# Vibration Signal Classification Using Stochastic Configuration Networks Ensemble
## 3 Fault Diagnosis Based on Ensemble SCNs
### SCNs Ensemble  
In order to improve the classification effect, we propose an ensemble method based on SCNs. The time-frequency image has been used to build the model, but the vectorization operation damages the image structure and cause a loss of information; the image data can be directly used via 2 DSCN modeling. As shown in Figure 3, the frequency spectrum, time-frequency images and the hybrid data are used to build sub-learners based on SCNs, respectively. And the ensemble learning is applied to combine the SCN learners and obtain the final classification result. Figure 2: The fault classification method based on the SCN model with hybrid data. The SCNs ensemble learning process is described as follows:  
(1) Data pre-processing. Given the dataset \(\{\mathbf{X},\mathbf{D},\mathbf{Z},\mathbf{T}\}\) with N samples, where \(\mathbf{X}=\{x_{1},x_{2},...,x_{N}\}\), \(\mathbf{D}=\{D_{1},D_{2},...,D_{N}\}\), \(\mathbf{Z}=\{z_{1},z_{2},...,z_{N}\}\) and target label \(\mathbf{T}=\{t_{1},t_{2},...,t_{N}\}\), all sample data are normalized to \([-1,1]\). (2) Select the base model. Because the samples from the training dataset have different dimensions, i.e., \(D_{i}\in R^{d1}\), \(Z_{i}\in R^{d}\), and \(x_{i}\in R^{d2\times d3}\), then two 1D-SCN learners and a 2D-SCN learner will be built on the datasets \(\mathbf{D}\), \(\mathbf{Z}\) and \(\mathbf{X}\), respectively. (3) Building the SCN learners. * SCN-D: Randomly assign \(w1_{L}\) from \([-\lambda,\lambda]^{d1}\), and calculate \(b1_{L}=-w1_{L}^{T}\mathbf{D}^{*}\); calculate \(h1_{L}\) by Equation (3) and \(\xi_{L,d}^{1}\) Equation (4); then select the proper weights under the supervision mechanism (5); find \  
supervision mechanism (5); find \(w3_{L}^{\star},b3_{L}^{\star}\) maximizing \(\xi_{L}^{(3)}\); and set the hidden output matrix \(\mathbf{H_{3}}=[h3_{1}^{\star},h3_{2}^{\star},...,h3_{L}^{\star}]\). (4) Ensemble learning. The hidden feature matrices of the model SCN-D, SCN-X and SCN-H are \(\mathbf{H_{1}},\mathbf{H_{2}}\), and \(\mathbf{H_{3}}\), respectively. Then, concatenate the hidden feature matrix \(\widetilde{\mathbf{H}}=[\mathbf{H_{1}},\mathbf{H_{2}},\mathbf{H_{3}}]\), and the corresponding parameter \(\beta\) is calculated by  
\[\beta^{\star}=\arg\min_{\beta}||\widetilde{\mathbf{H}}\beta-T||_{2}^{2}+ \alpha||\beta||_{1}. \tag{17}\]  
The output is obtained \(O_{escn}=\widetilde{\mathbf{H}}\beta^{\star}\). (5) Calculate the classification error. The residual error is calculated via \(e_{L}=||O_{escn}-T||_{F}\), then it is used to configure the weights of new hidden nodes by the supervised constraint. The SCNs ensemble model is built incrementally until the tolerance error \(\varepsilon\) is reached or the number of hidden nodes \(L\) equals to \(L_{max}\). Finally, the output \(O_{escn}\) is normalized to \([0,1]\) to predict the sample classes and obtain the resultant classification labels for fault identification. In the proposed ensemble method, the SCN learners (i.e., SCN-D, SCN-X and SCN-H) are built on different training datasets, including frequency spectrum data, time-frequency images, and their hybrid data, respectively. The pseudo code of the SCNs ensemble for fault recognition in this section is described in Algorithm 2. The goal of the ensemble learning is to combine the various features extracted from the sample data in different domains via the SCN based sub-learners for fault analysis. Because the stochastic configuration algorithm is incremental learning, the SCN model is built by adding new hidden nodes gradually. So in the ensemble learning process, the SCN learners are modeled synchronously and have the same number of hidden nodes with the weight \(\beta\), and the error is calculated with the configured parameters. In addition, the SCN model can also be built separately to obtain the optimal sub-learners, and then the global weight \(\beta\) is calculated, and the classification results are output. Figure 3: The ensemble method based on SCNs for fault diagnosis.