## 2 Brief Introductions to SSA and LS-SVM  
### Ssa  
Stationary Subspace Analysis (SSA) is a blind source separation algorithm which factorizes a multivariate time series into stationary and non-stationary sources by [PERSON] et al. [21]. In the SSA model, it is assumed that the observed multidimensional time series \(X(t)\in R^{D}\) is a linear superimposition of two types of sources: stationary sources and non-stationary sources. Its mathematical expression is as follows:  
\[X(t)=AS(t)=(A^{s}A^{n})\left( \begin{array}{c}S^{s}(t)\\ S^{n}(t)\end{array} \right) \tag{1}\]  
where \(A\) is an unknown but time-independent matrix, and is linearly superimposed by coefficient matrix \(A^{s}\) of the stationary sources and coefficient matrix \(A^{n}\) of the non-stationary sources; \(S(t)\) is the underlying intrinsic data of \(X(t)\), and is linearly superimposed by the stationary sources \(S^{s}(t)\in R^{d}\) and the non-stationary sources \(S^{n}(t)\in R^{D-d}\). The aim of the SSA is to estimate the inverse matrix \(\hat{A}^{-1}\) separating the stationary sources from non-stationary sources in \(X(t)\). That is, we want to find a demixing matrix, \(\hat{A}^{-1}=\left[ \begin{array}{c}B^{s}\\ B^{n}\end{array} \right]\) with \(B^{s}\in R^{d\times D}\) and \(B^{n}\in R^{(D-d)\times D}\) that consists of the matrixes \(B^{s}\) and \(B^{n}\) for the stationary and non-stationary directions, respectively. Thus, if we apply such an ideal demixing \(\hat{A}^{-1}\) to the observed data \(X(t)\),  
\[\left[ \begin{array}{c}\hat{S}^{s}_{t}\\ \hat{S}^{n}_{t}\end{array} \right]=\hat{A}^{-1}X(t)=\hat{A}^{-1}A\left[ \begin{array} []{c}S^{s}_{t}\\ S^{n}_{t}\end{array} \right]=\left[ \begin{array}{cc}B^{s}A^{s}&B^{s}A^{n}\\ B^{n}A^{s}&B^{n}A^{n}\end{array} \right]\left[ \begin{array}{c}S^{s}_{t}\\ S^{n}_{t}\end{array} \right] \tag{2}\]  
The SSA algorithm uses the optimal stationary signal recognition criteria, only the stationary source signal \(S^{s}_{t}\) and \(n\)-space can be uniquely determined, and the non-stationary source signal \(S^{n}_{t}\) can only get its estimate \(\hat{S}^{n}_{t}\) by maximizing the non-stationary. Among them, the stationary judgment is based on the weak stationary condition, that is, a time series is stationary if its first two moments are constant over time [25]. SSA needs multi-dimensional observed data that some situations cannot satisfy, owing to working conditions and experimental cost. In NC machine tool fault diagnosis, only one sensor can be used in some situations due to monitoring cost and processing environment, and the observed data are one-dimensional. To use SSA to extract stationary source and non-stationary source signals, the one-dimensional observed data should be expanded into multi-dimensional data. To accomplish this, phase space reconstruction (PSR) is introduced in this paper. PSR is a reconstructing space technology in Chaos theory according to the delay coordinate method [26]. PSR can be used to expand the dimensionality of a system for studying the characteristics of the system without prior knowledge about the structure and parameters of the system. The idea of PSR is that the evolution of any component in a system is determined by its interaction with other components. The time series implies the development process of each component. Considering a single component, PSR takes some measurement processing in fixed time delay point as a new dimension, the value of delay is considered as a new coordinate. By repeating this process at different time points, many of these points generate a new space. For one-dimensional time series \(X(t)\), different time delays \(0\), \(t\), \(2t\),..., \((m-1)t\) are used to construct a m-dimensional phase space vector:  
\[X_{i}(t)=\{x(t+i),x(t+i+\tau),x(t+i+2\tau),\ldots,x(t+i+(m-1)\tau)\},i=0,1, \ldots,m-1\]  
where \(\tau\) is the time delay and \(m\) is the embedding dimension. \[\mathrm{P}=\left[