# Eigenstate Transition of Multi-Channel Time Series Data around Earthquakes
## 4 Discussion  
We discuss the relation between the analytical results and earthquake locations. In the case of the Mie earthquake, we obtain an eigenstate transition of the covariance matrix such as a concentration of the contribution from specific channels immediately around the earthquakes. This shows that an increase in the occupancy rate corresponds to a preceding signal of the an earthquake. It seems that the proximity of this earthquake to the observatory KMDB1 results in this state change. In the case of the Kumamoto earthquake, we could not capture the explicit dominant state as shown in Figure 9A,B. However, the time in which the occupancy rate stays almost 100% is longer before the earthquake (Figure 9A,B) than after the earthquake (Figure 9E,F). This is possibly attributed to the distance between the earthquake and observatory. The observed waveform of DONET is the sum of the activity of various locations with weights depending on the distance. Thus, it is natural to assume that the weights are smaller when the distance is longer. Hence, even if earthquake activity of a far-away point becomes active, its effect on the eigenstate is not direct but reduced by the near-point activity. Thus, the eigenstate shown in Figure 8A,B is interpreted as the combination of a random contribution from the usual earthquake activity from the near point and the dominant state of specific channels before the Kumamoto earthquake. On the other hand, in the case of the Fukushima earthquake which is as far as Kumamoto case, we could not find such randomness. Namely, the concentration from specific channels in eigenstate already starts to occur at the beginning of the analyzed  
Figure 11: Occupancy rate and the first eigenvalue around the Fukushima earthquake for (**A**) 2016/326/12-16, (**B**) 2016/326/16-20, (**C**) 2016/326/20-24, (**D**) 2016/327/00-04, (**E**) 2016/327/04-08, (**F**) 2016/327/08-12. time as shown in Figure 10. Moreover, the eigenvalue increased before the earthquake (Figure 11). Even though we do not have a criteria value to judge earthquake occurrence about the eigenvalue, these behaviors could be interpreted as preceding signals. However, the reason why there is a difference between Kumamoto and Fukushima even though they are both far away from DONET station is an open question. We analyzed three different earthquakes in which DONET was able to observe the earthquake waveform obtained a common transition of the eigenstate,i.e., the concentration of contribution from the specific channel in the eigenstates. This transition possibly becomes a preceding signal because the concentration to some channels precedes earthquakes at a few hours' scale. However, using only qualitative analysis of this transition is not sufficient for being useful in preceding signals in practice. For future work, quantitative analysis of this transition is required such as the relation between the time concentration maintenance and the closeness or magnitude of earthquakes by analyzing many more earthquakes. Furthermore, it is expected that we are able to extract more information by comparing transitions of eigenstates developed from data of different observatories. Noting the relation between the results of our study and EWS methods, as described in the introduction, the EWS method results in an increase in the first eigenvalue and the weights of variables related to a transition. In our study, even though an increasing eigenvalue is confirmed only in the Fukushima earthquake, the concentration in specific channels, which precedes earthquakes, is shown. The data of typical earthquake activity fluctuate and include observational noise because of a requirement to maintain high sensitivity. Thus, it is more difficult to differentiate fluctuation before a transition and observational noise than that of other fields. Thus, it is possible that a difference between our results and the EWS method occurs.