# A Fault Detection Framework for Rotating Machinery with a Spectrogram and Convolutional Autoencoder
## 4 Experimental Study
Finally, the noise added to the spectrogram is generated from the Gaussian distribution, whose mean value is 0 and standard deviation is 0.1, because excessively large noises might deteriorate the patterns of the spectrogram, and the deteriorated spectrogram cannot generate appropriate latent features of input signals. In this comparative study, the spectrogram is used as the input for feature learning methods, except for the manually defined feature method. For the PCA and AE, each element of the spectrogram is regarded as a single variable. Thus, the spectrogram is \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Signal Representations** & **[PERSON]’s T\({}^{2}\) Statistic** & **MoG** & **PWDE** & **OCSVM** & **IsoForest** & **LOF** \\ \hline Time-domain + 1D-CAE & 0.608 & 0.609 & 0.608 & 0.656 & 0.638 & 0.790 \\ Frequency-domain + 1D-CAE & 0.853 & 0.853 & 0.799 & 0.720 & 0.823 & 0.930 \\ Spectrogram + 2D-CAE (Proposed) & **0.966** & **1.000** & **0.942** & **0.831** & **0.907** & **1.000** \\ \hline \hline \end{tabular}  
* The largest AUROC values are represented in bold and underline.  
\end{table} 
Table 4: Average AUROC values from three different signal representation schemes with CAE and six OCC algorithms. \begin{table}
\begin{tabular}{l l} \hline \hline
**Features** & **Definition** \\ \hline \multirow{5}{*}{\
\begin{tabular}{l} Time-related \\ features \\ \end{tabular} } & Root mean square & \(T_{rms}=\left[\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}^{2}\right]^{\frac{1}{2}}\) \\  & Skewness & \(T_{sk}=\frac{\sum_{i}(x_{i}-\overline{x})^{3}}{(n-1)\sigma^{3}}\) \\  & Kurtosis & \(T_{ku}=\frac{\sum_{i}(x_{i}-\overline{x})^{4}}{(n-1)\sigma^{4}}\) \\  & Crest factor & \(T_{cf}=\frac{x_{max}}{T_{rms}}\) \\  & Shape factor & \(T_{sf}=\frac{T_{mf}}{T^{2}}\) \\  & Impulse factor & \(T_{if}=\frac{x_{max}}{T^{2}}\) \\ \hline \multirow{5}{*}{
\begin{tabular}{l} Frequency-related \\ features \\ \end{tabular} } & Mean frequency & \(F_{mf}=\frac{N}{i}\frac{p_{i}}{N}\) \\  & Frequency center & \(F_{fc}=\frac{\sum_{i}^{N}f_{i}p_{i}}{\sum_{i+1}^{N}p_{i}}\) \\ \cline{1-1}  & Root mean square frequency & \(F_{msf}=\left[\frac{\sum_{i}^{N}f_{i}^{2}p_{i}^{2}}{\sum_{i+1}^{N}p_{i}}\right] ^{\frac{1}{2}}\) \\ \cline{1-1}  & Root variance frequency & \(F_{ref}=\left[\frac{\sum_{i}^{N}\left(f_{i}-F_{mf}\right)^{2}p_{i}}{\sum_{i+1}^ {N}p_{i}}\right]^{\frac{1}{2}}\) \\ \hline \hline \end{tabular}  
* In addition, for the kernel PCA, we adopted the radial basis function (RBF), whose bandwidth (\(\sigma\)) is one, as a kernel function. Finally, the noise added to the spectrogram is generated from the Gaussian distribution, whose mean value is 0 and standard deviation is 0.1, because excessively large noises might deteriorate the patterns of the spectrogram, and the deteriorated spectrogram cannot generate appropriate latent features of input signals.  
\end{table} 
Table 5: Summarization of 10 manually defined features considered in this study. transformed into a one-dimensional vector of length 16,384 from a square matrix of order 128. The spectrogram, a two-dimensional matrix, is directly used as the input for the 2D-CAE. For a fairer comparison with the manually defined features, we extracted 10 features for each feature learning method. Then, we applied the six OCC algorithms to the 10 extracted features for the fault detection. Table 6 presents AUROC values of each feature extraction scheme using six OCC algorithms. As presented in Table 6, the 2D-CAE tends to achieve the best or comparable performance among all feature extraction methods. Since the convolutional blocks, composed of convolutional and max-pooling layers, help to recognize local patterns from the image-formatted data, the 2D-CAE can successfully extract informative features from the spectrogram. In contrast, the manually defined features cannot demonstrate superior performance, as they are computed using simple statistics and fail to reflect the intrinsic properties of complicated signal data.