## 2 Brief Introductions to SSA and LS-SVM
### Least Squares SVM  
The support vector machine (SVM) is a powerful nonlinear classification method for overcoming the drawback of traditional linear classification methods. SVM is a kernel-based method. It maps the low-dimension nonlinear structures of signals to a high-dimensional (possibly infinite) linear feature space using the kernel trick. Least squares (LS)-SVM [29], as with the artificial neural network method, does not need training with large data sets to estimate corresponding parameters; it is numerically inexpensive and less overfitting in the training process [30]. Given a training data set \(\{(x(t),y(t)),\ t=1,2,\ldots,N\}\), \(x(t)\in R^{m}\) is the input data, \(y(t)\in R\) is the output corresponding to \(x(t)\), and \(N\) is the training samples number. In LS-SVM, The estimation model is \(y_{t}=w^{T}\varphi(x_{t})+b+e_{t}\), where \(\varphi(\ )\) is a nonlinear transformation that maps \(x(t)\) to the high-dimensional feature space, \(w\) is weight vector, \(b\) is intercept, and \(e_{t}\) is the error term that is assumed to be independent and identically distributed with zero mean and finite variance [31]. The unknowns \(w\) and \(b\) can be determined from a convex optimization problem as follows:  
\[\min\!\big{/}(w,e)=\ \frac{1}{2}(w^{T}w+c\sum\limits_{t=1}^{N}e_{t}^{2}) \tag{3}\]  
Such that \(y_{t}=w^{T}\varphi(x_{t})+b+e_{t}\), \(t=1,2,\ldots,N\). According to [29; 30], the LS-SVM model can be expressed in dual form as \(\hat{y}(z)=\sum_{t=1}^{N}a_{t}K(z,x_{t})+b\). Note that the unknown items \(w\) and \(\varphi(\;)\) are not included in the dual model; the mapping transformation operations are achieved implicitly through the use of kernel function \(K(,)\). That is, it is necessary to find the expression of mapping function \(\varphi(\;)\). For the kernel function \(K(,)\), a function satisfying the Mercer theorem can be used. There are several choices presently, including the linear kernel, the perceptron kernel, the spline kernel, or the RBF (Radial Basis Function) kernel. In the LS-SVM model, it is required to optimize the parameter \(c\) in Equation (3) and the kernel parameter according to the kernel function selected, e.g., \(\sigma\) in Gaussian RBF kernel (\(K(x_{p},x_{q})=\exp\left(-\frac{\|x_{p}-x_{q}\|^{2}}{2\sigma^{2}}\right)\)). A fast leave-one-out cross validation optimization [32] can be used for optimizing these parameters. However, LS-SVM cannot deal with time series directly. To generate the fault features training samples, some time- and frequency-domain indexes are selected from the non-stationary signals by SSA, and are taken as inputs to train the LS-SVM classifier model. In this paper, 10 dimensionless statistical indexes in time and frequency domains are chosen [33; 34]. A brief mathematical description of these indexes is summarized in Table 1.  
\(x_{i}\) is a signal series for \(i=1,2,\ldots,n\), \(n\) is the number of data points, \(\overline{x}\) and \(S\) are the mean and the standard deviation of \(x_{i}\), \(x_{rms}=\sqrt{\sum_{i=1}^{n}x_{i}^{2}/n}\); \(f_{i}\) is the frequency signal with \(x_{i}\) by FFT, \(P_{i}\) is the power spectrum of \(f_{i}\), \(\overline{f}=\sqrt{\sum_{i=1}^{n}f_{i}/n}\). The process of LS-SVM classifier according to [32] is described as Algorithm 2. Given data \((NT_{i},Y_{i}),i=1,\ldots,\ n\), \(NT_{i}\) is the feature indexes of the \(i\)-th signal and \(Y_{i}\) is the class corresponding to the \(i\)-th sample. 1. Calculate 10 dimensionless time-frequency indexes of each \(NT_{i}\), as inputs of LS-SVM for classification. 2. Standardize the input data, and select a kernel function. Find the optimal parameter \(c\) by solving the optimization problem (3). 3. Set candidate tuning sets for kernel parameter, perform fast leave-one-out test for each candidate with the optimal parameter \(c\) obtained in Step 2, and take the result that obtains the highest test accuracy as the optimal kernel parameter. 4. With the optimal kernel parameter obtained in Step 3, go back to Step 2 to refine parameter \(c\). \begin{table}