## 3 Fault Diagnosis Method
### Improved AdaBoost Algorithm
Therefore, the interval of the local frequency of each sample is obtained as follows:

\[\frac{j}{m}[Max(v_{ij})-Min(v_{ij})]+Min(v_{ij}),\ i=1,2,\ldots,80,\ \ \ j=0,1,\ldots,m-1 \tag{6}\]

The center frequency of the frequency interval of the top, having the highest local spectrum amplitude in the local spectrum of each sample, is extracted as the feature vectors of the sample, and the conversion of the sample data to the corresponding feature vectors is completed. Table 4 shows the top five center frequency data for the four types of samples in the standard data set. The frequency of the normal rolling bearing is mainly around 1200 Hz. The frequency of the inner fault mainly surrounds between 2500-2600 Hz. The frequency of the outer fault is mainly around 4800 Hz, and the frequency of the rolling fault is mainly concentrated at 3100 Hz. The distinction between the categories is obvious. Figure 7: Time-domain plot of standard dataset samples. (**a**) Inner fault; (**b**) Outer fault; (**c**) Rolling fault; (**d**) Normal bearings. The specific feature vectors of the rail train experimental data set with a rotational speed of 4 Hz and standard data set are, respectively, shown in Tables 5 and 6. As seen from the two tables, the feature vectors dimension of each sample is 100. That is, the conversion of the sample with the local spectrum and the center frequency of the top 100 frequency interval with the highest amplitude in the local spectrum are obtained as the eigenvector of the sample, and the number of samples per category is 80. The experiment classifies all the samples' feature vectors into four categories. The effectiveness and practicability of the algorithm are determined by the evaluation indexes given in Section 4.2. ### Experimental Target

Aiming to reflect the performance of the classification algorithm in an all-sided way, three evaluation indexes have been adopted in this paper. These are detection rate (DR), false alarm rate (FAR), and classification rate (CR), respectively. Their definitions are given here:

(1) Detection Rate (DR)

DR is applied to measure the classification accuracy of a certain variety of sample set, reflecting the percentage of detected number occupying the total number of a sample set of this variety. It is shown as follows:

\[DR=\frac{\text{Detected number of sample set of a variety}}{\text{total number of sample set of the variety}} \tag{7}\]

(2) False Alarm Rate (FAR)

FAR is for measuring the false alarm rate of a variety of sample set. This index is used to reflect the percentage of not belong to this variety being detected as the variety occupying the total not belong to this variety. It is shown as follows:

\[FAR=\frac{\text{Number of data not belong to the variety but detected as they are}}{\text{Number of real not belong to this variety data}} \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fault Type** & **Feature Vector Dimension** & **Feature Vector Label** \\ \hline Normal & \(80\times 100\) & 0 \\ Inner fault & \(80\times 100\) & 1 \\ Outer fault & \(80\times 100\) & 2 \\ Rolling fault & \(80\times 100\) & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The feature vector of standard data set. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Serial Number** & **Normal** & **Inner Fault** & **Outer Fault** & **Rolling Fault** \\ \hline
1 & 1202 & 2523 & 4822 & 3104 \\
2 & 1183 & 2558.5 & 4790 & 3085.5 \\
3 & 1267.5 & 2570 & 4831 & 3091 \\
4 & 1256 & 2585 & 4764 & 3169 \\
5 & 1162.5 & 2506 & 4814.5 & 3064 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The top five center frequency for the standard data set. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fault Type** & **Feature Vector Dimension** & **Feature Vector Label** \\ \hline
4 r/s Normal & \(80\times 100\) & 0 \\
4 r/s Inner fault & \(80\times 100\) & 1 \\
4 r/s Outer fault & \(80\times 100\) & 2 \\
4 r/s Rolling fault & \(80\times 100\) & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The feature vector of rail train experimental data.