Abstract
The detrended fluctuation analysis (DFA) is extensively useful in stochastic processes to unveil the long-term correlation. Here, we apply the DFA to point processes that mimick earthquake data. The point processes are synthesized by a model similar to the Epidemic-Type Aftershock Sequence model, and we apply the DFA to time series $N(t)$ of the point processes, where $N(t)$ is the cumulative number of events up to time $t$. Crossover phenomena are found in the DFA for these time series, and extensive numerical simulations suggest that the crossover phenomena are signatures of non-stationarity in the time series. We also find that the crossover time represents a characteristic time scale of the non-stationary process embedded in the time series. Therefore, the DFA for point processes is especially useful in extracting information of non-stationary processes when time series are superpositions of stationary and non-stationary signals. Furthermore, we apply the DFA to the cumulative number $N(t)$ of real earthquakes in Japan, and we find a crossover phenomenon similar to that found for the synthesized data.
Highlights
Stationarity is one of the most important properties in theoretical foundations of stochastic processes, nonstationary phenomena are rather ubiquitous in nature, ranging from disordered systems [1,2,3,4,5,6], to seismicity [7,8,9,10], to biological systems [11,12,13]
We apply the detrended fluctuation analysis (DFA) to the cumulative number N (t ) of real earthquakes included in the Japan Meteorological Agency (JMA) catalog; this catalog contains data of earthquakes with magnitude M 2 and the ones that occurred in the area of 25◦–50◦ N latitude and 125◦– 150◦ E longitude
We found that crossover phenomena in the DFA of N (t ), i.e., the number of earthquakes up to time t, are universally observed in earthquake data
Summary
Stationarity is one of the most important properties in theoretical foundations of stochastic processes, nonstationary phenomena are rather ubiquitous in nature, ranging from disordered systems [1,2,3,4,5,6], to seismicity [7,8,9,10], to biological systems [11,12,13]. Earthquakes are an example of the nonstationary point process of the first type. Plotting the diffusion coefficient as a function of the measurement time provides us information on how the process ages In another method of nonstationary data analysis of the second type, the interoccurrence times are utilized frequently [11,21,22]. The interoccurrence-time PDF analysis is based on the fact that the interoccurrence times are independent and identically distributed (IID) This assumption is not valid for nonstationary processes of the first type. We present analytical predictions of long-time behaviors in DFA for point processes.
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