In order to obtain the regularities of the periodicities and trends of the earthquake origin times and predict them, we considered to convert the earthquake origin times into the time intervals (days) that can better reflect the characteristics of the periodicities and trends of the earthquake origins, so the time series used for analysis were constructed. And based on the ARIMA model, a method for analyzing and predicting the earthquake origin times was designed. Its flow is shown in Fig. 2, and the specific process is as follows.
(1) Preparing for calculation
① Collect and sort out the natural earthquake catalog data in the target fault zone area. Extract the information such as the origin time and the magnitude of each earthquake, and the magnitude should be measured using the same method.
② Set the magnitude range of the earthquake to be predicted, such as M ≥ 2.5, M ≥ 3.0 and M ≥ 4.5.
③ According to the set of the magnitude range in step (1)-②, extract the earthquake origin times from the data information obtained from step (1)-① to obtain the series of the earthquake origin times, such as the earthquake series of M ≥ 2.5, M ≥ 3.0 and M ≥ 4.5.
④ Use the origin times of two adjacent earthquakes in the series obtained in step (1)-③ to calculate the earthquake origin time intervals, obtain the series of earthquake origin time intervals of the certain magnitude ranges.
⑤ Divide the series of earthquake origin time intervals into fitting-training set and prediction-verification set in a ratio (e.g. 9:1) to be used for the ARIMA models, and the orders of earthquake origin time intervals in the series remains unchanged in this operation.
(2) Analysis of the tendency and periodicity of the series of earthquake origin time intervals
① The center moving average method of different orders (periods) is used to remove the seasonal effects and extract the trend from the fitting-training set of the series of earthquake origin time intervals obtained in steps (1)-⑤.
② Observe and analyze the results obtained in step (2)-①, determine the approximate range of short, medium and long periods of the series of earthquake origin time intervals based on the effect of removing the periodicity, and determine the specific type of the ARIMA model based on the effect of extracting the trend (the seasonal addition model is adopted if there is no trend, otherwise the seasonal multiplication model is adopted).
(3) Fitting of ARIMA model
① Calculate the autocorrelation functions (ACF) and the partial autocorrelation functions (PACF) of the fitting-training set of the series of earthquake origin time intervals obtained from step (1)-⑤.
② Analyze and Determine the order p of non-seasonal autoregressive coefficient polynomial, the order q of non-seasonal moving average coefficient polynomial, the order P of seasonal autoregressive coefficient polynomial, and the order Q of seasonal moving average coefficient polynomial of the ARIMA model respectively based on the non-seasonal tailing, truncation and the seasonal (periodic) tailing, truncation of the ACF and PACF obtained in step (3)-①.
③ Set the non-seasonal difference d = 1, 2, ..., and the seasonal difference D = 1, 2, .... And so obtain several ARIMA(p,d,q)×(P,D,Q)S models, where s denotes the period length. And if the addition model is adopted, set P = 0, D = 1, Q = 0 in the models.
④ Fit all the ARIMA(p,d,q)×(P,D,Q)S models obtained from step (3)-③ with the fitting-training set of the series of earthquake origin time intervals obtained from (1)-⑤ respectively, calculate the autoregressive coefficients and moving average coefficients of the models, and then obtain the specific equation forms of the models.
⑤ Calculate the residuals between the fitted values of the earthquake origin time intervals by the ARIMA models in step (3)-④ and the true values. Then do the white noise Ljung Box (LB) tests on the series of residuals, and select the ARIMA models that can pass them.
⑥ Calculate the adjusted coefficients of determination (the adjusted R2) of the ARIMA models obtained from step (3)-⑤. Then select the optimal ARIMA models of the short, medium and long period with the maximum adjusted R2, and also determine the optimal values of d and D at the same time.
(4) Prediction of ARIMA model
① According to the demand for the predicted magnitude and period length, select the ARIMA model obtained from step (3)-⑥ to predictively calculate the earthquake origin time intervals of the prediction-verification set. Both direct prediction method and rolling prediction method can be used. The direct prediction method is using the model to calculate all subsequent earthquake origin time intervals that need to be predicted at one time. And the rolling prediction method is using the model to predict the next interval of the earthquake origin time interval series, adding them to the end of the series, fitting the ARIMA model again, and using the fitted model to predict the next interval until all time intervals were predicted.
② Calculate the root mean square error (RMSE) with the true value and the predicted value obtained from step (4)-①. So that obtain the basis for the predictive accuracies of the models, and the analysis and evaluation of the results.
③ Take the last earthquake origin time in the fitting-training set as the initial point of time, successively add the corresponding predicted values of the earthquake origin time intervals obtained from step (4)-①, the subsequent earthquake origin times are obtained.