# Eigenstate Transition of Multi-Channel Time Series Data around Earthquakes
## 1 Introduction  
Determining precursor signals of earthquakes is sought as earthquakes result in considerable damage including economic loss and fatalities. If these could be determined, the damage resulting from earthquakes could be reduced by countermeasures such as stopping the use of equipment, and evacuation to a safe location. Certainly, earthquake early warnings are effective for locations away from the epicenter. On the other hand, they are unhelpful for locations near the epicenter because they warn only after earthquakes occur. Thus, preceding earthquake signals are desired. However, no conclusive method has been developed owing to the complexity of earthquakes such as a variety of crustal hardness and applied stress inhomogeneity. There are studies determining preceding signals via defining feature quantities and analytical methods describing their behaviors. Some regard observable physical quantities as feature quantities, and adopt their change as preceding signals. Examples of such studies include those of earthquake lights [1], groundwater levels [2], earth current [3, 4], very low frequencies [5], crustal deformation measured using a global positioning system [6] and ionospheric anomalies [7, 8]. Other studies have extracted feature quantities from physical quantities via the use of statistical methods. The correlation between the b-value and earth tide [9] is found out by insight. Machine learning methods [10] are one trends in statical methods. There are example such as using the earth current [11] and radon monitoring [12]. Nevertheless, it cannot be said that the method to determine preceding earthquake signals is established. In the case of electro-magnetic phenomena and ionospheric anomalies, the relations between observed physical quantities and an earthquake are not direct. Earthquake lights and the change in the confined groundwater level are the consequences of a particular locality and do not have generality. Concerningthe studies using machine learning, they inherently need a large amount of data and they have difficulty showing generality, i.e., independence of locality owing to data. Crustal deformation measured using the global positioning system seems to be closely related to earthquakes but it does not show clear preceding signals. Feature quantities extracted from data that have a strong relation to earthquake activity such as velocity and acceleration on the surface of earth are a promising candidate for general preceding signals. Observational points are already established on the ground surface and observed time series data of several physical quantities are available. It is natural that these data have a direct relation to the state of the crust because of certain continuity between the observed point and epicenter. Thus, feature quantities extracted from these observed data are candidate for preceding signals if such signals exist. If we can treat earthquakes as one of the transition phenomena from one state to the other state, the methods termed \"early warning signals (EWSs)\" [13], which is one of the preceding signals, are available. EWSs are methods focusing on increasing fluctuations in variables and decreasing recovery rates against disturbances when the system reaches the vicinity of the transition point. EWSs based on a covariance matrix detect preceding signals from the relation between multiple types of time series data. It adopts the first eigenvalue as a preceding signal. In addition, the EWS termed \"dynamical network marker\" not only constructs preceding signals but also extracts variables contributing to the transition via the use of eigenstates [14; 15; 16]. In the case of earthquakes, in which the process of the transition is complex, the eigenstate is expected to play an important role as a factor estimation or as preceding signals themselves.