descending.) Samples of the waveform in rising were extracted, and their frequency response was obtained using FFT (Fast Fourier Transformation). As shown in Figure 2c,d, the converted results of the frequency characteristic have several peaks. These are caused by the structural resonance of each part of the lifter. It is suggested that the vibration components in the frequency band around the resonance frequencies are largely influenced by aging of the structure. Therefore, in this study, the remarkable peak frequencies that occur during normal operation are read from the spectrum and the energy levels of the signals in the 1/12 octave band range above and below these peak frequencies are calculated. These energy levels are sequentially calculated for each sample waveform generated by each rising. The calculation results for these energy levels are expressed as \(x\) in Figure 2c. For example, in the \(k\)th operation, the energy levels from \(x_{k2}\) to \(x_{km}\) in \(m\) bands are calculated. The unit space of the normal state is generated using the data output from all the energy levels from the first to the \(k\)th operation, as shown in Figure 2c. The output data are summarized in section (a-1) of Table 1.  
Figure 2: Simulation of Mahalanobis distance (MD) using the vibration acceleration data. The waveforms of the vibration as shown in (**b**) can be obtained by operating (**a**) the lifter. Then, by using (**c**) the generated unit space for normal data, finally (**d**) the MD can be obtained. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Model**} & Band: & \multirow{2}{*}{**Model**} & Band: & \multirow{2}{*}{**Model**} & Band: & \multirow{2}{*}{**Model**} \\  & & No. 1 & & & No. \(j\) & & No. \(m\) \\ \hline \multirow{4}{*}{**Model**} & Sample: No. 1 & \(x_{11}\) & \(\cdots\) & \(x_{ij}\) & \(\cdots\) & \(x_{1m}\) \\ \cline{2-7}  &... &... &... &... &... &... \\ \cline{2-7}  &... &... &... &... &... &... \\ \cline{2-7}  &... &... &... &... &... &... \\ \cline{2-7}  &... &... &... &... &... &... \\ \cline{2-7}  &... &... &... &... &... &... \\ \cline{2-7}  &... &... &... &... &... &... \\ \hline \multirow{4}{*}{**Model**} & Sample: No. \(k\) & \(x_{k1}\) & \(\cdots\) & \(x_{ij}\) & \(\cdots\) & \(x_{1m}\) \\ \cline{2-7}  & Average & \(\overline{x}_{1}\) & \(\cdots\) & \(\overline{x}_{j}\) & \(\cdots\) & \(\overline{  
To generate the unit space by using the \(x_{ij}\) (\(1\leq i\leq k\), \(1\leq j\leq m\)) data from \(k\) samples (no. 1 to no. \(k\)) with \(m\) bands (no. 1 through no.