# Eigenstate Transition of Multi-Channel Time Series Data around Earthquakes
## 2 Materials and Methods
### Methodology  
In reference to EWSs, we adopt the first eigenvalue and the corresponding eigenstate of the covariance matrix of the observed data as a feature quantity. To create a covariance matrix, we define a time window in which each data point is regarded as a sample for the matrix. The time dependence of the matrix is based on the feature quantities obtained by sliding the time window. We term the width of the slide a time step; the relation is shown in Figure 1.  
Here, we mention that the end time of the window corresponds to the present time because we can analyze only past data. Thus, we must extract preceding signals before a window includes the main shock data of an earthquake. We extract feature quantities based on the concept of an EWS from multi-channel time series data at a 1 Hz sampling as in the following procedures. 1. Prepare multi-channel time series data \(X_{i}(t_{q})\) with the indexes of the sensor channel \(i\) (\(i=1,2,\cdots,L\)) and discrete time \(q\) (\(q=1,2,\cdots\)). Then, set the time step width \(M\) and time window width \(N\). Analysis of the object for the \(p\)th time step is from \((p-1)M+1\) to \((p-1)M+N\). 2. Define a variable that represents the fluctuation from the average of the data in the time window. \[Y_{i}^{(p)}(t_{j})=X_{i}(t_{j})-\frac{1}{N}\sum_{j=1}^{N}X_{i}(t_{j}).\] (1)
3. Create normalized data \(Z_{i}^{(p)}\) with normalization constant \(k_{i}\) for channel \(i\) which is the average of the absolute value of \(Y_{i}^{(p_{\text{norm}})}(t_{j})\) at some time window \(p_{\text{norm}}\). \[Z_{i}^{(p)}(t_{j}) \equiv Y_{i}^{(p)}(t_{j})/k_{i},\] (2) \[k_{i} =\frac{1}{N}\sum_{j=1}^{N}|Y_{i}^{(p_{\text{norm}})}(t_{j})|.\] (3)
4. Define matrix \(\mathbf{Z}_{p}\) which has the element \(Z_{i}^{(p)}(t_{j})\) for the \(i\)th row and the \(j\)th column. \[\mathbf{Z}^{(p)}= \begin{pmatrix}Z_{1}^{(p)}(t_{1})&Z_{2}^{(p)}(t_{2})&\cdots&Z_{ 1}^{(p)}(t_{N})\\ Z_{2}^{(p)}(t_{1})&Z_{2}^{(p)}(t_{2})&\cdots&Z_{2}^{(p)}(t_{N})\\ \vdots&\vdots&\ddots&\vdots\\ Z_{L}^{(p)}(t_{1})&Z_{2}^{(p)}(t_{2})&\cdots&Z_{L}^{(p)}(t_{N})\\ \end{pmatrix} .\] (4)  
Figure 1: Setting of time windows (solid, dashed, and dotted lines) and time steps for an observed data. 5. Develop a following \(L\times L\) covariance matrix \(\mathbf{\Sigma}^{(p)}\) using \(\mathbf{Z}^{(p)}\) and its transverse matrix \(\left(\mathbf{Z}^{(p)}\right)^{\mathsf{T}}\). \[\mathbf{\Sigma}^{(p)}=\frac{1}{N-1}\mathbf{Z}^{(p)}\Big{(}\mathbf{Z}^{(p)}\Big{)}^{\mathsf{T}}.\] (5)
6. Calculate the sets of eigenvalues \(\lambda_{r}^{(p)}\) and eigenstate \(\mathbf{v}_{r}^{(p)}\) by means of the eigenvalue decomposition of \(\mathbf{\Sigma}^{(p)}\). Here, we define \(r\) (\(r=1,2,\cdots,L\)) as the index of the eigenvalue which has the \(r\)th largest value. 7. Calculate \(p\)-dependence, i.e., the time dependence of eigenstate \(\mathbf{v}_{r=1}^{(p)}\) corresponding to the first eigenvalue \(\lambda_{r=1}^{(p)}\). 8. Compare the time dependence to earthquakes focusing on the eigenvalue and the contribution weight to the eigenstate from the original channel. The contribution weights \(w_{i}^{(p)}\) are defined as follows: \[\mathbf{v}_{r=1}^{(p)}=\sum_{i=1}^{L}w_{i}^{(p)}\mathbf{u}_{i}.\] (6) Here, \(\mathbf{v}_{r=1}^{(p)}\) is the eigenstate, and \(\mathbf{u}_{1}\), \(\mathbf{u}_{2}\), \(\cdots\), \(\mathbf{u}_{L}\) are the bases that represent the original channel as used in Equation (4). According to the EWS concept, contributions from specific channels related to a transition will increase prior to an earthquakes. Thus, we define an occupancy rate \(\eta\), which is a ratio of the weights of specific \(R\) channels, named \(c_{1}\), \(\cdots\), \(c_{R}\):  
\[\eta=\sqrt{\sum_{i\in c}\Bigl{(}w_{i}^{(p)}\Bigr{)}^{2}}\qquad\quad(\,c=\{c_{ 1},\cdots,c_{R}\}\,). \tag{7}\]  
In typical EWS theory, both an increase in eigenvalue and occupancy rate from specific channels is expected.