## 2 Fundamental of Phase Space Reconstruction and Deep Neural Network  
### Phase Space Reconstruction  
Phase space reconstruction is a method proposed by [PERSON] to analyze the time series. The basic idea of the phase space reconstruction is to regard the time series as a component produced by a certain nonlinear dynamic system. The equivalent high-dimensional phase space of the power system can be reconstructed by the variation law of the component. Among them, the key to reconstruction is to determine the optimal embedding dimension \(m_{opt}\) and delay \(t_{opt}\).  
In this paper, the optimal embedding dimension \(m_{opt}\) and delay \(t_{opt}\) are simultaneously obtained by the C-C method. If a set of time series is \(\boldsymbol{x}~{}=~{}\{x_{1},x_{2}\cdots x_{N}\}\), the embedding dimension is \(m\)and the time delay is \(t\), then the set of points in the reconstructed phase space can be expressed by Formula (1), where \(M~{}=~{}N-(m-1)t\)[25].  
\[\left[\begin{array}{c}\mathbf{X}_{1}\\ \mathbf{X}_{2}\\ \vdots\\ \mathbf{X}_{M}\end{array}\right]~{}=~{}\left[\begin{array}{cccc}x_{1}&x_{1+ t}&\cdots&x_{1+(m-1)t}\\ x_{2}&x_{2+t}&\cdots&x_{2+(m-1)t}\\ \vdots&\vdots&\cdots&\vdots\\ x_{M}&x_{M+t}&\cdots&x_{N}\end{array}\right]. \tag{1}\]  
At this time, the correlation integral is Formula (2), where \(\theta(x)~{}=~{}\left\{\begin{array}{cc}0&x<0\\ 1&x\geq 0\end{array}\right.\). According to the statistical conclusion of BDS, the range of \(M\) and \(r_{k}\) can be obtained when \(N>3000\); \(m\in\{2,3,4,5\}\), and \(r_{k}~{}=~{}k\times 0.5\sigma\) which is a real number representing a given range of distances. \(\sigma\) is the standard deviation of time series, and \(k\in\{1,2,3,4\}\). Correlation integral indicates the probability that the distance between any two points in the phase space is less than \(r_{k}\).  
\[C(m,N,r_{k},t)~{}=~{}\frac{2}{M(M-1)}\sum\limits_{1\leq i<j\leq M}\theta\big{(} r_{k}-\|\mathbf{X}_{i}-\mathbf{X}_{j}\|\big{)}. \tag{2}\]  
We define the test statistic \(S\) and \(\Delta S\), and use the block averaging strategy, as shown in Formula (3).  
\[\left\{\begin{array}{c}S(m,N,r_{k},t)~{}=~{}\frac{1}{t}\sum\limits_{i=1}^{M} C_{i}\Big{(}m,\frac{N}{t},r_{k},t\Big{)}-C_{i}^{m}\Big{(}m,\frac{N}{t},r_{k},t \Big{)}\\ \Delta S(m,N,t)~{}=~{}\max[S(m,N,r_{k},t)]-\min[S(m,N,r_{k},t)]\end{array} \right.. \tag{3}\]  
Formula (4) is the average of \(S\) and \(\Delta S\). Rounding the \(t\) value of the first zero of \(\overline{S}\) or the first minimum of \(\Delta S\) is the optimal delay \(t_{opt}\).  
\[\left\{\begin{array}{c}\overline{S}(t)~{}=~{}\frac{1}{4\times 4}\sum\limits_{m =2}^{5}\sum\limits_{k=1}^{4}S(m,N,r_{k},t)\\ \overline{\Delta S}(t)~{}=~{}\frac{1}{4}\sum\limits_{m=2}^{5}\Delta S(m,N,t) \end{array}\right.. \tag{4}\]  
Formula (5) is the test statistic. The global minimum of \(S_{cor}(t)\) is the optimal embedded window \(t_{\omega}\).  
\[S_{cor}(t)~{}=~{}\overline{\Delta S}(t)+\big{|}\overline{S}(t)\big{|}. \tag{5}\]  
Then:  
\[t_{\omega}~{}=~{}(m_{opt}-1)t_{opt}. \tag{6}\]  
Therefore, the optimal delay \(t_{opt}\) determined by Equation (4) and the optimal embedded window \(t_{\omega}\) determined by Equation (5) can be substituted into Formula (6) and rounded to obtain the optimal embedding dimension \(m_{opt}\).