# Transmission Condition Monitoring of 3D Printers Based on the Echo State Network
## 2 Methodologies
### Feature Extraction  
Due to the high dimension and noise of the time-series data obtained by monitoring the 3D printer, it was very important to extract features from the time-series data. The data from the 3D printer can be extracted in different modalities such as time-domain features and time-frequency features. It should be noted that features in one domain are insufficient to reflect all aspects of the faults. Hence, both time-domain and time-frequency features are applied to fault diagnosis in this work.  
To obtain time-domain features from the raw data (i.e., monitoring data), 10 time-domain statistical features including mean, mean square, skewness, kurtosis, peak value, root mean square magnitude, crest factor, waveform factor, clearance factor, and impulse factor were used in this paper, which can reduce noise interference and increase the model's robustness [28]. These 10 features are described as follows:  
Given a set of data \(y_{i}\) (\(i=1\), 2,..., \(n\)), where \(y_{i}\) represents the data at the \(i\)-th time, the mean of the data can be expressed by Equation (1).  
\[\overline{Y}=\frac{\sum_{i=1}^{n}y_{i}}{n}, \tag{1}\]  
The mean square of the data, which can be calculated via Equation (2), is adopted to eliminate the influence of the time-series data on the sum of squares of errors.  
\[Y_{MS}=\frac{\sum_{i=1}^{n}y_{i}^{2}}{n}, \tag{2}\]  
Skewness of the data is used to reflect the degree of asymmetry of distribution, which is expressed in the following equation:  
\[T_{sk}=\frac{\sum_{i=1}^{n}(y_{i}-\overline{y})^{3}}{(n-1)Y_{sd}^{3}}, \tag{3}\]  
where \(Y_{sd}\) represents the standard deviation of the data. The kurtosis shown in Equation (4) is used to characterize the number of peaks of the probability density distribution curve at the average value.  
\[T_{ku}=\frac{\sum_{i=1}^{n}(y_{i}-\overline{y})^{4}}{(n-1)Y_{sd}^{4}}, \tag{4}\]  
The peak value reflects the range and intensity of the amplitude of the signal. The expression of peak value is defined as follows:  
\[Y_{peak}=\max\{y_{i}\}, \tag{5}\]  
The calculation method of the root mean square magnitude of the data is written in Equation (6).  
\[Y_{RMS}=\sqrt{\frac{\sum_{i=1}^{n}y_{i}^{2}}{n}}, \tag{6}\]  
The crest factor is the ratio of the peak to RMS magnitude, which can be calculated as follows:  
\[T_{cf}=\frac{Y_{peak}}{\overline{Y_{RMS}}}, \tag{7}\]  
The waveform factor is the ratio of RMS to absolute mean, which can be calculated via Equation (8).  
\[T_{wf}=\frac{Y_{RMS}}{\left|\overline{Y}\right|}, \tag{8}\]  
The clearance factor and the impulse factor are expressed in Equations (9) and (10), respectively.  
\[T_{cf}=\frac{Y_{peak}}{Y_{R}}, \tag{9}\]  
\[T_{if}=\frac{Y_{peak}}{\overline{Y}}, \tag{10}\]where \(Y_{R}\) is square root amplitude of \(y_{i}\).  
The wavelet packet energy is employed as a frequency feature. The expression of wavelet packet energy [29] is written in Equation (11).  
\[w_{i}=\frac{W_{3i}}{\sum W_{3i}}=\sum y(3,i), \tag{11}\]  
where \(w_{i}\) (\(i\) = 1, 2,..., 8) is the \(i\)-th wavelet packet energy, and \(W_{3i}\) means the \(i\)-th frequency range of three-level wavelet packets. In this paper, time-frequency features including the eight-wavelet packet energy was considered for monitoring 3D printers. In addition to the 10 time-domain features mentioned above, there are 18 features in total.