# Distributed Kernel Extreme Learning Machines for Aircraft Engine Failure Diagnostics
## 4 Verification on Benchmark Datasets
### Regression  
The selected seven regression datasets are shown in Table 1, and for each dataset, attributes number (#Attributes), training samples number (#Training) and testing samples number (#Testing) are given respectively. For regression problems, the following rooted mean square error (\(RMSE\)) is defined to describe the performance,  
\[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{N}(t_{i}-\hat{t}_{i})^{2}}{N}} \tag{21}\]  
where \(t_{i}\) denotes the output of testing sample, \(\hat{t}_{i}\) is the corresponding estimation, and \(N\) is the testing samples number. As a classic regression algorithm, n-support vector regression (n-SVR) algorithm would be utilized in this subsection for comparison experiments in the benchmark regression problem dataset with original KELM and proposed DKELMs algorithms. The regression performance comparison among the three algorithms is shown in Table 2, in which \(RMSE\), training time and testing time of each algorithm are listed respectively. As can be seen from Table 2, DKELMs has lower training time and testing time than KELM, and is of similar or higher accuracy. After adopting distributed structure, DKELMs does not reduce the accuracy of regression prediction. Moreover, the fact that each subnetwork is composed of a smaller training subset enables DKELMs to have a higher sparsity than KELM and maintain a similar or better regression accuracy. Since DKELMs takes the calculation of multiple smaller networks, replacing the calculation of large networks containing all samples, and replacing the matrix inverse calculation of \(n\times n\) matrix with the calculation of multiple small matrix inverses, the training time is greatly shortened, and the test time is also shortened on the whole. The reduction in computation is particularly evident in datasets with large sample sizes. The regression accuracy and testing time of n-SVR algorithm are close to KELM, but due to too many support vector points when dealing with regression problems, the training process is very time-consuming in most cases. In the application of failure diagnostics for aircraft engine, offline training and online testing are often adopted, so the testing time can better reflect the real-time performance than the training time. Therefore, in Figure 3, the proposed algorithm is compared with the original KELM and n-SVR algorithms in detail from the perspectives of \(RMSE\) and testing time. The horizontal and vertical coordinates in the figure both represent the ratio, and the denominator is the corresponding value of DKELMs algorithm, so all the green points are at the (1,1) coordinate. As can be seen from the figure, the (1,1) coordinate is almost in the bottom left corner, which means that in all seven regression datasets, for both KELM and n-SVR algorithms, their \(RMSE\)s and testing time are higher than or similar to DKELMs algorithm. Considering the precision, training time and test time comprehensively, the proposed DKELMs algorithm is obviously superior to KELM algorithm and n-SVR algorithm in terms of regression problems. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Datasets** & **\#Attributes** & **\#Train** & **\#Test** \\ \hline Abalone & 7 & 3133 & 1044 \\ Airfoil sound & 5 & 1127 & 376 \\ Auto mpg & 7 & 297 & 99 \\ Bike sharing & 11 & 549 & 183 \\ Boston housing & 13 & 380 & 126 \\ Cpusmall & 12 & 6144 & 2048 \\ White wine quality & 11 & 3674 & 1224 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the regression dataset.